Yourgrau P. A world without time.. the forgotten legacy of Goedel and Einstein (Basic Books, 2005)(ISBN 0465092934)(176s)_PPop_
Yourgrau P. A world without time.. the forgotten legacy of Goedel and Einstein (Basic Books, 2005)(ISBN 0465092934)(176s)_PPop_
Yourgrau P. A world without time.. the forgotten legacy of Goedel and Einstein (Basic Books, 2005)(ISBN 0465092934)(176s)_PPop_
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A World Without Time: The Forgotten Legacy <strong>of</strong> Godel <strong>and</strong> <strong>Einstein</strong><br />
Palle <strong>Yourgrau</strong><br />
<strong>2005</strong><br />
It is a widely known but little appreciated fact that Albert <strong>Einstein</strong>, <strong>the</strong> twentieth century's<br />
greatest physicist, <strong>and</strong> Kurt Godel, its greatest logician, were best friends for <strong>the</strong> last<br />
decade <strong>and</strong> a half <strong>of</strong> <strong>Einstein</strong>'s life. They walked home toge<strong>the</strong>r from Princeton's Institute<br />
for Advanced Study every day; <strong>the</strong>y shared ideas about physics, philosophy, politics, <strong>and</strong><br />
<strong>the</strong> lost <strong>world</strong> <strong>of</strong> German-Austrian science in which <strong>the</strong>y had grown up. What is not widely<br />
known is <strong>the</strong> discovery that grew out <strong>of</strong> this friendship. In 1949 Godel published a paper<br />
proving that <strong>the</strong>re exist possible <strong>world</strong>s described by <strong>the</strong> <strong>the</strong>ory <strong>of</strong> relativity in which <strong>time</strong>,<br />
as we ordinarily underst<strong>and</strong> it, does not exist. He went fur<strong>the</strong>r: if it is absent from those<br />
<strong>the</strong>oretical universes, he showed, <strong>time</strong> does not exist in our <strong>world</strong> ei<strong>the</strong>r. <strong>Einstein</strong>'s great<br />
work has not explained <strong>time</strong>, as most physicists <strong>and</strong> philosophers think, but explained it<br />
completely away.<br />
<strong>Einstein</strong> recognized Godel's paper as "an important contribution to <strong>the</strong> general <strong>the</strong>ory <strong>of</strong><br />
relativity." Physicists since <strong>the</strong>n have tried <strong>without</strong> success to find an error in Godel's<br />
physics or a missing element in relativity itself that would rule out <strong>world</strong> models like<br />
Godel's. Stephen Hawking went so far as to propose an ad hoc modification <strong>of</strong> <strong>the</strong> laws <strong>of</strong><br />
natureóa "chronology protection conjecture"óspecifically to negate Godel's contribution to<br />
relativity. Philosophers have been largely silentó<strong>and</strong> <strong>the</strong>ir silence, says <strong>Yourgrau</strong>, is one <strong>of</strong><br />
<strong>the</strong> intellectual sc<strong>and</strong>als <strong>of</strong> <strong>the</strong> past century.<br />
A World Without Time places Godel <strong>and</strong> <strong>Einstein</strong>'s epoch-making discoveries in <strong>the</strong> context<br />
<strong>of</strong> <strong>the</strong> great <strong>and</strong><br />
disturbing movements in physics, philosophy, logic, ma<strong>the</strong>matics, <strong>and</strong> <strong>the</strong> arts that<br />
dominated <strong>the</strong> twentieth century. It presents a poignant <strong>and</strong> intimate account <strong>of</strong> <strong>the</strong><br />
friendship between <strong>the</strong>se two magnificent thinkers, each put on <strong>the</strong> shelf by <strong>the</strong> scientific<br />
fashions <strong>of</strong> <strong>the</strong>ir dayó<strong>and</strong> oursó<strong>and</strong> attempts to rescue Godel's brilliant work from<br />
undeserved obscurity. Inspired by <strong>Einstein</strong>, Kurt Godel made clear for <strong>the</strong> first <strong>time</strong> <strong>the</strong><br />
truly revolutionary nature <strong>of</strong> <strong>the</strong> <strong>the</strong>ory <strong>of</strong> relativity, which to this day is hardly<br />
recognized.
PALLE YOURGRAU is associate pr<strong>of</strong>essor <strong>of</strong> Philosophy at Br<strong>and</strong>eis University. His 1999<br />
monograph, Godel Meets <strong>Einstein</strong>: Time Travel in <strong>the</strong> Godel Universe, which explores <strong>the</strong><br />
philosophical significance <strong>of</strong> Godel's cosmological ideas, sparked a resurgence <strong>of</strong> interest in<br />
Godel's ideas about <strong>time</strong> <strong>and</strong> relativity. He lives in Cambridge, Massachusetts.<br />
Copyright © <strong>2005</strong> by Palle <strong>Yourgrau</strong><br />
Published by <strong>Basic</strong> <strong>Books</strong>, A Member <strong>of</strong> <strong>the</strong> Perseus <strong>Books</strong> Group<br />
First Edition<br />
All rights reserved. Printed in <strong>the</strong> United States <strong>of</strong> America. No part<br />
<strong>of</strong> this book may be reproduced in any manner whatsoever <strong>without</strong><br />
written permission except in <strong>the</strong> case <strong>of</strong> brief quotations<br />
embodied in critical articles <strong>and</strong> reviews.<br />
Designed by Brent Wilcox Set in 11 point Sabon<br />
A CIP catalog record for this book is available from <strong>the</strong> Library <strong>of</strong> Congress. <strong>ISBN</strong> 0-465-<br />
09293-4 (he.)<br />
<strong>Books</strong> published by <strong>Basic</strong> <strong>Books</strong> are available at special discounts for bulk<br />
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050607 08/ 109 8 765432 1<br />
Contents<br />
Acknowledgments - vii<br />
1 A Conspiracy <strong>of</strong> Silence - 1<br />
2 A German Bias for Metaphysics - 9<br />
3 Vienna: Logical Circles - 21<br />
4 A Spy in <strong>the</strong> House <strong>of</strong> Logic - 51<br />
5 It's Hard to Leave Vienna - 77<br />
6 Amid <strong>the</strong> Demigods - 89<br />
7 The Sc<strong>and</strong>al <strong>of</strong> Big "T" <strong>and</strong> Little "t" - 119<br />
8 Twilight <strong>of</strong> <strong>the</strong> Gods - 145
9 In What Sense Is Godel (or Anyone Else) a Philosopher? - 161<br />
Notes - 185<br />
Works Cited - 195<br />
Index - 201<br />
Acknowledgments<br />
Having already written a book intended primarily for philosophers about Kurt Godel's<br />
attempt to make sense <strong>of</strong> <strong>Einstein</strong>'s <strong>the</strong>ory <strong>of</strong> relativity, I was intrigued when William<br />
Frucht <strong>of</strong> <strong>Basic</strong> <strong>Books</strong> suggested I write ano<strong>the</strong>r, this one accessible to normal readers.<br />
Such a book would focus on <strong>the</strong> sheer intellectual drama <strong>of</strong> <strong>the</strong> companionship <strong>of</strong> Godel<br />
<strong>and</strong> <strong>Einstein</strong>óa relationship sorely neglected in <strong>the</strong> literatureó <strong>and</strong> would place Godel's <strong>and</strong><br />
<strong>Einstein</strong>'s epoch-making discoveries in <strong>the</strong> context <strong>of</strong> <strong>the</strong> great intellectual movements <strong>of</strong><br />
<strong>the</strong> twentieth century, some <strong>of</strong> which, having helped to fa<strong>the</strong>r, <strong>the</strong>y tried, belatedly, to<br />
ab<strong>and</strong>on. It was an <strong>of</strong>fer too good to refuse, <strong>and</strong> I didn't. The task, however, turned out to<br />
be far from easy, <strong>and</strong> Frucht had to endure not only <strong>the</strong> late delivery <strong>of</strong> <strong>the</strong> final<br />
manuscript, but <strong>the</strong> drumbeat <strong>of</strong> my complaints about his editorial adventures at <strong>the</strong><br />
expense <strong>of</strong> my beloved prose; it may not have been much, but it was all mine. I am<br />
grateful to Frucht both for <strong>the</strong> initial invitation <strong>and</strong> for (what turned out to be) his wise<br />
editorial advice, at every stage, on how to improve <strong>the</strong> manuscript.<br />
I have greatly benefited from discussing <strong>the</strong> book with Mary Sullivan <strong>and</strong> Ben Callard.<br />
Sullivan, attentive as ever to <strong>the</strong> task <strong>of</strong> trying to keep me honest, read large parts <strong>of</strong> <strong>the</strong><br />
book <strong>and</strong> <strong>of</strong>fered acute, sobering advice, which I took to heart. Callard's critical remarks<br />
on every chapter were <strong>of</strong> great value <strong>and</strong> are reflected in <strong>the</strong> final draft. His strange<br />
affection for <strong>the</strong> work, moreover, kept my spirits up during <strong>the</strong> many dark moments when<br />
<strong>the</strong> project seemed to me ill-advised <strong>and</strong><br />
misbegotten. Mark van Atten also read <strong>the</strong> entire manuscript. His extraordinarily detailed<br />
remarks, on both substance <strong>and</strong> style, were a great boon. Robert Tragesser shared with me<br />
his deep underst<strong>and</strong>ing <strong>of</strong> Godel's <strong>the</strong>orem, <strong>and</strong> Eli Hirsch helped make certain that my<br />
discussions <strong>of</strong> logic <strong>and</strong> ma<strong>the</strong>matics were clear, accessible, <strong>and</strong> to <strong>the</strong> point. To each <strong>of</strong><br />
<strong>the</strong>se I owe a serious debt <strong>of</strong> gratitude, but especially to Callard <strong>and</strong> van Atten, for <strong>the</strong>ir<br />
extensive <strong>and</strong> thoughtful comments.
I would also like to express my appreciation for <strong>the</strong> conversations I had over a period <strong>of</strong><br />
years with <strong>the</strong> late Hao Wang, <strong>of</strong> Rockefeller University, whom I came to know when I<br />
taught at Barnard College in New York. With Wang, who was one <strong>of</strong> Godel's closest<br />
associates in his final years, I spent endless hours discussing Godel's ideas, published <strong>and</strong><br />
unpublished. The question <strong>of</strong> <strong>time</strong> was <strong>of</strong> particular interest to him. He confessed that,<br />
finding <strong>the</strong> topic uncongenial, he had resisted Godel's efforts to discuss his thoughts on this<br />
<strong>the</strong>me, <strong>and</strong> regretted it. I regretted it, too. It did not diminish, however, <strong>the</strong> fruitfulness<br />
<strong>of</strong> our conversations, nor <strong>the</strong> enjoyment we took in contemplating what Godel has said <strong>and</strong><br />
written.<br />
In addition to help from my friends, I have also depended on <strong>the</strong> kindness <strong>of</strong> strangers. My<br />
copy editor, David Kramer, in addition to improving <strong>the</strong> clarity <strong>and</strong> flow <strong>of</strong> <strong>the</strong> text, made<br />
a number <strong>of</strong> comments <strong>and</strong> suggestions <strong>of</strong> a more substantial nature on a wide variety <strong>of</strong><br />
topics, from music to ma<strong>the</strong>matics, which were an unexpected gift. AnnaLee Pauls,<br />
photoduplication coordinator at <strong>the</strong> Rare <strong>Books</strong> <strong>and</strong> Special Collections department <strong>of</strong><br />
Princeton University's Firestone Library, provided much needed assistance in <strong>the</strong> delicate<br />
task <strong>of</strong> choosing <strong>and</strong> reproducing <strong>the</strong> right photos from <strong>the</strong> Godel archives, housed in<br />
Firestone Library. I am grateful both for her assistance <strong>and</strong> for <strong>the</strong> spirit with which she<br />
provided it. I would also, finally, like to thank Arudra Burra, who is not a stranger, for his<br />
generous help, combined with fine judgment, in <strong>the</strong> search for <strong>the</strong> right photos to include<br />
in <strong>the</strong> book.<br />
A World Without Time<br />
1 A Conspiracy <strong>of</strong> Silence<br />
Godel was ... <strong>the</strong> only one <strong>of</strong> our colleagues who walked <strong>and</strong> talked on equal terms with<br />
<strong>Einstein</strong>.<br />
FREEMAN DYSON
In <strong>the</strong> summer <strong>of</strong> 1942, while German U-boats roamed in wolf packs <strong>of</strong>f <strong>the</strong> coast <strong>of</strong> Maine,<br />
residents in <strong>the</strong> small coastal town <strong>of</strong> Blue Hill were alarmed by <strong>the</strong> sight <strong>of</strong> a solitary<br />
figure, h<strong>and</strong>s clasped behind his back, hunched over like a comma with his eyes fixed on<br />
<strong>the</strong> ground, making his way along <strong>the</strong> shore in a seemingly endless midnight stroll. Those<br />
who encountered <strong>the</strong> man were struck by his deep scowl <strong>and</strong> thick German accent.<br />
Speculation mounted that he was a German spy giving secret signals to enemy warships.<br />
The dark stranger, however, was no German spy. He was Kurt Godel, <strong>the</strong> greatest logician<br />
<strong>of</strong> all <strong>time</strong>, a beacon in <strong>the</strong> intellectual l<strong>and</strong>scape <strong>of</strong> <strong>the</strong> last thous<strong>and</strong> years, <strong>and</strong> <strong>the</strong> prey<br />
he sought was not American ships bound for Britain but ra<strong>the</strong>r <strong>the</strong> so-called continuum<br />
hypo<strong>the</strong>sis, a conjecture made by <strong>the</strong> ma<strong>the</strong>matician Georg Cantor about <strong>the</strong> number <strong>of</strong><br />
points on a line. Godel was spending <strong>the</strong> summer vacationing at <strong>the</strong> Blue Hill Inn with his<br />
wife, Adele, although fellow visitors at <strong>the</strong> inn rarely saw ei<strong>the</strong>r <strong>of</strong> <strong>the</strong>m. They<br />
materialized for dinner, but were never observed actually eating. To <strong>the</strong> locals, Godel's<br />
scowl betrayed a dark disposition, but <strong>the</strong> innkeeper saw things differently. For her it was<br />
<strong>the</strong> expression <strong>of</strong> a man lost in thought. His last word to Blue Hill would not decide <strong>the</strong><br />
issue. He sent a letter accusing <strong>the</strong> innkeeper <strong>of</strong> stealing <strong>the</strong> key to his trunk.<br />
The place Godel would return to in <strong>the</strong> fall was a long way from Blue Hilló<strong>the</strong> prestigious<br />
Institute for Advanced Study in Princeton, New Jersey. There he would no longer have to<br />
walk alone, arousing <strong>the</strong> suspicions <strong>of</strong> neighbors. He had a walking companion, a colleague<br />
at <strong>the</strong> institute <strong>and</strong> his best friend. There was no danger that his reputation would<br />
intimidate his companion. For his friend, ano<strong>the</strong>r German-speaking refugee with a<br />
ma<strong>the</strong>matical bent, was <strong>the</strong> most famous scientist <strong>of</strong> all <strong>time</strong>, Albert <strong>Einstein</strong>, whose own<br />
meditative strolls already irritated <strong>the</strong> residents <strong>of</strong> Princeton.<br />
"From a distance," a biographer wrote, "<strong>the</strong> [residents <strong>of</strong> Princeton] chuckled discreetly<br />
over [<strong>Einstein</strong>'s] habit <strong>of</strong> licking an ice cream on Nassau Street on his way home from Fine<br />
Hall <strong>and</strong> were astonished by his utterly un-American long walks through <strong>the</strong> streets <strong>of</strong><br />
Princeton." Indeed, toward <strong>the</strong> end <strong>of</strong> his career, when he was more or less retired,<br />
<strong>Einstein</strong> commented that his own work no longer meant much to him <strong>and</strong> that he now went<br />
to his <strong>of</strong>fice "just to have <strong>the</strong> privilege <strong>of</strong> walking home with Kurt Godel." Ironically, it was<br />
not <strong>the</strong> scowling Godel but his smiling companion who had once given indirect aid to <strong>the</strong><br />
German U-boats, when, during World War I, although a courageous <strong>and</strong> committed pacifist,<br />
<strong>Einstein</strong> had helped improve <strong>the</strong> gyroscopes used by <strong>the</strong> German navy. Godel's research<br />
would also, in <strong>the</strong> end, relate to gyroscopes, but <strong>the</strong>se spun at <strong>the</strong> center <strong>of</strong> <strong>the</strong> universe,<br />
not in <strong>the</strong> dank bowels <strong>of</strong> submarines.<br />
Washed up onto America's shores by <strong>the</strong> storm <strong>of</strong> Nazism that raged in Europe in <strong>the</strong> 1930s,<br />
<strong>the</strong> two men had awakened to find <strong>the</strong>mselves str<strong>and</strong>ed in <strong>the</strong> same hushed academic<br />
retreat, Princeton's Institute for Advanced Study, an exclusive intellectual club, whose<br />
members had only one assigned duty: to think. But Godel <strong>and</strong> <strong>Einstein</strong> already belonged to<br />
an even more exclusive club. Toge<strong>the</strong>r with ano<strong>the</strong>r German-speaking <strong>the</strong>orist, Werner<br />
Heisenberg, <strong>the</strong>y were <strong>the</strong> authors <strong>of</strong> <strong>the</strong> three most fundamental scientific results <strong>of</strong> <strong>the</strong><br />
century. Each man's discovery, moreover, established a pr<strong>of</strong>ound <strong>and</strong> disturbing limitation.<br />
<strong>Einstein</strong>'s <strong>the</strong>ory <strong>of</strong> relativity set a limit -<strong>the</strong> speed <strong>of</strong> light -to <strong>the</strong> flow <strong>of</strong> any<br />
3
information-bearing signal. And by defining <strong>time</strong> in terms <strong>of</strong> its measurement with clocks,<br />
he set a limit to <strong>time</strong> itself. It was no longer absolute but henceforth limited or relative to<br />
a frame <strong>of</strong> measurement. Heisenberg's uncertainty principle in quantum mechanics set a<br />
limit on our simultaneous knowledge <strong>of</strong> <strong>the</strong> position <strong>and</strong> momentum <strong>of</strong> <strong>the</strong> fundamental<br />
particles <strong>of</strong> matter. This was not just a restriction on what we can know: for Heisenberg it<br />
signified a limit to reality. Finally, Godel's incompleteness <strong>the</strong>oremó"<strong>the</strong> most significant<br />
ma<strong>the</strong>matical truth <strong>of</strong> <strong>the</strong> century," as it would soon be described in a ceremony at<br />
Harvard Universityóset a permanent limit on our knowledge <strong>of</strong> <strong>the</strong> basic truths <strong>of</strong><br />
ma<strong>the</strong>matics: The complete set <strong>of</strong> ma<strong>the</strong>matical truths will never be captured by any<br />
finite or recursive list <strong>of</strong> axioms that is fully formal. Thus, no mechanical device, no<br />
computer, will ever be able to exhaust <strong>the</strong> truths <strong>of</strong> ma<strong>the</strong>matics. It follows immediately,<br />
as Godel was quick to point out, that if we are able somehow to grasp <strong>the</strong> complete truth<br />
in this domain, <strong>the</strong>n we, or our minds, are not machines or computers. (Enthusiasts <strong>of</strong><br />
artificial intelligence were not amused.)<br />
<strong>Einstein</strong>, Godel, Heisenberg: three men whose fundamental scientific results opened up<br />
new horizons, paradoxically, by setting limits to thought or reality. Toge<strong>the</strong>r <strong>the</strong>y<br />
embodied <strong>the</strong> Zeitgeist, <strong>the</strong> spirit <strong>of</strong> <strong>the</strong> age. Mysteriously, each had reached an<br />
ontological conclusion about reality through <strong>the</strong> employment <strong>of</strong> an epistemic principle<br />
concerning knowledge. The dance or dialectic <strong>of</strong> knowledge <strong>and</strong> realityó <strong>of</strong> limit <strong>and</strong><br />
limitlessnessówould become a dominant <strong>the</strong>me <strong>of</strong> <strong>the</strong> twentieth century. Yet Godel's <strong>and</strong><br />
<strong>Einstein</strong>'s relations to <strong>the</strong>ir century were more uneasy than Heisenberg's.<br />
The Zeitgeist took root most famously in quantum mechanics. Here Godel <strong>and</strong> <strong>Einstein</strong><br />
would find <strong>the</strong>mselves in lonely opposition to Heisenberg, who, on <strong>the</strong> wrong side in <strong>the</strong><br />
war <strong>of</strong> nations, chose <strong>the</strong> winning team in <strong>the</strong> wars <strong>of</strong> physics. Heisenberg was a champion<br />
<strong>of</strong> <strong>the</strong> school <strong>of</strong> positivism, in quantum physics known as <strong>the</strong> Copenhagen interpretation in<br />
deference to Heisenberg's mentor, <strong>the</strong> Danish physicist Niels Bohr. What had been a mere<br />
heuristic principle in <strong>Einstein</strong>'s special relativityódeducing <strong>the</strong> nature <strong>of</strong> reality from<br />
limitations on what can be knownóbecame for Heisenberg a kind <strong>of</strong> religion, a religion that<br />
Godel <strong>and</strong> <strong>Einstein</strong> had no wish to join. Some, however, claimed to see in Godel's <strong>the</strong>orem<br />
itself an echo <strong>of</strong> Heisenberg's uncertainty principle. That group did not include Godel.<br />
<strong>Einstein</strong>, himself one <strong>of</strong> <strong>the</strong> great pioneers <strong>of</strong> quantum mechanics, had known <strong>and</strong> inspired<br />
Heisenberg in Germany. In 1911, in Prague, years before Heisenberg came on <strong>the</strong> scene,<br />
<strong>Einstein</strong> once pointed out to his colleague Philipp Frank <strong>the</strong> insane asylum in <strong>the</strong> park<br />
below his study <strong>and</strong> remarked, "Here you see that portion <strong>of</strong> lunatics who do not concern<br />
<strong>the</strong>mselves with quantum <strong>the</strong>ory." By <strong>Einstein</strong>'s lights, a bad situation had become even<br />
worse after Heisenberg. In an early encounter, Heisenberg, on <strong>the</strong> defensive against<br />
<strong>Einstein</strong>'s harangue against quantum mechanics, fought back: "When I objected that in [my<br />
approach] I had merely been applying <strong>the</strong> type <strong>of</strong> philosophy that he, too, had made <strong>the</strong><br />
basis <strong>of</strong> his special <strong>the</strong>ory <strong>of</strong> relativity, [<strong>Einstein</strong>] answered simply, 'Perhaps I did use such<br />
philosophy earlier, <strong>and</strong> also wrote it, but it is nonsense all <strong>the</strong> same.'"<br />
The two parted before <strong>the</strong> war, <strong>Einstein</strong> emigrating to <strong>the</strong> United States, Heisenberg<br />
remaining in Germany, to which he would remain loyal to <strong>the</strong> end. In Princeton,<br />
<strong>Einstein</strong>ópacifist, bohemian, socialist <strong>and</strong> Jewówas a man apart. To be sure, he found<br />
Godel, but toge<strong>the</strong>r <strong>the</strong>y remained isolated <strong>and</strong> alone, not least because <strong>of</strong> <strong>the</strong>ir<br />
opposition to Heisenberg's positivist <strong>world</strong>view, which ruled <strong>the</strong> intellectual scene even as
Heisenberg's fa<strong>the</strong>rl<strong>and</strong> was attempting to dominate <strong>the</strong> <strong>world</strong>. Godel <strong>and</strong> <strong>Einstein</strong> were<br />
not merely intellectual engineers, as so many <strong>of</strong> <strong>the</strong>ir brethren, inspired by positivism, had<br />
become, but philosopher-scientists. Ironically, while <strong>the</strong>ir stars had begun to wane, <strong>the</strong><br />
sheer size <strong>of</strong> <strong>the</strong>ir reputations had made <strong>the</strong>m unapproachable. Not to each o<strong>the</strong>r,<br />
however. "Godel," wrote <strong>the</strong>ir colleague Freeman Dyson, "was <strong>the</strong> only one <strong>of</strong> our<br />
colleagues who walked <strong>and</strong> talked on equal terms with <strong>Einstein</strong>."<br />
Their tastes, however, remained distinct. <strong>Einstein</strong>, a violinist, could never bring his friend<br />
to subject himself to <strong>the</strong> likes <strong>of</strong> Beethoven <strong>and</strong> Mozart. Godel, in turn, had no more<br />
success, surely, in dragging <strong>Einstein</strong> to Snow White <strong>and</strong> <strong>the</strong> Seven Dwarfs, his favorite<br />
movie. History,<br />
5<br />
sadly, does not record which <strong>of</strong> <strong>the</strong> seven dwarfs was Godel's favorite, but we do know why<br />
he favored fairy tales: "Only fables," he said, "present <strong>the</strong> <strong>world</strong> as it should be <strong>and</strong> as if it<br />
had meaning." (That meaning, <strong>of</strong> course, may be dark. It is not known whe<strong>the</strong>r Alan Turing<br />
acquired an affection for Snow White from Godel when he visited <strong>the</strong> institute in <strong>the</strong><br />
1930s, but some have heard an echo <strong>of</strong> <strong>the</strong> dark side <strong>of</strong> that tale in Turing's decision to end<br />
his life by eating a poisoned apple when, as a reward for his having broken <strong>the</strong> Enigma<br />
code <strong>of</strong> <strong>the</strong> German navy, <strong>the</strong> British government ordered him to receive hormone<br />
injections as a "cure" for his homosexuality.)<br />
<strong>Einstein</strong>, before fleeing Germany, had already become a refugee from ma<strong>the</strong>matics. He<br />
later said that he could not find, in that garden <strong>of</strong> many paths, <strong>the</strong> one to what is<br />
fundamental. He turned to <strong>the</strong> more earthly domain <strong>of</strong> physics, where <strong>the</strong> way to <strong>the</strong><br />
essential was, he thought, clearer. His disdain for ma<strong>the</strong>matics earned him <strong>the</strong> nickname<br />
"lazy dog" from his teacher Hermann Minkowski (who would soon recast <strong>the</strong> lazy dog's<br />
special relativity into its characteristic four-dimensional form). "You know, once you start<br />
calculating," <strong>Einstein</strong> would quip, "you shit yourself up before you know it." Godel's<br />
journey, in contrast, was in <strong>the</strong> opposite direction. Having befriended Godel, <strong>Einstein</strong><br />
commented that he knew now, at last, that in ma<strong>the</strong>matics too one could find a path to<br />
<strong>the</strong> fundamental. In befriending <strong>Einstein</strong>, Godel was reawakened to his early interest in<br />
physics. On <strong>the</strong>ir long walks home from <strong>the</strong> <strong>of</strong>fice, <strong>Einstein</strong>, forever cheerful, would<br />
attempt to raise <strong>the</strong> spirits <strong>of</strong> <strong>the</strong> gloomy <strong>and</strong> pessimistic Godel by recounting his latest<br />
insights on general relativity. Sadly, however, pessimism blossomed into paranoia. The<br />
economist Oskar Morgenstern, calling one day on his good friend, was shocked to find <strong>the</strong><br />
great Godel hiding in <strong>the</strong> cellar behind <strong>the</strong> furnace.<br />
From those long walks that <strong>Einstein</strong> <strong>and</strong> Godel shared, from <strong>the</strong>ir endless discussions,<br />
something beautiful would soon be born. The scene was pregnant with possibility. Time,<br />
which has taunted thinkers from Plato to Saint Augustine to Kant, had finally met its match<br />
in <strong>Einstein</strong>. While <strong>the</strong> U-boats <strong>of</strong> his former fa<strong>the</strong>rl<strong>and</strong> were stalking <strong>the</strong> Allied fleet,<br />
this most un-German <strong>of</strong> Germans was hunting a more elusive prey. He had amazed <strong>the</strong><br />
<strong>world</strong> decades earlier when he alone succeeded in capturing <strong>time</strong> itself in <strong>the</strong> equations <strong>of</strong>
elativity. "Every boy in <strong>the</strong> streets <strong>of</strong> Gottingen," his countryman David Hilbert wrote,<br />
"underst<strong>and</strong>s more about four-dimensional geometry than <strong>Einstein</strong>. Yet, in spite <strong>of</strong> that,<br />
<strong>Einstein</strong> did <strong>the</strong> work <strong>and</strong> not <strong>the</strong> ma<strong>the</strong>maticians." Relativity had rendered <strong>time</strong>, <strong>the</strong> most<br />
elusive <strong>of</strong> beings, manageable <strong>and</strong> docile by transforming it into a fourth dimension <strong>of</strong><br />
space, or ra<strong>the</strong>r, <strong>of</strong> relativistic space-<strong>time</strong>. Sharing with Godel his latest thoughts on <strong>the</strong><br />
four-dimensional universe <strong>of</strong> space-<strong>time</strong> that he himself had conjured into being, <strong>Einstein</strong><br />
was sowing <strong>the</strong> seeds <strong>of</strong> relativity in <strong>the</strong> mind <strong>of</strong> a thinker who would later be described as<br />
a combination <strong>of</strong> <strong>Einstein</strong> <strong>and</strong> Kafka.<br />
If <strong>Einstein</strong> had succeeded in transforming <strong>time</strong> into space, Godel would perform a trick yet<br />
more magical: He would make <strong>time</strong> disappear. Having already rocked <strong>the</strong> ma<strong>the</strong>matical<br />
<strong>world</strong> to its foundations with his incompleteness <strong>the</strong>orem, Godel now took aim at <strong>Einstein</strong><br />
<strong>and</strong> relativity. Wasting no <strong>time</strong>, he announced in short order his discovery <strong>of</strong> new <strong>and</strong><br />
unsuspected cosmological solutions to <strong>the</strong> field equations <strong>of</strong> general relativity, solutions in<br />
which <strong>time</strong> would undergo a shocking transformation. The ma<strong>the</strong>matics, <strong>the</strong> physics <strong>and</strong><br />
<strong>the</strong> philosophy <strong>of</strong> Godel's results were all new. In <strong>the</strong> possible <strong>world</strong>s governed by <strong>the</strong>se<br />
new cosmological solutions, <strong>the</strong> so-called rotating or Godel universes, it turned out that<br />
<strong>the</strong> space-<strong>time</strong> structure is so greatly warped or curved by <strong>the</strong> distribution <strong>of</strong> matter that<br />
<strong>the</strong>re exist <strong>time</strong>like future-directed paths by which a spaceship, if it travels fast<br />
enoughó<strong>and</strong> Godel worked out <strong>the</strong> precise speed <strong>and</strong> fuel requirements, omitting only <strong>the</strong><br />
lunch menuócan penetrate into any region <strong>of</strong> <strong>the</strong> past, present or future.<br />
Godel, <strong>the</strong> union <strong>of</strong> <strong>Einstein</strong> <strong>and</strong> Kafka, had for <strong>the</strong> first <strong>time</strong> in human history proved,<br />
from <strong>the</strong> equations <strong>of</strong> relativity, that <strong>time</strong> travel was not a philosopher's fantasy but a<br />
scientific possibility. Yet again he had somehow contrived, from within <strong>the</strong> very heart <strong>of</strong><br />
ma<strong>the</strong>matics, to drop a bomb into <strong>the</strong> laps <strong>of</strong> <strong>the</strong> philosophers. The fallout, however, from<br />
this ma<strong>the</strong>matical bomb was even more perilous than that from<br />
7<br />
<strong>the</strong> incompleteness <strong>the</strong>orem. Godel was quick to point out that if we can revisit <strong>the</strong> past,<br />
<strong>the</strong>n it never really "passed." But a <strong>time</strong> that fails to pass is no <strong>time</strong> at all. <strong>Einstein</strong> saw at<br />
once that if Godel was right, he had not merely domesticated <strong>time</strong>: he had killed it. Time,<br />
"that mysterious <strong>and</strong> seemingly self-contradictory being," as Godel put it, "which, on <strong>the</strong><br />
o<strong>the</strong>r h<strong>and</strong>, seems to form <strong>the</strong> basis <strong>of</strong> <strong>the</strong> <strong>world</strong>'s <strong>and</strong> our own existence," turned out in<br />
<strong>the</strong> end to be <strong>the</strong> <strong>world</strong>'s greatest illusion. In a word, if <strong>Einstein</strong>'s relativity was real, <strong>time</strong><br />
itself was merely ideal. The fa<strong>the</strong>r <strong>of</strong> relativity was shocked. Though he praised Godel for<br />
his great contribution to <strong>the</strong> <strong>the</strong>ory <strong>of</strong> relativity, he was fully aware that <strong>time</strong>, that elusive<br />
prey, had once again slipped his net.<br />
But now something truly amazing took place: nothing. Although in <strong>the</strong> immediate<br />
aftermath <strong>of</strong> Godel's discoveries a few physicists bestirred <strong>the</strong>mselves to refute him <strong>and</strong>,<br />
when this failed, tried to generalize <strong>and</strong> explore his results, this brief flurry <strong>of</strong> interest<br />
soon died down. Within a few years <strong>the</strong> deep footprints in intellectual history traced by<br />
Godel <strong>and</strong> <strong>Einstein</strong> in <strong>the</strong>ir long walks home had disappeared, dispersed by <strong>the</strong> harsh winds<br />
<strong>of</strong> fashion <strong>and</strong> philosophical prejudice. A conspiracy <strong>of</strong> silence descended on <strong>the</strong> <strong>Einstein</strong>-<br />
Godel friendship <strong>and</strong> its scientific consequences.
An association no less remarkable than <strong>the</strong> friendship between Michelangelo <strong>and</strong><br />
Leonardoóif such had occurredóhas simply vanished from sight. To this day, not only is <strong>the</strong><br />
man on <strong>the</strong> street unaware <strong>of</strong> <strong>the</strong> intimate relationship between <strong>the</strong>se two giants <strong>of</strong> <strong>the</strong><br />
twentieth century, even <strong>the</strong> most exhaustive intellectual biographies <strong>of</strong> <strong>Einstein</strong> ei<strong>the</strong>r<br />
omit all mention <strong>of</strong> this friendship or at best begrudge a sentence or two. Whereas a whole<br />
industry has grown up in search <strong>of</strong> Lieserl, <strong>the</strong> "love child" <strong>of</strong> <strong>Einstein</strong>'s first marriage, <strong>the</strong><br />
child <strong>of</strong> <strong>the</strong> imagination that was born <strong>of</strong> <strong>the</strong> friendship <strong>of</strong> <strong>Einstein</strong> <strong>and</strong> Godel has been<br />
ab<strong>and</strong>oned.<br />
Only in <strong>the</strong> last few years has this child, <strong>the</strong> Godel universe, received any glimmer <strong>of</strong><br />
recognition. This comes from <strong>the</strong> redoubtable Stephen Hawking. Revisiting <strong>the</strong> rotating<br />
Godel universe, Hawking was moved to deliver <strong>the</strong> highest <strong>of</strong> compliments. So threatening<br />
did<br />
he find results like Godel's demonstrating <strong>the</strong> consistency <strong>of</strong> <strong>time</strong> travel with <strong>the</strong> laws <strong>of</strong><br />
relativity, that he put forward what amounts to an anti-Godel postulate. If accepted,<br />
Hawking's famous "chronology protection conjecture" would precisely negate Godel's<br />
contribution to relativity. So physically unacceptable did Hawking find conclusions like<br />
Godel's that he felt compelled to propose what looks like an ad hoc modification <strong>of</strong> <strong>the</strong><br />
laws <strong>of</strong> nature that would have <strong>the</strong> effect <strong>of</strong> ruling out <strong>the</strong> Godel universe as a genuine<br />
physical possibility.<br />
Hawking's attempt to neutralize <strong>the</strong> Godel universe shows how dangerous it is to break <strong>the</strong><br />
conspiracy <strong>of</strong> silence that has shrouded <strong>the</strong> Godel-<strong>Einstein</strong> connection. Not only does this<br />
mysterious silence hide from <strong>the</strong> <strong>world</strong> one <strong>of</strong> <strong>the</strong> most moving <strong>and</strong> consequential<br />
friendships in <strong>the</strong> history <strong>of</strong> science, it also keeps <strong>the</strong> <strong>world</strong> from realizing <strong>the</strong> full<br />
implications <strong>of</strong> <strong>the</strong> <strong>Einstein</strong> revolution. It is one thing to overturn, as <strong>Einstein</strong> did, Newton's<br />
centuries-old conception <strong>of</strong> <strong>the</strong> absoluteness <strong>and</strong> independence <strong>of</strong> space <strong>and</strong> <strong>time</strong>. It is<br />
quite ano<strong>the</strong>r to demonstrate that <strong>time</strong> is not just relative but ideal. Unlike <strong>Einstein</strong>, a<br />
classicist who forever sought continuity with <strong>the</strong> past, Godel was at heart an ironist, a<br />
truly subversive thinker. With his incompleteness <strong>the</strong>orem he had shaken <strong>the</strong> foundations<br />
<strong>of</strong> ma<strong>the</strong>matics, prompting <strong>the</strong> great ma<strong>the</strong>matician David Hilbert to propose a new law <strong>of</strong><br />
logic just to refute Godel's results. The Godel universe, correctly understood, shares with<br />
<strong>the</strong> incompleteness <strong>the</strong>orem an underlying methodology <strong>and</strong> purpose. It is a bomb, built<br />
from cosmology's most cherished materials, lobbed into <strong>the</strong> foundations <strong>of</strong> physics.<br />
In <strong>the</strong> footsteps <strong>of</strong> Godel <strong>and</strong> <strong>Einstein</strong>, <strong>the</strong>n, can be heard an echo <strong>of</strong> <strong>the</strong> Zeitgeist, a clue<br />
to <strong>the</strong> secret <strong>of</strong> <strong>the</strong> great <strong>and</strong> terrible twentieth century, a century that, like <strong>the</strong><br />
seventeenth, will go down in history as one <strong>of</strong> genius. The residents <strong>of</strong> Blue Hill,<br />
preoccupied with war <strong>and</strong> <strong>the</strong> enemy out at sea, had failed to take <strong>the</strong> full measure <strong>of</strong><br />
<strong>the</strong>ir man.<br />
2 A German Bias for Metaphysics
The German man <strong>of</strong> science was a philosopher. J.T. MERZ<br />
It is a remarkable fact.. . that at least in one point relativity <strong>the</strong>ory has furnished a very<br />
striking confirmation <strong>of</strong> Kantian doctrines.<br />
KURT GODEL<br />
Physically <strong>the</strong>y were opposites. Godel, thin to <strong>the</strong> point <strong>of</strong> emacia-tion, his spectral body<br />
hidden, even in <strong>the</strong> heat <strong>of</strong> summer, in overcoat <strong>and</strong> scarf, gaunt, harrowed, <strong>and</strong> haunted,<br />
peering through thick glasses like an owl from ano<strong>the</strong>r dimension, could not fail to arouse<br />
suspicion. Early in life he had come to <strong>the</strong> conclusion that <strong>the</strong> less food one ate <strong>the</strong> better.<br />
This dubious insight he carried out with ruthless consistency, unencumbered by <strong>the</strong> excess<br />
baggage <strong>of</strong> common sense, a faculty he approached life <strong>without</strong>. His preconception, fueled<br />
by hypochondria that grew out <strong>of</strong> childhood rheumatic fever <strong>and</strong> by paranoia about <strong>the</strong><br />
intentions <strong>of</strong> doctors, developed into a neurosis that would eventually take his life. During<br />
several periods <strong>of</strong> extreme stress he was confined to sanatoria, from one <strong>of</strong> which, by<br />
some accounts, he enlisted <strong>the</strong> services <strong>of</strong> his wife to escape. At his death he weighed a<br />
mere sixty-five pounds.<br />
<strong>Einstein</strong>, in contrast, whose sanity was never in question, was as satisfied by a good<br />
sausage as by a good <strong>the</strong>orem. He had a taste for solid German cooking, which he<br />
consumed with relish, topped <strong>of</strong>f by his omnipresent pipe. Friends <strong>and</strong> wives would be<br />
swept aside in <strong>the</strong> current <strong>of</strong><br />
his turbulent life, but his pipe never left him. Late in life he was <strong>the</strong> proud owner <strong>of</strong> a<br />
respectable pr<strong>of</strong>essorial paunch. "I have firmly resolved," he wrote his wife Elsa, "to bite<br />
<strong>the</strong> dust, when my <strong>time</strong> comes, with <strong>the</strong> minimum <strong>of</strong> medical assistance, <strong>and</strong> until <strong>the</strong>n to<br />
sin cheerfully ... smoke like a chimney, work like a beaver, eat <strong>without</strong> thought or choice,<br />
<strong>and</strong> walk only in agreeable company, in o<strong>the</strong>r words, rarely."<br />
With brown hair <strong>and</strong> blue eyes, Godel measured barely five feet six. This number came as<br />
a surprise to his colleagues. His intellectual presence was so great that his modest height<br />
<strong>of</strong>ten went unnoticed. His frailty, however, was obvious. "Of course he has no children,"<br />
<strong>the</strong> proprietor <strong>of</strong> <strong>the</strong> Blue Hill Inn said <strong>of</strong> Godel; "he hasn't <strong>the</strong> strength to make babies."<br />
He did, however, have in his youth <strong>the</strong> strength to pursue women. "There is no doubt,"<br />
wrote a college friend, Olga Taussky-Todd, "about <strong>the</strong> fact that Godel had a liking for<br />
members <strong>of</strong> <strong>the</strong> opposite sex, <strong>and</strong> he made no secret about this fact." Godel, she went on,<br />
was not beyond showing <strong>of</strong>f his acquaintance with a pretty face. Taussky-Todd herself, to<br />
her dismay, was once enlisted to come to <strong>the</strong> ma<strong>the</strong>matical aid <strong>of</strong> one such young woman<br />
who in turn was trying to make an impression on Godel. Was this interest in women<br />
confined to Godel's youth? Not if his wife, Adele, is to believed. Teasing her husb<strong>and</strong>, she<br />
quipped that <strong>the</strong> Institute for Advanced Studyó which she liked to call an<br />
Altersversorgungsheim, or home for elderly pensionersówas packed with pretty female<br />
students who lined up outside <strong>the</strong> <strong>of</strong>fice doors <strong>of</strong> <strong>the</strong> great pr<strong>of</strong>essors. <strong>Einstein</strong>, who with<br />
well-knit limbs <strong>and</strong> hardy disposition measured five feet nine, did actually make babies, in<br />
<strong>and</strong> out <strong>of</strong> wedlock. Early <strong>and</strong> late, <strong>the</strong> constraints <strong>of</strong> marriage did not hamper him, even
as his discoveries in physics were unconstrained by <strong>the</strong> conventions <strong>of</strong> classical physics.<br />
The event itself <strong>of</strong> entering into <strong>the</strong> institution <strong>of</strong> marriage bore <strong>the</strong> unmistakable stamp<br />
<strong>of</strong> unconventionality: Though <strong>Einstein</strong> wished to marry his cousin Elsa, he desired even<br />
more strongly to marry her twenty-year-old daughter, Ilsa. "Albert himself," wrote <strong>the</strong><br />
flustered daughter to a friend, "is refusing to take any decision; he is prepared to marry<br />
ei<strong>the</strong>r Mama or me."<br />
11<br />
Clothing too, like marriage, he considered a bourgeois affectation whose strictures he did<br />
his best to circumvent, spurning socks, tie, <strong>and</strong> belt whenever possible. Hair uncut <strong>and</strong><br />
unkempt, he could embarrass a female guest when his robe, with nothing underneath, fell<br />
open, <strong>and</strong> <strong>the</strong>n express surprise at her consternation. Bursting with <strong>the</strong> juices <strong>of</strong> life, he<br />
was an indefatigable optimist whose faith in common sense <strong>and</strong> human nature survived<br />
even <strong>the</strong> Holocaust.<br />
Godel, in contrast, was in <strong>the</strong> fullest sense <strong>of</strong> <strong>the</strong> phrase "buttoned up." Dressed severely<br />
even in <strong>the</strong> summer heat, he was <strong>the</strong> very model <strong>of</strong> dour reserve: gloomy, pessimistic,<br />
averse to all human contact except for <strong>the</strong> closest <strong>of</strong> friends <strong>and</strong> <strong>the</strong> direst <strong>of</strong> intellectual<br />
necessities. The institute still echoes with stories <strong>of</strong> Godel's foolpro<strong>of</strong> method for evading a<br />
rendezvous. He would carefully arrange a precise location in space <strong>and</strong> <strong>time</strong> for <strong>the</strong><br />
projected meeting. With <strong>the</strong>se coordinates in place, he confided to friends, he had<br />
achieved certainty as to where not to be when <strong>the</strong> appointed <strong>time</strong> arrived. Yet this<br />
method had its limita-tions. Finding himself trapped at an unavoidable institute tea, he<br />
negotiated <strong>the</strong> territory between guests, noted <strong>the</strong> ma<strong>the</strong>matician Paul Halmos in his<br />
memoirs, with maximum attention to <strong>the</strong> goal <strong>of</strong> avoiding any possibility <strong>of</strong> physical<br />
contact.<br />
Against every stereotype <strong>of</strong> <strong>the</strong> pure ma<strong>the</strong>maticianó<strong>and</strong> particularly one who, like Godel,<br />
had studied <strong>and</strong> taught in ViennaóGodel was all but allergic to <strong>the</strong> masters <strong>of</strong> classical<br />
music, preferring instead light classics <strong>and</strong> operettas, <strong>and</strong> was even more so to <strong>the</strong><br />
abstractions <strong>of</strong> modern art. He was untouched by intellectual snobbery <strong>and</strong> made plain his<br />
love <strong>of</strong> fairy tales. His fondness for Walt Disney cartoons was no secret to his friends.<br />
Comedies, however, he disliked.<br />
<strong>Einstein</strong> was consumed by his passion for <strong>the</strong> great Austrian-German classicists, Bach,<br />
Mozart <strong>and</strong> Beethoven, but especially Mozart. His friend <strong>and</strong> biographer, Philipp Frank,<br />
<strong>of</strong>fered some shrewd observations about what made Mozart special. What passed for many<br />
as a sign <strong>of</strong> <strong>Einstein</strong>'s cynicism was for Frank an expression ra<strong>the</strong>r <strong>of</strong> <strong>Einstein</strong>'s urge "to<br />
make <strong>the</strong> serious things in <strong>the</strong> <strong>world</strong> tolerable by means <strong>of</strong> a playful guise." But this also<br />
characterizes much <strong>of</strong> Mozart's music,<br />
"which might also be called 'cynical.' It does not take our tragic <strong>world</strong> very seriously."<br />
<strong>Einstein</strong> was always ready to perform Mozart at a moment's notice on his beloved violin,<br />
which he played, myths notwithst<strong>and</strong>ing, very well. He "was an experienced sight reader,"<br />
wrote <strong>the</strong> pr<strong>of</strong>essional violinist Boris Schwartz, "with a steady rhythm, excellent
intonation, a clear <strong>and</strong> pure tone, <strong>and</strong> a minimum <strong>of</strong> vibrato." Only his pipe was as familiar<br />
a companion. Violin <strong>and</strong> pipe: <strong>the</strong>se will be forever <strong>the</strong> icons <strong>of</strong> <strong>the</strong> great scientist,<br />
toge<strong>the</strong>r with his tousled hair.<br />
Godel, as is clear from photographs, was meticulously clean-shaven, every hair combed in<br />
place, whereas as every schoolchild knows, a small brush <strong>of</strong> a moustache floated above<br />
<strong>Einstein</strong>'s full lips. Combs, moreover, in <strong>the</strong> <strong>Einstein</strong> household were verboten. With <strong>the</strong><br />
visual signature comes <strong>the</strong> acoustic: When something or someone struck <strong>Einstein</strong> funny, a<br />
huge belly laugh welled up inside <strong>the</strong> scientist <strong>and</strong> erupted like a volcano that shook his<br />
entire body. More than a few <strong>time</strong>s it shook up as well <strong>the</strong> surprised object <strong>of</strong> this<br />
laughter, who realized too late its full meaning. Godel, in contrast, had a s<strong>of</strong>t, high<br />
pitched chuckle, more a musing to himself on <strong>the</strong> ironies <strong>of</strong> <strong>the</strong> universe than a fullthroated<br />
laugh. Raising <strong>the</strong> pitch <strong>of</strong> his voice at <strong>the</strong> end <strong>of</strong> each sentence <strong>and</strong> trailing <strong>of</strong>f<br />
into silence, he left his audience with a feeling <strong>of</strong> detached query. (As a child <strong>of</strong> four he<br />
had been nicknamed Herr Warum, Mr. Why. "Why is your nose so large?" he asked an<br />
embarrassed guest.)<br />
By age a generation apart, <strong>Einstein</strong> <strong>and</strong> Godel shared an anniversary by one degree <strong>of</strong><br />
separation. The year <strong>of</strong> <strong>Einstein</strong>'s birth, 1879, was that <strong>of</strong> Godel's mo<strong>the</strong>r, Marianne. (It<br />
was also <strong>the</strong> year that saw publication <strong>of</strong> Gottlob Frege's masterpiece, Begriffsschrift, <strong>and</strong><br />
thus <strong>the</strong> birth <strong>of</strong> modern ma<strong>the</strong>matical logic, a field Godel would raise to unparalleled<br />
heights.) They were born into different religions, <strong>Einstein</strong> a Jew, Godel baptized a<br />
Lu<strong>the</strong>ran. Skeptical <strong>of</strong> <strong>the</strong> faith <strong>of</strong> his fa<strong>the</strong>rs in his youth, with <strong>the</strong> rise <strong>of</strong> Nazism <strong>Einstein</strong><br />
rediscovered what he called his "tribal companions" <strong>and</strong> became a passionate, if thorny,<br />
Zionist. He never did, however, embrace <strong>the</strong> transcendent God <strong>of</strong> his people, accounting<br />
himself ra<strong>the</strong>r a "deeply religious unbeliever." His hero was not Moses but Spinoza, <strong>the</strong><br />
pan<strong>the</strong>ist <strong>and</strong> excommunicant, <strong>and</strong> he re-<br />
13<br />
fleeted this predilection throughout a scientific career in which such seemingly<br />
transcendent, untouchable things as space, <strong>time</strong> <strong>and</strong> light were revealed to be fully<br />
immanent <strong>and</strong> subject to physical causality.<br />
Godel was not a pan<strong>the</strong>ist but ra<strong>the</strong>r a self-described <strong>the</strong>ist, "following Leibniz," he said,<br />
"not Spinoza." Spinoza's God, he said, "is less than a person. Mine is more than a person. . .<br />
. He can play <strong>the</strong> role <strong>of</strong> a person." He noted <strong>the</strong> <strong>of</strong>t-neglected fact that <strong>the</strong> founders <strong>of</strong><br />
modern science were not a<strong>the</strong>ists. More radical than <strong>Einstein</strong>, he belonged to a rare breed<br />
<strong>of</strong> thinker: <strong>the</strong> true believers. Whereas "ninety per cent <strong>of</strong> philosophers <strong>the</strong>se days," he<br />
would say, "consider it <strong>the</strong> business <strong>of</strong> philosophy to knock religion out <strong>of</strong> people's heads,"<br />
he would exploit <strong>the</strong> machinery <strong>of</strong> modern logic to reconstruct Leibniz's famous<br />
"ontological argument" for <strong>the</strong> existence <strong>of</strong> God. Though not a Jew, he was never<strong>the</strong>less<br />
taken for one. In a Vienna teeming with Nazis, his wife once employed her umbrella to<br />
fend <strong>of</strong>f a group <strong>of</strong> rowdies who were jostling Godel, mistaking him for a Jew.
The misattribution was not confined to Nazis. While at <strong>the</strong> Insti-tute for Advanced Study in<br />
Princeton, Godel was for a <strong>time</strong> a member <strong>of</strong> an eliteóa very eliteódiscussion group,<br />
consisting <strong>of</strong> himself, <strong>Einstein</strong>, <strong>the</strong> German physicist Wolfgang Pauli, <strong>and</strong> Bertr<strong>and</strong> Russell,<br />
one <strong>of</strong> <strong>the</strong> founders <strong>of</strong> modern "analytical" philosophy. Russell reacted badly to <strong>the</strong><br />
discussions, finding <strong>the</strong>m too philosophical in <strong>the</strong> "old-fashioned sense." (The failings <strong>of</strong> an<br />
entire century are crystallized in this fact.) In an unpleasant aside he vented his<br />
frustration: "All three ol <strong>the</strong> o<strong>the</strong>rs were Jews <strong>and</strong> exiles, <strong>and</strong> in intention, cosmopolitans,"<br />
he wrote later, "[who shared] a German bias for metaphysics." "I am not a Jew," Godel<br />
would respond later, "even though I don't think this question is <strong>of</strong> any importance." He<br />
admired <strong>the</strong> tenacity <strong>of</strong> <strong>the</strong> Jewish people. "Kurt had a friendly attitude toward people <strong>of</strong><br />
<strong>the</strong> Jewish faith," said his friend Olga Taussky-Todd. "And once he said out <strong>of</strong> <strong>the</strong> blue that<br />
it was a miracle how, <strong>without</strong> a country, <strong>the</strong>y were able to survive for thous<strong>and</strong>s <strong>of</strong> years,<br />
almost like a nation, merely by <strong>the</strong>ir faith." <strong>Einstein</strong>, wishing to eliminate <strong>the</strong> Jewish need<br />
for miracles, pushed hard for most <strong>of</strong> his life for a homel<strong>and</strong> for <strong>the</strong> nation that had<br />
survived so many years <strong>without</strong> a home. (Never too concerned with consistencyóunlike his<br />
logician companionóhe was undisturbed by his earlier briefs against nationalism.)<br />
Seeing <strong>the</strong> h<strong>and</strong>writing on <strong>the</strong> wall, <strong>Einstein</strong> <strong>and</strong> Godel ab<strong>and</strong>oned comfortable university<br />
positions in Berlin <strong>and</strong> Vienna when <strong>the</strong> Nazis came to power in <strong>the</strong> 1930s. At <strong>the</strong> zenith <strong>of</strong><br />
<strong>the</strong>ir powers, <strong>the</strong>y were snatched up by <strong>the</strong> newly formed Institute for Advanced Study in<br />
Princeton, good European root stock for <strong>the</strong> vineyards <strong>of</strong> <strong>the</strong> new <strong>world</strong>. Toge<strong>the</strong>r <strong>the</strong>y<br />
w<strong>and</strong>ered <strong>the</strong> narrow streets <strong>of</strong> a cloistered <strong>and</strong> provincial academic town, <strong>the</strong>y who once<br />
strode <strong>the</strong> boulevards <strong>of</strong> <strong>the</strong> great capitals <strong>of</strong> Europe, centerpieces <strong>of</strong> a once great<br />
civilization crashing down in ruins. Strangers to each o<strong>the</strong>r in Europe, it was not until 1942<br />
that <strong>the</strong>y began <strong>the</strong> friendship that lasted until <strong>Einstein</strong>'s death in 1955, a loss from which<br />
Godel never recovered. <strong>Einstein</strong>, a German Jew in a nest <strong>of</strong> Wasps, felt out <strong>of</strong> place in<br />
Princeton. Godel, already a recluse, resented less <strong>the</strong> isolation, although his wife, Adele,<br />
suffered. A cafe dancer in Vienna, Adele was out <strong>of</strong> her element in <strong>the</strong> elite college town.<br />
When an opportunity opened up to move to Harvard, she pleaded for <strong>the</strong> more<br />
cosmopolitan Cambridge, Massachusetts. But Godel was not prepared to accept an <strong>of</strong>fer<br />
where teaching was required.<br />
What attraction could have drawn toge<strong>the</strong>r such opposites as <strong>Einstein</strong> <strong>and</strong> Godel? Certainly<br />
not scientific agreement. This was not a case <strong>of</strong> <strong>the</strong> strong force uniting like-charged<br />
protons in <strong>the</strong> atomic nucleus. The charges here were opposite. Godel opined, in fact, that<br />
one <strong>of</strong> <strong>the</strong> reasons <strong>Einstein</strong> enjoyed his company was precisely because he made no<br />
attempt to hide his very different views, not just in politics <strong>and</strong> philosophy but in physics.<br />
"I frequently held an opinion," Godel said, "counter to <strong>Einstein</strong>'s <strong>and</strong> made no attempt to<br />
conceal my disagreement." <strong>Einstein</strong>'s failed search, for example, for a unified field <strong>the</strong>ory<br />
to unite <strong>the</strong> domain <strong>of</strong> quantum mechanics with general relativity, which occupied much <strong>of</strong><br />
<strong>the</strong>ir discussions, was a particular target <strong>of</strong> Godel's skepticism.<br />
Indeed, Godel was skeptical <strong>of</strong> <strong>the</strong> ultimate significance <strong>of</strong> natural science itself, despite<br />
its great success in enabling us (as he put it) to<br />
15
uild TVs <strong>and</strong> bombs. At a faculty dinner at <strong>the</strong> institute <strong>the</strong> young John Bahcall, having<br />
introduced himself as a new astrophysicist on <strong>the</strong> faculty, was taken aback when Godel<br />
replied flatly that he didn't believe in natural science. By Godel's lights, physics had taken<br />
<strong>the</strong> wrong turn centuries ago when it chose to follow <strong>the</strong> path laid by <strong>the</strong> naturalistically<br />
minded British empiricist Isaac Newton, ra<strong>the</strong>r than that <strong>of</strong> <strong>the</strong> German idealist Gottfried<br />
Leibniz. Godel's fascination with Leibniz was boundless, prompting a ma<strong>the</strong>matical<br />
colleague, Paul Erdos, to <strong>of</strong>fer a rebuke: "You became a ma<strong>the</strong>matician," he told Godel,<br />
"so that people should study you, not that you should study Leibniz." Godel even succeeded<br />
in transferring his own paranoia to Leibniz, arguing at length that some <strong>of</strong> his hero's crucial<br />
manuscripts had been secretly destroyed by "those who do not want man to become more<br />
intelligent." "You have a vicarious persecution complex," replied his friend Karl Menger, "on<br />
Leibniz's behalf." Menger, like most intellectuals a child <strong>of</strong> <strong>the</strong> Enlightenment, went on to<br />
ask why none <strong>of</strong> Voltaire's papers had been destroyed. "Who ever became more<br />
intelligent," Godel answered, "by reading Voltaire?"<br />
Fur<strong>the</strong>r separating <strong>Einstein</strong> from Godel was <strong>the</strong> fact that <strong>Einstein</strong> never fully resolved his<br />
native suspicion <strong>of</strong> ma<strong>the</strong>matics. To <strong>the</strong> end, <strong>the</strong> great physicist favored his cherished<br />
physical intuitions. Even though it was precisely Minkowski's ma<strong>the</strong>matical reworking <strong>of</strong><br />
special relativity in terms <strong>of</strong> four-dimensional geometry (which <strong>Einstein</strong> resented at <strong>the</strong><br />
<strong>time</strong>) that led to <strong>the</strong> ma<strong>the</strong>matical abstractions <strong>of</strong> general relativity, <strong>the</strong> physicist<br />
remained forever wary <strong>of</strong> being led by <strong>the</strong> nose by ma<strong>the</strong>maticians. He confessed once to<br />
being suspicious <strong>of</strong> a new move in general relativity that he said he could reach only<br />
ma<strong>the</strong>matically (i.e., not intuitively). Godel, in contrast, always felt most secure when he<br />
had formulated a problem in symbolic, ma<strong>the</strong>matical terms. "If you had a particular<br />
problem in mind," wrote Taussky-Todd, "he would start by writing it down in symbols." Yet<br />
Godel also believed, famously, that in ma<strong>the</strong>matics too <strong>the</strong>re are intuitions (a doctrine for<br />
which logicians still have not forgiven him). For Godel <strong>the</strong> equations <strong>of</strong> ma<strong>the</strong>matics, as<br />
opposed to <strong>the</strong> counsels <strong>of</strong> common<br />
sense, would lead us into <strong>the</strong> promised l<strong>and</strong> <strong>of</strong> new insights, whereas for <strong>Einstein</strong>, it was<br />
precisely common sense that was <strong>the</strong> final touchstone for assessing what <strong>the</strong><br />
ma<strong>the</strong>maticians had to <strong>of</strong>fer.<br />
Beneath <strong>the</strong>se disagreements, however, or beyond <strong>the</strong>m, <strong>the</strong>re was much that united <strong>the</strong><br />
two minds. Both had grown to maturity in <strong>the</strong> ancient capitals <strong>of</strong> Europe. They were heirs<br />
to <strong>the</strong> great Austrian-Germanic philosophical tradition, with "philosophy" understood here<br />
in its widest sense. Prejudice aside, Russell's comment on <strong>the</strong> "German bias for<br />
metaphysics" had not really missed its mark. Raised in this culture, <strong>the</strong> composer Gustav<br />
Mahler had kept, quite naturally, in his "composing hut," volumes by both Wolfgang Goe<strong>the</strong><br />
<strong>and</strong> Immanuel Kant. It comes as no surprise, <strong>the</strong>n, that Godel <strong>and</strong> <strong>Einstein</strong> cut <strong>the</strong>ir<br />
philosophical teeth on <strong>the</strong> great works <strong>of</strong> Kant, whose fingerprints can be clearly discerned<br />
throughout <strong>the</strong> work <strong>of</strong> each. For Godel, his writings on <strong>Einstein</strong> were as much an<br />
expression <strong>of</strong> his interest in Kant's <strong>and</strong> Leibniz's ideas <strong>of</strong> <strong>time</strong> as <strong>of</strong> his personal association<br />
with <strong>Einstein</strong>. He would characterize his own contributions to relativity <strong>the</strong>oryóto <strong>Einstein</strong>'s<br />
consternationóas showing that relativity had "verified" Kant's philosophical idealism.
<strong>Einstein</strong>'s own reading <strong>of</strong> Kant, in turn, did much to free him from <strong>the</strong> excessive reliance<br />
on immediate sensory data to which many <strong>of</strong> his contemporaries, especially Ernst Mach,<br />
were susceptible. At <strong>the</strong> tender age <strong>of</strong> sixteen <strong>Einstein</strong> had reread Kant's weighty<br />
masterpiece, The Critique <strong>of</strong> Pure Reasonó<strong>the</strong> same age at which Godel too read Kantó<strong>and</strong><br />
as a student at <strong>the</strong> Technical Institute in Zurich he had enrolled in a course on Kant. Still,<br />
he <strong>of</strong>ten made light <strong>of</strong> <strong>the</strong> tendency, especially strong in Germany, to venerate <strong>the</strong><br />
German master. "Kant," he said, "is a sort <strong>of</strong> highway with lots <strong>and</strong> lots <strong>of</strong> milestones. Then<br />
all <strong>the</strong> little dogs come <strong>and</strong> each deposits his bit at <strong>the</strong> milestones."<br />
"At <strong>the</strong> Institute in Princeton," Gerald Holton has noted, "[<strong>Einstein</strong>'s] favorite topic <strong>of</strong><br />
discussion with his friend Kurt Godel was . . . Kant." Kant, deeply impressed by<br />
Newtonómuch <strong>of</strong> his Critique, indeed, was intended to provide a philosophical foundation<br />
for Newton <strong>and</strong> Euclidóhad made famous <strong>the</strong> doctrine that science is fundamen-<br />
17<br />
tal <strong>and</strong> rigorous exactly to <strong>the</strong> degree to which it is ma<strong>the</strong>matical. <strong>Einstein</strong> <strong>and</strong> Godel, in<br />
turn, each in his own way, approached <strong>the</strong> <strong>world</strong> ma<strong>the</strong>matically. For both, ma<strong>the</strong>matics<br />
was a window onto ultimate reality, not, as for many <strong>of</strong> <strong>the</strong>ir scientific colleagues, a mere<br />
tool for intellectual bookkeeping.<br />
Huddled over a desk in Fine Hall or walking home from <strong>the</strong> insti-tute, <strong>the</strong>y were a model <strong>of</strong><br />
ma<strong>the</strong>matical companionship. A chance photograph taken by a visiting ma<strong>the</strong>matician<br />
finds <strong>the</strong> two friends toge<strong>the</strong>r on <strong>the</strong> road, each sporting a white straw hat, <strong>Einstein</strong><br />
beaming tor <strong>the</strong> camera, his convex body bursting from rumpled, baggy pants held up by<br />
an ancient pair <strong>of</strong> suspenders, while <strong>the</strong> white linen <strong>of</strong> Godel's fitted coat holds him<br />
closely, his eyes fixed in a cold stare. (Two gentlemen farmers from a Faulkner novel,<br />
commented one observer.) Each had found in <strong>the</strong> o<strong>the</strong>r a rare companion who could resist<br />
<strong>the</strong> charms <strong>of</strong> <strong>the</strong> "new physics" <strong>of</strong> Bohr <strong>and</strong> Heisenberg, according to whom ma<strong>the</strong>matics<br />
could no longer provide for science a picture <strong>of</strong> <strong>the</strong> <strong>world</strong> as it actually is in itselfóa<br />
<strong>world</strong>viewóbut could serve only as a tool for calculation, a means for predicting <strong>the</strong><br />
outcome <strong>of</strong> experiments. An impossible prescription to follow for "Mr. Why," <strong>and</strong> no less so<br />
for <strong>Einstein</strong>. For a signature <strong>of</strong> <strong>Einstein</strong>ian science is <strong>the</strong> Socratic search tor "definitions,"<br />
for what something "really is," in itself (a favorite expression <strong>of</strong> Plato's). <strong>Einstein</strong>, after all,<br />
was <strong>the</strong> man who had taught Kant what <strong>time</strong> "really was" (<strong>the</strong> fourth dimension <strong>of</strong><br />
relativistic space-<strong>time</strong>), taught Newton what gravity "really was" (<strong>the</strong> curvature <strong>of</strong> fourdimensional<br />
space-<strong>time</strong>), <strong>and</strong> taught everyone what energy "really was" (as every<br />
schoolchild knows, E = mc2).<br />
As students <strong>of</strong> Kant, <strong>Einstein</strong> <strong>and</strong> Godel were well aware that although space <strong>and</strong> <strong>time</strong> are<br />
<strong>the</strong> two fundamental forms <strong>of</strong> human experienceóspace, as Kant had it, <strong>the</strong> form <strong>of</strong><br />
intuition <strong>of</strong> "outer sense," <strong>time</strong> <strong>the</strong> form <strong>of</strong> "inner sense"óit was space that was <strong>the</strong> natural<br />
object <strong>of</strong> scientific inquiry. And it was space that was first captured by <strong>the</strong> Greek<br />
ma<strong>the</strong>matician Euclid, whose axiomatic-deductive system <strong>of</strong> geometryó<strong>the</strong> bane <strong>of</strong> every<br />
high school studentóbecame <strong>the</strong> paradigm <strong>of</strong> science, a model from Newton to <strong>Einstein</strong>.<br />
Even in his new
physics <strong>of</strong> space, <strong>Einstein</strong> had simply generalized geometry from Euclid to <strong>the</strong> new non-<br />
Euclidean geometries, in which <strong>the</strong> angles <strong>of</strong> a triangle could sum to less, or more, than<br />
180 degrees. (To <strong>the</strong> end <strong>of</strong> his life, <strong>Einstein</strong> could wax nostalgic about a boyhood gift that<br />
had turned his life around, his "holy geometry booklet.")<br />
Yet as <strong>Einstein</strong> <strong>and</strong> Godel well knew, it is not space but <strong>time</strong> that in <strong>the</strong> end poses <strong>the</strong><br />
greatest challenge to science. The dynamic nature <strong>of</strong> <strong>time</strong>, <strong>the</strong> fact that it flows, is<br />
obviously its most striking feature. But it is ano<strong>the</strong>r thing entirely to make sense <strong>of</strong> this<br />
seemingly obvious truth. After all, to flow is to flow in <strong>time</strong>. What sense can one attach,<br />
<strong>the</strong>n, to <strong>the</strong> idea <strong>of</strong> <strong>the</strong> flow <strong>of</strong> <strong>time</strong> itself? Saint Augustine, in his Confessions, tied himself<br />
in knots over such conundrums. Western thought as such, one might say, is characterized<br />
by a kind <strong>of</strong> geometrical Midas touch. Whatever science touches becomes subject to<br />
geometry, <strong>the</strong> science <strong>of</strong> space. "Time," Kant himself had said, "is nothing but <strong>the</strong> form <strong>of</strong><br />
inner sense, that is, <strong>of</strong> <strong>the</strong> intuition <strong>of</strong> ourselves <strong>and</strong> our inner states . . . <strong>and</strong> just because<br />
this inner intuition yields no [geometrical] shape, we endeavor to make up for this want by<br />
analogies." The analogy, for Kant, is to think <strong>of</strong> <strong>time</strong>, which is not space, as spatial! "We<br />
cannot," said Kant, "obtain for ourselves a representation <strong>of</strong> <strong>time</strong> which is not an object <strong>of</strong><br />
outer intuition [i.e., <strong>of</strong> sensory experience] except under <strong>the</strong> [spatial] image <strong>of</strong> a line."<br />
Thus when <strong>Einstein</strong> in 1905 captured <strong>time</strong> in special relativity, he once again transformed<br />
it into space, this <strong>time</strong>, into <strong>the</strong> fourth, temporal component <strong>of</strong> <strong>the</strong> geometrical structure<br />
<strong>of</strong> four-dimensional "space-<strong>time</strong>." Not for nothing did G.J. Whitrow write, "<strong>the</strong> primary<br />
object <strong>of</strong> <strong>Einstein</strong>'s pr<strong>of</strong>ound researches on <strong>the</strong> forces <strong>of</strong> nature has been well epitomized<br />
in <strong>the</strong> slogan, '<strong>the</strong> geometrization <strong>of</strong> physics,' <strong>time</strong> being completely absorbed into <strong>the</strong><br />
geometry <strong>of</strong> a hyper-space." The universe, however, not being empty <strong>of</strong> matter, is not<br />
governed by <strong>the</strong> matter-free idealization <strong>of</strong> special relativity but ra<strong>the</strong>r by <strong>Einstein</strong>'s next<br />
brainchild, <strong>the</strong> general <strong>the</strong>ory <strong>of</strong> relativity, <strong>the</strong> subject <strong>of</strong> <strong>Einstein</strong>'s free tutorials with<br />
Godel. Worse, <strong>the</strong> <strong>world</strong> <strong>of</strong> general relativity, much to <strong>Einstein</strong>'s displeasure, was actually<br />
"exp<strong>and</strong>ing," that is, ex-<br />
19<br />
p<strong>and</strong>ing over <strong>time</strong>. (God, apparently, had for once failed to consult first with <strong>Einstein</strong>.)<br />
But special relativity had taught <strong>the</strong> <strong>world</strong> that simultaneity, <strong>and</strong> thus <strong>time</strong>, is not, as<br />
Newton thought, <strong>world</strong>wide <strong>and</strong> absolute, but ra<strong>the</strong>r local <strong>and</strong> relative. In what sense <strong>of</strong><br />
<strong>time</strong>, <strong>the</strong>n, could <strong>the</strong> universe itself be exp<strong>and</strong>ing, absolutely, over <strong>time</strong>? Time itself must<br />
have been smiling over <strong>the</strong> puzzle it had created. Appearances notwithst<strong>and</strong>ing, <strong>Einstein</strong><br />
had not after all succeeded in trapping this elusive prey in <strong>the</strong> net <strong>of</strong> general relativity. As<br />
Hubble showed, <strong>the</strong> universe really is exp<strong>and</strong>ing! The problem could not be avoided. But if<br />
even <strong>Einstein</strong> had run aground on <strong>the</strong>se rocky shoals, who was left to take <strong>the</strong> lead? Whom<br />
could one compare with <strong>Einstein</strong> if not his traveling companion in general relativity, Kurt<br />
Godel? But what made Godel <strong>the</strong> logician, whose universe consisted <strong>of</strong> <strong>the</strong> <strong>time</strong>less<br />
ma<strong>the</strong>matical realm <strong>of</strong> sets <strong>and</strong> numbers, <strong>the</strong> right person to carry forward <strong>Einstein</strong>'s torch<br />
into <strong>the</strong> uncertainties <strong>of</strong> <strong>the</strong> new space-<strong>time</strong>?
3 Vienna: Logical Circles<br />
After one session in which Schlick, Hahn, Neurath <strong>and</strong> Waismann had talked about<br />
language, but in which nei<strong>the</strong>r Godel nor I had spoken a word, I said on <strong>the</strong> way home,<br />
"Today we out-Wittgensteined <strong>the</strong>se Wittgensteinians: we kept silent."<br />
KARL MENGER<br />
Born into <strong>the</strong> Austrian-German minority <strong>of</strong> Brno, a city now in <strong>the</strong> Czech Republic, <strong>the</strong><br />
place where Mendel laid <strong>the</strong> foundations <strong>of</strong> <strong>the</strong> science <strong>of</strong> genetics, <strong>the</strong> Godel bro<strong>the</strong>rs,<br />
Rudolf <strong>and</strong> Kurt, took it as a given that <strong>the</strong>y would undertake <strong>the</strong>ir final academic studies<br />
at <strong>the</strong> storied University <strong>of</strong> Vienna. Vienna remained even after <strong>the</strong> Great War one <strong>of</strong> <strong>the</strong><br />
premier intellectual centers <strong>of</strong> <strong>the</strong> <strong>world</strong>, distinguished in law, medicine (Rudolf would<br />
become a radiologist), physics, ma<strong>the</strong>matics, social sciences, economics, philosophy, <strong>and</strong><br />
<strong>the</strong>ology. In those years <strong>the</strong>re passed through <strong>the</strong> city many <strong>of</strong> <strong>the</strong> individuals who created<br />
<strong>the</strong> twentieth century, including Sigmund Freud, <strong>the</strong> founder <strong>of</strong> psychoanalysis; <strong>the</strong><br />
composers Richard Strauss <strong>and</strong> Gustav Mahler as well as Arnold Schoenberg, <strong>the</strong> inventor <strong>of</strong><br />
twelve-tone music; <strong>the</strong> painters Gustav Klimt <strong>and</strong> Oscar Kokoschka, as well as <strong>the</strong><br />
revolutionary architect Adolf Loos, who presaged <strong>the</strong> famous Bauhaus school; <strong>the</strong> physicistphilosophers<br />
Ludwig Boltzmann <strong>and</strong> Ernst Mach; <strong>and</strong> <strong>the</strong> philosophers Karl Popper <strong>and</strong><br />
Ludwig Wittgenstein. The list could be extended indefinitely. Wittgenstein, himself a kind<br />
<strong>of</strong> minimalist, was an admirer <strong>of</strong> <strong>the</strong> minimalism practiced by Loos, <strong>and</strong><br />
harbored architectural designs <strong>of</strong> his own. The attraction was mutual: "You are me!" said<br />
Loos to Wittgenstein when <strong>the</strong>y met in 1914. The ratio <strong>of</strong> intellectual genius to square<br />
footage in Godel's Vienna takes one's breath away.<br />
Among those who were privileged to think <strong>the</strong> unthinkable, however, <strong>the</strong>re is ano<strong>the</strong>r<br />
name that belongs here. Adolf Hitler's path to Vienna began in Linz, <strong>the</strong> city <strong>of</strong> his birth,<br />
where in 1904 he attended <strong>the</strong> same realschule as Wittgenstein. Though <strong>the</strong> same age as<br />
young Ludwig, young Adolf was two years behind him at school. There exists a class<br />
photograph in which Wittgenstein appears to be placed near Hitler.<br />
Both <strong>of</strong> <strong>the</strong> Godel boys excelled in secondary school in Brno, but Kurt's gifts were clearly<br />
exceptional. He was a st<strong>and</strong>out in all subjects, from science <strong>and</strong> ma<strong>the</strong>matics to<br />
languages, <strong>and</strong> is said never to have made a single error in his Latin exercises. (It was in<br />
ma<strong>the</strong>matics, ironically, that he received his only less than perfect grade.) Arriving in<br />
Vienna in 1924, Godel decided at first to concentrate in physics, a choice that would serve<br />
him well. He also received a solid grounding in philosophy, especially <strong>the</strong> history <strong>of</strong><br />
philosophy, with Heinrich Gomperz, <strong>and</strong> excelled in all his classes in ma<strong>the</strong>matics, a<br />
subject in which he acquired by graduation a remarkable degree <strong>of</strong> depth as well as<br />
breadth, from geometry to number <strong>the</strong>ory <strong>and</strong> ma<strong>the</strong>matical logic. It would soon emerge<br />
that he was embarked on an intellectual journey in <strong>the</strong> direction <strong>of</strong> increased rigor <strong>and</strong><br />
precision, from ma<strong>the</strong>matical physics to ma<strong>the</strong>matics, from <strong>the</strong>re to ma<strong>the</strong>matical logic,<br />
<strong>and</strong> finally from ma<strong>the</strong>matical logic to ma<strong>the</strong>matical philosophy.
As an undergraduate, Godel was particularly impressed by <strong>the</strong> lectures on number <strong>the</strong>ory,<br />
attended by hundreds <strong>of</strong> students, given by Philip Furtwangler, a cousin <strong>of</strong> <strong>the</strong> legendary<br />
orchestral conductor Wilhelm Furtwangler, whose fame in those years would turn to<br />
infamy when he declined to leave Germany during <strong>the</strong> next <strong>world</strong> war. Godel claimed later<br />
that Philip Furtwangler, who was paralyzed from <strong>the</strong> neck down, gave <strong>the</strong> best lectures he<br />
had ever heard. It was Furtwangler whom Godel credited with his turn to ma<strong>the</strong>matics.<br />
The drama <strong>of</strong><br />
his lectures was heightened by <strong>the</strong> fact that Furtwangler lectured from his wheelchair,<br />
<strong>without</strong> notes, while an assistant wrote equations on <strong>the</strong> blackboard. The feeling <strong>of</strong><br />
disembodiment this engendered fit <strong>the</strong> subject <strong>of</strong> <strong>the</strong> lectures perfectly. The natural<br />
numbers 0, 1, 2, 3,. . . seem to possess <strong>the</strong> kind <strong>of</strong> independent existence <strong>and</strong> "geometry"<br />
usually reserved for concrete physical objects. Unsurprisingly, <strong>the</strong>refore, this branch <strong>of</strong><br />
ma<strong>the</strong>matics is a breeding ground for Platonists, who like Plato believe in <strong>the</strong> objective,<br />
independent existence <strong>of</strong> ideal, disembodied "forms," <strong>of</strong> which <strong>the</strong> natural numbers are a<br />
paradigm. These are no more subject to <strong>the</strong> arbitrary manipulations <strong>of</strong> <strong>the</strong> human will than<br />
<strong>the</strong> distant stars, which we observe but cannot touch. (As <strong>the</strong> minimalist ma<strong>the</strong>matician<br />
Kronecker put it, "God made <strong>the</strong> natural numbers; all else is <strong>the</strong> work <strong>of</strong> man." For Godel,<br />
all numbers are "<strong>the</strong> work <strong>of</strong> God.")<br />
Godel's journey from physics to ma<strong>the</strong>matical logic took place just as <strong>the</strong> new field was<br />
coming into its own as a well-established intellectual enterprise, although, truth be told,<br />
logic remains to this day in <strong>the</strong> eyes <strong>of</strong> many ma<strong>the</strong>maticians a poor relation, not quite<br />
ma<strong>the</strong>matics, not quite philosophy. Having for centuries been <strong>the</strong> province <strong>of</strong> rhetoricians<br />
<strong>and</strong> grammarians, logic emerged as a branch <strong>of</strong> ma<strong>the</strong>matics at <strong>the</strong> turn <strong>of</strong> <strong>the</strong> century,<br />
due in large part to <strong>the</strong> work <strong>of</strong> <strong>the</strong> German philosopher-ma<strong>the</strong>matician Gottlob Frege, an<br />
acquaintance <strong>of</strong> both Russell <strong>and</strong> Wittgenstein <strong>and</strong> a seminal influence on <strong>the</strong>ir thinking.<br />
Frege's early masterpiece, Begriffsscbrift (Concept Script), published in 1879, succeeded in<br />
simultaneously axiomatizing logic <strong>and</strong> formalizing it, that is, formulating it in an artificially<br />
constructed, purely symbolic language, prefiguring today's computer programming<br />
languages. The rules <strong>of</strong> such a language are unambiguous <strong>and</strong> can be followed<br />
"mechanically," <strong>without</strong> <strong>the</strong> need to underst<strong>and</strong> <strong>the</strong> meaning <strong>of</strong> <strong>the</strong> symbols. Not content<br />
with this, Frege employed this new math-ematized logicówhich for him was not a mere<br />
calculating device, but a proper science, with its own content <strong>and</strong> subject matteróas itself<br />
a foundation for ma<strong>the</strong>matics, in particular for arithmetic, or number <strong>the</strong>ory.<br />
Ano<strong>the</strong>r German ma<strong>the</strong>matician, David Hilbert, in turn developed a ma<strong>the</strong>matical <strong>the</strong>ory<br />
devoted to <strong>the</strong> study <strong>of</strong> <strong>the</strong> new symbolic logical systems like Frege's, a field that came to<br />
be known as metama<strong>the</strong>matics, since it involved <strong>the</strong> ma<strong>the</strong>matical study <strong>of</strong> ma<strong>the</strong>matical<br />
systems <strong>the</strong>mselves. (Hilbert had been introduced to <strong>the</strong> idea <strong>of</strong> metama<strong>the</strong>maticsó<strong>and</strong><br />
more generally to <strong>the</strong> notion <strong>of</strong> a metalanguageóby <strong>the</strong> Dutch ma<strong>the</strong>matician L. E. J.<br />
Brouwer when <strong>the</strong>y shared a holiday in 1909 at <strong>the</strong> resort <strong>of</strong> Scheveningen, <strong>the</strong><br />
Ne<strong>the</strong>rl<strong>and</strong>s. In <strong>the</strong> course <strong>of</strong> <strong>time</strong>, Brouwer would become Hilbert's nemesis.) This went<br />
beyond Frege, who believed that <strong>the</strong>re is only one genuine logical system, <strong>the</strong> one he had<br />
developed, <strong>and</strong> that <strong>the</strong>re was no "stepping outside" it to compare it with o<strong>the</strong>r systems or<br />
with <strong>the</strong> objects <strong>the</strong>mselves, i.e., ma<strong>the</strong>matical modelsóa conception outside <strong>the</strong><br />
ma<strong>the</strong>matical mainstream that would eventually be taken up by Frege's admirer<br />
Wittgenstein. It was indeed in this new ma<strong>the</strong>matical field <strong>of</strong> metama<strong>the</strong>matics that <strong>the</strong><br />
question <strong>of</strong> completeness was raised by Hilbert, who asked whe<strong>the</strong>r a given symbolic
logical system, such as <strong>the</strong> one developed by Frege in Begriffsschrift, given its axioms <strong>and</strong><br />
pro<strong>of</strong> procedures, was both internally consistent (<strong>the</strong> axioms <strong>and</strong> pro<strong>of</strong> procedures could<br />
not be used to prove two statements that contradicted each o<strong>the</strong>r) <strong>and</strong> complete (<strong>the</strong><br />
pro<strong>of</strong> procedures sufficed to prove every true statement in <strong>the</strong> system). It was in<br />
answering just such questions that Godel discovered his famous incompleteness <strong>the</strong>orem.<br />
The development <strong>of</strong> ma<strong>the</strong>matical logic also went beyond Frege ins<strong>of</strong>ar as it replaced<br />
Frege's use <strong>of</strong> concepts (<strong>the</strong> Begriffe <strong>of</strong> his Begriffsschrift) with <strong>the</strong> extensions <strong>of</strong> <strong>the</strong>se<br />
conceptsói.e., <strong>the</strong> set <strong>of</strong> objects described by <strong>the</strong> conceptsówhich came to be known as<br />
sets or classes. But whereas for Frege <strong>the</strong> <strong>the</strong>ory <strong>of</strong> concepts <strong>and</strong> <strong>the</strong>ir extensions had<br />
been contained in logic itselfó<strong>the</strong> very part <strong>of</strong> Frege's <strong>the</strong>ory that Bertr<strong>and</strong> Russell would<br />
later show contained an inconsistencyóas <strong>the</strong> field developed in <strong>the</strong> early years <strong>of</strong> <strong>the</strong><br />
twentieth century, set <strong>the</strong>ory came into being as a new field unto itself, with its own<br />
axioms. This new axiomatic set <strong>the</strong>ory, developed by such thinkers as Ernst Zermelo<br />
<strong>and</strong> Abraham Fraenkel, replaced both <strong>the</strong> axiomatized <strong>the</strong>ory <strong>of</strong> concepts <strong>and</strong> <strong>the</strong>ir<br />
extensions <strong>of</strong> Frege, as well as <strong>the</strong> earlier, unaxioma-tized, "naive" set <strong>the</strong>ory <strong>of</strong> Cantor.<br />
Frege's work gave birth, <strong>the</strong>n, to two new subfields <strong>of</strong> ma<strong>the</strong>matics: ma<strong>the</strong>matical logic<br />
<strong>and</strong> axiomatic set <strong>the</strong>ory.<br />
Godel's ma<strong>the</strong>matical advisor, Hans Hahn, kept abreast <strong>of</strong> all <strong>the</strong>se new developments in<br />
ma<strong>the</strong>matical logic <strong>and</strong> set <strong>the</strong>ory. Indeed, he directed a seminar devoted to <strong>the</strong> classic <strong>of</strong><br />
modern ma<strong>the</strong>matical logic, Principia Matbematica, by Bertr<strong>and</strong> Russell <strong>and</strong> Albert North<br />
Whitehead. Godel did not participate in that seminar, but he did attend one given by <strong>the</strong><br />
philosopher Moritz Schlick on Bertr<strong>and</strong> Russell's later work, Introduction to Ma<strong>the</strong>matical<br />
Philosophy (written while Russell was in jail in Engl<strong>and</strong> for his protests against British<br />
participation in World War I). Godel also attended a seminar on <strong>the</strong> foundations <strong>of</strong><br />
ma<strong>the</strong>matics <strong>of</strong>fered by Rudolph Carnap, who had been a student <strong>of</strong> Frege's at <strong>the</strong><br />
University <strong>of</strong> Jena. Carnap would shortly become one <strong>of</strong> <strong>the</strong> most influential members <strong>of</strong><br />
Schlick's "Vienna Circle." In a city full <strong>of</strong> cliques, salons <strong>and</strong> discussion groups on every<br />
conceivable topic, <strong>the</strong> Vienna Circle was <strong>the</strong> most exclusive.<br />
Outside <strong>the</strong> circle, Godel's life was not unlike those <strong>of</strong> o<strong>the</strong>r well-<strong>of</strong>f Viennese<br />
intellectuals. With his bro<strong>the</strong>r, Rudolf, his senior by four years, Kurt lived in a comfortable<br />
apartment in which <strong>the</strong>y were joined regularly by <strong>the</strong>ir mo<strong>the</strong>r, Marianne. Toge<strong>the</strong>r with<br />
her, <strong>the</strong> two bro<strong>the</strong>rs enjoyed automotive vacations in Rudolf's new Chrysler, one <strong>of</strong> <strong>the</strong><br />
first in <strong>the</strong> region, to spots as far away as Marienbad. Though <strong>the</strong> family employed a<br />
chauffeur, on vacations <strong>the</strong> bro<strong>the</strong>rs preferred doing <strong>the</strong> driving. Kurt liked to drive fast.<br />
This, combined with his penchant for indulging in abstract reverie while behind <strong>the</strong> wheel,<br />
led his future wife, Adele, to put an end to his driving career. In town, Marianne made<br />
certain that her two academic sons did not neglect <strong>the</strong> full cultural <strong>of</strong>ferings laid before<br />
<strong>the</strong>m by beautiful Vienna. These included plays, with box seats at Max Reinhardt's famous<br />
Josefstadt Theater, <strong>and</strong> concerts, especially light opera, <strong>of</strong> which Godel was especially<br />
fond.<br />
Godel's exceptionally clear mind made him a much sought after intellectual companion<br />
among his fellow students. He was generous with his <strong>time</strong> <strong>and</strong> patient with his<br />
interlocutors. His friend Karl Menger writes that Godel "always grasped problematic points
quickly <strong>and</strong> his replies <strong>of</strong>ten opened new perspectives for <strong>the</strong> enquirer. He expressed all<br />
his insights . . . with a certain shyness <strong>and</strong> a charm that awoke warm <strong>and</strong> personal feelings<br />
for him in many a listener." But Godel's shyness should not be mistaken for timidity. When<br />
<strong>the</strong> already distinguished Carnap suggested to his young student that he write some<br />
encyclopedia entries to gain recognition, Godel responded that he had no need for such<br />
devices to achieve renown. Nor was he timid with women, or above a little showing <strong>of</strong>f.<br />
His fellow student Olga Taussky-Todd would later describe one particular encounter that<br />
impressed her, albeit negatively. In a classroom near <strong>the</strong> ma<strong>the</strong>matical seminar, "<strong>the</strong> door<br />
opened <strong>and</strong> a very small, very young girl entered. She was good-looking . . . <strong>and</strong> wore a<br />
beautiful, quite unusual summer dress. Not much later Kurt entered <strong>and</strong> . .. <strong>the</strong> two <strong>of</strong><br />
<strong>the</strong>m left toge<strong>the</strong>r. It seemed a clear show <strong>of</strong>f on <strong>the</strong> part <strong>of</strong> Kurt." Later, this same young<br />
woman sought <strong>the</strong> reluctant Taussky-Todd's ma<strong>the</strong>matical assistance in an attempt,<br />
apparently, to impress Godel. (She complained to Taussky-Todd, however, about Godel<br />
being spoiled, inclined to rise late in <strong>the</strong> morning, <strong>and</strong> so on.) According to Rudolf, his<br />
bro<strong>the</strong>r developed a particular fondness for a family-run restaurant near <strong>the</strong>ir apartment,<br />
an attraction he attributed to Kurt's interest in <strong>the</strong> attractive twenty-year-old daughter,<br />
who served as a waitress.<br />
Generally, however, Godel seems to have preferred <strong>the</strong> company <strong>of</strong> older women. His very<br />
first romantic interest, <strong>the</strong> daughter <strong>of</strong> friends who used to visit his family, was described<br />
as an "eccentric beauty," but she was also ten years his senior, <strong>and</strong> his parents put an end<br />
to <strong>the</strong> relationship. More serious was <strong>the</strong> attachment he formed at <strong>the</strong> age <strong>of</strong> twenty-one<br />
for Adele Porkert, a nightclub performeróself described as a "ballet dancer," a pr<strong>of</strong>ession<br />
at that <strong>time</strong> only marginally more acceptableóemployed at Der Nachtfalter (The Moth). Six<br />
years his<br />
senior, Adele was married, her face partially disfigured by a "port wine stain," <strong>and</strong> Catholic<br />
(a religion for which nei<strong>the</strong>r Godel nor his parents had any sympathy). Her marriage,<br />
however, was brief <strong>and</strong> unhappy, <strong>and</strong> a prolonged romance ensued between <strong>the</strong> dancer<br />
<strong>and</strong> <strong>the</strong> ma<strong>the</strong>matician. Georg Kreisel, who used to visit Kurt <strong>and</strong> Adele, described her as<br />
lacking formal education but possessing "a real flair for le mot juste." In addition,<br />
according to Kreisel, Adele liked to tease Kurt by constructing "farfetched grounds for<br />
jealousy," <strong>and</strong> also by making fun <strong>of</strong> his curious interest in ghosts <strong>and</strong> demons. Kurt's<br />
parents, for reasons that are obvious if not admirable, were not amused by Adele <strong>and</strong><br />
objected strenuously to <strong>the</strong> relationship. It was only after his fa<strong>the</strong>r's death in 1929, at <strong>the</strong><br />
age <strong>of</strong> fifty-four, that marriage to Adele became a possibility, though it took nine more<br />
years for <strong>the</strong> deal to be clinched. (The delay, according to Godel's friend in later life, Hao<br />
Wang, may have been partly responsible lor <strong>the</strong> fact that <strong>the</strong> Godels never had children, a<br />
circumstance that would weigh heavily on <strong>the</strong> increasingly sad <strong>and</strong> lonely Adele.)<br />
The bro<strong>the</strong>rs were separated not only by Kurt's interest in womenó his bro<strong>the</strong>r never<br />
marriedóbut by <strong>the</strong> fact that while Rudolf spent <strong>the</strong> day at <strong>the</strong> hospital, Kurt attended <strong>the</strong><br />
university. If not at <strong>the</strong> university or with a woman or attending a play, he would repair<br />
frequently to one <strong>of</strong> Vienna's famous c<strong>of</strong>feehouses, generally to one <strong>of</strong> <strong>the</strong> cafes that were<br />
<strong>the</strong> preferred haunts <strong>of</strong> <strong>the</strong> Vienna Circle, such as <strong>the</strong> Reicbsrat, Schattentor, or Arkaden.<br />
Though far from gregarious, Godel developed close friendships with several colleagues <strong>and</strong><br />
pro-lessors, including Carnap <strong>and</strong> Hahn, as well as Herbert Feigl <strong>and</strong> Marcel Natkin <strong>of</strong> <strong>the</strong><br />
Schlick circle. Von Neumann, too, became a friend, <strong>and</strong> a close one at that. Feigel, for his<br />
part, recalled long walks with Godel through <strong>the</strong> parks <strong>of</strong> Vienna <strong>and</strong> c<strong>of</strong>feehouse<br />
discussions <strong>of</strong> matters philosophical, logical, ma<strong>the</strong>matical, <strong>and</strong> scientific that continued<br />
late into <strong>the</strong> night. And Karl Menger, who was reported to have been <strong>the</strong> favorite student
<strong>of</strong> Godel's advisor, Hahn, also became a good friend <strong>of</strong> Godel's, inviting him to participate<br />
inó <strong>and</strong> eventually edit <strong>the</strong> proceedings <strong>of</strong>ó<strong>the</strong> ma<strong>the</strong>matical colloquium<br />
he founded. The most organized <strong>and</strong> regular philosophical interactions, however, between<br />
Godel <strong>and</strong> o<strong>the</strong>r minds were doubtless <strong>the</strong> weekly discussions conducted in Schlick's Vienna<br />
Circle, <strong>of</strong> which he became a regular member in 1926, having been introduced into <strong>the</strong><br />
circle by Hahn.<br />
Godel's Vienna was a city <strong>of</strong> c<strong>of</strong>feehouses, each devoted to a particular intellectual<br />
<strong>the</strong>meóthose with white table tops, convenient for writing formulas, being especially<br />
favored by ma<strong>the</strong>maticiansóas well as <strong>of</strong> intellectual circles, especially philosophical ones.<br />
The <strong>the</strong>me <strong>of</strong> <strong>the</strong> Vienna Circle was logical positivism. Though a guest in <strong>the</strong> house <strong>of</strong><br />
Schlick, Godel was hardly enamored <strong>of</strong> <strong>the</strong> circle's credo <strong>of</strong> positivism, nor <strong>of</strong> <strong>the</strong> hero <strong>of</strong><br />
this cult, Ludwig Wittgenstein. The bible <strong>of</strong> <strong>the</strong> Vienna Circle was Wittgenstein's Tractatus<br />
(Tractatus Logico-Pbilosophicus was <strong>the</strong> full title, suggested by Wittgenstein's friend <strong>and</strong><br />
former teacher, G. E. Moore, emulating Spinoza's Tractatus Tbeo-logico-Politicus),<br />
completed while <strong>the</strong> author was a prisoner <strong>of</strong> war. But Wittgenstein's true war, like that <strong>of</strong><br />
<strong>the</strong> Vienna Circle, was not against <strong>the</strong> Allies but against metaphysics. Positivism, a<br />
particularly severe br<strong>and</strong> <strong>of</strong> intellectual minimalismóa spirit that thrived in Godel's<br />
Viennaóis an antiphilosophical philosophy dedicated to <strong>the</strong> belief that most <strong>of</strong> what has<br />
passed for deep metaphysical thinking over <strong>the</strong> centuries is nothing more than confusion<br />
based on an inadequate underst<strong>and</strong>ing <strong>of</strong> language, which, through artifice, leads <strong>the</strong> mind<br />
by <strong>the</strong> nose in all <strong>the</strong> wrong directions.<br />
Godel did not share <strong>the</strong> positivist credo that philosophy begins <strong>and</strong> ends with an analysis <strong>of</strong><br />
language <strong>and</strong> its limitations, nor Wittgenstein-ian's doctrine that <strong>the</strong> subject matter <strong>of</strong><br />
traditional philosophy, as opposed to that <strong>of</strong> physical science, is precisely that which<br />
cannot be expressed in language. He had no sympathy for <strong>the</strong> famous line with which <strong>the</strong><br />
Tractatus concludes, that "what we cannot speak about we must pass over in silence," as<br />
shown in a reminiscence by Menger after <strong>the</strong> two had attended a session <strong>of</strong> <strong>the</strong> Vienna<br />
Circle: "Today we . . . out-Wittgensteined <strong>the</strong>se Wittgensteinians; we kept silent."<br />
Apparently, Godel <strong>and</strong> Wittgenstein never met, though Godel said that he<br />
29<br />
saw him once, when both attended a lecture in Vienna by <strong>the</strong> Dutch anti-Platonist,<br />
"intuitionist" ma<strong>the</strong>matician, L.E. J. Brouwer.<br />
In <strong>the</strong> meetings <strong>of</strong> <strong>the</strong> Vienna Circle, Godel rarely spoke, signaling his agreement or<br />
disagreement only by a slight inclination <strong>of</strong> <strong>the</strong> head. Participation in <strong>the</strong>se meetings was<br />
by invitation only, <strong>and</strong> membership hovered between ten <strong>and</strong> twenty. The regular<br />
participants included Schlick <strong>and</strong> Carnap, <strong>the</strong> philosophers Carl Hempel, Otto Neu-rath,<br />
Friedrich Waismann <strong>and</strong> Feigl, <strong>and</strong> finally, Menger, Hahn <strong>and</strong> Godel. Conspicuous by <strong>the</strong>ir<br />
absence were <strong>the</strong> philosophers Popper <strong>and</strong> Wittgenstein, <strong>the</strong> former because he had not<br />
been invited due to his views about <strong>the</strong> latter, <strong>the</strong> latter because he had declined <strong>the</strong><br />
invitation. The meetings took place in a dingy room filled with rows <strong>of</strong> chairs <strong>and</strong> long
tables on <strong>the</strong> ground floor <strong>of</strong> <strong>the</strong> building in <strong>the</strong> Boltz-manngasse that housed <strong>the</strong><br />
ma<strong>the</strong>matical <strong>and</strong> physical institutes. Early arrivals were expected to clear <strong>the</strong> chairs away<br />
from <strong>the</strong> blackboard to allow <strong>the</strong> day's speaker room to maneuver. One table was reserved<br />
for smokers. Intellectually, <strong>the</strong> circle was devoted to <strong>the</strong> <strong>the</strong>me <strong>of</strong> positivism, <strong>the</strong> doctrine<br />
that physical science, whose ultimate basis is sensory experience, exhausts what can be<br />
known, leaving philosophy <strong>the</strong> task primarily <strong>of</strong> policing <strong>the</strong> ever-present tendency <strong>of</strong><br />
thought to pretend to more knowledge than can be delivered by science. (Wittgenstein<br />
himself, though <strong>the</strong>ir hero, was not a positivist. What separated him from <strong>the</strong>m was this:<br />
what must be "passed over in silence" was for Wittgenstein precisely what had value.)<br />
The note struck by <strong>the</strong> positivists was hardly new. Immanuel Kant had declared centuries<br />
earlier that "reason," as such, st<strong>and</strong>s in need <strong>of</strong> an internal "critique." In <strong>the</strong> "Dialectic" <strong>of</strong><br />
his Critique <strong>of</strong> Pure Reason he described in detail reason's attempts to fly through thin<br />
airóa region, he noted, heavily populated by philosophers. Never<strong>the</strong>less, Kant himself<br />
proceeded to develop a system <strong>of</strong> philosophy that pretended to a kind <strong>of</strong> knowledge not<br />
derivable from science. This <strong>the</strong> new positivism rejected. What gave it its "logical" twist<br />
were <strong>the</strong> recent efforts by Frege, Russell, Hilbert, <strong>and</strong> o<strong>the</strong>rs to develop logic both as an<br />
instrument that served to formalize <strong>the</strong> physical sciencesó<strong>and</strong><br />
thus to assist in <strong>the</strong>ir policingó<strong>and</strong> as a new branch <strong>of</strong> ma<strong>the</strong>matics that was<br />
simultaneously a foundation for <strong>the</strong> rest <strong>of</strong> ma<strong>the</strong>matics <strong>and</strong> a close cousin to what was<br />
worth preserving in <strong>the</strong> philosophical tradition. It was unsurprising that Schlick's logical<br />
positivists chose as <strong>the</strong>ir patron saint <strong>the</strong> Wittgenstein <strong>of</strong> <strong>the</strong> Tractatus, since it was one <strong>of</strong><br />
<strong>the</strong> <strong>the</strong>mes <strong>of</strong> that slender but potent work that a primary task <strong>of</strong> philosophy is to separate<br />
clearly <strong>and</strong> forever what can be said from what cannot, to clear out centuries <strong>of</strong><br />
philosophical clutter <strong>and</strong> render a clear path for science.<br />
For Wittgenstein, <strong>the</strong> new logic <strong>of</strong> Frege <strong>and</strong> Russell provided <strong>the</strong> tool that made not just<br />
<strong>the</strong> attempt but <strong>the</strong> completion <strong>of</strong> this task possible: "I. . . believe myself to have found,"<br />
he declared modestly in his preface, "on all essential points, <strong>the</strong> final solution <strong>of</strong> <strong>the</strong><br />
problems." Wittgenstein, moreover, had a line on a problem that had haunted <strong>the</strong><br />
positivists. The physical sciences that served as <strong>the</strong>ir model for all thought were rigorous<br />
precisely to <strong>the</strong> degree that <strong>the</strong>y were ma<strong>the</strong>matical, yet ma<strong>the</strong>matics itself is not a<br />
physical science. It appears altoge<strong>the</strong>r immune to <strong>the</strong> touchstone <strong>of</strong> sensory experience<br />
that forms <strong>the</strong> very basis <strong>of</strong> physical science. Without an account <strong>of</strong> ma<strong>the</strong>matics, <strong>the</strong>n,<br />
<strong>the</strong> new minimalist edifice <strong>of</strong> logical positivism threatened to crumble under its own<br />
weight. The Tractatus was a gift from God, for if Wittgenstein was right, ma<strong>the</strong>matics was<br />
not a science at all. Strictly speaking, <strong>the</strong>re is no proper knowledge in ma<strong>the</strong>matics, no<br />
truth. Ra<strong>the</strong>r, <strong>the</strong> systems <strong>of</strong> equations represent conventional rules for <strong>the</strong> manipulation<br />
<strong>of</strong> abstract symbols that make possible <strong>the</strong> genuine knowledge <strong>of</strong>fered by physical science.<br />
With this approach, ma<strong>the</strong>matics as such is merely a calculus, a calculating device, not a<br />
language <strong>of</strong> thought, as it was for Frege. As Frege's former student Carnap put it,<br />
ma<strong>the</strong>matics is not a genuine language that can express thoughts but ra<strong>the</strong>r <strong>the</strong> "logical<br />
syntax <strong>of</strong> language." This was a doctrine that Godel, <strong>the</strong> true heir to Frege, would spend<br />
<strong>the</strong> rest <strong>of</strong> his life working to defeat.<br />
Wittgenstein was <strong>the</strong> patron saint <strong>of</strong> <strong>the</strong> Vienna Circle. "I can testify to this . . . ," wrote<br />
Olga Taussky-Todd. "Wittgenstein was <strong>the</strong>
idol <strong>of</strong> this group. ... An argument could be settled by citing his Tractates." The un<strong>of</strong>ficial<br />
saint, however, was Godel's future friend <strong>Einstein</strong>, considered by many to be <strong>the</strong> greatest<br />
scientist since Newton. (Wittgenstein himself once stated that in a sense he was a follower<br />
<strong>of</strong> <strong>Einstein</strong>.) It was not only that <strong>Einstein</strong>'s <strong>the</strong>ories had revolutionized <strong>the</strong> scientific image<br />
<strong>of</strong> <strong>the</strong> <strong>world</strong>. The philosophical aspects <strong>of</strong> <strong>Einstein</strong>'s <strong>the</strong>ory <strong>of</strong> relativity held a special<br />
appeal. In <strong>the</strong> special <strong>the</strong>ory <strong>of</strong> relativity, <strong>Einstein</strong> had rejected Newton's "metaphysical"<br />
postulates <strong>of</strong> absolute space <strong>and</strong> <strong>time</strong>, which resisted any direct empirical confirmation.<br />
Time, <strong>Einstein</strong> insisted, was physically real only to <strong>the</strong> extent that it could be measured by<br />
a clock. (As <strong>the</strong> positivists would put it, <strong>the</strong> meaning <strong>of</strong> a term consists in its method <strong>of</strong><br />
verification.) Since physical experimentation demonstrated that not all clocks could be<br />
definitively synchronized, <strong>Einstein</strong> declared that <strong>time</strong> was not after all absolute, as<br />
Newton had believed, but ra<strong>the</strong>r relative to <strong>the</strong> frame <strong>of</strong> reference <strong>of</strong> <strong>the</strong> clock by which it<br />
was measured. Similarly, since <strong>the</strong>re existed no definitive empirical method to detect<br />
whe<strong>the</strong>r an object's motion through space was absolute, <strong>Einstein</strong> declared that all spatial<br />
relations were also relative to a given reference frame chosen, by convention, as <strong>the</strong> "rest<br />
frame." For <strong>the</strong> positivists, <strong>the</strong> success <strong>of</strong> this <strong>the</strong>ory meant that <strong>the</strong> tenets <strong>of</strong> <strong>the</strong>ir credo<br />
made for good science, while <strong>the</strong>ir rejection could lead to bad philosophy or a scientific<br />
dead end.<br />
For <strong>Einstein</strong>, <strong>the</strong> rejection <strong>of</strong> absolute space <strong>and</strong> <strong>time</strong> was merely a statement about <strong>the</strong><br />
physical <strong>world</strong>. It was much more, however, to <strong>the</strong> Vienna Circle. As much a religion as a<br />
scientific methodology, positivism denied that science or philosophy, with <strong>the</strong> help <strong>of</strong><br />
ma<strong>the</strong>matics, could provide a Weltbild, or <strong>world</strong>view, a picture or account <strong>of</strong> ultimate<br />
reality. They could only supply <strong>the</strong> scientist with a method <strong>of</strong> calculation <strong>and</strong> prediction.<br />
In addition to rejecting <strong>the</strong> task <strong>of</strong> providing a metaphysical picture <strong>of</strong> realityó<strong>the</strong> very<br />
goal <strong>of</strong> philosophy itself as it has been historically practicedó<strong>the</strong> positivist confined his<br />
episte-mology to direct sensory experience, rejecting any claims for insight into, or<br />
intuition <strong>of</strong>, <strong>the</strong> concepts <strong>of</strong> <strong>the</strong> <strong>the</strong>orist or <strong>the</strong> abstract objects<br />
<strong>of</strong> <strong>the</strong> ma<strong>the</strong>matician. Godel <strong>and</strong> <strong>Einstein</strong>, in contrast, held <strong>the</strong> faculty <strong>of</strong> intuition in <strong>the</strong><br />
highest regard. "I put my faith in organization," John D. Rockefeller Jr. once said on<br />
meeting <strong>Einstein</strong>. "I put my faith in intuition," <strong>the</strong> physicist replied. Godel was even more<br />
explicit. In perhaps his most (in)famous philosophical remark, he laid down <strong>the</strong> gauntlet<br />
against positivism: "Despite <strong>the</strong>ir remoteness from sense-experience, we do have<br />
something like a perception <strong>of</strong> <strong>the</strong> objects <strong>of</strong> set <strong>the</strong>ory, as is seen from <strong>the</strong> fact that <strong>the</strong><br />
axioms force <strong>the</strong>mselves upon us as being true. I don't see any reason why we should have<br />
less confidence in this kind <strong>of</strong> perception, i.e., in ma<strong>the</strong>matical intuition, than in senseperception."<br />
For <strong>the</strong> positivist, however, <strong>the</strong>re is nei<strong>the</strong>r an abstract realm <strong>of</strong> concepts nor<br />
a human faculty <strong>of</strong> intuition that could provide insight into this realm. The concepts<br />
invoked by <strong>the</strong> philosopher must give way to <strong>the</strong> techniques employed by <strong>the</strong> engineer.<br />
Thus Wittgenstein, a some<strong>time</strong> aeronautical engineer, in <strong>the</strong> Tractatus: Ma<strong>the</strong>matics is<br />
not, as Frege had it, a science <strong>of</strong> <strong>the</strong> platonic realm <strong>of</strong> ma<strong>the</strong>matical concepts <strong>and</strong><br />
objects, but ra<strong>the</strong>r a system <strong>of</strong> techniques for <strong>the</strong> manipulation <strong>of</strong> ma<strong>the</strong>matical signs.<br />
The War <strong>of</strong> <strong>the</strong> Titans
That it was Vienna that gave birth to this extreme antiphilosophical philosophy was no<br />
accident, nor was it an accident that <strong>the</strong> Vienna Circle was its cradle. Moritz Schlick, <strong>the</strong><br />
founder <strong>of</strong> <strong>the</strong> Circle, had in 1922 assumed <strong>the</strong> chair in <strong>the</strong> philosophy <strong>of</strong> <strong>the</strong> inductive<br />
sciences formerly occupied by Ludwig Boltzmann, <strong>and</strong> before that Ernst Mach. The<br />
philosopher-physicist Mach was a prominent figure at <strong>the</strong> University <strong>of</strong> Vienna, occupying<br />
<strong>the</strong> chair in <strong>the</strong> history <strong>and</strong> philosophy <strong>of</strong> <strong>the</strong> inductive sciences from 1895 to 1901. In<br />
1864, he had been pr<strong>of</strong>essor <strong>of</strong> ma<strong>the</strong>matics at Graz, in 1867, pr<strong>of</strong>essor <strong>of</strong> physics at<br />
Prague. He made important contributions to acoustics, electricity, hydrodynamics,<br />
mechanics, optics <strong>and</strong> <strong>the</strong>rmodynamics. It was Mach who in 1887 laid <strong>the</strong> basis for <strong>the</strong><br />
principles <strong>of</strong> supersonics,<br />
which demonstrated that a material object traveling past <strong>the</strong> speed <strong>of</strong> sound would have<br />
an effect that is now called a "sonic boom." An object's speed relative to <strong>the</strong> speed <strong>of</strong><br />
sound is today called its "Mach number," Mach 2, for example, indicating a velocity <strong>of</strong><br />
twice <strong>the</strong> speed <strong>of</strong> sound.<br />
Mach had an enormous influence on his contemporaries. He was an early devotee <strong>of</strong> an<br />
extreme version <strong>of</strong> positivism <strong>and</strong> was a passionate advocate for his doctrines. His polemics<br />
succeeded in making a generation <strong>of</strong> scientists skeptical not only <strong>of</strong> <strong>the</strong>orists' speculations<br />
about <strong>the</strong> microscopic <strong>world</strong>, but even <strong>of</strong> <strong>the</strong> extended use <strong>of</strong> abstract ma<strong>the</strong>matics as an<br />
element <strong>of</strong> ma<strong>the</strong>matical <strong>the</strong>ories. He was a successful polemicist <strong>and</strong> popularizer, <strong>and</strong> he<br />
acquired a kind <strong>of</strong> cult following among intellectuals <strong>of</strong> various stripes. His admirers<br />
included <strong>the</strong> young poet Hugo von H<strong>of</strong>mannsthal, a member <strong>of</strong> <strong>the</strong> artistic circle Young<br />
Vienna, who would become famous as <strong>the</strong> librettist for <strong>the</strong> operas <strong>of</strong> <strong>the</strong> composer Richard<br />
Strauss.<br />
Mach advocated a "critique," somewhat in Kant's sense, <strong>of</strong> physical science, attempting to<br />
purge it <strong>of</strong> all elements not verifiable by sensory experience. "Pseudoproblems" he called<br />
<strong>the</strong> traditional concerns <strong>of</strong> philosophy (a term for which Carnap developed a strong<br />
attachment); "an<strong>time</strong>taphysical" he subtitled <strong>the</strong> introductory remarks to one <strong>of</strong> his books.<br />
For Mach, <strong>the</strong> ultimate foundation <strong>of</strong> science was <strong>the</strong> data <strong>of</strong>fered by <strong>the</strong> five senses. He<br />
was ruthless in his rejection <strong>of</strong> any conceptions that resisted empirical confirmation.<br />
Newton's absolute space <strong>and</strong> <strong>time</strong> were for him ana<strong>the</strong>ma. Is <strong>the</strong> sun "really st<strong>and</strong>ing still" '<br />
while <strong>the</strong> earth revolves around it, or is it <strong>the</strong> earth that is "really stationary"? The<br />
question for Mach was nonsensical, since physical science contains no detector for one's<br />
position in "absolute space." <strong>Einstein</strong>, clearly, was impressed. He cited Mach explicitly as a<br />
seminal influence on his special <strong>the</strong>ory <strong>of</strong> relativity. But Mach's influence on him was<br />
primarily negative, clearing <strong>the</strong> way for <strong>Einstein</strong> to find, with no help from Mach, a<br />
positive <strong>the</strong>ory to replace Newton's. "Mach's way," wrote <strong>Einstein</strong> to his old friend Michele<br />
Besso, "cannot give birth to anything living; it can only exterminate harmful vermin."<br />
Mach's o<strong>the</strong>r maxims could prove positively harmful to an attempt to find a successful<br />
scientific <strong>the</strong>ory. For one, he was constitutionally opposed to <strong>the</strong> forming <strong>of</strong> "models" in<br />
physics, simplified abstract simulacra <strong>of</strong> real, complex phenomena, fruitful in forming<br />
<strong>the</strong>oretical constructions that can be tested against empirical facts. He labored under <strong>the</strong><br />
illusion that science could be built upon <strong>the</strong> basis <strong>of</strong> "inductions," generalizations based on<br />
patterns in observed phenomena. <strong>Einstein</strong>'s proclivity was for gedankenexperiments,<br />
thought experiments, in which <strong>the</strong> imagination manipulated images to see what would<br />
happen if a hypo<strong>the</strong>sis not necessarily derived from sensory experience were true. He was
more sympa<strong>the</strong>tic, however, to ano<strong>the</strong>r <strong>of</strong> Mach's maxims, against <strong>the</strong> excessive use <strong>of</strong><br />
abstract ma<strong>the</strong>matics in <strong>the</strong>oretical physics. Mach's preoccupation with <strong>the</strong> data <strong>of</strong> sensory<br />
experience made him suspicious <strong>of</strong> high flights <strong>of</strong> <strong>the</strong> ma<strong>the</strong>matical imagination. He<br />
suspected ma<strong>the</strong>maticians <strong>of</strong> substituting <strong>the</strong> artful manipulation <strong>of</strong> symbols for <strong>the</strong> honest<br />
work <strong>of</strong> empirical testing <strong>and</strong> confirmation. <strong>Einstein</strong> was sympa<strong>the</strong>tic. The ma<strong>the</strong>matics <strong>of</strong><br />
his revolutionary paper on special relativity was relatively elementary, <strong>and</strong> at first he<br />
resisted its reformulation in terms <strong>of</strong> four-dimensional space-<strong>time</strong> by his former teacher<br />
Hermann Minkowski, complaining that "since <strong>the</strong> ma<strong>the</strong>maticians pounced on <strong>the</strong> relativity<br />
<strong>the</strong>ory I no longer underst<strong>and</strong> it myself." Unlike Mach, however, he quickly came to his<br />
senses, <strong>and</strong> fortunately so, since <strong>the</strong> progression from special to general<br />
relativityó<strong>Einstein</strong>'s crowning achievementówould have been impossible <strong>without</strong><br />
Minkowski's ma<strong>the</strong>matical reformulation.<br />
<strong>Einstein</strong> believed that physical reality contained more than what we can derive from <strong>the</strong><br />
data <strong>of</strong> sensory experience. The real <strong>world</strong>, for him, was what corresponded to physical<br />
<strong>the</strong>ory. It consisted <strong>of</strong> entities like atoms <strong>and</strong> force fields, in <strong>the</strong>mselves undetectable by<br />
<strong>the</strong> senses, but indirectly discernible by <strong>the</strong>ir effects on systems that can affect human or<br />
artificial receptors. Mach, in contrast, had no truck with <strong>the</strong>oretical constructs <strong>and</strong><br />
remained skeptical <strong>of</strong> anything that could not be reduced to a law based on a combination<br />
<strong>of</strong> sensory experiences. This bias had several consequences, each individually devastating<br />
to<br />
scientific inquiry. First, Mach remained to his final days violently opposed to <strong>the</strong> new<br />
scientific view that much <strong>of</strong> <strong>the</strong> real <strong>world</strong> consists <strong>of</strong> entities, like atoms, forever invisible<br />
to <strong>the</strong> unaided human senses. This stance by such a powerful figure hindered scientists, in<br />
<strong>the</strong>ir research <strong>and</strong> in <strong>the</strong>ir careers, who did not share Mach's prejudice, <strong>and</strong> ensured that<br />
Mach's own scientific <strong>world</strong>view would become increasingly irrelevant. As <strong>Einstein</strong> wrote<br />
later, "<strong>the</strong> antipathy <strong>of</strong> <strong>the</strong>se scholars [Oswald, Mach] towards atomic <strong>the</strong>ory can<br />
indubitably be traced back to <strong>the</strong>ir positivistic philosophical attitude. This is an interesting<br />
example <strong>of</strong> <strong>the</strong> fact that even scholars <strong>of</strong> audacious spirit <strong>and</strong> fine instinct can be<br />
obstructed in <strong>the</strong> interpretation <strong>of</strong> facts by philosophical prejudices."<br />
Second, Mach, by intention <strong>the</strong> most empiricist <strong>of</strong> thinkers, was rendered indistinguishable<br />
from <strong>the</strong> philosophical "idealists" who believe that <strong>the</strong> real <strong>world</strong> is simply a fiction created<br />
by <strong>the</strong> human mind. The sensations that for Mach formed <strong>the</strong> very basis <strong>of</strong> science are<br />
after all individual, private, subjective mental phenomena that cannot be shared, in direct<br />
contrast to <strong>the</strong> shared, objective, independent physical <strong>world</strong> that constitutes <strong>the</strong><br />
mainstay <strong>of</strong> <strong>the</strong> empiricist <strong>world</strong>view. Ironically, it is <strong>the</strong> ma<strong>the</strong>matical Platonist Frege<br />
who turns out to be a greater empirical realist than <strong>the</strong> supposedly hardnosed Mach. In<br />
"Thought," a late essay, Frege wrote, "We really experience only [our mental] ideas, not<br />
<strong>the</strong>ir causes. And if <strong>the</strong> scientist wants to avoid all mere hypo<strong>the</strong>sis, <strong>the</strong>n he is left just<br />
with ideas; everything dissolves into ideas, even <strong>the</strong> light rays [<strong>and</strong>] nerve fibers . . . from<br />
which he started. So he finally undermines <strong>the</strong> foundations <strong>of</strong> his own construction." Frege<br />
was not <strong>the</strong> only prominent figure to be alarmed by Mach's tendency toward idealism. In<br />
Materialism <strong>and</strong> Empirio-Criticism, in 1909, V. I. Lenin took <strong>time</strong> out from <strong>the</strong> revolution to<br />
launch a spirited critique <strong>of</strong> Mach's idealistic tendencies. That <strong>the</strong> busy Lenin thought it<br />
necessary to refute him is an indication <strong>of</strong> Mach's reach.
In direct opposition to Mach stood his contemporary Ludwig Boltzmann, <strong>the</strong> founder <strong>of</strong> <strong>the</strong><br />
statistical <strong>the</strong>ory <strong>of</strong> mechanics. A gifted pianist, Boltzmamn loved to play Beethoven's<br />
symphonies in <strong>the</strong>ir piano transcriptions by Franz Liszt. He was also fond <strong>of</strong> Wagner<br />
<strong>and</strong> at one point took piano lessons from <strong>the</strong> composer <strong>and</strong> organist Anton Bruckner. The<br />
lessons came to an abrupt end when Boltz-mann's mo<strong>the</strong>r discovered a wet raincoat <strong>the</strong><br />
composer had left on <strong>the</strong> bed. Boltzmann had an unstable personality but a warm <strong>and</strong> s<strong>of</strong>t<br />
heart. Short <strong>and</strong> stocky, with thick eyeglasses, curly hair, an equally curly beard, <strong>and</strong> a<br />
surprisingly high-pitched voice, he was called by his fiancee "my sweet fat darling." Mach<br />
felt o<strong>the</strong>rwise. The enmity between <strong>the</strong> two was such that after Mach accepted <strong>the</strong> chair<br />
in <strong>the</strong> philosophy <strong>of</strong> <strong>the</strong> inductive sciences in Vienna in 1895 <strong>and</strong> began giving lectures to<br />
large enthusiastic audiences, Boltzmann resigned <strong>the</strong> chair <strong>of</strong> <strong>the</strong>oretical physics <strong>and</strong><br />
moved to Leipzig, where he immediately encountered ano<strong>the</strong>r scientific enemy in Wilhelm<br />
Oswald. Depressed by continual arguments with Oswald, Boltzmann attempted suicide. He<br />
was pleased to return to Vienna in 1901 when Mach, having received an appointment to<br />
<strong>the</strong> Austrian parliament, resigned <strong>the</strong> chair in philosophy, leaving it open for Boltzmann.<br />
When Boltzmann began lecturing on philosophy, his classes became so popular that <strong>the</strong><br />
university's largest lecture room could not contain <strong>the</strong>m.<br />
It was Boltzmann who introduced probabilistic thinking as essential to physics. The<br />
behavior <strong>of</strong> <strong>the</strong> molecules <strong>of</strong> a gas could be seen to obey precise laws only to <strong>the</strong> extent to<br />
which <strong>the</strong>ir motions were considered in <strong>the</strong> aggregate, statistically. It is one <strong>of</strong> <strong>the</strong> great<br />
paradoxes <strong>of</strong> physics that <strong>the</strong> principles <strong>of</strong> mechanics are <strong>time</strong>-symmetricó<strong>the</strong>y operate<br />
identically if run in reverseó<strong>and</strong> yet disorder increases with <strong>time</strong>. Boltzmann was able to<br />
demonstrate that <strong>the</strong> second law <strong>of</strong> <strong>the</strong>rmodynamicsóthat an independent physical system<br />
always moves over <strong>time</strong> in <strong>the</strong> direction <strong>of</strong> "maximum entropy" (i.e., maximum disorder)ó<br />
holds if <strong>the</strong> system is given a statistical or probabilistic interpretation. Since <strong>the</strong> direction<br />
<strong>of</strong> maximum entropy has been held to account for <strong>the</strong> "direction <strong>of</strong> <strong>time</strong>" <strong>and</strong> for <strong>the</strong><br />
existence <strong>of</strong> irreversible physical processes like breaking an egg, Boltzmann's contribution<br />
was enormous. It was certainly not lost on <strong>the</strong> young <strong>Einstein</strong>, who embraced probability<br />
eagerly, <strong>and</strong> it later paved <strong>the</strong> way for <strong>the</strong> use <strong>of</strong> proba-<br />
bilistic methods in quantum mechanics (which an older <strong>Einstein</strong> found far less palatable).<br />
Again in opposition to Mach, Boltzmann was a firm believer in <strong>the</strong> importance <strong>of</strong><br />
ma<strong>the</strong>matical methods in <strong>the</strong> physical sciences, pointing <strong>the</strong> way to <strong>the</strong> future <strong>of</strong> physics,<br />
while Mach faced toward <strong>the</strong> past. In ma<strong>the</strong>matizing a physical <strong>the</strong>ory, not only did one<br />
escape <strong>the</strong> idealism <strong>and</strong> subjectivity <strong>of</strong> Mach's "sensationalism," one could also use <strong>the</strong><br />
power <strong>of</strong> ma<strong>the</strong>matical manipulation itself to achieve new insights that would never have<br />
been discovered through mere empirical observation. Boltzmann, unlike Mach, believed in<br />
<strong>the</strong> power <strong>of</strong> <strong>the</strong>oretical models to increase <strong>the</strong> scope <strong>of</strong> physical <strong>the</strong>ory, <strong>and</strong> yet again in<br />
opposition to Mach, he held that <strong>the</strong>se models, if successful, establish a genuine<br />
conception <strong>of</strong> <strong>the</strong> <strong>world</strong> that describes physical reality itselfó a realm, like <strong>the</strong> <strong>world</strong> <strong>of</strong><br />
microscopic atoms, invisible to <strong>the</strong> unaided human senses.<br />
A war ensued between <strong>the</strong> titans Mach <strong>and</strong> Boltzmann over whe<strong>the</strong>r atoms were genuine<br />
features <strong>of</strong> <strong>the</strong> physical <strong>world</strong> or merely useful fictions formulated to assist <strong>the</strong> physicist.<br />
Though a powerful <strong>and</strong> dynamic teacher, Boltzmann was an inferior polemicist compared
to Mach, who carried <strong>the</strong> day on that battlefield, whe<strong>the</strong>r in person or in print <strong>and</strong><br />
whe<strong>the</strong>r in Vienna with Boltzmann or waging battle from afar. In 1897, in one particularly<br />
unpleasant encounter at a meeting <strong>of</strong> <strong>the</strong> Imperial Academy <strong>of</strong> Sciences in Vienna, Mach,<br />
following Boltzmann's talk, rose to voice his objections, declaring bluntly, "I don't believe<br />
that atoms exist!" Boltzmann never did re- cover from such polemics, in spite <strong>of</strong> <strong>the</strong> fact<br />
that in 1905 <strong>Einstein</strong> would deliver a terminal blow to Mach's epistemological critique <strong>of</strong><br />
<strong>the</strong> atomic hypo<strong>the</strong>sis.<br />
In <strong>the</strong> same year, indeed, in <strong>the</strong> same volume <strong>of</strong> <strong>the</strong> same journal, Annalen der Physik, in<br />
which he propounded <strong>the</strong> <strong>the</strong>ory <strong>of</strong> relativity, <strong>Einstein</strong> published a paper on "Brownian<br />
motion," <strong>the</strong> r<strong>and</strong>om dance long observed in <strong>the</strong> behavior <strong>of</strong> microparticles, in which he<br />
made explicit use <strong>of</strong> <strong>the</strong> hypo<strong>the</strong>sis <strong>of</strong> <strong>the</strong> reality <strong>of</strong> atoms to explain this phenomenon <strong>and</strong><br />
to make precise <strong>and</strong> verifiable predictions about <strong>the</strong><br />
behavior <strong>of</strong> particles based on measurements <strong>of</strong> real, nonfictional, atoms. In <strong>the</strong> same<br />
year, <strong>Einstein</strong> had managed to make pr<strong>of</strong>ound physical discoveries employing both Mach's<br />
verificationist critique <strong>of</strong> Newton (to produce <strong>the</strong> <strong>the</strong>ory <strong>of</strong> relativity) <strong>and</strong> Boltzmann's<br />
ma<strong>the</strong>matical <strong>and</strong> model-<strong>the</strong>oretic realism (to produce <strong>the</strong> <strong>the</strong>ory <strong>of</strong> Brown-ian motion <strong>and</strong><br />
establish <strong>the</strong> existence <strong>of</strong> invisible atoms). This was nei<strong>the</strong>r <strong>the</strong> first nor <strong>the</strong> last <strong>time</strong><br />
<strong>Einstein</strong> would succeed in overthrowing not just <strong>the</strong> <strong>the</strong>ories but <strong>the</strong> very <strong>world</strong>view <strong>of</strong> a<br />
great thinker who had once been his inspiration or fa<strong>the</strong>r figure. He would write years<br />
later that "<strong>the</strong> scientist must appear to <strong>the</strong> systematic epistemologist as a type <strong>of</strong><br />
unscrupulous opportunist; he appears as a realist ins<strong>of</strong>ar as he seeks to describe a <strong>world</strong><br />
independent <strong>of</strong> <strong>the</strong> acts <strong>of</strong> perception . . . as a positivist ins<strong>of</strong>ar as he considers concepts<br />
<strong>and</strong> <strong>the</strong>ories justified only to <strong>the</strong> extent to which <strong>the</strong>y furnish a logical representation <strong>of</strong><br />
relations among sensory experiences."<br />
Boltzmann's rescue by <strong>Einstein</strong>, however, came too late. Worn down by years <strong>of</strong> verbal<br />
battles with <strong>the</strong> ruthless Mach, Boltzmann succumbed to his wildly unstable moods. In<br />
1906, on holiday at <strong>the</strong> Bay <strong>of</strong> Duino near Trieste, <strong>the</strong> great physicist tried once again to<br />
take his own life, while his wife <strong>and</strong> daughters were swimming, this <strong>time</strong> by hanging. This<br />
second attempt succeeded. A great influence, not only on <strong>Einstein</strong>, had come to a sudden<br />
<strong>and</strong> tragic end. Ludwig Wittgenstein, who had been hoping to make a career in physics <strong>and</strong><br />
engineering by following Boltzmann's great model, now embarked on a new course that<br />
would ultimately lead him, via research in aeronautical engineering in Manchester,<br />
Engl<strong>and</strong>, to Frege <strong>and</strong> Russell <strong>and</strong> <strong>the</strong> foundations <strong>of</strong> ma<strong>the</strong>matics. Boltzmann's influence,<br />
however, had already left its mark. The physicist's prophetic idea <strong>of</strong> describing a physical<br />
system by locating it in a logical framework in various dimensions <strong>of</strong> physical significance<br />
would have a pr<strong>of</strong>ound effect not only on <strong>the</strong> future <strong>of</strong> quantum mechanics but on <strong>the</strong><br />
bible <strong>of</strong> <strong>the</strong> Schlick circle. For it was in Boltzmann's conception that Wittgenstein found<br />
<strong>the</strong> germ <strong>of</strong> his idea <strong>of</strong> locating an objectóany objectóby its position in what in <strong>the</strong><br />
Tractatus he called "logical space." Where Boltzmann chose to lo-<br />
cate a physical body by specifying its position in terms <strong>of</strong> a set <strong>of</strong> coordinates, three<br />
spatial <strong>and</strong> one temporal, as well as a fifth coordinate, temperature, <strong>and</strong> a sixth, pressure,<br />
<strong>and</strong> so on, which gave <strong>the</strong> "ensemble <strong>of</strong> possible states" <strong>of</strong> a physical system, Wittgenstein<br />
wrote that "<strong>the</strong> facts in logical space are <strong>the</strong> <strong>world</strong>. ... A picture presents a situation in<br />
logical space. ..."
It was no accident that Wittgenstein's Tractatus was required reading in Schlick's Vienna<br />
Circle <strong>of</strong> philosophically minded scientists <strong>and</strong> scientifically minded philosophers. Moritz<br />
Schlick himself had been a student <strong>of</strong> Max Planck, <strong>the</strong> seminal figure in quantum<br />
mechanics. He was deeply impressed by <strong>Einstein</strong>'s <strong>the</strong>ory <strong>of</strong> relativity <strong>and</strong> published<br />
extensively on it, as well as on general epistemology. His most influential work was<br />
probably The General Theory <strong>of</strong> Knowledge. He was clear, quiet, <strong>and</strong> s<strong>of</strong>t-spoken,<br />
politically liberal like most <strong>of</strong> his associates, but not as likely as <strong>the</strong>y were to mix poli-tics<br />
with science <strong>and</strong> philosophy. He did not strike Karl Menger as a German from Berlin. Yet,<br />
when asked by Menger at a party if he was really from Berlin, Schlick's answer was, "Sad,<br />
but true." Schlick tended to idolize intellectual figures, though only, as Menger put it,<br />
"figures <strong>of</strong> <strong>the</strong> first order." A fascination with <strong>Einstein</strong> was succeeded by one for Russell, to<br />
be succeeded in turn by an even stronger worship <strong>of</strong> Wittgenstein. It was in <strong>the</strong> Vienna<br />
Circle that Schlick tried to bring toge<strong>the</strong>r <strong>the</strong> str<strong>and</strong>s <strong>of</strong> physics, philosophy <strong>and</strong><br />
ma<strong>the</strong>matics that had emerged in recent decades.<br />
Unlike Boltzmann, Schlick was hardly an exciting speaker. His lectures, delivered in a<br />
barely audible monotone, were characterized more by precision than by passion. Silverhaired<br />
<strong>and</strong> sporting an elegant vest, he was a model <strong>of</strong> sober dignity. Philosophically, he<br />
rejected Mach's skepticism about <strong>the</strong> existence <strong>of</strong> sense-transcendent entities like atoms,<br />
cleaving, ra<strong>the</strong>r, to Boltzmann's line. But he agreed with Mach that verification is <strong>the</strong><br />
lifeblood <strong>of</strong> physical <strong>the</strong>ories, <strong>and</strong> toge<strong>the</strong>r with Carnap, Hempel <strong>and</strong> o<strong>the</strong>rs, he raised <strong>the</strong><br />
methodological maxims that had guided <strong>Einstein</strong>'s first steps in relativity to <strong>the</strong> level <strong>of</strong> a<br />
formal <strong>the</strong>oretical postulate: <strong>the</strong> meaning <strong>of</strong> a scientific term is<br />
exhausted by its method <strong>of</strong> verification (<strong>the</strong>- "verifiability <strong>the</strong>ory <strong>of</strong> meaning"). The Vienna<br />
Circle made a serious <strong>and</strong> honest attempt to come to grips with <strong>the</strong> extraordinary<br />
developments that had taken place at <strong>the</strong> turn <strong>of</strong> <strong>the</strong> century in physics, philosophy <strong>and</strong><br />
ma<strong>the</strong>matics, but under Schlick's guidance, positivism reigned supreme, even if not in <strong>the</strong><br />
crude form it had taken under Mach. It was precisely <strong>the</strong> hegemony <strong>of</strong> positivism, Godel<br />
wrote later, that allowed <strong>the</strong> members <strong>of</strong> <strong>the</strong> circle to mistake <strong>Einstein</strong> for an ally <strong>and</strong> to<br />
underestimate <strong>the</strong> difficulty <strong>of</strong> rendering ma<strong>the</strong>matics empirically acceptable by<br />
reconstructing it as a system for <strong>the</strong> formal manipulation <strong>of</strong> signs. <strong>Einstein</strong> himself would<br />
awaken <strong>the</strong> positivists from <strong>the</strong>ir misconceptions about <strong>the</strong> ultimate relationship between<br />
his thought <strong>and</strong> <strong>the</strong>irs. And Godel, in short order, would surprise everyone by striking a<br />
fatal blow to <strong>the</strong> most rigorous attempt to reconstitute ma<strong>the</strong>matics as a formal <strong>the</strong>ory <strong>of</strong><br />
signs.<br />
Schlick himself was never fully awakened. In 1936, when he was fifty-four years old, a<br />
mentally unstable former doctoral student <strong>of</strong> his named Hans Nelbock began to hound <strong>and</strong><br />
threaten him. Nelbock had been spurned by a fellow student, Sylvia Borowicka, whose<br />
romantic inclinations were reserved for <strong>the</strong> leader <strong>of</strong> <strong>the</strong> Vienna Circle. Whe<strong>the</strong>r her<br />
affections were reciprocated is not known. In addition, Nelbock's attempts to find work<br />
had come to nothing, something he also held against Schlick, whose complaints against him<br />
had been <strong>the</strong> original source <strong>of</strong> his treatment for mental disorder. Nelbock would show up<br />
in class, glaring over his spectacles at <strong>the</strong> mild pr<strong>of</strong>essor, <strong>and</strong> later make menacing phone<br />
calls when <strong>the</strong> lecturer had retired to his home. Deeply concerned, Schlick notified <strong>the</strong><br />
police <strong>and</strong> acquired a bodyguard, but to no avail. Early on <strong>the</strong> morning <strong>of</strong> June 21, 1936,<br />
<strong>the</strong> deranged student encountered him on <strong>the</strong> steps <strong>of</strong> <strong>the</strong> philosophy building <strong>and</strong> with an<br />
automatic pistol fired four rounds point blank, killing him instantly. "Now you damned
astard, <strong>the</strong>re you have it!" he is reported to have screamed as he stood over Schlick's<br />
body, <strong>the</strong> gun smoking in his h<strong>and</strong>. Thus was <strong>the</strong> Vienna Circle abruptly closed, its history<br />
framed by a suicide <strong>and</strong> a murder.<br />
Long considered a haven for a<strong>the</strong>ism, communism, materialism <strong>and</strong> assorted o<strong>the</strong>r crimes<br />
<strong>of</strong> its Jewish pr<strong>of</strong>essors, Schlick's circle had become a target for <strong>the</strong> increasingly virulent<br />
strain <strong>of</strong> anti-Semitism in Austrian nationalism. Schlick's death was taken to be a promising<br />
development. "It is to be hoped," wrote one newspaper, "that <strong>the</strong> terrible murder at <strong>the</strong><br />
University <strong>of</strong> Vienna will quicken efforts to find a truly satisfactory solution to <strong>the</strong> Jewish<br />
question." The writer, apparently, did not know or care that Schlick was nei<strong>the</strong>r Jewish nor<br />
an a<strong>the</strong>ist but ra<strong>the</strong>r a German Protestant. He had, in fact, but one Jewish assistant,<br />
Friedrich Waismann, who had already been dismissed as part <strong>of</strong> <strong>the</strong> university's attempt to<br />
rid itself <strong>of</strong> Jews. Nelbock's sentencing upon conviction was a ra<strong>the</strong>r lenient ten years,<br />
whereas hanging was <strong>the</strong> customary penalty. The court cited his mental instability as a<br />
mitigating factor. He was, however, forced to sleep on a hard bed, with a new one<br />
delivered every three months. After <strong>the</strong> Anschluss, Nelbock became a kind <strong>of</strong> folk hero. He<br />
was released on probation <strong>and</strong> spent <strong>the</strong> war as a geological technician for <strong>the</strong> Third<br />
Reich. At long last, Nelbock had found work.<br />
The Meaning <strong>of</strong> Relativity<br />
What Schlick had been seeking in <strong>the</strong> Vienna Circle was a unified epis-temologyóa<br />
systematic account <strong>of</strong> what can really be knownóon <strong>the</strong> basis <strong>of</strong> a philosophically coherent<br />
interpretation <strong>of</strong> what <strong>Einstein</strong> had achieved in physics <strong>and</strong> what Frege, Russell, Hilbert<br />
<strong>and</strong> <strong>the</strong>ir predecessors had achieved in <strong>the</strong> foundations <strong>of</strong> ma<strong>the</strong>matics. What exactly had<br />
<strong>Einstein</strong> accomplished, however, <strong>and</strong> what was <strong>the</strong> meaning <strong>of</strong> <strong>the</strong> new direction in <strong>the</strong><br />
foundations <strong>of</strong> ma<strong>the</strong>matics? As <strong>the</strong> century was turning, a perceived tension had arisen<br />
between Newton's laws <strong>of</strong> motionóaccording to which measurement should always be<br />
relative to, <strong>and</strong> affected by, <strong>the</strong> state <strong>of</strong> motion <strong>of</strong> <strong>the</strong> observeró<strong>and</strong> <strong>the</strong> equations <strong>of</strong> <strong>the</strong><br />
Scottish physicist James Clerk Maxwell, one <strong>of</strong> <strong>Einstein</strong>'s heroes, who had unified <strong>the</strong><br />
<strong>the</strong>ories <strong>of</strong> electromagnetism <strong>and</strong> optics.<br />
Maxwell's equations gave as <strong>the</strong> speed <strong>of</strong> light a velocity, denoted by c, relative to a<br />
reference system at rest in Newton's absolute space, known o<strong>the</strong>rwise as <strong>the</strong> e<strong>the</strong>r, a<br />
substance whose existence was <strong>the</strong>orized from first principles ra<strong>the</strong>r than empirical<br />
evidence. This invariance suggested that one could determine whe<strong>the</strong>r a given reference<br />
system was at rest or in motion relative to <strong>the</strong> e<strong>the</strong>r by testing <strong>the</strong> speed <strong>of</strong> light relative<br />
to this system. Any deviation from c would signal motion relative to absolute space. All<br />
efforts at measurement, however, such as those performed in <strong>the</strong> famous (because<br />
exquisitely precise) Michelson-Morley experiments, failed to detect any variation in <strong>the</strong><br />
speed <strong>of</strong> light, which seemed impossible: since <strong>the</strong> reference frames tested were in motion<br />
relative to each o<strong>the</strong>r, <strong>the</strong>y could not all be at rest in <strong>the</strong> e<strong>the</strong>r! Amazingly, although<br />
bullets shot from a moving train have a velocity that is increased by <strong>the</strong> speed <strong>of</strong> <strong>the</strong><br />
engine, <strong>the</strong> measurable velocity <strong>of</strong> a light beam sent out from this same engine is<br />
unaffected by <strong>the</strong> train's speed. The classical principle <strong>of</strong> "addition <strong>of</strong> velocities" was in<br />
peril.
Into <strong>the</strong> breach stepped <strong>the</strong> Dutch physicist Hendrik Antoon Lorentz, <strong>Einstein</strong>'s fa<strong>the</strong>r<br />
figure <strong>and</strong> ano<strong>the</strong>r <strong>of</strong> his heroes. It was Lorentz who, having already perfected <strong>the</strong> form <strong>of</strong><br />
Maxwell's equations, appeared now to save <strong>the</strong> day by supplying <strong>the</strong> exact equations, <strong>the</strong><br />
"Lorentz transformations," that made measurements in one reference, or inertial, frame<br />
equivalent to those obtained in ano<strong>the</strong>r, including <strong>the</strong> "absolute" rest frame <strong>of</strong> <strong>the</strong><br />
postulated e<strong>the</strong>r. In one stroke, Lorentz had succeeded in crystallizing <strong>and</strong> rendering<br />
harmless <strong>the</strong> mismatch between Newton's account <strong>and</strong> Maxwell's. Once <strong>the</strong> Lorentz<br />
transformations were applied, <strong>the</strong> same physical laws could be seen to hold in <strong>the</strong> absolute<br />
rest frame <strong>of</strong> <strong>the</strong> e<strong>the</strong>r <strong>and</strong> in any o<strong>the</strong>r inertial frame.<br />
For most serious thinkers, including Lorentz himself, <strong>the</strong>se "transformations" signaled at<br />
most our inability to measure <strong>the</strong> "real" velocity <strong>of</strong> light or our real velocity through<br />
Newton's postulated e<strong>the</strong>r <strong>of</strong> "absolute space." It was assumed that <strong>the</strong>re were simply<br />
unavoidable distortions in <strong>the</strong> measurement <strong>of</strong> light. lor <strong>the</strong> young <strong>Einstein</strong>, in<br />
43<br />
contrast, <strong>the</strong> mismatch demonstrated that space <strong>and</strong> <strong>time</strong> <strong>the</strong>mselvesó what Kant had<br />
called <strong>the</strong> fundamental "forms <strong>of</strong> intuition"óneeded to be re-created or redefined.<br />
Henceforth, with <strong>the</strong> <strong>Einstein</strong> revolution, <strong>time</strong> itself, not just its measurement with clocks,<br />
would be understood as something essentially relative to <strong>the</strong> motion <strong>of</strong> <strong>the</strong> observer <strong>and</strong><br />
his or her frame <strong>of</strong> reference, as something in its essence related to <strong>the</strong> speed <strong>of</strong> light.<br />
Instead <strong>of</strong> trying to explain our inability to detect <strong>the</strong> "true" speed <strong>of</strong> light, <strong>the</strong>n, <strong>Einstein</strong><br />
incorporated that speed into <strong>the</strong> very definition <strong>of</strong> <strong>time</strong> <strong>and</strong> motion. Lorentzian<br />
engineering had been replaced by <strong>Einstein</strong>ian metaphysics.<br />
In a move that contained as much philosophy as it did science, <strong>Einstein</strong> had succeeded in<br />
combining <strong>the</strong> letter <strong>of</strong> positivismórejecting any properties <strong>of</strong> space <strong>and</strong> <strong>time</strong> that could<br />
not be determined through measurement with rods <strong>and</strong> clocksówith <strong>the</strong> spirit <strong>of</strong> German<br />
metaphysics, determining what kind <strong>of</strong> things space <strong>and</strong> <strong>time</strong> are. Though generations <strong>of</strong><br />
physicists, not least <strong>the</strong> redoubtable Heisenberg, would conclude that <strong>Einstein</strong> had become<br />
<strong>the</strong> self-appointed st<strong>and</strong>ard-bearer <strong>of</strong> positivism, <strong>the</strong> truth lay elsewhere. He was, if<br />
anything, an opportunist.<br />
<strong>Einstein</strong> was merely exploiting, for his own philosophical purposes, certain elements <strong>of</strong><br />
positivism that in <strong>the</strong> particular case <strong>of</strong> special relativity were justified. Godel too would<br />
come to exploit elements <strong>of</strong> <strong>the</strong> positivist methodologyóin his case, <strong>the</strong> formalism <strong>of</strong> <strong>the</strong><br />
Hilbert school <strong>of</strong> ma<strong>the</strong>maticsóto serve his own antipositivist, Pla-tonist ends. Far<br />
removed, also, from <strong>the</strong> positivist creed was <strong>Einstein</strong>'s masterpiece, general relativity,<br />
which went beyond <strong>the</strong> special <strong>the</strong>ory <strong>of</strong> relativity by providing an account <strong>of</strong> gravity.<br />
Fur<strong>the</strong>r removed still was his lifelong opposition to <strong>the</strong> positivistic Copenhagen<br />
interpretation <strong>of</strong> quantum mechanics championed by Heisenberg <strong>and</strong> Bohr. In <strong>Einstein</strong>, <strong>the</strong><br />
positivists would soon discover, <strong>the</strong>y had acquired not a friend but an enemy.
The same lesson was learned, <strong>the</strong> hard way, by David Hilbert, a lowering figure in<br />
ma<strong>the</strong>matics, who, inspired by <strong>Einstein</strong>, had formulated <strong>the</strong> equarioiis <strong>of</strong> general relativity<br />
five days before <strong>Einstein</strong> himself<br />
succeeded, a situation which led, unsurprisingly, to some uncomfortable moments in <strong>the</strong>ir<br />
relationship. The positivistic creedóby its own nature as opposed to <strong>the</strong> spirit <strong>of</strong><br />
ma<strong>the</strong>matics as to philosophyóhad in <strong>the</strong> course <strong>of</strong> <strong>time</strong> found a home in ma<strong>the</strong>matics as<br />
well. As <strong>the</strong> pos-itivists would have it, <strong>the</strong> hierarchy <strong>of</strong> transfinite numbers discovered by<br />
Georg Cantor, a surprising consequence <strong>of</strong> his <strong>the</strong>ory <strong>of</strong> sets, was cast into disrepute for<br />
bearing <strong>the</strong> stain <strong>of</strong> Platonism, for pointing to infinite horizons beyond <strong>the</strong> frame <strong>of</strong> <strong>the</strong><br />
natural realm. The great Hilbert, however, defended Cantor's set <strong>the</strong>ory, proclaiming, "No<br />
one shall expel us from <strong>the</strong> paradise that Cantor has created," <strong>and</strong> calling it "one <strong>of</strong> <strong>the</strong><br />
supreme achievements <strong>of</strong> purely intellectual human activity."<br />
Cantor's paradise was a lush tropical domain <strong>of</strong> infinities that he claimed to have<br />
encountered at <strong>the</strong> very heart <strong>of</strong> ma<strong>the</strong>matics. The importance <strong>of</strong> a sound <strong>the</strong>ory <strong>of</strong><br />
infinity was lost on nei<strong>the</strong>r ma<strong>the</strong>maticians nor physicists. Ma<strong>the</strong>matics, a tool<br />
indispensable to physicists, had been undergoing a gradual development <strong>of</strong> increased rigor<br />
<strong>and</strong> clarification <strong>of</strong> foundations, a process that came to fruition in <strong>the</strong> second half <strong>of</strong> <strong>the</strong><br />
nineteenth century <strong>and</strong> <strong>the</strong> first years <strong>of</strong> <strong>the</strong> twentieth. Infinity played an essential role.<br />
Once <strong>and</strong> for all, it seemed, a firm foundation had been laid for <strong>the</strong> calculus invented by<br />
Newton <strong>and</strong> Leibniz, in which so-called infinitesimals (infinitely small quantities) enjoyed<br />
an ambiguous twilight existence between finitude <strong>and</strong> infinity. Weierstrass, Cauchy, Cantor<br />
<strong>and</strong> o<strong>the</strong>rs developed <strong>the</strong> modern <strong>the</strong>ory <strong>of</strong> limits <strong>of</strong> infinite sequences, which for <strong>the</strong> first<br />
<strong>time</strong> made rigorous sense <strong>of</strong> Newtonian concepts like "point" <strong>and</strong> "instantaneous velocity."<br />
Fur<strong>the</strong>r, Cantor, Frege, Dedekind <strong>and</strong> o<strong>the</strong>rs put forward a convincing <strong>the</strong>ory <strong>of</strong> real<br />
numbersórational numbers as infinite sequences <strong>of</strong> natural numbers, <strong>and</strong> irrational<br />
numbers as infinite sequences <strong>of</strong> rational numbersówhich was crucial, since <strong>the</strong> physical<br />
continuum <strong>of</strong> space <strong>and</strong> <strong>time</strong> could be fully described only by <strong>the</strong> real numbers. (Frege also<br />
advanced an account <strong>of</strong> <strong>the</strong> natural numbers in terms <strong>of</strong> infinite aggregates <strong>of</strong> concepts,<br />
but this fell on deaf ears.) All <strong>of</strong> this required, however, a comprehensive ma<strong>the</strong>matical<br />
<strong>the</strong>ory <strong>of</strong> sequences, or more generally groupings, sets or classes <strong>of</strong> numbers, as well as a<br />
ma<strong>the</strong>matical account <strong>of</strong> infinity. Cantor, in a single bold move, developed precisely what<br />
was needed, a set <strong>the</strong>ory that provided a rigorous account <strong>of</strong> infinite sets.<br />
His first discovery was that <strong>the</strong> requisite infinity had to be "actual," which went against a<br />
two-thous<strong>and</strong>-year tradition in ma<strong>the</strong>matics, from Aristotle to Gauss, which held that<br />
infinity is merely "potential." Before Cantor, it was axiomatic that infinity was not to be<br />
considered a definite number. To say, for example, that <strong>the</strong> natural numbers are infinite in<br />
number was taken to mean not that <strong>the</strong>re is an actual number, infinity, that numbers <strong>the</strong><br />
set <strong>of</strong> natural numbers, but ra<strong>the</strong>r that <strong>the</strong> set <strong>of</strong> natural numbers goes on forever, <strong>and</strong><br />
that <strong>the</strong> most that one can say is that no natural number is big enough to number <strong>the</strong><br />
entire set. Cantor, in contrast, produced a powerful argument for <strong>the</strong> <strong>the</strong>sis that <strong>the</strong>re is<br />
an actual number, which he called Xo (aleph null), that numbers <strong>the</strong> set <strong>of</strong> natural<br />
numbers. Naturally, he emphasized a fact that we can put as follows: The number that<br />
numbers <strong>the</strong> natural numbers cannot itself be a natural number. It must be an unnatural,<br />
or supranatural, or (as Cantor characterized it) transfinite number. The king cannot arise<br />
from <strong>the</strong> class <strong>of</strong> peasants. What established <strong>the</strong> significance <strong>of</strong> such a transfinite number
was Cantor's second great discovery, a pro<strong>of</strong> that Xo is not <strong>the</strong> only transfinite number but<br />
only <strong>the</strong> first, <strong>the</strong> smallest. His pro<strong>of</strong> is one <strong>of</strong> <strong>the</strong> wonders <strong>of</strong> <strong>the</strong> <strong>world</strong>, like <strong>the</strong> hanging<br />
gardens <strong>of</strong> Babylon or <strong>the</strong> pyramids <strong>of</strong> Egypt.<br />
Cantor, along with Frege, had introduced a rigorous <strong>and</strong> ma<strong>the</strong>matically fruitful definition<br />
<strong>of</strong> when two sets are <strong>the</strong> same size, namely, when <strong>the</strong>ir elements can be put into one-toone<br />
correspondence. In his pro<strong>of</strong>, he assumed, by way <strong>of</strong> contradiction, that <strong>the</strong>re exists a<br />
one-to-one pairing between <strong>the</strong> natural numbers <strong>and</strong> <strong>the</strong> real numbers. Using this pairing,<br />
he succeeded in constructing, via what came to be called a "diagonal argument," a real<br />
number that differed from every o<strong>the</strong>r real number in <strong>the</strong> supposedly complete list. Any<br />
attempt to pair <strong>of</strong>f <strong>the</strong> natural numbers with <strong>the</strong> real numbers will fail. It follows that<br />
<strong>the</strong>re are more real numbers than natural numbers, even though <strong>the</strong>re<br />
are infinitely many natural numbers <strong>and</strong> infinitely many real numbers. Since it can be<br />
shown that <strong>the</strong> number <strong>of</strong> real numbers is <strong>the</strong> same as <strong>the</strong> number <strong>of</strong> subsets <strong>of</strong> <strong>the</strong><br />
natural numbers, namely, 2xo, it follows that 2xo > X0. By generalizing his argument,<br />
Cantor was able to show that <strong>the</strong> power set <strong>of</strong> any set is always larger than <strong>the</strong> original<br />
set, <strong>and</strong> <strong>the</strong>refore, for any number, including a transfinite one, <strong>the</strong>re will always exist<br />
ano<strong>the</strong>r that is strictly greater. Thus, not only is infinity an actual number, but <strong>the</strong>re is an<br />
infinity <strong>of</strong> infinities.<br />
Needless to say, infinity is not accessible to <strong>the</strong> five senses, <strong>and</strong> an infinity <strong>of</strong> infinities was<br />
clearly too much for any self-respecting empirically minded positivist to bear. Yet as a<br />
ma<strong>the</strong>matician as well as a physicist, Hilbert realized that ma<strong>the</strong>matics could not do<br />
<strong>without</strong> Cantor's new foundation for <strong>the</strong> <strong>the</strong>ory <strong>of</strong> sets <strong>and</strong> infinities, <strong>and</strong> <strong>the</strong>reby for real<br />
numbers <strong>and</strong> <strong>the</strong> calculus, indispensable for physics. He <strong>the</strong>refore took it upon himself to<br />
make certain, as he put it, that no one would ever be driven from Cantor's paradise. But a<br />
defense was needed for <strong>the</strong> set <strong>the</strong>ory that Cantor had constructed.<br />
From its inception, set <strong>the</strong>ory was haunted by paradoxes <strong>and</strong> conundrums, which served<br />
only to make <strong>the</strong> skeptics more skeptical. For one, as Cantor himself proved, <strong>the</strong> very<br />
"universe" <strong>of</strong> set <strong>the</strong>ory could not itself be a set. There is, provably, no universal set, no<br />
set <strong>of</strong> all sets. The reason, in a nutshell, is that if <strong>the</strong>re were such a set, its number would<br />
have to be larger than any o<strong>the</strong>r transfinite number. As we have already seen, however,<br />
Cantor proved that for any transfinite number, <strong>the</strong>re is always a larger one. Set <strong>the</strong>ory,<br />
conceived <strong>of</strong> as a foundational science, was unable to account for itself. It failed even to<br />
tell us how many sets <strong>the</strong>re are.<br />
The coup de grace, however, to unreconstructed, or "naive," set <strong>the</strong>ory was <strong>the</strong> paradox<br />
that Bertr<strong>and</strong> Russell discovered in its very foundations. Every concept, every property, it<br />
was thought, determines <strong>the</strong> set <strong>of</strong> things that have this property. The property <strong>of</strong> being a<br />
horse determines <strong>the</strong> set <strong>of</strong> horses; <strong>the</strong> property <strong>of</strong> being a prime number, <strong>the</strong> set <strong>of</strong> prime<br />
numbers; <strong>and</strong> <strong>the</strong> property <strong>of</strong> being a small set, <strong>the</strong> set <strong>of</strong> small sets. At worst, <strong>the</strong> set or<br />
class <strong>of</strong> tilings determined by a given
property is empty. As things turned out, however, this was by no means <strong>the</strong> worst that<br />
could happen. Russell, annoyingly, asked us to consider <strong>the</strong> property <strong>of</strong> being a set that is<br />
not a member <strong>of</strong> itself. The set <strong>of</strong> small sets, for example, is not a member <strong>of</strong> itself (since<br />
it is clearly not a small set), whereas <strong>the</strong> set <strong>of</strong> big sets surely is. Russell was able to show,<br />
however, that <strong>the</strong>re could not be such a thing as <strong>the</strong> set <strong>of</strong> all sets that are not members<br />
<strong>of</strong> <strong>the</strong>mselves. If <strong>the</strong>re were such a set, it would have to be <strong>and</strong> also not be a member <strong>of</strong><br />
itself. It follows that it is not true that every property determines <strong>the</strong> set <strong>of</strong> things that<br />
have that property. But <strong>the</strong>n, which properties do determine sets, <strong>and</strong> more generally,<br />
exactly which sets actually exist?<br />
Russell's paradox was disarmingly simple. It left ma<strong>the</strong>maticians breathless. How, one<br />
wonders, did Russell ever come up with his dangerous idea? Historical research has<br />
revealed that he invented his paradox in <strong>the</strong> course <strong>of</strong> trying to refute Cantor's pro<strong>of</strong>,<br />
rehearsed above, that <strong>the</strong>re are more real numbers than natural numbers. His arrow<br />
missed Cantor but struck Frege squarely in <strong>the</strong> chest, toppling his formal development <strong>of</strong><br />
set <strong>the</strong>ory <strong>and</strong> shattering his life's work. We still have <strong>the</strong> polite <strong>and</strong> lethal letter Russell<br />
sent to Frege in 1902: "Dear Colleague, ... I find myself in agreement with you in all<br />
essentials. ... I find in your work discussions <strong>and</strong> distinctions . . . one seeks in vain in <strong>the</strong><br />
works <strong>of</strong> o<strong>the</strong>r logicians. There is just one point where I have encountered a difficulty. ..."<br />
Russell's paradox threw not just set <strong>the</strong>ory but ma<strong>the</strong>matics itself into a crisis, <strong>the</strong> third<br />
great crisis in <strong>the</strong> history <strong>of</strong> ma<strong>the</strong>matics. The first had taken place when <strong>the</strong> Pythagorean<br />
<strong>the</strong>orem revealed to <strong>the</strong> ancient Greeks <strong>the</strong> existence <strong>of</strong> irrational numbers, those that<br />
cannot be expressed as a ratio <strong>of</strong> two natural numbers. The second came when Newton<br />
<strong>and</strong> Leibniz founded <strong>the</strong> infinitesimal calculus on <strong>the</strong> basis <strong>of</strong> infinitesimal numbers, which<br />
were supposed somehow to be simultaneously nonzero <strong>and</strong> yet count for nothing. The<br />
crises had a common cause: ma<strong>the</strong>maticians found <strong>the</strong>mselves confronted with a<br />
paradoxical new kind <strong>of</strong> number. If a way could not be found to incorporate this new entity<br />
into <strong>the</strong>ir thinking, <strong>the</strong>y were faced with <strong>the</strong> prospect <strong>of</strong> seeing <strong>the</strong>ir edifice crumble.<br />
"The sole possible foundations <strong>of</strong> arithmetic seem to vanish," Frege wrote, when<br />
confronted with Russell's paradox.<br />
With <strong>the</strong> third crisis, <strong>the</strong> positivists' star had risen. Ma<strong>the</strong>matics itself, by its very nature as<br />
an a priori, rationalistic science, had always been a thorn in <strong>the</strong> side <strong>of</strong> empiricists. But<br />
now, with Cantor, ma<strong>the</strong>matics had seemingly overreached itself. It had tried to fly too<br />
high in <strong>the</strong> thin air <strong>of</strong> infinity <strong>and</strong> was in danger <strong>of</strong> crashing down on <strong>the</strong> solid earth below,<br />
<strong>the</strong> empirical soil on which natural science is based. For ma<strong>the</strong>maticians like Hilbert who<br />
were also, in spirit, positivists, this engendered a crisis <strong>of</strong> divided loyalties. A way must be<br />
found somehow to preserve Cantor's ma<strong>the</strong>matical paradise. The answer, for Hilbert, was<br />
to reconstruct ma<strong>the</strong>matics itself along <strong>the</strong> lines <strong>of</strong> positivism. The formal pro<strong>of</strong> <strong>of</strong> <strong>the</strong><br />
ma<strong>the</strong>matician would serve as an analogue <strong>of</strong> <strong>the</strong> measuring apparatus <strong>of</strong> <strong>the</strong> empirical<br />
scientist. Formal ma<strong>the</strong>matical pro<strong>of</strong>sówhich can be written down on a blackboard <strong>and</strong><br />
perceived with <strong>the</strong> sensesóare, no less than <strong>the</strong> instruments <strong>of</strong> <strong>the</strong> physicist, things you<br />
can actually "get your h<strong>and</strong>s on." Hilbert, <strong>the</strong>n, was <strong>the</strong> Moses who would lead<br />
ma<strong>the</strong>maticians through <strong>the</strong> desert <strong>of</strong> positivism back to Cantor's paradise. He would<br />
preserve <strong>the</strong> letter if not <strong>the</strong> spirit <strong>of</strong> Cantor's <strong>the</strong>ory <strong>of</strong> infinite sets, in a manner that<br />
satisfied <strong>the</strong> strict epistemological requirements <strong>of</strong> positivism.
Some years later, Kurt Godel would describe positivism as but one element, along with<br />
skepticism <strong>and</strong> materialism, in what he called <strong>the</strong> dominant "leftward" <strong>world</strong>view. The<br />
"rightward" view, in contrast, was characterized by spiritualism, idealism <strong>and</strong> <strong>the</strong>ology (or<br />
metaphysics). "The development <strong>of</strong> philosophy since <strong>the</strong> Renaissance," said Godel, "has by<br />
<strong>and</strong> large gone from right to left," reaching a high water mark in <strong>the</strong> positivistic<br />
Copenhagen interpretation <strong>of</strong> quantum mechanics. "Ma<strong>the</strong>matics," however, "by its nature<br />
as an a priori science, has in <strong>and</strong> <strong>of</strong> itself an inclination toward <strong>the</strong> right." But ma<strong>the</strong>matics<br />
could not escape <strong>the</strong> Zeitgeist, <strong>and</strong> so "around <strong>the</strong> turn <strong>of</strong> <strong>the</strong> century its hour struck: in<br />
particular, it was <strong>the</strong> antinomies <strong>of</strong> set <strong>the</strong>ory, contradictions whose significance was<br />
exaggerated by<br />
skeptics <strong>and</strong> empiricists <strong>and</strong> which were employed as a pretext for a leftward upheaval."<br />
The result <strong>of</strong> this upheaval was Hilbert's brainchild, <strong>the</strong> ma<strong>the</strong>matical program known as<br />
"formalism," in which <strong>the</strong> intuitive notion <strong>of</strong> ma<strong>the</strong>matical truth was to be replaced by <strong>the</strong><br />
formula game <strong>of</strong> pro<strong>of</strong> from a list <strong>of</strong> axioms according to a set <strong>of</strong> rules <strong>of</strong> derivation. "Thus<br />
came into being," wrote Godel, "that curious hermaphroditic thing that Hilbert's formalism<br />
represents, which sought to do justice both to <strong>the</strong> Zeitgeist <strong>and</strong> to <strong>the</strong> nature <strong>of</strong><br />
ma<strong>the</strong>matics." Justice was to be done to <strong>the</strong> spirit <strong>of</strong> <strong>the</strong> <strong>time</strong> by refusing to acknowledge<br />
<strong>the</strong> fundamental axioms <strong>of</strong> ma<strong>the</strong>matics as in any sense true, <strong>and</strong> by declaring that <strong>the</strong><br />
inferences to be drawn from <strong>the</strong>se axioms are, in Godel's words, to be "construed as a<br />
mere game with symbols according to certain rules, likewise not [supported] by insight."<br />
The "implicit definitions" <strong>of</strong> such a formula game did not aim at a true account <strong>of</strong> <strong>the</strong><br />
fundamental entities, such as points, lines, or numbers; ra<strong>the</strong>r, a point, line, or number<br />
was "by definition" simply anything that satisfied <strong>the</strong> axioms. To this conception <strong>the</strong><br />
positivist attached himself like glue, <strong>and</strong> refused to consider <strong>the</strong> possibility <strong>of</strong> an<br />
alternative picture according to which <strong>the</strong>re exists a priori insight into an objective<br />
ma<strong>the</strong>matical <strong>world</strong>. But such stubbornness on <strong>the</strong> part <strong>of</strong> self-described empiricists,<br />
Godel remarked, was really no more than "an a priorism with <strong>the</strong> sign reversed."<br />
Just as Mach's supposedly hard-headed empiricism left him adrift on an ocean <strong>of</strong> purely<br />
mental phenomena, so <strong>the</strong> formalist rejection <strong>of</strong> <strong>the</strong> very idea <strong>of</strong> ma<strong>the</strong>matical truth<br />
turned ma<strong>the</strong>matics into a purely mental construct, a mere game with formulas, with no<br />
intrinsic connection to <strong>the</strong> physical <strong>world</strong>. As Schlick wrote, "A system <strong>of</strong> truths created<br />
with <strong>the</strong> aid <strong>of</strong> implicit definitions does not at any point rest on <strong>the</strong> ground <strong>of</strong> reality." This<br />
had <strong>the</strong> effect <strong>of</strong> making what Eugene Wigner, decades later, would describe as "<strong>the</strong><br />
unreasonable effectiveness <strong>of</strong> ma<strong>the</strong>matics in <strong>the</strong> physical sciences" even more<br />
unreasonable.<br />
Hilbert's formalism, so beloved <strong>of</strong> strict empiricists, was a crowning achievement <strong>of</strong><br />
positivism, <strong>the</strong> most articulate <strong>and</strong> well-developed<br />
attempt to put <strong>the</strong> positivist's money where his mouth is, an attempt to prove once <strong>and</strong> for<br />
all that <strong>the</strong> apparent Platonism <strong>of</strong> ma<strong>the</strong>maticsó <strong>the</strong> very foundation, from Plato on, <strong>of</strong> all<br />
philosophical Platonismsó could be sidestepped. Godel encountered this achievement <strong>of</strong><br />
Hilbert's in Vienna, <strong>the</strong> city in which he lived <strong>and</strong> worked, <strong>the</strong> home <strong>of</strong> <strong>the</strong> Vienna Circle.<br />
Indeed, his own dissertation advisor at <strong>the</strong> University <strong>of</strong> Vienna, <strong>the</strong> ma<strong>the</strong>matician Hans<br />
Hahn, was a positivist who concluded a famous essay, "The Crisis in Intuition," with <strong>the</strong>se
words: "It is not true, as Kant urged, that intuition is a pure a priori means <strong>of</strong> knowledge,<br />
but ra<strong>the</strong>r that it is a force <strong>of</strong> habit rooted in psychological inertia." And <strong>the</strong> patron saint,<br />
as we have seen, <strong>of</strong> <strong>the</strong> Vienna Circle was Wittgenstein, its bible, <strong>the</strong> Tractatus, which<br />
declared that <strong>the</strong> only real facts are <strong>of</strong> <strong>the</strong> empirical variety. "The essence <strong>of</strong> this view,"<br />
Godel noted, "is that <strong>the</strong>re exists no such thing as a ma<strong>the</strong>matical fact." Ma<strong>the</strong>matics, for<br />
Wittgenstein, consists merely <strong>of</strong> <strong>the</strong> transformations <strong>of</strong> formulas to obtain ma<strong>the</strong>matical<br />
identities. In particular, it is not derived from ma<strong>the</strong>matical facts, whereas physics is<br />
derived from facts about <strong>the</strong> physical <strong>world</strong>.<br />
Between Hilbert <strong>and</strong> Wittgenstein, it seemed, <strong>the</strong> positivists had finally laid to rest <strong>the</strong><br />
ghost <strong>of</strong> ma<strong>the</strong>matics, which had seemed to resist incorporation into <strong>the</strong>ir fold. The ghost,<br />
however, was hardly dead, as Godel would demonstrate from within <strong>the</strong> heart <strong>of</strong><br />
Wittgenstein country, Vienna. His incompleteness <strong>the</strong>orem was a grenade aimed at Hilbert<br />
that l<strong>and</strong>ed in <strong>the</strong> very laps <strong>of</strong> <strong>the</strong> positivists. Vienna's circle could not after all be<br />
completed.<br />
4 A Spy in <strong>the</strong> House <strong>of</strong> Logic<br />
Every spy's life has ended in ignominious death. ANAIS NIN<br />
Russell's Paradox, communicated to Frege in 1902, engendered widespread fear in<br />
ma<strong>the</strong>matical circles. As Godel put it, in retrospect, "[Russell] brought to light <strong>the</strong> amazing<br />
fact that our logical intuitions . . . are self-contradictory." On pain <strong>of</strong> contradiction, it<br />
could no longer be assumed that every property determines <strong>the</strong> class <strong>of</strong> things that have<br />
that property. It could still be trusted that <strong>the</strong> property <strong>of</strong> redness sufficed to determine<br />
<strong>the</strong> set <strong>of</strong> all red things, but such confidence was no longer justified for every property.<br />
The concept <strong>of</strong> class or set, in particular, which had assumed increasing importance in<br />
ma<strong>the</strong>matics <strong>and</strong> logic <strong>and</strong> which seemed intuitively clear, turned out to be so poorly<br />
understood that in Frege's epoch-making formulation <strong>of</strong> modern logic it led to a<br />
straightforward contradiction. The concern was not a communist under every bed but a<br />
paradox asleep on every ma<strong>the</strong>matical bedspread.<br />
There was no place to hide. Frege had built a contradiction into <strong>the</strong> very foundation stone<br />
<strong>of</strong> <strong>the</strong> mighty edifice <strong>of</strong> logic. Cantor, <strong>the</strong> creator <strong>of</strong> modern set <strong>the</strong>ory, had also<br />
constructed, if not contradictions, <strong>the</strong>n at least paradoxes aplenty in his dizzying hierarchy<br />
<strong>of</strong> transfinite numbers. Even Euclidean geometry, <strong>the</strong> ancestor <strong>of</strong> all models <strong>of</strong><br />
ma<strong>the</strong>matical rigor <strong>and</strong> certainty, <strong>the</strong> s<strong>of</strong>t pillow for two thous<strong>and</strong> years <strong>of</strong> sweet<br />
ma<strong>the</strong>matical dreams, had become suspect due to<br />
<strong>the</strong> recent demonstration <strong>of</strong> <strong>the</strong> logical consistency <strong>of</strong> non-Euclidean geometries. If<br />
Euclidean geometry could be rejected <strong>without</strong> contradiction as <strong>the</strong> truth about space (as<br />
arithmetic is <strong>the</strong> truth about natural numbers), wherein did its truth lie? Everywhere,<br />
intuition was under suspicion. If <strong>the</strong> house <strong>of</strong> logic was not itself secure, where else could<br />
safety be found? Something had to be done.
In Gottingen, <strong>the</strong> dean <strong>of</strong> ma<strong>the</strong>maticians, David Hilbert, declared, "Where else would<br />
reliability <strong>and</strong> truth be found if even ma<strong>the</strong>matical thinking fails?" Cantor's paradise, in<br />
particular, had to be shored up. "The definitive clarification <strong>of</strong> <strong>the</strong> nature <strong>of</strong> <strong>the</strong> infinite,"<br />
said Hilbert, "has become necessary, not merely for <strong>the</strong> special interests <strong>of</strong> <strong>the</strong> individual<br />
sciences, but ra<strong>the</strong>r for <strong>the</strong> honor <strong>of</strong> human underst<strong>and</strong>ing itself." Having spent <strong>the</strong> first<br />
years <strong>of</strong> <strong>the</strong> century in wide-ranging research, he turned his attention in <strong>the</strong> century's<br />
second decade to <strong>the</strong> crises infecting <strong>the</strong> foundations <strong>of</strong> his mighty edifice, devoting his<br />
considerable resources to a solution.<br />
In Vienna, Kurt Godel was becoming part <strong>of</strong> <strong>the</strong> problem, though in 1930 he did not yet<br />
know it. Even Hans Hahn, his <strong>the</strong>sis advisor <strong>and</strong> a true believer in <strong>the</strong> positivist credo,<br />
discovered too late <strong>the</strong> full extent <strong>of</strong> Godel's heresy. "As concerns <strong>the</strong> <strong>world</strong>," Hahn<br />
declared, "<strong>the</strong> only possible st<strong>and</strong>point seems to me to be <strong>the</strong> empiricist one. . ..<br />
Knowledge concerning reality can in no way be obtained through pure thought." Yet<br />
ma<strong>the</strong>matics <strong>and</strong> logic, <strong>the</strong> tools <strong>of</strong> scientific empiricism, were not cooperating. As Hahn<br />
put it, "A very simple fact now seems to st<strong>and</strong> in <strong>the</strong> way <strong>of</strong> realizing this empiricist<br />
st<strong>and</strong>point, namely, <strong>the</strong> existence <strong>of</strong> logic <strong>and</strong> ma<strong>the</strong>matics." These words were uttered at<br />
<strong>the</strong> very conference at which Godel would rise to defeat <strong>the</strong> last best hope for <strong>the</strong><br />
positivists to incorporate ma<strong>the</strong>matics within <strong>the</strong>ir religion <strong>of</strong> ultraempiricism. And he<br />
would do so by exploiting Hilbert's own weapon <strong>of</strong> choice: <strong>the</strong> formalism <strong>of</strong> ma<strong>the</strong>matical<br />
logic. Godel, in short, would destroy from within. This is <strong>the</strong> reason <strong>the</strong>y shoot spies.<br />
In <strong>the</strong> case <strong>of</strong> Godel, <strong>the</strong> formalisms he employed, though in <strong>the</strong>mselves acceptable to <strong>the</strong><br />
positivists, were Janus-faced by design. On one h<strong>and</strong>, <strong>the</strong>y were irrefutable ma<strong>the</strong>matical<br />
<strong>the</strong>orems. On <strong>the</strong> o<strong>the</strong>r, <strong>the</strong>ir<br />
most natural, indeed irresistible philosophical implications would undermine <strong>the</strong> very spirit<br />
<strong>of</strong> positivism. It was a case <strong>of</strong> using <strong>the</strong> letter <strong>of</strong> a false doctrine to overthrow its spirit.<br />
And this was a specialty <strong>of</strong> Godel's, as it was <strong>of</strong> his future friend <strong>Einstein</strong>: constructing<br />
ma<strong>the</strong>matical formalisms pregnant with philosophical meaning. It was a talent guaranteed<br />
to arouse ire on both sides: <strong>the</strong> friends <strong>of</strong> formalism would be hard pressed to reject<br />
philosophical implications derived from inescapable premises, while <strong>the</strong> enemies <strong>of</strong><br />
formalism would be forced to admit that something philosophically significant could after<br />
all be achieved within <strong>the</strong> narrow confines <strong>of</strong> <strong>the</strong> formal.<br />
The Leitmotif <strong>of</strong> <strong>the</strong> Twentieth Century<br />
who pays any attention to <strong>the</strong> syntax <strong>of</strong> things will never wholly kiss you.<br />
e.e. cummings<br />
(iodel's incompleteness <strong>the</strong>orem <strong>of</strong> 1931 began innocently, as an attempt not to refute but<br />
to fulfill Hilbert's program. Hilbert's idea was to safeguard ma<strong>the</strong>matics from hidden<br />
contradictions by replacing <strong>the</strong> intuitive ma<strong>the</strong>matics <strong>of</strong> each ma<strong>the</strong>matical domain with a
system <strong>of</strong> axioms written in a pure formula language that, although having a st<strong>and</strong>ard<br />
semantic interpretation, could be manipulated according to <strong>the</strong> mechanical rules <strong>of</strong> pure<br />
syntax (much like a computer program <strong>of</strong> today). Hilbert's program consisted in finding a<br />
system <strong>of</strong> primitive formulas called axioms from which, according to fixed rules <strong>of</strong><br />
pro<strong>of</strong>órules <strong>of</strong> syntaxóone could derive all <strong>the</strong> <strong>the</strong>orems <strong>of</strong> <strong>the</strong> given ma<strong>the</strong>matical<br />
domain. Two features <strong>of</strong> such a formal system were crucial: consistency <strong>and</strong> completeness.<br />
As a prophylactic against unwelcome surprises, a formal system had to be consistent: two<br />
<strong>the</strong>orems that contradict each o<strong>the</strong>r should not be able to be derived from <strong>the</strong> axioms.<br />
And <strong>the</strong> system should be complete, in <strong>the</strong><br />
sense that all true statements expressible within <strong>the</strong> system (under a suitable<br />
interpretation) should be derivable from <strong>the</strong> axioms. To prevent circularity, <strong>the</strong> system in<br />
which consistency is to be proved must not itself employ any ma<strong>the</strong>matically suspect or<br />
controversial procedures that could render its own consistency suspect. It must be, to use<br />
Hilbert's invented term, not exactly finite but ra<strong>the</strong>r "finitary," in <strong>the</strong> sense that its pro<strong>of</strong>s<br />
must be in principle surveyable by sense experience <strong>and</strong> must not at any point appeal to an<br />
abstract, completed infinity <strong>of</strong> <strong>the</strong> kind proposed by Cantor.<br />
Hilbert's formalism was just one exampleó<strong>the</strong> most rigorous, ma<strong>the</strong>matical oneó<strong>of</strong> <strong>the</strong><br />
spirit that lies at <strong>the</strong> very heart <strong>of</strong> <strong>the</strong> twentieth century. At its core it involves <strong>the</strong><br />
dominance <strong>of</strong> form over content, syntax over semantics, pro<strong>of</strong> over truth. It is no surprise<br />
that <strong>the</strong> principal embodiment <strong>of</strong> a formal system, <strong>the</strong> computer, a pure syntax machine,<br />
would become <strong>the</strong> century's dominant mechanical device. But <strong>the</strong> computer was still just<br />
one element <strong>of</strong> <strong>the</strong> Zeitgeist. In art, science, philosophy, ma<strong>the</strong>matics, music,<br />
architecture <strong>and</strong> linguistics, formalism in its most general sense became <strong>the</strong> dominant<br />
<strong>the</strong>me. In painting, for instance, Cezanne's realism was a hidden case <strong>of</strong> <strong>the</strong> free play <strong>of</strong><br />
geometrical forms, a fact his subjects came increasingly to appreciate as <strong>the</strong>y realized<br />
that <strong>the</strong> geometrical constraints <strong>of</strong> his canvases dominated any attempt to capture <strong>the</strong><br />
shape or spirit <strong>of</strong> those who sat before him. This paved <strong>the</strong> way for <strong>the</strong> explicit rule <strong>of</strong> <strong>the</strong><br />
play <strong>of</strong> free forms by <strong>the</strong> cubists, led by Gris, Braque, <strong>and</strong> Picasso. The Cezanne <strong>of</strong> music<br />
was Brahms, whose post-Romantic chromaticism hid <strong>the</strong> dominance <strong>of</strong> pure logical<br />
structure at its core. Wittgenstein, in whose Vienna home Brahms performed, put <strong>the</strong><br />
matter darkly: In Brahms, he said, "I can begin to hear <strong>the</strong> sound <strong>of</strong> machinery." This<br />
hidden formalism, this logical machinery, was not lost on Brahms's admirer Schoenberg,<br />
who would soon champion <strong>the</strong> freely constructed forms <strong>of</strong> serial music, <strong>the</strong> most explicitly<br />
conventional, ma<strong>the</strong>matical method ever undertaken in music. The principal st<strong>and</strong>ardbearer<br />
<strong>of</strong> Schoenberg's piano music, Glenn Gould, would speak <strong>of</strong> producing a structural "xray"<br />
<strong>of</strong> <strong>the</strong><br />
score in his performances. As <strong>the</strong> high priest <strong>of</strong> bones <strong>without</strong> flesh, Gould made it clear<br />
that his first god was Bach, whom Schoenberg also worshipped, his gospel, <strong>the</strong> fugue.<br />
Nor was physics left behind. On <strong>the</strong> contrary, it was in <strong>the</strong> vanguard. In special relativity,<br />
<strong>Einstein</strong> had ab<strong>and</strong>oned <strong>the</strong> Kantian intuitions <strong>of</strong> space <strong>and</strong> <strong>time</strong> for <strong>the</strong> ma<strong>the</strong>matical<br />
formalism <strong>of</strong> space-<strong>time</strong>, constrained only by <strong>the</strong> formal requirement <strong>of</strong> Lorentz invariance<br />
<strong>and</strong> <strong>the</strong> physical postulate <strong>of</strong> <strong>the</strong> limiting value <strong>of</strong> <strong>the</strong> speed <strong>of</strong> electromagnetic signals.<br />
General relativity, <strong>the</strong> more inclusive <strong>the</strong>ory, would yield an abstract structure governed<br />
by yet more general logical constraints. And <strong>the</strong> cognitive <strong>and</strong> social sciences would follow<br />
physics' lead. Later in <strong>the</strong> century, Noam Chomsky would re-create linguistics as a
structural science modeled on Frege's logic, with syntax explicitly dominant over<br />
semantics, while Claude Levi-Strauss would stitch <strong>the</strong> abstractions <strong>of</strong> structuralism into <strong>the</strong><br />
many-colored quilt <strong>of</strong> anthropology. In ma<strong>the</strong>matics itself, <strong>the</strong> trend was increasingly<br />
toward <strong>the</strong> reduction <strong>of</strong> a domain into <strong>the</strong> structural relationships that obtained between<br />
its elements. What mattered, Hilbert insisted in his reconstruction <strong>of</strong> Euclid's geometry,<br />
was not what points, lines <strong>and</strong> planes are, i.e., <strong>the</strong> semantics <strong>of</strong> <strong>the</strong> fundamental terms,<br />
but <strong>the</strong> logical relationships that obtain between <strong>the</strong> basic elements, i.e., <strong>the</strong> syntax <strong>of</strong><br />
<strong>the</strong> formal system. As Hilbert put it, a point in Euclidean space could be a beer mug for all<br />
he cared, as long as it obeyed <strong>the</strong> rules <strong>of</strong> his formal system. (This idea <strong>of</strong> a so-called<br />
implicit definition <strong>of</strong> a concept via its relationships with o<strong>the</strong>r concepts was firmly<br />
rejected by <strong>the</strong> fa<strong>the</strong>r <strong>of</strong> <strong>the</strong> formal system, Gottlob Frege. In his account <strong>of</strong> <strong>the</strong> natural<br />
numbers, Frege objected that his own preliminary contextual or implicit definitions failed<br />
to determine what <strong>the</strong> individual numbers actually are, in particular whe<strong>the</strong>r some<br />
collection could have <strong>the</strong> number Julius Caesar attached to it, nor "whe<strong>the</strong>r that same<br />
familiar conqueror <strong>of</strong> Gaul is a number or not." This was a remark as crazy as it was<br />
beautiful. No one, <strong>of</strong> course, is likely to confuse Julius Caesar with a number. But that is<br />
not, Frege pointed out, due to <strong>the</strong> power <strong>of</strong> <strong>the</strong> implicit definitions.)<br />
In <strong>the</strong> division to reduce all to syntax, to focus on form alone, could be found <strong>the</strong> freedom<br />
<strong>of</strong> <strong>the</strong> creative imagination. There could also be found safety. Because <strong>the</strong> rules <strong>of</strong> <strong>the</strong><br />
formal system are our own creation, we would be able to police <strong>the</strong>m, to examine <strong>the</strong>m as<br />
pure signs to see that <strong>the</strong>y did not lead to inconsistency, to contradiction. Consistency, not<br />
truth, increasingly became <strong>the</strong> goal <strong>of</strong> <strong>the</strong> formal systems <strong>of</strong> science, just as au<strong>the</strong>nticity<br />
became <strong>the</strong> battle cry <strong>of</strong> <strong>the</strong> systems <strong>of</strong> ethics or forms <strong>of</strong> life. (Sartre's existentialists, for<br />
example, attempted bravely, <strong>and</strong> perhaps foolishly, to replace conscience with<br />
au<strong>the</strong>nticity. The problem, <strong>of</strong> course, was that Hitler too was au<strong>the</strong>ntic. It was <strong>the</strong> content<br />
<strong>of</strong> his beliefs that was <strong>the</strong> problem, not <strong>the</strong>ir consistency.) Since ma<strong>the</strong>matics is <strong>the</strong><br />
language <strong>of</strong> formal relationships, it became increasingly clear that <strong>the</strong> central formalism<br />
was that <strong>of</strong> ma<strong>the</strong>matics itself, <strong>and</strong> that if this could not be rendered secure from<br />
inconsistency, nothing else could. If, <strong>the</strong>n, formalism is <strong>the</strong> leitmotif <strong>of</strong> <strong>the</strong> twentieth<br />
century, <strong>and</strong> Hilbert's ma<strong>the</strong>matical formalism captures <strong>the</strong> essence <strong>of</strong> all o<strong>the</strong>r<br />
formalisms, <strong>the</strong>n Godel's incompleteness <strong>the</strong>orem, which dramatically <strong>and</strong> inescapably<br />
refutes Hilbert's program, can well be considered <strong>the</strong> most significant intellectual<br />
accomplishment <strong>of</strong> <strong>the</strong> twentieth century.<br />
Godel took up Hilbert's momentous project by attempting first to see whe<strong>the</strong>r one could<br />
prove <strong>the</strong> consistency <strong>and</strong> completeness <strong>of</strong> a formal axiom system for ma<strong>the</strong>matical<br />
analysis. He began with <strong>the</strong> task <strong>of</strong> proving consistency <strong>and</strong> completeness for <strong>the</strong> weaker<br />
axiom system <strong>of</strong> arithmetic, or number <strong>the</strong>ory, a subsystem <strong>of</strong> analysis. Here <strong>the</strong><br />
conditions were propitious. For thous<strong>and</strong>s <strong>of</strong> years, geometry had been <strong>the</strong> paradigm <strong>of</strong> an<br />
axiomatic system, but in <strong>the</strong> late nineteenth century, Frege, Dedekind <strong>and</strong> Peano had<br />
achieved <strong>the</strong> same result for arithmetic. They constructed a system <strong>of</strong> axioms, or<br />
postulatesó known today, for no good reason, simply as <strong>the</strong> five Peano postulatesófrom<br />
which it was believed that all truths about <strong>the</strong> natural numbers could be derived. But<br />
derived how? By logic alone, <strong>the</strong> logic that Frege had invented in his Begriffsschrift.<br />
Whereas <strong>the</strong> axioms had, semantically speaking, genuine ma<strong>the</strong>matical content, <strong>the</strong> rules<br />
<strong>of</strong> in-<br />
ference were a matter <strong>of</strong> pure syntax, a series <strong>of</strong> mechanical instructions that could be<br />
followed blindly, with no reference to truth or ma<strong>the</strong>matical content, followed, as <strong>the</strong><br />
logician John Myhill put it, by an imbecile or a computer.
What Godel discovered, however, was that not only are <strong>the</strong> Peano postulates in fact<br />
incomplete, any system <strong>of</strong> axioms or postulates (even if infinitely large) from which<br />
arithmetic can be derived that satisfies any reasonable ma<strong>the</strong>matical criteria <strong>of</strong><br />
surveyability by a finite mind is <strong>of</strong> necessity incomplete. (An infinite mind, like God's,<br />
which can grasp all <strong>the</strong> numbers at once, presumably has no need <strong>of</strong> axioms.) So <strong>the</strong><br />
simplest <strong>and</strong> most basic domain <strong>of</strong> ma<strong>the</strong>matics, <strong>the</strong> arithmetic <strong>of</strong> <strong>the</strong> natural numbers,<br />
<strong>the</strong> rock on which <strong>the</strong> gr<strong>and</strong> edifice <strong>of</strong> ma<strong>the</strong>matics st<strong>and</strong>s, turns out to be, from a formal<br />
axiomatic point <strong>of</strong> view, incomplete, <strong>and</strong> even worse, incompletable. Indeed, since a<br />
computer can prove only <strong>the</strong>orems based on <strong>the</strong> axioms its programmer has fed into itóit<br />
cannot, as Godel emphasized, create new axioms on its ownóit follows that in principle no<br />
computer or fully specified system <strong>of</strong> computers, even if infinite, will ever capture all <strong>the</strong><br />
truths <strong>of</strong> arithmetic (never mind <strong>the</strong> rest <strong>of</strong> ma<strong>the</strong>matics). As Godel put it, "Continued<br />
appeals to ma<strong>the</strong>matical intuition are necessary . . . for <strong>the</strong> solution <strong>of</strong> <strong>the</strong> problems <strong>of</strong><br />
finitary number <strong>the</strong>ory."<br />
The ma<strong>the</strong>matical fact <strong>of</strong> <strong>the</strong> incompleteness <strong>of</strong> formal arithmetic, moreover, is accessible<br />
not only to us, thinkers with minds <strong>and</strong> ma<strong>the</strong>matical intuitions; ironically, a computer can<br />
be programmed to prove Godel's <strong>the</strong>orems, <strong>the</strong> very <strong>the</strong>orems that establish <strong>the</strong> intrinsic<br />
limitations <strong>of</strong> computers. The truths <strong>of</strong> arithmetic, <strong>the</strong>n, cannot in principle be confined to<br />
a formal system. Here is a crucial difference between truth <strong>and</strong> pro<strong>of</strong>: a ma<strong>the</strong>matical<br />
pro<strong>of</strong>, in <strong>the</strong> sense in which we are discussing it here, is always a pro<strong>of</strong> in, <strong>and</strong> relative to,<br />
a given formal system, whereas truth, as such, is absolute. What Godel proved is that<br />
ma<strong>the</strong>matical truth is not reducible to (formal or mechanical) pro<strong>of</strong>. Syntax cannot<br />
supplant semantics. The leitmotif <strong>of</strong> <strong>the</strong> twentieth century, it turns out, st<strong>and</strong>s in need <strong>of</strong><br />
revision. Mechanical rules cannot obviate <strong>the</strong> need for meaning, <strong>and</strong> what gives us access<br />
to meaning,<br />
namely, intuition, cannot be dispensed with even in ma<strong>the</strong>matics, indeed, even in<br />
arithmetic. This was <strong>the</strong> first nail in Hilbert's c<strong>of</strong>fin.<br />
The second nail was not long in coming. Godel soon proved his second incompleteness<br />
<strong>the</strong>orem, which demonstrated, with yet fur<strong>the</strong>r irony, that if a given system <strong>of</strong> axioms for<br />
arithmetic were in fact consistent, <strong>the</strong>n it could not be proved consistent by <strong>the</strong> system<br />
itself. Put o<strong>the</strong>rwise, only an inconsistent formal system can prove its own consistency!<br />
Von Neumann, <strong>the</strong> quickest <strong>of</strong> <strong>the</strong> quick, having heard Godel announce his incompleteness<br />
results, derived, shortly <strong>the</strong>reafter, <strong>the</strong> unprovability <strong>of</strong> consistency. "I would be very<br />
much interested," he wrote Godel, "to hear your views on this. ... If you are interested, I<br />
will send you <strong>the</strong> pro<strong>of</strong> details." One can imagine his disappointment when Godel informed<br />
him that <strong>the</strong> manuscript for <strong>the</strong> second <strong>the</strong>orem was already on its way to <strong>the</strong> editors. It<br />
was Von Neumann, however, who argued, against Godel himself, that <strong>the</strong> unprovability <strong>of</strong><br />
consistency, as Godel had demonstrated it, left no wiggle room for <strong>the</strong> Hilbert program.<br />
Whereas for several years, Godel was cautious not to prejudge <strong>the</strong> question <strong>of</strong> whe<strong>the</strong>r<br />
Hilbert might discover a finitary pro<strong>of</strong> <strong>of</strong> consistency to which Godel's second <strong>the</strong>orem did<br />
not apply, Von Neumann, from <strong>the</strong> beginning, was confident that this could never happen.<br />
Assuming that one rejects Russell's controversial "axiom <strong>of</strong> reducibility," he said, "one<br />
cannot obtain a foundation for classical ma<strong>the</strong>matics via logical means." Von Neumann's<br />
striking prescience, however, concerning <strong>the</strong> full significance <strong>of</strong> what Godel had<br />
discovered may well have served only to deepen his regret that he had not been <strong>the</strong> first<br />
to make <strong>the</strong>se discoveries. Even <strong>the</strong> fact that he was one <strong>of</strong> <strong>the</strong> fa<strong>the</strong>rs <strong>of</strong> <strong>the</strong> modern
computer <strong>and</strong> a chief architect <strong>of</strong> <strong>the</strong> atomic bomb in Los Alamos did not suffice to<br />
assuage his disappointment.<br />
The incompleteness <strong>the</strong>orems sent a shock wave through <strong>the</strong> <strong>world</strong> <strong>of</strong> ma<strong>the</strong>matics.<br />
Hermann Weyl, one <strong>of</strong> <strong>the</strong> first permanent members <strong>of</strong> <strong>the</strong> ma<strong>the</strong>matical faculty at <strong>the</strong><br />
Institute for Advanced Study, spoke <strong>of</strong> <strong>the</strong> Godel "debacle," <strong>the</strong> Godel "catastrophe." The<br />
two-thous<strong>and</strong>-year-old ideal <strong>of</strong> axiomatization inaugurated by Euclidó<strong>the</strong> paradigm<br />
<strong>of</strong> captured rationalityóhad been shattered, <strong>and</strong> <strong>the</strong> blow had been struck, annoyingly,<br />
just when Frege <strong>and</strong> Hilbert had succeeded in perfecting <strong>the</strong> very idea <strong>of</strong> a formal system<br />
<strong>of</strong> axioms. Not only <strong>the</strong> results but <strong>the</strong> very methods employed in Godel's <strong>the</strong>orem were so<br />
unexpected that it was years before ma<strong>the</strong>maticians <strong>and</strong> logicians began to grasp <strong>the</strong>ir full<br />
significance. Godel had carried even fur<strong>the</strong>r than 1 filbert <strong>the</strong> idea <strong>of</strong> treating formal<br />
systems <strong>of</strong> ma<strong>the</strong>matics as ma<strong>the</strong>matical objects in <strong>the</strong>ir own right, which resulted in<br />
conclusions exactly opposite to what Hilbert had intended. (This is an instance <strong>of</strong> what<br />
Hegel called <strong>the</strong> "cunning <strong>of</strong> history," whereby history itself con-trives, somehow, to<br />
subvert <strong>the</strong> intentions <strong>of</strong> its most dramatic actors.)<br />
In <strong>the</strong> case <strong>of</strong> Godel's <strong>the</strong>orem, as with his later writings on rela-tivity, <strong>the</strong> difficulty in<br />
taking <strong>the</strong> true measure <strong>of</strong> its significance was due not just to <strong>the</strong> ma<strong>the</strong>matical but to <strong>the</strong><br />
philosophical hurdles that had to be overcome. In <strong>the</strong> next few paragraphs, we will take a<br />
firsth<strong>and</strong> look at some <strong>of</strong> <strong>the</strong> details <strong>of</strong> Godel's mysterious <strong>and</strong> beautiful pro<strong>of</strong>. If <strong>the</strong> going<br />
gets tough, <strong>the</strong> tough may ei<strong>the</strong>r get going or <strong>the</strong>y may, <strong>without</strong> loss, simply skim lightly<br />
over <strong>the</strong> details to get <strong>the</strong> gist <strong>of</strong> (lodel's argument, or <strong>the</strong>y may even take <strong>of</strong>f, fly over<br />
<strong>and</strong> pick up <strong>the</strong> thread at <strong>the</strong> clearing after <strong>the</strong> forest. Do not, in any case, be<br />
intimidated; to switch images, you can admire <strong>the</strong> music <strong>without</strong> attending to <strong>the</strong> words.<br />
To appreciate Godel's <strong>the</strong>orem is your birthright; let no one, including <strong>the</strong> ma<strong>the</strong>matical<br />
police, deprive you <strong>of</strong> what you have a right to enjoy.<br />
Bear in mind also what Godel proved <strong>and</strong> what he did not. He did not discover some deep<br />
<strong>and</strong> mysterious ma<strong>the</strong>matical proposition that no formal system was powerful enough to<br />
count among its <strong>the</strong>orems. That would have demonstrated <strong>the</strong> existence <strong>of</strong> an absolutely<br />
unprovable ma<strong>the</strong>matical proposition, something that Godel, like Hilbert, was deeply<br />
skeptical <strong>of</strong>. Ra<strong>the</strong>r, what he showed is that in any particular formal system <strong>of</strong> sufficient<br />
strength, given <strong>the</strong> limitations imposed on such a system ins<strong>of</strong>ar as it is truly formal, <strong>the</strong>re<br />
would always be some formula which, while intuitively true, could not be proved in or<br />
relative to that system. And <strong>the</strong> same holds for its negation. But <strong>the</strong><br />
formula would be a perfectly ordinary, though complex, ma<strong>the</strong>matical proposition, which<br />
never<strong>the</strong>less, because <strong>of</strong> its form, slipped through <strong>the</strong> net <strong>of</strong> <strong>the</strong> given formal system. That<br />
very formula, however, could always be proved in a more inclusive formal system; only<br />
that new formal system, in turn, would be unable to prove some new formula, which was<br />
never<strong>the</strong>less intuitively true. And so on. There was, <strong>the</strong>n, no "supervirus" that affected all<br />
formal systems. Instead, for each particular formal system, <strong>the</strong>re would be some perfectly<br />
ordinary bug or virus that rendered that system incomplete.
Triple Fugue: Intuitive Ma<strong>the</strong>matics, Formal Ma<strong>the</strong>matics, <strong>and</strong> Metama<strong>the</strong>matics<br />
Godel's beautiful fugue was constructed from three distinguishable ma<strong>the</strong>matical<br />
languages or <strong>the</strong>ories. The beauty was to be found in <strong>the</strong> pattern <strong>of</strong> relationships woven<br />
from <strong>the</strong> three parts. To begin with, <strong>the</strong>re was intuitive arithmetic, <strong>the</strong> arithmetic found<br />
in ma<strong>the</strong>matical textbooks written in <strong>the</strong> language <strong>of</strong> ordinary ma<strong>the</strong>matics. Call this<br />
language or <strong>the</strong>ory IA (for intuitive arithmetic). The propositions <strong>of</strong> IA are sentences with<br />
content: <strong>the</strong>y express truths or falsehoods about numbers. Next <strong>the</strong>re was a formal<br />
deductive system for arithmeticóin Godel's pro<strong>of</strong>, a system <strong>of</strong> pure syntax put forward by<br />
Bertr<strong>and</strong> Russell, modeled on <strong>the</strong> original by Fregeówith a specified set <strong>of</strong> axioms <strong>and</strong><br />
explicit rules <strong>of</strong> deduction that determined which formulas were <strong>the</strong>orems. Call this<br />
system FA (for formal arithmetic). The "sentences" <strong>of</strong> FA are simply formulas <strong>without</strong><br />
semantic content. In <strong>the</strong>mselves, <strong>the</strong>y are nei<strong>the</strong>r true nor false. They are, however,<br />
ei<strong>the</strong>r provable from <strong>the</strong> axioms <strong>of</strong> FA or not. If provable, <strong>the</strong>y are called <strong>the</strong>orems. FA,<br />
however, is so designed that we can give it an interpretation, a semantics, under which it<br />
can be read as corresponding to IA. That is, FA is designed to mirror IA, so that if all goes<br />
well, <strong>the</strong>re will be an exact one-to-one correspondence between <strong>the</strong> numerals in FA <strong>and</strong><br />
<strong>the</strong> numbers in IA, <strong>and</strong> a similar correspondence between <strong>the</strong> true sentences<br />
<strong>of</strong> IA <strong>and</strong> <strong>the</strong> <strong>the</strong>orems or FA. Pur succinctly, FA is designed to represent IA.<br />
The third language or <strong>the</strong>ory is <strong>the</strong> meta<strong>the</strong>ory <strong>of</strong> formal arithmetic, <strong>the</strong> framework in<br />
which <strong>the</strong> syntactic rules, <strong>the</strong> pro<strong>of</strong> <strong>the</strong>ory, <strong>of</strong> FA is spelled out. Call this language MFA<br />
(for <strong>the</strong> meta<strong>the</strong>ory <strong>of</strong> formal arithmetic). If FA is <strong>the</strong> machine, MFA is <strong>the</strong> owner's manual<br />
that specifies how <strong>the</strong> machine works. Like IA, it consists <strong>of</strong> meaningful sentences that<br />
have truth values. MFA specifies which formulas <strong>of</strong> FA are "well-formed formulas," meaning<br />
that <strong>the</strong>y satisfy <strong>the</strong> <strong>of</strong>ficial rules <strong>of</strong> formula construction. Crucially, it also specifies what<br />
it means to be a pro<strong>of</strong> in FA.<br />
Godel's insight was to see that FA could be used to represent not only IAóto <strong>the</strong> extent to<br />
which this is possibleóbut also MFA. He proved <strong>the</strong> latter by showing how MFA could be<br />
represented in IA, via a revolutionary device known today as <strong>the</strong> "arithmetization <strong>of</strong> metama<strong>the</strong>matics."<br />
But <strong>the</strong>n if FA can represent IA, it can also, via <strong>the</strong> interpretation <strong>of</strong> MFA in<br />
IA, represent MFA. That is, FA can represent its own meta<strong>the</strong>ory. The trick, <strong>the</strong>n, was to<br />
construct a formula <strong>of</strong> FA that would have two simultaneous meanings in two languages,<br />
MFA <strong>and</strong> IA. Godel was able to exhibit just such a formula <strong>and</strong> to prove that it was<br />
simultaneously unprovable in FA <strong>and</strong>, intuitively, true in IA <strong>and</strong> MFA. This would be a<br />
formula provably unprovable in FA, <strong>and</strong> yet expressing a true proposition in IA about <strong>the</strong><br />
natural numbers as well as a true proposition in MFA about its own unprovability.<br />
Nothing like this had ever been seen before. Godel had skirted around <strong>the</strong> deadly liar's<br />
paradox, substituting for it an unproblematic unprovability paradox (which was not really a<br />
paradox at all); established <strong>the</strong> possibility <strong>and</strong> harmlessness <strong>of</strong> self-reference;<br />
demonstrated representability relationships among three distinct languages; arithme-tized<br />
<strong>the</strong> syntax <strong>of</strong> one <strong>of</strong> those languages; <strong>and</strong> finally, exhibited a formula <strong>of</strong> one language that<br />
was provably unprovable <strong>and</strong> simultaneously true. This was logic, it was ma<strong>the</strong>matics, but<br />
it didn't look like logic or ma<strong>the</strong>matics. It looked more like Kafka. Indeed, when <strong>the</strong><br />
ma<strong>the</strong>matician Paul Cohen, a Fields medalist who proved <strong>the</strong> independence
<strong>of</strong> <strong>the</strong> continuum hypo<strong>the</strong>sis, first encountered Godel's <strong>the</strong>orem, he was skeptical,<br />
remarking that it seemed more like philosophy than ma<strong>the</strong>matics. After discussing <strong>the</strong><br />
<strong>the</strong>orem with <strong>the</strong> logician Stephen Kleene, however, his doubts evaporated. Still, "I was<br />
ra<strong>the</strong>r depressed," he commented later, "when I realized Godel was right."<br />
Reusable Numbers:<br />
The Arithmetization <strong>of</strong> Syntax<br />
Godel's first task was <strong>the</strong> easy one, showing that FA could be usedóat least a fortiorióto<br />
represent IA. The hard work had already been done by Frege, Dedekind <strong>and</strong> Peano in<br />
establishing <strong>the</strong> Peano postulates, finding <strong>the</strong> crucial definitions <strong>of</strong> number-<strong>the</strong>oretic<br />
concepts needed to employ <strong>the</strong>se postulates to represent <strong>the</strong> facts <strong>of</strong> number <strong>the</strong>ory, <strong>and</strong><br />
constructing a system <strong>of</strong> strict logical rules that would specify precisely which inferences<br />
were permitted in FA. The second task, showing that FA could st<strong>and</strong> in for MFA, was <strong>the</strong><br />
real challenge. Not only its implementation but <strong>the</strong> very suspicion that it could be done<br />
required a stroke <strong>of</strong> genius.<br />
In effect, Godel was borrowing a leaf from Descartes's book. Descartes, by assigning<br />
numbers to figures in geometrical space through what are now known as Cartesian<br />
coordinates, was able, as we would now say, to "arithmetize geometry." In this system <strong>of</strong><br />
so-called analytic geometry, statements about geometrical figures are translated into<br />
statements about numbers, <strong>and</strong> <strong>the</strong> powerful rules <strong>of</strong> manipulation <strong>of</strong> numbers, in turn,<br />
can be exploited to make discoveries about geometry. Godel's insight was to realize that<br />
<strong>the</strong> elements <strong>of</strong> <strong>the</strong> formal system FAóprimitive signs, sequences <strong>of</strong> <strong>the</strong>se signs (such as<br />
formulas), <strong>and</strong> sequences <strong>of</strong> formulas (including pro<strong>of</strong>s)ócould also be assigned a numerical<br />
representation. Through a system now known as Godel numbering, Godel assigned unique<br />
natural numbers systematically to every primitive symbol <strong>of</strong> FA, <strong>and</strong> <strong>the</strong>n showed how to<br />
construct, again uniquely, <strong>the</strong> number assigned to a sequence <strong>of</strong><br />
63<br />
primitive symbols, i.e., a formula, <strong>and</strong> to a sequence <strong>of</strong> formulas, including a sequence<br />
that constituted a pro<strong>of</strong>. To ensure uniqueness in <strong>the</strong> system <strong>of</strong> numbering, Godel relied on<br />
well-known facts about natural numbers such as <strong>the</strong> fundamental <strong>the</strong>orem <strong>of</strong> arithmetic:<br />
every positive integer can be resolved uniquely into a product <strong>of</strong> prime numbers. This<br />
arithmetization <strong>of</strong> syntax established, in effect, a language translation scheme between<br />
symbols <strong>of</strong> FA <strong>and</strong> natural numbers, so that a statement about <strong>the</strong> syntax <strong>of</strong> FA could be<br />
translated, via <strong>the</strong> rule book <strong>of</strong> Godel numbering, into a statement about natural numbers,<br />
just as in <strong>the</strong> Cartesian system, a fact about figures in <strong>the</strong> geometrical plane could be<br />
translated into one about real numbers.
So far, <strong>the</strong> system <strong>of</strong> Godel numbering as we have described it only sets up a<br />
correspondence between numbers <strong>and</strong> symbols, sequences <strong>of</strong> symbols, <strong>and</strong> sequences <strong>of</strong><br />
sequences <strong>of</strong> symbols <strong>of</strong> FA. But <strong>the</strong>re is more to <strong>the</strong> syntax <strong>of</strong> FA than this. There is also<br />
<strong>the</strong> question whe<strong>the</strong>r a given formula <strong>of</strong> FA is a well-formed formula, <strong>and</strong>, crucially, <strong>the</strong><br />
question whe<strong>the</strong>r a sequence <strong>of</strong> formulas constitutes a pro<strong>of</strong>. What Godel proved is that all<br />
<strong>the</strong> crucial functions needed to describe <strong>the</strong> complete syntax <strong>of</strong> FA, including being a wellformed<br />
formula <strong>and</strong> being a pro<strong>of</strong> <strong>of</strong> FA, corresponded to certain recursive functions in IA.<br />
A recursive function is one that, intuitively speaking, can be mechanically computed. This<br />
kind <strong>of</strong> function can also be characterized strictly ma<strong>the</strong>matically, <strong>and</strong> this Godel<br />
proceeded to do. An example <strong>of</strong> a so-called primitive recursive function, <strong>the</strong> "+ function,"<br />
also known as <strong>the</strong> "addition function," will illustrate what is meant by recursivity. Let us<br />
call <strong>the</strong> number that comes right after a natural number x <strong>the</strong> successor <strong>of</strong> x, or s(x). The<br />
"+" function, <strong>the</strong>n, is given by two rules:<br />
(a) x + 0 = x;<br />
(b) x + s(y) = s(x + y). (This can be read aloud as "x plus <strong>the</strong> successor <strong>of</strong> y equals <strong>the</strong><br />
successor <strong>of</strong> x-plus-y.")<br />
The successor <strong>of</strong> x, namely, s(x), can be defined as x + 1. This kind <strong>of</strong> recursive definition<br />
can be used to compute mechanically, by a<br />
kind <strong>of</strong> "bootstrapping," <strong>the</strong> sum <strong>of</strong> any two natural numbers, since every natural number is<br />
ei<strong>the</strong>r 0 or <strong>the</strong> successor <strong>of</strong> some o<strong>the</strong>r natural number.<br />
Recursive definitions were studied by Dedekind, Peano, Skolem <strong>and</strong> o<strong>the</strong>rs, <strong>and</strong> recursive<br />
functions had been used implicitly throughout <strong>the</strong> history <strong>of</strong> ma<strong>the</strong>matics, but <strong>the</strong> first to<br />
elaborate a precise <strong>and</strong> forceful account <strong>of</strong> such functions was Godel, who cited his young<br />
French colleague Jacques Herbr<strong>and</strong> as having influenced his underst<strong>and</strong>ing <strong>of</strong> <strong>the</strong>se ideas.<br />
Herbr<strong>and</strong> had written to Godel on hearing <strong>of</strong> his incompleteness results from Von<br />
Neumann. Godel wrote a detailed <strong>and</strong> deeply respectful response, at <strong>the</strong> end <strong>of</strong> which he<br />
suggested that in <strong>the</strong> future <strong>the</strong>y correspond each in his mo<strong>the</strong>r tongue. (Godel was very<br />
good at languages.) He never received a reply. What he did receive was a touching letter<br />
from Herbr<strong>and</strong>'s fa<strong>the</strong>r informing him that <strong>the</strong> reason for his son's silence was that he had<br />
fallen to his death while climbing in <strong>the</strong> Alps. Jacques Herbr<strong>and</strong> had been just twentythree<br />
years old.<br />
Godel demonstrated, <strong>the</strong>n, that <strong>the</strong> fundamental concepts <strong>of</strong> MFA, in which was found <strong>the</strong><br />
metama<strong>the</strong>matics, or pro<strong>of</strong> <strong>the</strong>ory, <strong>of</strong> FA, corresponded to certain recursive functions in<br />
IA. In particular, <strong>the</strong> function Bew(x, y), i.e., x is a pro<strong>of</strong> <strong>of</strong> y (from <strong>the</strong> German for pro<strong>of</strong>,<br />
Beweis), when coded into natural numbers, yields a recursive function. This was important<br />
because in proving that FA can represent IA, Godel had already shown that any recursive<br />
function contained in IA could be represented in FA. Specifically, if <strong>the</strong>re was a truth<br />
about a recursive function in IA, <strong>the</strong>re would be a corresponding formula that was a<br />
<strong>the</strong>orem <strong>of</strong> FA. Once he had demonstrated that <strong>the</strong> basic functions in MFA when coded into<br />
natural numbers yield recursive functions, he could conclude that MFA, just like IA, could
e represented in FA. Fur<strong>the</strong>r, via Godel numbering he had already arithmetized <strong>the</strong><br />
syntax <strong>of</strong> FA, so a fact about <strong>the</strong> syntax <strong>of</strong> FA would correspond to a fact in IA about <strong>the</strong><br />
natural numbers. Godel had shown, <strong>the</strong>n, that <strong>the</strong> <strong>the</strong>orems <strong>of</strong> FA could represent<br />
simultaneously <strong>the</strong> arithmetic truths <strong>of</strong> IA as well as, via Godel numbering, <strong>the</strong><br />
syntactic truths <strong>of</strong> MFA. That is, a given <strong>the</strong>orem <strong>of</strong> FA would represent a ma<strong>the</strong>matical<br />
truth in IA that would itself, via Godel numbering, represent a syntactic truth about FA.<br />
Godel had succeeded in proving, <strong>the</strong>n, that FA, though in itself a system <strong>of</strong> formal,<br />
meaningless signs, could be "double stuffed" with meaning, i.e., assigned meanings that<br />
ensured that it could be used to represent, simultaneously, number <strong>the</strong>ory <strong>and</strong> <strong>the</strong> syntax<br />
or pro<strong>of</strong> <strong>the</strong>ory <strong>of</strong> FA itself.<br />
In o<strong>the</strong>r words, FA could speak about itself via <strong>the</strong> natural numbers. The natural numbers,<br />
Godel had shown, were "reusable," in <strong>the</strong> spirit <strong>of</strong> Descartes: <strong>the</strong>y could be used as<br />
elements <strong>of</strong> arithmetic <strong>and</strong> at <strong>the</strong> same <strong>time</strong> as representatives <strong>of</strong> <strong>the</strong> syntax or pro<strong>of</strong><br />
<strong>the</strong>ory <strong>of</strong> formalized arithmetic.<br />
I Cannot Be Proved<br />
With <strong>the</strong>se elements in place, <strong>the</strong> coordination <strong>of</strong> <strong>the</strong> three languages IA, FA, <strong>and</strong> MFA <strong>and</strong><br />
<strong>the</strong> implementation <strong>of</strong> Godel numbering, Godel made his move. He constructed a formula<br />
<strong>of</strong> FAóknown, familiarly, today, as <strong>the</strong> Godel formula, or Gówhose interpretation in IA was<br />
a statement, ei<strong>the</strong>r true or false, about <strong>the</strong> natural numbers. But G also had, via Godel<br />
numbering <strong>and</strong> <strong>the</strong> arithmetization <strong>of</strong> <strong>the</strong> syntax <strong>of</strong> FA, ano<strong>the</strong>r meaning. On this<br />
interpretation it asserted a pro<strong>of</strong>-<strong>the</strong>oretic fact about a certain formula <strong>of</strong> FA, via <strong>the</strong><br />
Godel number assigned to that formula, to <strong>the</strong> effect that <strong>the</strong> formula with that Godel<br />
number g was unprovable in FA. But Godel had set it up that <strong>the</strong> formula whose Godel<br />
number was g was none o<strong>the</strong>r than G! In effect, <strong>the</strong>n, on one interpretation, what G stated<br />
was, "I cannot be proved." The question <strong>the</strong>n was, Is G provable in FA? (Equivalently, is G a<br />
<strong>the</strong>orem <strong>of</strong> FA?)<br />
Clearly, this question was a close cousin <strong>of</strong> <strong>the</strong> ancient liar paradox in which a sentence S<br />
says <strong>of</strong> itself, "I am not true," whereupon one asks, Is S true? But <strong>the</strong> liar paradox really is a<br />
paradox, or more precisely an antinomy, ins<strong>of</strong>ar as <strong>the</strong> natural answer to <strong>the</strong> question, "Is<br />
S<br />
true?" seems to imply both that S is true <strong>and</strong> that it is not. What Godel realized is that <strong>the</strong><br />
question posed about G, which concerned not its truth but its provability, did not lead to<br />
paradox or antinomy. The Godel formula, after all, is simply a formula in <strong>the</strong> formal<br />
system, <strong>and</strong> as such it was a clear-cut, unproblematic question whe<strong>the</strong>r or not G was<br />
provable, i.e., was a <strong>the</strong>orem. Similarly, interpreted as a statement about numbers via <strong>the</strong><br />
coordination <strong>of</strong> FA with IA, G was a statement not about itself but about natural numbers,<br />
<strong>and</strong> as such it said something unproblematically ei<strong>the</strong>r true or false, but not both, about<br />
<strong>the</strong> natural numbers.
So is G a <strong>the</strong>orem <strong>of</strong> FA? We assume first that FA is consistent. If it were not consistent, it<br />
would obviously be useless for ma<strong>the</strong>matics, <strong>and</strong> <strong>the</strong> game would be over. Assume next<br />
that G is provable in FA. That means that some sequence <strong>of</strong> formulas S <strong>of</strong> FA is a pro<strong>of</strong> <strong>of</strong><br />
G. Using <strong>the</strong> abbreviation we introduced earlier, that means that Bew(S, G). But we saw<br />
earlier that Godel had shown that <strong>the</strong> function, Bew(x, y), when coded into natural<br />
numbers, yields a recursive function <strong>and</strong> is thus rep-resentable in FA. The representation<br />
occurs via <strong>the</strong> arithmetization <strong>of</strong> <strong>the</strong> syntax <strong>of</strong> FA, so corresponding to a given syntactical<br />
truth Bew(x, y) <strong>of</strong> MFA, <strong>the</strong>re is an arithmetical truth Bew(x, y) <strong>of</strong> IA that corresponds to a<br />
formula Bew(x, y) in FA that can be interpreted as saying that <strong>the</strong> sequence <strong>of</strong> formulas<br />
with Godel number x is a pro<strong>of</strong> <strong>of</strong> <strong>the</strong> formula with Godel number y, <strong>and</strong> this formula,<br />
Bew(x, y), is a <strong>the</strong>orem <strong>of</strong> FA. Thus if G is provable in FA for some s, <strong>the</strong>n Bew(s, g) is a<br />
<strong>the</strong>orem <strong>of</strong> FA (where s is <strong>the</strong> numeral in FA for s, <strong>the</strong> Godel number <strong>of</strong> <strong>the</strong> sequence <strong>of</strong><br />
formulas S that constitutes <strong>the</strong> pro<strong>of</strong> <strong>of</strong> G, <strong>and</strong> g is <strong>the</strong> numeral in FA for g, <strong>the</strong> Godel<br />
number <strong>of</strong> G). But as we saw at <strong>the</strong> beginning, what G states is that formula number g<br />
(i.e., G) is unprovable, that is, for any sequence x, it is not <strong>the</strong> case that Bew(x, g). Using<br />
a little logic, however, it follows from Bew(s, g) that <strong>the</strong>re exists a sequence <strong>of</strong> formulas x<br />
such that Bew(x, g), hence, using a bit more logic, that it is not <strong>the</strong> case that for any<br />
sequence x it is not <strong>the</strong> case that Bew(x, g). That is, if something is a pro<strong>of</strong> <strong>of</strong> formula<br />
number g, it can't be that <strong>the</strong>re is no pro<strong>of</strong> <strong>of</strong> g. But since G can be interpreted as saying<br />
that <strong>the</strong>re is no pro<strong>of</strong> <strong>of</strong> g, this last<br />
conclusion is evidently equivalent to not-G, <strong>the</strong> negation <strong>of</strong> G, <strong>and</strong> since it was derived,<br />
logically, from Bew(s, g), which, by hypo<strong>the</strong>sis, is a <strong>the</strong>orem, it too is a <strong>the</strong>orem. That is,<br />
not-G is also a <strong>the</strong>orem. The assumption that G is provable, <strong>the</strong>n, leads to <strong>the</strong> conclusion<br />
that both G <strong>and</strong> not-G are <strong>the</strong>orems, i.e., that both are provable in FA. Hence FA is<br />
inconsistent. But <strong>the</strong> assumption was that FA is consistent. If FA is consistent, <strong>the</strong>n, G is<br />
not provable. That is <strong>the</strong> first part <strong>of</strong> GodePs <strong>the</strong>orem.<br />
How about not-G? Godel assumed for this half <strong>of</strong> his <strong>the</strong>orem that FA is not just consistent<br />
but "omega-consistent." (Omega is <strong>the</strong> traditional symbol for <strong>the</strong> natural numbers.) If a<br />
system is omega-consistent, <strong>the</strong> following cannot happen: some statement F( ) is provable<br />
individually <strong>of</strong> each natural number, i.e., F(0), F(l), F(2), etc., but it cannot be proved<br />
that for all x, <strong>the</strong> statement F(x) is true. Clearly, if FA is to represent all <strong>the</strong> natural<br />
numbers via its coordination with IA, omega-consistency if nonnegotiable. If not-G is a<br />
<strong>the</strong>orem, <strong>the</strong>n, <strong>of</strong> FA, given what G says, it follows that <strong>the</strong> following is provable: it is not<br />
<strong>the</strong> case that every sequence <strong>of</strong> formulas number x does not constitute a pro<strong>of</strong> <strong>of</strong> formula<br />
number g. But we have already learned that formula number g, i.e., G, is not provable in<br />
FA. Thus, for every individual sequence <strong>of</strong> formulas x, not-Bew(x, g) holds. Given <strong>the</strong><br />
representability <strong>of</strong> <strong>the</strong> function Bew(x, y) in FA, it follows that for every natural number x,<br />
it is provable in FA that not-Bew(x, g). Since we are assuming that FA is omega-consistent,<br />
it follows that it cannot also be provable that it is not <strong>the</strong> case that for all sequences <strong>of</strong><br />
formulas x, not-Bew(x, g) holds. But <strong>the</strong> latter is precisely what we assumed when we<br />
agreed for <strong>the</strong> sake <strong>of</strong> argument that not-G is a <strong>the</strong>orem. If FA is omega-consistent, <strong>the</strong>n it<br />
follows that not-G is also not provable. This is <strong>the</strong> second half <strong>of</strong> Godel's <strong>the</strong>orem. Barkley<br />
Rosser later showed that one can construct an unprovable statement assuming only<br />
consistency, not omega-consistency.<br />
But Godel wasn't done yet. He was able to prove, quickly, his second incompleteness<br />
<strong>the</strong>orem, also on <strong>the</strong> assumption that FA is consistent. Using techniques similar to his pro<strong>of</strong><br />
<strong>of</strong> <strong>the</strong> first <strong>the</strong>orem, Godel proved easily that if FA is consistent, it cannot prove that it is
consistent. Not only was truth not fully representable in a formal <strong>the</strong>ory, consistency, too,<br />
could not be formally represented. The Hilbert program had suffered a fatal blow. Godel<br />
had proved that <strong>the</strong>re was in principle no method by which a ma<strong>the</strong>matician, regarding his<br />
<strong>the</strong>ories simply as uninterpreted formula games, could prove <strong>the</strong>m free from hidden<br />
inconsistency. There simply was no such thing as a magic shield that would resolve all a<br />
ma<strong>the</strong>matician's fears <strong>of</strong> an assault from some unsuspected inconsistency. (Similarly, as a<br />
direct consequence <strong>of</strong> Godel's incompleteness <strong>the</strong>orems, <strong>the</strong>re can never be a foolpro<strong>of</strong><br />
antivirus computer program that we can be certain will not alter <strong>the</strong> program being<br />
protected but that will detect <strong>the</strong> presence <strong>of</strong> any o<strong>the</strong>r program that is attempting to<br />
alter <strong>the</strong> protected program.)<br />
Yet lest we be tempted to complain that GodePs <strong>the</strong>orem brought only bad news for our<br />
cherished computers, we should recall that <strong>the</strong> very existence <strong>of</strong> <strong>the</strong> modern computer,<br />
which is after all no more than a sophisticated calculation or deduction machine, is a<br />
direct consequence <strong>of</strong> <strong>the</strong> isolation <strong>and</strong> clarification by Godel <strong>and</strong> Herbr<strong>and</strong> <strong>of</strong> those<br />
recursive ma<strong>the</strong>matical functions that are at <strong>the</strong> heart <strong>of</strong> <strong>the</strong> incompleteness <strong>the</strong>orem. It<br />
was Alan Turing, who developed his idea <strong>of</strong> what came to be called a Turing machine on<br />
<strong>the</strong> basis <strong>of</strong> <strong>the</strong> concept <strong>of</strong> recursive functions, who became <strong>the</strong> modern computer's most<br />
direct parent. What is crucial to a computer, after all, is that <strong>the</strong> instructions that are<br />
programmed into it can be followed mechanically, syntactically, <strong>without</strong> recourse to<br />
meaning, <strong>and</strong> in a manner that is "iterative" or "bootstrapping," so that one instruction<br />
leads directly to ano<strong>the</strong>r. This is <strong>the</strong> heart <strong>of</strong> <strong>the</strong> idea <strong>of</strong> a Turing machine.<br />
These same recursive functions served ultimately to render GodePs <strong>the</strong>orem a yet more<br />
decisive refutation <strong>of</strong> Hilbert's program. For <strong>the</strong> ma<strong>the</strong>matical characterization <strong>of</strong> a<br />
function as recursive, developed by Godel <strong>and</strong> Herbr<strong>and</strong>, was closely followed by a similar<br />
characterization, by Turing, <strong>of</strong> Turing computable functions, <strong>and</strong> a likeminded<br />
characterization <strong>of</strong> a class <strong>of</strong> ma<strong>the</strong>matical functions introduced by <strong>the</strong> American logician<br />
Alonzo Church in his calculus <strong>of</strong> lambda conversion. Soon, Church's students Kleene <strong>and</strong><br />
Rosser (Turing too, it should<br />
he noted, was a some<strong>time</strong> student) proved that <strong>the</strong>se three classes <strong>of</strong> ma<strong>the</strong>matical<br />
functions are in a strict sense equivalent. This prompted Church to propound what is today<br />
called Church's <strong>the</strong>sis, which states that this class <strong>of</strong> ma<strong>the</strong>matical functions, <strong>the</strong> recursive<br />
functions, corresponds exactly to <strong>the</strong> intuitively characterized class <strong>of</strong> functions that are<br />
mechanically or effectively calculable.<br />
Church's <strong>the</strong>sis is today widely accepted, <strong>and</strong> it gives Godel's <strong>the</strong>orems yet greater force in<br />
two respects. First, it made definitive <strong>the</strong> characterization <strong>of</strong> a formal system like FA as<br />
one whose syntactic rules must be mechanically specifiable, <strong>and</strong> thus specifiable by<br />
recursive functions. Since Godel's pro<strong>of</strong> clearly could be extended to any formal system<br />
whose syntax was fully characterized by recursive functions, that meant that his<br />
incompleteness <strong>the</strong>orem applied with equal force to any system characterizable as a<br />
genuinely formal one. Second, Church's <strong>the</strong>sis added force to Godel's second <strong>the</strong>orem,<br />
since it could plausibly be argued that <strong>the</strong> methods that Hilbert would find acceptably<br />
"finitary" were exactly those that could be characterized as mechanically specifiable, <strong>and</strong><br />
once <strong>the</strong>se were identified with <strong>the</strong> precisely specified class <strong>of</strong> recursive ma<strong>the</strong>matical
functions, it was an inescapable consequence <strong>of</strong> <strong>the</strong> second <strong>the</strong>orem that it directly<br />
refuted Hilbert's program <strong>of</strong> establishing a finitary pro<strong>of</strong> <strong>of</strong> consistency for formal<br />
arithmetic. Through <strong>the</strong> work <strong>of</strong> Church <strong>and</strong> Turing, <strong>the</strong>n, <strong>the</strong> full force <strong>of</strong> Godel's <strong>the</strong>orem<br />
began to be appreciated.<br />
The Formal <strong>and</strong> <strong>the</strong> Intuitive<br />
God's mercy preserves ma<strong>the</strong>matics from being drowned in mere technique<br />
SIMONE WEIL<br />
But it was not always so. The ma<strong>the</strong>matical shot heard round <strong>the</strong> <strong>world</strong> began as a<br />
whisper. Godel may have been a spy in <strong>the</strong> house <strong>of</strong> logic, a revolutionary, an intellectual<br />
bomb thrower, but he was also a<br />
citizen <strong>of</strong> Vienna, <strong>the</strong> city <strong>of</strong> c<strong>of</strong>feehouses. So he first communicated his momentous<br />
discoveries to Rudolf Carnap in <strong>the</strong> Cafe Reichsrat, shortly before <strong>the</strong>y both left for a<br />
conference on <strong>the</strong> foundations <strong>of</strong> ma<strong>the</strong>matics in Konigsberg (today, Kaliningrad, in which<br />
both Kant <strong>and</strong> Hilbert were born), where Godel would announce his results publicly. The<br />
first to be apprised <strong>of</strong> what <strong>the</strong> young man had done, however, seems to have been <strong>the</strong><br />
first to fail to appreciate it. At <strong>the</strong> Konigsberg conference a few days later, Carnap<br />
proceeded, in good positivist style, to recommend consistency once again as <strong>the</strong><br />
touchstone <strong>of</strong> formal ma<strong>the</strong>matical <strong>the</strong>ories.<br />
Carnap had been a student <strong>of</strong> Frege <strong>and</strong> <strong>of</strong> <strong>the</strong> philosopher Edmund Husserl <strong>and</strong> by 1931<br />
was an established logician. If Frege can with justice be called <strong>the</strong> fa<strong>the</strong>r <strong>of</strong> modern<br />
"analytical" philosophy, Husserl can be considered <strong>the</strong> fa<strong>the</strong>r <strong>of</strong> <strong>the</strong> "continental" branch.<br />
The two reviewed each o<strong>the</strong>r's books <strong>and</strong> corresponded, but <strong>the</strong>ir followers assembled into<br />
armed camps that exchanged intellectual gunfire. The influence Husserl came to have on<br />
Godel late in his life was an anomaly; to his colleagues, it was an embarrassment. In 1931,<br />
however, it was not an embarrassment for Carnap to have studied with Husserl. That such<br />
a man as Carnap failed to grasp <strong>the</strong> force <strong>of</strong> GodePs accomplishment is an indication <strong>of</strong> <strong>the</strong><br />
break with tradition that Godel was inaugurating (a phenomenon his future friend <strong>Einstein</strong><br />
had already encountered with relativity <strong>the</strong>ory). Even a year later, Carnap confided that<br />
he still found Godel's results hard to underst<strong>and</strong>.<br />
Von Neumann alone immediately grasped <strong>the</strong> force <strong>of</strong> <strong>the</strong> discovery. He was present at <strong>the</strong><br />
meeting as one <strong>of</strong> <strong>the</strong> principal speakers, <strong>and</strong> after this encounter he would become a<br />
lifelong friend <strong>and</strong> admirer <strong>of</strong> Godel. But Von Neumann, as was so <strong>of</strong>ten <strong>the</strong> case, was <strong>the</strong><br />
exception. So little impact did GodePs announcement have on his immediate listenersó<strong>the</strong><br />
cream <strong>of</strong> <strong>the</strong> foundationalist ma<strong>the</strong>maticians <strong>and</strong> formal logiciansóthat when Hans<br />
Reichenbach, ano<strong>the</strong>r prominent member <strong>of</strong> <strong>the</strong> Vienna Circle, wrote up an account <strong>of</strong> <strong>the</strong><br />
conference for <strong>the</strong> journal Erkenntnis, he did not even mention Godel. The ma<strong>the</strong>matical<br />
community, however, was quick to recover from this
71<br />
slight. Immediately after <strong>the</strong> conference, when news <strong>of</strong> <strong>the</strong> paper Godel had submitted for<br />
publication began to circulate, he was invited by <strong>the</strong> editors <strong>of</strong> Erkenntnis to add a<br />
postscript summarizing his soon-to-be published results.<br />
Often, a ma<strong>the</strong>matical resultólike Fermat's last <strong>the</strong>oremóis believed even though its pro<strong>of</strong><br />
is unexpected, so that <strong>the</strong> appearance <strong>of</strong> a genuine pro<strong>of</strong> creates a sensation. But in <strong>the</strong><br />
case <strong>of</strong> Godel's <strong>the</strong>orem, <strong>the</strong> result itself was unexpected, so <strong>the</strong> appearance <strong>of</strong> a pro<strong>of</strong>óa<br />
pro<strong>of</strong> like nothing seen beforeówas explosive. But fame does not bring with it<br />
comprehension, <strong>and</strong> Godel's methods as much as his results were so revolutionary that it<br />
would be years before <strong>the</strong> ma<strong>the</strong>matical <strong>and</strong> philosophical communities could fully digest<br />
<strong>the</strong>m.<br />
Paul Bernays, Hilbert's assistant <strong>and</strong> a great logician in his own right, who would go on to<br />
become one <strong>of</strong> GodePs closest associates, was perplexed by some <strong>of</strong> <strong>the</strong> details <strong>of</strong> GodePs<br />
pro<strong>of</strong>s <strong>and</strong> asked for Hilbert's assistance. Hilbert's response was at first merely anger,<br />
followed by denial. He became <strong>the</strong> first, but by no means <strong>the</strong> last, to propose, in effect,<br />
an "anti-Godel" principle: a formal principle, concocted ad hoc, to be appended to formal<br />
ma<strong>the</strong>matics simply in order to block <strong>the</strong> application <strong>of</strong> GodePs <strong>the</strong>orem. Godel, usually<br />
unflappable even in <strong>the</strong> face <strong>of</strong> opposition or incomprehension, was genuinely irritated by<br />
this. Hilbert's ideaówhich he would give up soon enoughówas to propose that <strong>the</strong>re be<br />
adjoined to his formal systems a new rule <strong>of</strong> deductive inference that would allow for <strong>the</strong><br />
employ- ment <strong>of</strong> infinitely many premises. As Godel pointed out, however, this proposal<br />
violated <strong>the</strong> very idea <strong>of</strong> a formal system, an idea Hilbert, following Frege, had been at<br />
pains to develop. The proposed cure would kill <strong>the</strong> patient.<br />
If Hilbert did not come <strong>of</strong>f well in his first response to GodePs <strong>the</strong>orem, he displayed<br />
charm <strong>and</strong> fairness elsewhere. Two characteristic incidents bring this out clearly. Before<br />
World War I, he stood out by his willingness to take on as a doctoral student Jakob<br />
Grommer, who had attended a Talmudic school in Eastern Europe <strong>and</strong> lacked a gymnasium<br />
certificate. In addition, Grummer's h<strong>and</strong>s <strong>and</strong> feet were de-<br />
formed, which led <strong>the</strong> daughter <strong>of</strong> <strong>the</strong> rabbi he was to replace to reject him. Hilbert,<br />
however, did not reject him, declaring that "if students <strong>without</strong> <strong>the</strong> gymnasium diploma<br />
will always write such dissertations as Grommer's, it will be necessary to make a law<br />
forbidding <strong>the</strong> taking <strong>of</strong> <strong>the</strong> examination for <strong>the</strong> diploma." The second incident concerns<br />
<strong>the</strong> physicist Max Born, who began his academic career in ma<strong>the</strong>matics. Before his<br />
examination, he asked Hilbert for advice, <strong>and</strong> was asked what <strong>the</strong> area was in which he<br />
was <strong>the</strong> least prepared. It was ideal <strong>the</strong>ory. To Born's dismay, in <strong>the</strong> examination that<br />
followed, just this was <strong>the</strong> topic on which Hilbert chose to focus his questions. His<br />
explanation to Born? "I was just interested to find out what you know about things about<br />
which you know nothing."
Hilbert soon recovered from <strong>the</strong> shock <strong>of</strong> GodePs discovery <strong>and</strong> proceeded to incorporate<br />
<strong>and</strong> develop it in his <strong>and</strong> Bernays's new textbook on ma<strong>the</strong>matical logic. The same cannot<br />
be said, however, for Ernst Zermelo, <strong>the</strong> ma<strong>the</strong>matician who had inaugurated <strong>the</strong><br />
axiomatic development <strong>of</strong> set <strong>the</strong>ory after Russell's paradox had demonstrated that <strong>the</strong><br />
naive set <strong>the</strong>ory developed by Cantor lacked a coherent philosophical foundation. Even<br />
today, <strong>the</strong> axioms <strong>of</strong> Zermelo-Fraenkel set <strong>the</strong>ory are <strong>the</strong> most widely used <strong>and</strong> accepted<br />
in <strong>the</strong> field. Yet Zermelo, from beginning to end, was unable to underst<strong>and</strong> or accept<br />
GodePs results. He became <strong>the</strong>ir principal ma<strong>the</strong>matical opponent. (Wittgenstein bears <strong>the</strong><br />
honor <strong>of</strong> being <strong>the</strong>ir chief philosophical detractor.)<br />
Zermelo's difficulties were underst<strong>and</strong>able. GodePs <strong>the</strong>orems traded on crucial distinctions<br />
such as truth versus pro<strong>of</strong>, semantics versus syntax, <strong>and</strong> completeness versus formal<br />
consistency, distinctions that, though in <strong>the</strong> air, became fully clarified for <strong>the</strong> first <strong>time</strong><br />
only after GodePs pro<strong>of</strong>s had appeared. It was not that Hilbert, <strong>the</strong> founder <strong>of</strong> formalism,<br />
distinguished carefully between truth <strong>and</strong> pro<strong>of</strong> <strong>and</strong> simply opted for <strong>the</strong> latter. Ra<strong>the</strong>r, as<br />
Godel himself put <strong>the</strong> matter years later, "formalists considered formal demonstrability to<br />
be an analysis <strong>of</strong> <strong>the</strong> concept <strong>of</strong> ma<strong>the</strong>matical truth <strong>and</strong>, <strong>the</strong>refore, were <strong>of</strong> course not in<br />
a position to distinguish <strong>the</strong> two." In <strong>the</strong> realm <strong>of</strong> ma<strong>the</strong>matics, pro<strong>of</strong>, for <strong>the</strong><br />
formalist, was indistinguishable from truth, <strong>and</strong> so any attempt to draw distinctions<br />
between <strong>the</strong>m was simply incomprehensible. Zermelo's philosophical framework, in turn,<br />
though different from Hilbert's, was so contrary to GodePs that reconciliation was<br />
impossible.<br />
Fate brought <strong>the</strong> two men toge<strong>the</strong>r at ano<strong>the</strong>r ma<strong>the</strong>matical meeting, this <strong>time</strong> in Bad<br />
Elster, a year after <strong>the</strong> conference at Konigsberg. When it was suggested to Zermelo after<br />
<strong>the</strong> talks were over that he meet with Godel for lunch on a nearby hill, he demurred,<br />
complaining first that he "did not like GodePs looks," <strong>the</strong>n that <strong>the</strong> supply <strong>of</strong> food was<br />
insufficient, <strong>and</strong> finally that <strong>the</strong> climb would defeat him. Zermelo should have trusted his<br />
instincts. He was finally talked into meeting with Godel, but <strong>the</strong> encounter, though polite,<br />
was fruitless. He would soon write to Godel that he had a discovered a "major gap" in his<br />
argument, <strong>and</strong> a lengthy replyórunning to ten h<strong>and</strong>written pagesóby Godel did little to<br />
disabuse him <strong>of</strong> his doubts. Having once failed to enlighten Zermelo, Godel apparently<br />
gave it up as a lost cause, declining to respond even when Zermelo published his criticisms.<br />
Carnap, when shown Zermelo's letters, agreed that he had "completely misunderstood"<br />
GodePs achievement.<br />
If Zermelo's intransigence was to be expected, Bertr<strong>and</strong> Russell's ambivalence was not. The<br />
coauthor <strong>of</strong> <strong>the</strong> monumental Principia Ma<strong>the</strong>matical, which provided <strong>the</strong> actual formal<br />
system for GodePs pro<strong>of</strong>, continued, late in life, to refer to GodePs results only guardedly.<br />
In a letter written in 1963, Russell, while acknowledging <strong>the</strong> greatness <strong>of</strong> Godel's<br />
achievement, did not conceal that he remained puzzled by it, asking rhetorically "are we<br />
to think that 2 + 2 is not 4, but 4.001 ?" This suggests that Godel had purported to have<br />
demonstrated a flaw in classical ma<strong>the</strong>matics, which precisely misses <strong>the</strong> point <strong>of</strong> Godel's<br />
<strong>the</strong>orem. Russell knew, however, that he had not yet fully thought this through. He<br />
commented, dryly, that he was "glad [I] was no longer working at ma<strong>the</strong>matical logic."<br />
(Apparently, Godel was too. In a letter to a colleague he wrote that "Russell evidently<br />
misinterprets my result; however, he does so in a very interesting manner. ... In<br />
contradistinction
Wittgenstein, in his posthumous book, advances a completely trivial <strong>and</strong> uninteresting<br />
misinterpretation.")<br />
Still, most logicians <strong>and</strong> ma<strong>the</strong>maticians came to appreciate Godel's achievements, which<br />
eventually assumed <strong>the</strong>ir rightful place as part <strong>of</strong> <strong>the</strong> new orthodoxy. Hilbert's program<br />
was largely ab<strong>and</strong>oned, in company with o<strong>the</strong>r ma<strong>the</strong>matical misadventures like squaring<br />
<strong>the</strong> circle <strong>and</strong> proving Euclid's parallel postulate. There simply was no safe method by<br />
which <strong>the</strong> security <strong>of</strong> formal ma<strong>the</strong>matical systems powerful enough to represent <strong>the</strong><br />
natural numbers could be ensured. And <strong>the</strong>re simply was no such thing as a formal system<br />
that could adequately <strong>and</strong> completely represent <strong>the</strong> natural numbers. GodePs<br />
ma<strong>the</strong>matical methods, too, found <strong>the</strong>ir way into daily ma<strong>the</strong>matics, including <strong>the</strong><br />
employment <strong>of</strong> recursive functions, <strong>the</strong> arithmetization <strong>of</strong> metama<strong>the</strong>matics, <strong>and</strong> <strong>the</strong><br />
construction <strong>of</strong> "Godel formulas" to establish, via self-reference, <strong>the</strong> incompleteness <strong>of</strong><br />
formal systems. Never again would syntax be substituted for semantics, pro<strong>of</strong> for truth.<br />
But <strong>the</strong> wider significance <strong>of</strong> GodePs achievement, its true meaning, was something else.<br />
One would have thought that after GodePs incompleteness <strong>the</strong>orems, which established<br />
<strong>the</strong> essential limitations <strong>of</strong> formalization, <strong>the</strong> very enterprise <strong>of</strong> formalizing ma<strong>the</strong>matical<br />
domains would have been reconsidered. Yet nothing <strong>of</strong> <strong>the</strong> kind happened. The American<br />
logician Emil Post was one <strong>of</strong> <strong>the</strong> few to take note <strong>of</strong> this curious fact. In an essay<br />
submitted to a ma<strong>the</strong>matical journal in 1941 (<strong>and</strong> rejected), Post observed that as a result<br />
<strong>of</strong> Godel's <strong>the</strong>orems we know that "ma<strong>the</strong>matical thinking is, <strong>and</strong> must be, essentially<br />
creative." He went on to remark, however, that "it is to <strong>the</strong> writer's continuing amazement<br />
that ten years after Godel's remarkable achievement current views on <strong>the</strong> nature <strong>of</strong><br />
ma<strong>the</strong>matics are <strong>the</strong>reby affected only to <strong>the</strong> point <strong>of</strong> seeing <strong>the</strong> need <strong>of</strong> many formal<br />
systems, instead <strong>of</strong> [a single] universal one." One would have expected, he went on to say,<br />
that <strong>the</strong> fascination with formal systems, with pro<strong>of</strong> <strong>and</strong> syntax, would give way to "a<br />
return to meaning <strong>and</strong> truth." Yet this never happened. The<br />
zeitgeist would not be denied, (Even <strong>the</strong> members <strong>of</strong> <strong>the</strong> explicitly an-tiformalist<br />
"intuitionist" school <strong>of</strong> ma<strong>the</strong>matics championed by L.E.J. Brouwer would in <strong>the</strong> course <strong>of</strong><br />
<strong>time</strong> expend a great deal <strong>of</strong> energy constructing formal systems for parts <strong>of</strong> intuitionist<br />
ma<strong>the</strong>matics.)<br />
A more general aspect <strong>of</strong> Godel's results remains neglected to this day. "Godel's program,"<br />
though closely related to Hilbert's, was different in an important way. Whereas Hilbert<br />
wished to do for certain parts <strong>of</strong> ma<strong>the</strong>matics what <strong>the</strong> positivists wished to do<br />
everywhereóreplace <strong>the</strong> intuitive with <strong>the</strong> formalóGodel's overarching ambition throughout<br />
his career consisted in <strong>the</strong> attempt to establish, by formal means, <strong>the</strong> limits <strong>of</strong> formal<br />
methods in capturing intuitive concepts. The goal <strong>of</strong> his incompleteness <strong>the</strong>orem was thus<br />
to establish, by <strong>the</strong> most formal <strong>of</strong> methodsómethods that could be programmed into a<br />
computeró<strong>the</strong> limits <strong>of</strong> formal systems <strong>of</strong> pro<strong>of</strong> in capturing <strong>the</strong> intuitive concept <strong>of</strong><br />
ma<strong>the</strong>matical truth. Church's <strong>the</strong>sis, in turnówhich made precise <strong>the</strong> concept <strong>of</strong> a formal<br />
system, <strong>and</strong> thus demonstrated that Godel's results applied to any formal system<br />
whatsoeveróitself constituted a chapter in Godel's program, ins<strong>of</strong>ar as <strong>the</strong> intuitive notion<br />
<strong>of</strong> effective calculability or mechanical solvability, an epistemological concept, concerning<br />
what we can come to know using mere calculation, was to be identified with <strong>the</strong> formal,
ma<strong>the</strong>matical concept <strong>of</strong> a (general) recursive function. As Church himself made clear, his<br />
<strong>the</strong>sis was based on an idea <strong>of</strong> Godel's, inspired by a suggestion <strong>of</strong> Herbr<strong>and</strong>'s, that it might<br />
be possible to identify <strong>the</strong> intuitive concept with <strong>the</strong> formal one. What was remarkable<br />
was that here, unlike <strong>the</strong> case <strong>of</strong> Godel's <strong>the</strong>orem, <strong>the</strong>re turned out to be no essential<br />
limitation on <strong>the</strong> effort to find a formal characterization <strong>of</strong> an intuitive concept. In this<br />
instance, <strong>the</strong> "Godel program" achieved a positive result. Godel himself found this<br />
astonishing. With <strong>the</strong> concept <strong>of</strong> [general] recursiveness or Turing computability, he said<br />
later, we have " for <strong>the</strong> first <strong>time</strong> succeeded in giving an absolute definition <strong>of</strong> an<br />
interesting epistemological notion." It was, he said, "a kind <strong>of</strong> miracle" that <strong>the</strong> diagonal<br />
procedure (as in his incompleteness <strong>the</strong>orem) "does not lead outside <strong>the</strong> defined notion."<br />
What was striking in <strong>the</strong> case <strong>of</strong> Godel's incompleteness <strong>the</strong>orems, however, was not just<br />
that his program was satisfied negatively, but that this result was proved formally.<br />
(Church's <strong>the</strong>sis, though convincing to most, including, eventually, Godel, cannot be<br />
proved formally, since one <strong>of</strong> <strong>the</strong> concepts involved, effective calculability, is <strong>of</strong> course<br />
precisely not formal, but ra<strong>the</strong>r, intuitive.) That is what made his results so irresistible <strong>and</strong><br />
so aggravating. This was <strong>the</strong> secret <strong>of</strong> Godel's strategy: where possible, he would establish<br />
<strong>the</strong> limits <strong>of</strong> <strong>the</strong> formal from within <strong>the</strong> formalism itself. He was a ma<strong>the</strong>matician one <strong>of</strong><br />
whose principal tasks was proving, ma<strong>the</strong>matically, what formal ma<strong>the</strong>matics can <strong>and</strong><br />
cannot accomplish.<br />
This overarching methodology <strong>of</strong> Godel's, however, while practiced in plain sight, has<br />
proved invisible. No one noticed that Godel's later contribution to relativity <strong>the</strong>ory<br />
provided yet ano<strong>the</strong>r example <strong>of</strong> his program <strong>of</strong> discovering <strong>the</strong> limits <strong>of</strong> formal methods in<br />
capturing intuitive concepts, this <strong>time</strong>, however, with a reversed conclusion. It was not<br />
only Godel <strong>the</strong> logician who was a spy. Godel <strong>the</strong> philosopher proved to be a figure yet<br />
more difficult to discern.<br />
5 It's Hard to Leave Vienna<br />
In <strong>the</strong> midst <strong>of</strong> <strong>the</strong> exultant joy that is pervading our country . .. you will be very happy if.<br />
. . in accord with <strong>the</strong> true will <strong>of</strong> <strong>the</strong> Fiihrer you may be allowed to support <strong>the</strong> decision <strong>of</strong><br />
his now united people with all your strength.<br />
ERWIN SCHRODINGER, AUSTRIA, 1938, "CONFESSION TO THE FUHRER"<br />
Having announced his incompleteness results at Konigsberg <strong>and</strong> published <strong>the</strong>m shortly<br />
<strong>the</strong>reafter, Godel rocketed to international fame in ma<strong>the</strong>matical <strong>and</strong> logical circles. In<br />
<strong>the</strong> months <strong>and</strong> years that tollowed, he presented <strong>the</strong>se results, <strong>and</strong> a slew <strong>of</strong> o<strong>the</strong>rs, in a<br />
host <strong>of</strong> ma<strong>the</strong>matical colloquia, including those presided over by his advisor Hans Hahn <strong>and</strong><br />
by Hahn's friend <strong>and</strong> colleague Karl Menger. In Menger's colloquium, in 1933, Godel<br />
delivered a paper on <strong>the</strong> rela-tionship <strong>of</strong> classical to intuitionist logic. In <strong>the</strong> audience was<br />
<strong>the</strong> American ma<strong>the</strong>matician Oswald Veblen, newly recruited by Abraham Flexner to join<br />
<strong>the</strong> ma<strong>the</strong>matical faculty <strong>of</strong> <strong>the</strong> Institute for Advanced Study in Princeton, New Jersey.<br />
Veblen, impressed, assisted Flexner by doing some recruiting <strong>of</strong> his own.
Abraham Flexner Goes Shopping<br />
In 1929, just moments before <strong>the</strong> stock market crashed, <strong>the</strong> owners <strong>of</strong> <strong>the</strong> New Jersey<br />
department store Bamberger's, Louis Bamberger <strong>and</strong> his<br />
sister Mrs. Felix Fuld, sold <strong>the</strong> business to R.H. Macy <strong>and</strong> Co. with <strong>the</strong> intention <strong>of</strong> using<br />
<strong>the</strong>ir pr<strong>of</strong>its to found an institution <strong>of</strong> higher learning. To assist <strong>the</strong>m in this endeavor <strong>the</strong>y<br />
enlisted <strong>the</strong> services <strong>of</strong> Abraham Flexner, a distinguished educational reformer. Flexner<br />
proposed an Institute devoted entirely to <strong>the</strong> exercise <strong>of</strong> pure thought <strong>of</strong> <strong>the</strong> highest<br />
order, where <strong>the</strong> faculty, <strong>the</strong> elite <strong>of</strong> <strong>the</strong> elite, would be unencumbered with <strong>the</strong> usual<br />
burdens <strong>of</strong> teaching <strong>and</strong> administration. (The burden <strong>of</strong> publication, <strong>of</strong> course, would<br />
remain.) He quickly convinced Bamberger <strong>and</strong> Fuld that this Institute for Advanced Study<br />
should be located in Princeton, New Jersey, where it could take advantage <strong>of</strong> <strong>the</strong><br />
resources <strong>and</strong> <strong>the</strong> tradition <strong>of</strong> academic culture provided by Princeton University. Flexner<br />
argued as well that <strong>the</strong> institute's first <strong>and</strong> fundamental faculty should be in <strong>the</strong> area <strong>of</strong><br />
ma<strong>the</strong>maticsóincluding ma<strong>the</strong>matical physicsówhich combined purity, significance, <strong>and</strong><br />
universally accepted canons <strong>of</strong> objectivity. Wasting no <strong>time</strong>, he began by recruiting, in<br />
1932, Oswald Veblen <strong>and</strong> Albert <strong>Einstein</strong> as permanent full pr<strong>of</strong>essors. Ve-blen he stole<br />
from Princeton University, <strong>the</strong> first <strong>of</strong> several <strong>the</strong>fts that would strain <strong>the</strong> relationship<br />
between <strong>the</strong> two institutions. A vigorous <strong>and</strong> enthusiastic shopper with an eye for<br />
geniuses, within a year Flexner acquired <strong>the</strong> ma<strong>the</strong>maticians Hermann Weyl, James<br />
Alex<strong>and</strong>er, <strong>and</strong> John Von Neumann.<br />
Getting <strong>Einstein</strong>, clearly, was a coup. In California in 1931, trolling <strong>the</strong> halls at Caltech for<br />
his new institute, Flexner learned that Albert <strong>Einstein</strong> happened to be visiting. The great<br />
physicist was intrigued but cautious, having already received <strong>of</strong>fers from prestigious <strong>and</strong><br />
well-established centers <strong>of</strong> learning around <strong>the</strong> <strong>world</strong>. He advised Flexner to approach him<br />
again in Oxford, Engl<strong>and</strong>, where he would be spending <strong>the</strong> spring semester <strong>of</strong> 1932.<br />
Flexner did, but <strong>Einstein</strong> remained ambivalent, suggesting that Flexner approach him again<br />
that summer, when <strong>Einstein</strong> would be at his country home in Caputh, Germany. The third<br />
<strong>time</strong> was <strong>the</strong> charm. Ambivalence was replaced with enthusiasm: "I am fire <strong>and</strong> flame for<br />
it," announced <strong>Einstein</strong>.<br />
If Flexner's shopping trip had been productive, so was Veblen's. A guest at Menger's<br />
ma<strong>the</strong>matical colloquium in Vienna, 1933, Veblen<br />
was so impressed with Godel's presentation, which confirmed, no doubt, what he had<br />
already heard from o<strong>the</strong>rs, that he invited <strong>the</strong> young logician to visit <strong>the</strong> Institute for<br />
Advanced Study during its inaugural year, 1933-1934. Godel would not be a permanent<br />
pr<strong>of</strong>essor, like Veblen himself, but one <strong>of</strong> twenty-four "workers," as <strong>the</strong>y were <strong>the</strong>n called.<br />
(They would soon acquire a more genteel title, "temporary member.") Veblen enticed<br />
Godel by <strong>of</strong>fering him <strong>the</strong> opportunity to attend a seminar on quantum mechanics to be<br />
<strong>of</strong>fered at <strong>the</strong> Institute by Von Neumann. Godel responded that he had a "lively interest" in<br />
quantum mechanics <strong>and</strong> would welcome <strong>the</strong> opportunity. (His interest in physics was<br />
indeed lively. He returned from <strong>the</strong> library in those years with volumes by Schrodinger,<br />
Dirac, Planck, Mach, Born, <strong>and</strong> Lorentz.)
If he came to Princeton, Veblen pointed out, Godel would have <strong>the</strong> opportunity to work<br />
with <strong>the</strong> American logician Alonzo Church, who had taken his doctorate under Veblen <strong>and</strong><br />
who had just developed a new system <strong>of</strong> formal logic, including what would later be known<br />
as <strong>the</strong> lambda calculus. It would be interesting, Veblen hinted, to see whe<strong>the</strong>r Godel's<br />
incompleteness <strong>the</strong>orem applied with equal force to Church's system <strong>of</strong> logic. Godel agreed<br />
that it would indeed be very interesting. So, it turned out, did Church, who believed,<br />
however, that his formal system was sufficiently different from <strong>the</strong> one Godel had focused<br />
on in his pro<strong>of</strong>, <strong>the</strong> Principia Matbematica <strong>of</strong> Russell <strong>and</strong> Whitehead, that it would be<br />
untouched by Godel's reasoning. (Church, it turned out, was mistaken. Worse still, his own<br />
students, Kleene <strong>and</strong> Rosser, proved his system inconsistent. In <strong>time</strong> he would become a<br />
champion <strong>of</strong> attempts to prove Godel's results inescapable by any genuinely formal<br />
system.) A third benefit that would attend a visit was suggested by Godel himself: <strong>the</strong><br />
opportunity to improve his English. The <strong>of</strong>fer was too good to refuse. So, in <strong>the</strong> course <strong>of</strong><br />
<strong>time</strong>, was a second invitation, <strong>and</strong> <strong>the</strong>n a third. Yet in spite <strong>of</strong> <strong>the</strong> nightmare that was <strong>the</strong><br />
political scene in Austria, Godel remained hesitant about a permanent move to <strong>the</strong><br />
institute <strong>and</strong> America. Even on <strong>the</strong> eve <strong>of</strong> <strong>the</strong> Nazi occupation (a conquest welcomed in<br />
<strong>the</strong> streets with patriotic fervor), he streng<strong>the</strong>ned his commitment to Vienna.<br />
The Vienna Syndrome<br />
O<strong>the</strong>r Viennese intellectuals suffered from <strong>the</strong> same syndrome: an attachment to <strong>the</strong> city<br />
<strong>of</strong> charm <strong>and</strong> culture so unreasonably strong that even <strong>the</strong> rumblings <strong>of</strong> <strong>the</strong> approaching<br />
German war machine could not dislodge <strong>the</strong>m. In its worst hour, <strong>the</strong> German-speaking<br />
<strong>world</strong> <strong>of</strong> Austria-Hungary <strong>and</strong> Germany still <strong>of</strong>fered such intellectual depth <strong>and</strong> warm<br />
collegiality to likeminded thinkers that its luminaries feared, perhaps rightly, that nowhere<br />
else would <strong>the</strong>ir light ever again burn so bright. Erwin Schrodinger, one <strong>of</strong> <strong>the</strong> pioneers <strong>of</strong><br />
quantum mechanics, certainly thought so. Having ab<strong>and</strong>oned Germany in <strong>the</strong> tumultuous<br />
early 1930s, he became disenchanted with life in Britain <strong>and</strong> America <strong>and</strong> decided in 1936<br />
to return to his native Austria, where, <strong>and</strong> only where, he could flourish. He would later<br />
describe this decision as "an unprecedented stupidity."<br />
Having deliberately climbed into a hole, Schrodinger proceeded to dig it deeper. A selfadvertised<br />
apolitical man, he was especially allergic to Nazis. Newly appointed to <strong>the</strong> Karl<br />
Franzen University <strong>of</strong> Graz, he couldn't resist <strong>the</strong> opportunity, in giving a lecture in Vienna<br />
on "World Structure in <strong>the</strong> Large <strong>and</strong> in <strong>the</strong> Small," introduced by Hans Thirring, who had<br />
taught physics to Godel, to append at <strong>the</strong> conclusion a political remark. "When one returns<br />
again from <strong>the</strong> kingdom <strong>of</strong> <strong>the</strong> stars," he said, "to our <strong>world</strong>, one finds <strong>the</strong>re a liking for a<br />
concept that wants to place one <strong>of</strong> <strong>the</strong> nations that live in this <strong>world</strong> over or under ano<strong>the</strong>r<br />
one."<br />
The audience cheered wildly; <strong>the</strong> Nazis took note. When <strong>the</strong> Anschluss occurred, it became<br />
clear that if he were to remain at Graz, Schrodinger would have to make an equally public<br />
retraction. This he proceeded to do in a "Confession to <strong>the</strong> Fiihrer" on March 30, 1938,<br />
published in all <strong>the</strong> Austrian <strong>and</strong> German papers. "In <strong>the</strong> midst <strong>of</strong> <strong>the</strong> exultant joy that is<br />
pervading our country," he wrote, ". . . you will be very happy if ... in accord with <strong>the</strong> true<br />
will <strong>of</strong> <strong>the</strong> Fuhrer you may be allowed to support <strong>the</strong> decision <strong>of</strong> his now united people<br />
with all your strength." This embarrassing statement he tried to explain afterward
to his friend Albert <strong>Einstein</strong>. He knew, <strong>of</strong> course, that his return to Austria would be<br />
dangerous, but he had not anticipated that "<strong>the</strong> fortress would be surrendered <strong>without</strong><br />
striking a blow." "I hope you have not seriously taken amiss," he wrote to <strong>Einstein</strong>, "my<br />
certainly quite cowardly statement... I wanted to remain free <strong>and</strong> could not do so <strong>without</strong><br />
great duplicity." The duplicity paid <strong>of</strong>f, but its effect was brief. Within months he received<br />
a short note from <strong>the</strong> interior ministry dismissing him from <strong>the</strong> university, "with no right to<br />
any legal recourse to this dismissal." The reason for <strong>the</strong> action: "political unreliability." An<br />
embarrassment had turned into a disaster.<br />
Not all who suffered from <strong>the</strong> Vienna syndrome were so unsuccessful. The Wittgenstein<br />
family was one <strong>of</strong> <strong>the</strong> richest <strong>and</strong> most prominent in Vienna. Assimilated Jews for<br />
generations, <strong>the</strong>y like many o<strong>the</strong>rs were awakened by <strong>the</strong> Anschluss to <strong>the</strong> fact that in <strong>the</strong><br />
eyes <strong>of</strong> <strong>the</strong> new regime <strong>the</strong>y were not Austrians or Germans but Jews. Ludwig<br />
Wittgenstein, <strong>the</strong> philosopher, was safely in Engl<strong>and</strong>, his bro<strong>the</strong>r Paul, <strong>the</strong> pianist, in<br />
America, where <strong>the</strong>ir sister Margarete would also live out <strong>the</strong> war. The sisters Hermine <strong>and</strong><br />
Helene, however, could not be cured <strong>of</strong> <strong>the</strong> Vienna syndrome. Nazis or no Nazis, <strong>the</strong> city<br />
was <strong>the</strong>ir home, <strong>and</strong> <strong>the</strong>y were not about to leave it. The problem was that <strong>the</strong> Nuremberg<br />
laws declared Jewish anyone with at least three Jewish gr<strong>and</strong>parents, a criterion <strong>the</strong><br />
Wittgenstein sisters appeared to satisfy. There was, however, a way out. The Nazi regime<br />
had a procedure for reclassification <strong>of</strong> Jews, a Befreiung. The Wittgensteins were <strong>of</strong><br />
"mixed race," Mischlinge, but if <strong>the</strong>y could prove that in fact <strong>the</strong>y had only two Jewish<br />
gr<strong>and</strong>parents, <strong>the</strong>nóassuming <strong>the</strong>y had not been so unwise as to actually practice Judaism<br />
or to marry a Jewó<strong>the</strong>y could be relabeled "Mischlinge <strong>of</strong> first degree" <strong>and</strong> permitted to<br />
remain, albeit tenuously, in <strong>the</strong> new Reich.<br />
Successful attempts at Befreiung were extraordinarily rare. The year <strong>the</strong> Wittgensteins<br />
made <strong>the</strong>ir attempt, 1939, saw 2,100 petitions for reclassification. The Fiihrer approved<br />
twelve. It seemed, however, that <strong>the</strong> Wittgensteins had a chance, since service in <strong>the</strong><br />
Great War counted loward reclassification, <strong>and</strong> Ludwig <strong>and</strong> Paul had accumulated <strong>the</strong>ir<br />
share <strong>of</strong> medals in service to <strong>the</strong> fa<strong>the</strong>rl<strong>and</strong> (Paul losing his right arm in <strong>the</strong> process,<br />
Ludwig working on <strong>the</strong> Tractatus in his spare <strong>time</strong> in <strong>the</strong> army). But a case would have to<br />
be made to <strong>the</strong> Fiihrer, who declared himself deluged by requests forwarded by <strong>the</strong><br />
interior ministry: "I get buckets <strong>and</strong> buckets <strong>of</strong> such applications . . . my fellow party<br />
members! Obviously you know <strong>of</strong> more decent Jews than <strong>the</strong>re are Jews in <strong>the</strong> whole <strong>of</strong><br />
<strong>the</strong> German Reich." In <strong>the</strong> end, however, Ludwig <strong>and</strong> Paul were able to assist <strong>the</strong>ir sisters.<br />
Medals did not impress <strong>the</strong> Reich, but gold did, <strong>and</strong> <strong>the</strong> Wittgenstein wealth, much <strong>of</strong> it<br />
invested abroad, came to <strong>the</strong> attention <strong>of</strong> <strong>the</strong> German ministry. With <strong>the</strong> active assistance<br />
(if not <strong>the</strong> approval) <strong>of</strong> <strong>the</strong> bro<strong>the</strong>rs Paul <strong>and</strong> Ludwig, <strong>and</strong> a Viennese lawyer who<br />
specialized in such things, Dr. Arthur Seyss-Inquart, who would later be hanged as a war<br />
criminal, it was arranged, after extensive negotiations, to transfer a vast amount <strong>of</strong> <strong>the</strong><br />
Wittgenstein fortune into <strong>the</strong> c<strong>of</strong>fers <strong>of</strong> <strong>the</strong> Reichsbank in Berlin. In exchange, <strong>the</strong> ministry<br />
agreed to accept as true <strong>the</strong> story that <strong>the</strong> Wittgensteins' paternal gr<strong>and</strong>fa<strong>the</strong>r, Hermann<br />
Christian, was in fact <strong>the</strong> illegitimate son <strong>of</strong> <strong>the</strong> "princely" house <strong>of</strong> Waldeck, leaving <strong>the</strong><br />
Wittgenstein sisters with just two Jewish gr<strong>and</strong>parents <strong>and</strong> permitting <strong>the</strong>ir <strong>of</strong>ficial<br />
reclassification as Mischlinge <strong>of</strong> first degree. No price, it seems, was too high to pay if it<br />
enabled <strong>the</strong> Wittgenstein sisters to remain in Vienna. Her-mine <strong>and</strong> Helene survived <strong>the</strong><br />
Vienna syndrome <strong>without</strong> having to undergo a cure.
The Viennese disease also had a German strain. It unmistakably stamped Wilhelm<br />
Furtwanglerócousin <strong>of</strong> Philip Furtwangler, <strong>the</strong> ma<strong>the</strong>matician whose lectures had so<br />
inspired Godelólegendary conductor <strong>of</strong> <strong>the</strong> Berlin Philharmonic, with whom Paul<br />
Wittgenstein had once performed with left h<strong>and</strong> alone. Forced to dismiss musician after<br />
talented musician from his beloved philharmonic to satisfy <strong>the</strong> dem<strong>and</strong>s <strong>of</strong> racial purity,<br />
compelled for <strong>the</strong> same reason to delete composers from Mendelssohn to Mahler from his<br />
concert programs, Furtwangler at no point considered <strong>the</strong> possibility <strong>of</strong> ab<strong>and</strong>oning his<br />
post or his fa<strong>the</strong>rl<strong>and</strong>. He believed it his duty to preserve <strong>the</strong> sacred <strong>legacy</strong> <strong>of</strong> German<br />
music,<br />
evenóindeed, especiallyóin a <strong>time</strong> <strong>of</strong> political madness. His contempt for Nazis, his<br />
unceasing attempts to assist Jewsó"Can you name me a Jew on whose behalf Furtwangler<br />
has not intervened?" bemoaned Josef Goebbelsóput him on a permanent short list for <strong>the</strong><br />
concentration camps. On <strong>the</strong> knife edge <strong>of</strong> execution throughout <strong>the</strong> war, he was<br />
preserved only by <strong>the</strong> patronage <strong>of</strong> his greatest fan, Adolf Hitler. (The attempt by <strong>the</strong><br />
Nazis to replace Furtwangler with a brilliant young upstart named Herbert von<br />
Karajanówho enthusiastically joined <strong>the</strong> Nazi party not once but twiceófell on deaf ears;<br />
<strong>the</strong> suggestion was dismissed by Hitler with contempt.)<br />
Hated by many Germans for his opposition to <strong>the</strong> Nazis, Furtwangler was yet more hated<br />
by <strong>the</strong> allies for his refusal to leave Germany, a twin fate he shared with his compatriot,<br />
<strong>the</strong> physicist Werner Heisen-berg. Heisenberg was, with Schrodinger, a founding fa<strong>the</strong>r <strong>of</strong><br />
<strong>the</strong> "new" quantum <strong>the</strong>ory: Heisenberg <strong>of</strong> <strong>the</strong> abstract, algebraic matrix <strong>the</strong>ory beloved <strong>of</strong><br />
positivists, Schrodinger <strong>of</strong> <strong>the</strong> more metaphysical <strong>and</strong> intuitive wave mechanics. (It was<br />
Godel's friend Von Neumann who would prove <strong>the</strong>m, in a strong sense, equivalent.)<br />
Between <strong>the</strong> Austrian <strong>and</strong> <strong>the</strong> German no love was lost. "I. .. felt discouraged, not to say<br />
repelled, by [Heisenberg's] methods <strong>of</strong> transcendental algebra," wrote Schrodinger, "which<br />
appeared very difficult to me, <strong>and</strong> by <strong>the</strong> lack <strong>of</strong> visualizability." Heisenberg, for his part,<br />
complained that "<strong>the</strong> more I reflect on <strong>the</strong> physical portion <strong>of</strong> Schrodinger's <strong>the</strong>ory <strong>the</strong><br />
more disgusting I find it... . What Schrodinger writes on <strong>the</strong> visualizability <strong>of</strong> his <strong>the</strong>ory ...<br />
I consider trash." Yet <strong>the</strong>y shared more than <strong>the</strong>ir creation <strong>of</strong> quantum mechanics, for<br />
Heisenberg too suffered from <strong>the</strong> Vienna syndrome, albeit, like Furtwangler, from <strong>the</strong><br />
German strain. He too could not bring himself to leave his fa<strong>the</strong>rl<strong>and</strong> during <strong>the</strong> war,<br />
despite his contempt for <strong>the</strong> Nazis <strong>and</strong> all <strong>the</strong>y stood for, despite his longst<strong>and</strong>ing<br />
admiration for <strong>the</strong> "Jewish physics" <strong>of</strong> his friend <strong>and</strong> hero Albert <strong>Einstein</strong>, which rendered<br />
him, like Furtwangler, a German outcast in <strong>the</strong> fa<strong>the</strong>rl<strong>and</strong>. At war's end, Heisenberg, like<br />
Furtwangler, was a hunted man in his own country; to <strong>the</strong> outside <strong>world</strong>, a traitor.<br />
Red Roses from Dr. <strong>Einstein</strong><br />
<strong>Einstein</strong> himself appeared to be immune to <strong>the</strong> Vienna syndrome. Appearances, however,<br />
can be deceptive. Though his exit from Germany in <strong>the</strong> early 1930s, before many <strong>of</strong> his<br />
colleagues realized it was <strong>time</strong> to leave, has been widely noted, what has achieved less<br />
attention is <strong>Einstein</strong>'s decision in <strong>the</strong> first place to take up residence in Berlin after his<br />
epoch-making papers in <strong>the</strong> "miraculous year" <strong>of</strong> 1905 catapulted him to <strong>world</strong> fame. (This<br />
neglect has begun to be remedied.) Having taken <strong>the</strong> extraordinary step <strong>of</strong> ab<strong>and</strong>oning his<br />
German homel<strong>and</strong> in his teens to take up residency <strong>and</strong> citizenship in <strong>the</strong> less bellicose <strong>and</strong>
academically rigid domain <strong>of</strong> Switzerl<strong>and</strong>ó where he received his degree in physics, where<br />
in seven years as a patent clerk he established himself as "<strong>the</strong> new Copernicus," <strong>and</strong> where<br />
he became established as a pr<strong>of</strong>essor in Zurichówhy did he decide in 1913 to return not<br />
just to Germany, but to Berlin, <strong>the</strong> very heart <strong>of</strong> Prussia?<br />
Max Planck <strong>and</strong> Wal<strong>the</strong>r Nernst approached <strong>the</strong> young Dr. <strong>Einstein</strong> in Zurich <strong>and</strong> gave him a<br />
day to think over <strong>the</strong>ir <strong>of</strong>fer <strong>of</strong> a pr<strong>of</strong>essorship in Berlin. He would meet <strong>the</strong>m at <strong>the</strong> train<br />
station, bearing white roses if he declined, red if he accepted. The roses were red. "It is<br />
not entirely clear why <strong>Einstein</strong> accepted <strong>the</strong> invitation to Berlin," writes his biographer<br />
Albrecht Folsing, nor why he chose to stay <strong>the</strong>re after <strong>the</strong> Great War for which he held his<br />
own country to blame. Folsing <strong>of</strong>fers some suggestions: Planck was after all <strong>the</strong> first to<br />
recognize, <strong>and</strong> publicize, <strong>the</strong> importance <strong>of</strong> <strong>the</strong> <strong>the</strong>ory <strong>of</strong> relativity; <strong>the</strong> position in Berlin<br />
was free <strong>of</strong> any teaching obligations; Elsa Lowenthal, <strong>Einstein</strong>'s cousin <strong>and</strong> soon his<br />
paramour, would be <strong>the</strong>re. Each persuasive, but none conclusive. He seems to have come<br />
under <strong>the</strong> German strain <strong>of</strong> <strong>the</strong> Vienna syndrome. This would help to explain why <strong>Einstein</strong><br />
remained in Germany during <strong>and</strong> after World War I, when he was a lone pacifist in a sea <strong>of</strong><br />
not merely German citizens but Prussian scientists caught up in <strong>the</strong> hysteria <strong>of</strong> <strong>the</strong> war,<br />
<strong>and</strong> why he continued to reside in Germany during <strong>the</strong> early years <strong>of</strong> <strong>the</strong> N.v/.i madness.<br />
The h<strong>and</strong>writing, after all, had been on <strong>the</strong> wall a long <strong>time</strong>. As early as 1920, an<br />
"antirelativity," i.e., "anti-<strong>Einstein</strong>," club had been formed in Berlin, bearing <strong>the</strong> name<br />
"Study Group <strong>of</strong> German Natural Philosophers." This group was devoted to denouncing<br />
<strong>Einstein</strong>'s "Jewish physics," even <strong>of</strong>fering money to those who would join <strong>the</strong>ir cause. On<br />
August 24, 1920, <strong>the</strong> insanity found a home. In Berlin's Philharmonic Hall, where<br />
Furtwangler had conducted <strong>the</strong> Berlin Philharmonic <strong>and</strong> Paul Wittgenstein had performed<br />
concertos for <strong>the</strong> left h<strong>and</strong>, <strong>the</strong> antirelativists staged a meeting to make <strong>the</strong>ir voices<br />
heard. <strong>Einstein</strong> himself attended, <strong>and</strong> amazingly found <strong>the</strong> <strong>time</strong> afterwards to publish a<br />
response. The debate was a tar baby, <strong>and</strong> <strong>Einstein</strong>, toge<strong>the</strong>r with his unhappy friends,<br />
became chagrined that he had allowed himself to be drawn into this fray. Yet it would<br />
take well over a decade's worth <strong>of</strong> such events to induce <strong>Einstein</strong> to leave Germany. The<br />
Vienna syndrome indeed, in its Berlin strain.<br />
The Eleventh Hour Plus One<br />
If <strong>Einstein</strong> left Berlin not quite at <strong>the</strong> eleventh hour but perhaps at <strong>the</strong> tenth, Godel left<br />
Vienna not at <strong>the</strong> eleventh hour but even later, just before midnight. He would visit <strong>the</strong><br />
Institute for Advanced Study not once but three <strong>time</strong>s before finally making <strong>the</strong> decision to<br />
leave Vienna. He accepted invitations to be a "worker" at <strong>the</strong> IAS from 1933 to 1934, 1935<br />
to 1936, <strong>and</strong> 1936 to 1937. Though <strong>the</strong> visits were on <strong>the</strong> whole successfulówith <strong>the</strong><br />
exception <strong>of</strong> <strong>the</strong> second, which Godel terminated abruptly, citing health reasonsó<strong>the</strong> years<br />
immediately following his great discoveries took a heavy toll on his mental <strong>and</strong> physical<br />
well-being. The lengthy ocean voyages to <strong>the</strong> States were a great strain, <strong>and</strong> <strong>the</strong> work<br />
itself (which he engaged in excessively, quickly producing a string <strong>of</strong> important<br />
ma<strong>the</strong>matical results) was also clearly a burden; but <strong>the</strong> need to defend his work against<br />
critical misunderst<strong>and</strong>ings no doubt had <strong>the</strong> most devastating effect. Unlike <strong>Einstein</strong>, who<br />
would have toasted Franklin Roosevelt's statement that "I am an
old campaigner <strong>and</strong> love a good fight," Godel throughout his life was deeply averse to<br />
dispute <strong>and</strong> controversy. This had an unfortunate effect on his published papers, which<br />
were carefully designed to put forward not all that he knew or believed but only what he<br />
could establish beyond all reasonable doubt, what even his opponents would be forced to<br />
accept. The manuscripts <strong>of</strong> his publications, sadly, contain vast amounts <strong>of</strong> valuable<br />
material crossed out, no doubt because <strong>the</strong>y failed to meet his excessive st<strong>and</strong>ards <strong>of</strong><br />
acceptability <strong>without</strong> controversy. In <strong>the</strong> end, ironically, this strategy <strong>of</strong> safety failed<br />
completely. All along, Godel was understood to be a man apart from his <strong>time</strong>s, whose<br />
beliefs on a host <strong>of</strong> topicsófrom truth <strong>and</strong> pro<strong>of</strong>, to language-centered philosophy, to God<br />
<strong>and</strong> spiritsówere wildly out <strong>of</strong> step with those <strong>of</strong> his contemporaries.<br />
With fame came not only friends but enemies, <strong>and</strong> this Godel was unprepared for. It was<br />
during <strong>the</strong>se years that he suffered his first bouts <strong>of</strong> depression <strong>and</strong> began to show signs <strong>of</strong><br />
<strong>the</strong> eating disorder that would eventually kill him. Three visits to sanatoria are<br />
documented: 1934 at Sanatorium Westend in Purkersdorf, near Vienna; 1935 at<br />
Breitenstein am Semmering, <strong>and</strong> 1936 at a sanitorium at Rekawinkel, near Vienna. These<br />
inner struggles were complemented by <strong>the</strong> outward turmoil <strong>of</strong> his beloved Vienna, which<br />
was coming down around him in ruins. It was in <strong>the</strong>se years that Germany invaded<br />
Czechoslovakia <strong>and</strong> that Austria succumbed to <strong>the</strong> Anschluss, in this period that Schlick, <strong>of</strong><br />
<strong>the</strong> Vienna Circle, was murdered <strong>and</strong> Herbr<strong>and</strong> fell to his death while mountain climbing.<br />
In addition, Godel's teacher Hahn had also died, <strong>and</strong> his friends Menger <strong>and</strong> Carnap had left<br />
for America. He also had more immediate concerns. His mo<strong>the</strong>r had been forced by<br />
financial circumstances to return to Brno, where her vociferous objections to <strong>the</strong> Nazis put<br />
her at considerable risk.<br />
Godel himself was eerily silent about <strong>the</strong> political events surrounding him, both in person<br />
<strong>and</strong> in correspondence, which resulted in a considerable cooling <strong>of</strong> his friendship with<br />
Menger. If this silence sprang from Godel's excessive caution, <strong>the</strong>n once again it was<br />
ineffective. The Nazi authorities noted his association with "<strong>the</strong> Jewish pro-<br />
fessor Hahn," adding that "it redounded to his discredit" that he "traveled in liberal-Jewish<br />
circles." The signs were all around him. His cleaning lady presented him with a bill at <strong>the</strong><br />
bottom <strong>of</strong> which were appended <strong>the</strong> words, "Heil Hitler!" Yet in spite <strong>of</strong> everything, he<br />
made no decision to leave. On <strong>the</strong> contrary, as late as 1939 he <strong>and</strong> Adele, whom he had<br />
married in 1938 in a civil ceremonyósurprising his bro<strong>the</strong>r, who had only just been<br />
introduced to herómoved from a rented apartment to one <strong>the</strong>y had purchased in <strong>the</strong> city<br />
<strong>of</strong> Vienna. The Vienna syndrome, in spades. More still needed to happen before Godel<br />
could be persuaded to flee.<br />
More did. One day late in 1939, Godel was accosted by a gang <strong>of</strong> youths who took him for a<br />
Jewóor at least someone who associated with Jewsó<strong>and</strong> roughed him up, knocking <strong>of</strong>f his<br />
glasses. Adele, fortunately, was able to beat <strong>the</strong>m <strong>of</strong>f with her umbrella. This incident,<br />
however, seems to have done <strong>the</strong> trick, especially when combined with <strong>the</strong> order he had<br />
recently received to take a physical examination to determine whe<strong>the</strong>r he was fit for<br />
military service with <strong>the</strong> German army, an examination which, amazingly, <strong>the</strong> frail, thirtytwo-year-old<br />
logician, who had been in <strong>and</strong> out <strong>of</strong> sanatoria, passed. With <strong>the</strong> considerable<br />
assistance <strong>of</strong> <strong>the</strong> new director <strong>of</strong> <strong>the</strong> IAS, Frank Aydelotte, who wrote to <strong>the</strong> authorities<br />
that Godel should be given special treatment as an Aryan who was a <strong>world</strong>-famous<br />
ma<strong>the</strong>matician, Godel succeeded, in December 1939, in securing a visa to travel to
America. The great logician escaped from Austria just as <strong>the</strong> door was closing behind him.<br />
"I am told in all steamship bureaus," he wrote to Aydelotte, "that <strong>the</strong> danger for German<br />
citizens to be arrested by <strong>the</strong> English is very great on <strong>the</strong> Atlantic." He decided, <strong>the</strong>refore,<br />
to travel with Adele via <strong>the</strong> trans-Siberian railway to Japan, from <strong>the</strong>re to voyage across<br />
<strong>the</strong> Pacific to San Francisco, <strong>and</strong> <strong>the</strong>nce by rail to Princeton. A grueling journey about<br />
which Godel, typically, made no comment. Adele, however, remarked afterward that <strong>the</strong>y<br />
traveled frequently at night, in constant fear <strong>of</strong> being detained <strong>and</strong> returned to Austria.<br />
The journey itself, however, was <strong>without</strong> incident, <strong>and</strong> Godel arrived at last in San<br />
Francisco harbor on March 4, 1940. Soon he<br />
would be at <strong>the</strong> IAS with ano<strong>the</strong>r survivor, Albert <strong>Einstein</strong>, whose friendship would be a<br />
watershed in two lives that had already marked some <strong>of</strong> <strong>the</strong> greatest intellectual<br />
achievements <strong>of</strong> <strong>the</strong> twentieth or any o<strong>the</strong>r century. Godel <strong>and</strong> <strong>Einstein</strong> would discover<br />
that what <strong>the</strong>y had left behind in Vienna <strong>and</strong> Berlin <strong>the</strong>y would never find again. Each in<br />
his own way would become increasingly isolated <strong>and</strong> lonely, a creature <strong>of</strong> ano<strong>the</strong>r <strong>time</strong> <strong>and</strong><br />
ano<strong>the</strong>r culture, <strong>the</strong>ir native language a constant reminder <strong>of</strong> <strong>the</strong>ir origins in <strong>the</strong> l<strong>and</strong> that<br />
had become <strong>the</strong>ir adopted country's mortal enemy. A historic empire had dwindled to <strong>the</strong><br />
company <strong>of</strong> two, what Kurt Vonnegut would describe in his novel Mo<strong>the</strong>r Night as "das<br />
Reich der Zwei." But what a two! While lesser souls might look only to <strong>the</strong> glories <strong>of</strong> <strong>the</strong>ir<br />
past, Godel <strong>and</strong> <strong>Einstein</strong> in <strong>the</strong>ir remarkable friendship would explore a new <strong>world</strong> <strong>of</strong><br />
ideas. More <strong>and</strong> more, <strong>the</strong>ir thoughts would turn to a topic at <strong>the</strong> very center <strong>of</strong> <strong>Einstein</strong>'s<br />
relativity, at <strong>the</strong> core as well <strong>of</strong> Godel's preoccupation with Kant <strong>and</strong> Leibniz, <strong>the</strong> German<br />
idealists. The question was <strong>time</strong>. The answer would be yet ano<strong>the</strong>r surprise from Dr.<br />
Godel.<br />
Participants at <strong>Einstein</strong>'s 70th birthday celebration. Left to right: Eugene Wigner, Hermann<br />
Weyl, Godel, I.I. Rabi, <strong>Einstein</strong>, Rudolf Ladenburg, J. Robert Oppenheimer.<br />
Photo by Howard Schrader<br />
Kurt Godel with his<br />
bro<strong>the</strong>r, Rudolf, <strong>and</strong><br />
his mo<strong>the</strong>r, Marianne.<br />
Courtesy <strong>of</strong> <strong>the</strong> Godel Archives, Firestone Library, Princeton University, <strong>and</strong> <strong>the</strong> Institute<br />
for Advanced Study<br />
Kurt <strong>and</strong> Adele Godel dining outdoors.
Courtesy <strong>of</strong> <strong>the</strong> Godel Archives, Firestone Library, Princeton University, <strong>and</strong> <strong>the</strong> Institute<br />
for Advanced Study<br />
Kurt <strong>and</strong> Adele<br />
Godel dining<br />
indoors.<br />
Courtesy <strong>of</strong> <strong>the</strong><br />
Godel Archives,<br />
Firestone Library,<br />
Princeton University,<br />
<strong>and</strong> <strong>the</strong> Institute for<br />
Advanced Study<br />
Godel, dressed for driving.<br />
Courtesy <strong>of</strong> <strong>the</strong> Godel Archives, Firestone Library, Princeton University, <strong>and</strong> <strong>the</strong> Institute<br />
for Advanced Study<br />
Godel, dressed for hiking.<br />
Courtesy <strong>of</strong> <strong>the</strong> Godel Archives, Firestone Library, Princeton University, <strong>and</strong> <strong>the</strong> Institute<br />
for Advanced Study<br />
Young Adele Porkert at <strong>the</strong> piano.
Courtesy <strong>of</strong> <strong>the</strong> Godel Archives, Firestone Library, Princeton University, <strong>and</strong> <strong>the</strong> Institute<br />
for Advanced Study<br />
6 Amid <strong>the</strong> Demigods<br />
When pygmies cast such long shadows, it must be very late in <strong>the</strong> day. GIAN-CARLO ROTA<br />
Princeton is not Vienna. Having fled Nazi-occupied Austria, Godel <strong>and</strong> Adele found<br />
<strong>the</strong>mselves in an Ivy League college town, small, provincial <strong>and</strong> inbred, dominated by <strong>the</strong><br />
imposing presence <strong>of</strong> Princeton University, itself outclassed by <strong>the</strong> still more prestigious<br />
Institute for Advanced Study. Princeton's students may have touted <strong>the</strong>mselves as <strong>the</strong><br />
creme de la creme; <strong>the</strong> institute could boast that mere students, ol whatever caliber,<br />
were not welcome. In <strong>the</strong>se streets it was hard to avoid rubbing shoulders with <strong>the</strong><br />
intellectual elite, <strong>and</strong> with those who thought <strong>of</strong> <strong>the</strong>mselves as such. Bertr<strong>and</strong> Russell was<br />
unimpressed. He found Princeton "full <strong>of</strong> new Gothic, <strong>and</strong> ... as like Oxford as monkeys can<br />
make it." <strong>Einstein</strong> was more delicate: "Princeton is a wonderful piece <strong>of</strong> earth <strong>and</strong> at <strong>the</strong><br />
same <strong>time</strong> an exceedingly amusing ceremonial backwater <strong>of</strong> tiny spindle-shanked<br />
demigods." The pygmies on stilts would have to make way for <strong>the</strong> entrance <strong>of</strong> two giants.<br />
Godel, however, was content with his new home. Unlike Adele, who struggled with English,<br />
he had long been fluent, <strong>and</strong> he found <strong>the</strong> people <strong>and</strong> culture in Princeton "ten <strong>time</strong>s more<br />
congenial" than those in Vienna. He admired also <strong>the</strong> "prompt functioning <strong>of</strong> government<br />
<strong>of</strong>ficials in America," which he said made life "10 x 10 x . . ." better than in <strong>the</strong> old<br />
country. The very fact that intellectual life was more narrowly focused, centered at <strong>the</strong><br />
institute on ma<strong>the</strong>matics <strong>and</strong><br />
ma<strong>the</strong>matically oriented sciences, could not fail to please him. But in such a rarefied<br />
atmosphere, it is no surprise that Adele, lacking academic pretensions, found it hard to<br />
brea<strong>the</strong>. In <strong>the</strong> cafe, opera, club, <strong>and</strong> cabaret scene <strong>of</strong> prewar Vienna, <strong>the</strong> easy charm <strong>of</strong><br />
social life had blended smoothly with <strong>the</strong> life <strong>of</strong> <strong>the</strong> mind, pr<strong>of</strong>essors mingled with<br />
dancers, <strong>and</strong> composers dined with philosophers. In Princeton, such mixing would be hard<br />
to imagine. (There were, in any case, few cabarets or opera houses to be found.)<br />
Without children, Adele sought refuge in a menagerie <strong>of</strong> pets that would eventually<br />
encompass a pair <strong>of</strong> love birds, a dog named Penny <strong>and</strong> a cat, though not one <strong>of</strong> <strong>the</strong><br />
tailless Manx variety <strong>of</strong> which she had become inordinately fond. (Only <strong>the</strong> desperate pleas<br />
<strong>of</strong> her friend <strong>and</strong> neighbor, Bobbie Brown, dissuaded her from removing <strong>the</strong> tail <strong>of</strong> <strong>the</strong> cat<br />
she did purchase.) In <strong>time</strong>, <strong>the</strong> Godels would also sponsor a foster child abroad. Their first<br />
home in Princeton was an apartment at 245 Nassau Street. Later, <strong>the</strong>y moved into an<br />
apartment at 108 Stockton Street, where <strong>the</strong>ir neighbor, George Brown, already knew<br />
Godel. During one <strong>of</strong> Godel's early visits to <strong>the</strong> institute, Brown, a graduate student in<br />
ma<strong>the</strong>matical statistics who had studied logic at Harvard with W.V.O. Quine, had been<br />
given <strong>the</strong> task <strong>of</strong> taking notes on Godel's lectures to prepare <strong>the</strong>m for publication. He <strong>and</strong><br />
his wife Bobbie quickly befriended <strong>the</strong> new arrivals. But during <strong>the</strong>ir visits with <strong>the</strong>
eclusive Godels, <strong>the</strong>y were put <strong>of</strong>f by <strong>the</strong>ir hosts' decision to remove <strong>the</strong> screens from <strong>the</strong><br />
windows. Godel claimed that this allowed him to brea<strong>the</strong> properly, but it also allowed easy<br />
access to dust <strong>and</strong> insects, which considerably dampened Bobbie Brown's enthusiasm for<br />
coming over.<br />
This residence, too, was temporary. The Godels moved again, this <strong>time</strong> to an apartment at<br />
120 Alex<strong>and</strong>er Street, near <strong>the</strong> train station. Godel was fond <strong>of</strong> his new abode, remarking<br />
that it was located "directly opposite <strong>the</strong> most elegant hotel in town" (<strong>the</strong> Princeton Inn).<br />
They occupied <strong>the</strong> entire upper floor <strong>of</strong> <strong>the</strong> building, with windows on all sides, which<br />
Godel found helpful in surviving <strong>the</strong> hot Princeton summers. Adele, however, was unhappy<br />
with <strong>the</strong> poor condition <strong>of</strong> <strong>the</strong><br />
91<br />
premises, which she considered unhygienic, <strong>and</strong> she found <strong>the</strong> neighborhood dreary.<br />
Her disappointment turned to enthusiasm when she discovered a house for sale at 129<br />
(later 145) Linden Lane, at <strong>the</strong> outskirts <strong>of</strong> town. Built just a few years earlier, it was a<br />
small one-story structure <strong>of</strong> "sturdy cinder block" with an automatic oil furnace <strong>and</strong> built-in<br />
air conditioning, <strong>and</strong> included a wood-burning fireplace. Adele would not rest until she had<br />
persuaded her husb<strong>and</strong> to purchase it. The house was beyond <strong>the</strong>ir means, but Godel was<br />
able to secure a mortgage for three-fourths <strong>of</strong> <strong>the</strong> purchase price, while <strong>the</strong> director <strong>of</strong><br />
<strong>the</strong> institute, J. Robert Oppenheimer, arranged for a salary advance to cover <strong>the</strong> rest. It<br />
was, to GodePs mind, a "somewhat shaky" arrangement, but he went along with it, <strong>and</strong> in<br />
August 1949 <strong>the</strong>y moved in to stay. Adele arranged <strong>the</strong> furnishings, which were for <strong>the</strong><br />
most part modest, but she did indulge her weakness for oriental rugs <strong>and</strong> ch<strong>and</strong>eliers. Over<br />
<strong>the</strong> years, she would oversee <strong>the</strong> construction <strong>of</strong> a sun porch, refinish a room to serve as<br />
GodePs study, <strong>and</strong> plant a flower garden. Evergreens would also be planted in front <strong>of</strong><br />
<strong>the</strong>ir home, as a barrier to shield <strong>the</strong> Godels from passersby. Adele engaged in traditional<br />
domestic activities. She was, friends noted, a good cook, though with a penchant for heavy<br />
German fare. In <strong>the</strong> summer<strong>time</strong>, she was relieved to escape <strong>the</strong> confines <strong>of</strong> Princeton by<br />
accompanying Godel to resorts in Maine.<br />
Still, it was a sad life, centered on <strong>the</strong> principal occupation <strong>of</strong> tending to her fragile<br />
husb<strong>and</strong>, a task Adele shared with a succession <strong>of</strong> Princeton luminaries. In <strong>the</strong> early years<br />
it was Oswald Veblen, <strong>the</strong> one who had first moved to acquire Godel for <strong>the</strong> institute, who<br />
took it on himself to look after him. The job <strong>the</strong>n passed to <strong>Einstein</strong>, <strong>and</strong> after <strong>Einstein</strong>'s<br />
death to <strong>the</strong> economist Oskar Morgenstern. Shepherding <strong>the</strong> logician became increasingly<br />
necessary, for Godel's eccentricities, already evident in Vienna, had blossomed to such an<br />
extent that as early as 1941, Frank Aydelotte, director <strong>of</strong> <strong>the</strong> institute before<br />
Oppenheimer, took <strong>the</strong> extraordinary step <strong>of</strong> inquiring <strong>of</strong> GodePs doctor whe<strong>the</strong>r "<strong>the</strong>re is<br />
any danger <strong>of</strong> his malady taking a violent form<br />
which might involve his doing injury ci<strong>the</strong>r to himself or to o<strong>the</strong>rs ..." He was assured that<br />
violence would not he a problem.
These eccentricities included Godel's conviction that "bad air" was emanating from his<br />
refrigerator as well as from <strong>the</strong> heating system. Fear <strong>of</strong> air was accompanied by an<br />
aversion to cold. Even in midsummer, his gaunt figure could be seen enshrouded in a<br />
winter coat, along with hat <strong>and</strong> gloves. His nascent hypochondria developed into a<br />
fullblown syndrome in which imaginary maladies were given <strong>the</strong> same status as real ones.<br />
Worse, fear <strong>of</strong> disease was accompanied by fear <strong>and</strong> mistrust <strong>of</strong> doctors. There was also<br />
Godel's increasingly reclusive behavior. He endeavored to abstain from all "unnecessary"<br />
social <strong>and</strong> intellectual interaction <strong>and</strong> was at pains to avoid, at all costs, actual physical<br />
contact with o<strong>the</strong>r human beings. The telephone became his preferred method <strong>of</strong><br />
communication. (His penchant for waking up long-suffering friends at all hours <strong>of</strong> <strong>the</strong> night<br />
to engage in endless telephone calls puts one in mind <strong>of</strong> ano<strong>the</strong>r reclusive genius, <strong>the</strong><br />
Canadian pianist Glenn Gould, who could also be seen in <strong>the</strong> summer heat bundled up in<br />
coat, hat <strong>and</strong> gloves.) Fear <strong>of</strong> disease finally grew into a generalized fear <strong>of</strong> o<strong>the</strong>rs, a<br />
paranoia that may or may not have evolved into psychosis. The significance <strong>of</strong> his mental<br />
instability had been presaged by his great ma<strong>the</strong>matics pr<strong>of</strong>essor at <strong>the</strong> University <strong>of</strong><br />
Vienna, Philip Furtwangler, who, on learning <strong>of</strong> Godel's incompleteness <strong>the</strong>orems, asked,<br />
"Is his [mental] illness a consequence <strong>of</strong> proving <strong>the</strong> unprovability [<strong>of</strong> <strong>the</strong> consistency <strong>of</strong><br />
formal arithmetic], or is his illness necessary for such an occupation?" Godel's eccentricities<br />
would eventually contribute to his death, cited by his doctor as due to "inanition" brought<br />
on by personality disorder.<br />
<strong>Einstein</strong>'s eccentricities also flourished in his new homel<strong>and</strong>, but <strong>the</strong>y were <strong>of</strong> an<br />
altoge<strong>the</strong>r milder order <strong>and</strong> affected his family <strong>and</strong> friends far more than himself. The<br />
great scientist's colleagues were embarrassed by his unpr<strong>of</strong>essorial penchant for strolling<br />
down Nassau Street, Princeton's leafy main thoroughfare, while licking an ice cream cone,<br />
<strong>and</strong> female friends were taken aback by his sexual frankness <strong>and</strong> disdain for traditional<br />
mores. More generally, <strong>Einstein</strong>'s attitude to-<br />
ward women was nothing to boast <strong>of</strong>. Peter Bucky, who had plenty <strong>of</strong> opportunities to view<br />
<strong>the</strong> great physicist up close while serving as chauffeur <strong>and</strong> companion for his fa<strong>the</strong>r's<br />
friend, has written that "in today's terms, <strong>Einstein</strong> would have been considered a classic<br />
male chauvinist. He once wrote a letter to a friend, a Dr. Muesham in Haifa, that his<br />
definition <strong>of</strong> a good wife was someone who stood somewhere between a pig <strong>and</strong> a chronic<br />
cleaner." In Princeton, a bastion <strong>of</strong> bourgeois intellectual respectability, <strong>Einstein</strong> raised his<br />
bohemian lifestyle to an art. His unkempt hair became longer with each succeeding year.<br />
He wore shoes <strong>without</strong> socks <strong>and</strong> a lea<strong>the</strong>r jacket, not from a sense <strong>of</strong> fashion but from <strong>the</strong><br />
belief that what doesn't wear out won't need to be replaced. His son Hans, who came to<br />
America already estranged from his famous fa<strong>the</strong>r, was especially embarrassed by<br />
<strong>Einstein</strong>'s manner <strong>of</strong> dress <strong>and</strong> became, in protest, a d<strong>and</strong>y. The complete neglect <strong>of</strong> all<br />
surface appearances may be <strong>the</strong> signature <strong>of</strong> a deep thinker, but it does little to smooth<br />
over <strong>the</strong> already fractious relations among human beings. The one companion that suffered<br />
no harm was <strong>the</strong> omnipresent pipe. His friend Gariella Oppenheim-Errara recalled an<br />
occasion when his sailboat, Tannef, capsized, leaving <strong>the</strong> great physicist paddling in <strong>the</strong><br />
water, puffing contentedly on his faithful pipe. There is no doubt that <strong>Einstein</strong>'s pipe was<br />
his closest associate, while o<strong>the</strong>rsóincluding wife <strong>and</strong> familyówere never permitted <strong>the</strong><br />
illusion that <strong>the</strong>y would ever be at <strong>the</strong> center <strong>of</strong> his life.<br />
<strong>Einstein</strong> <strong>the</strong> bohemian was also a Jew in a town whose university still enforced Jewish<br />
quotas for its students, something <strong>Einstein</strong> had personally protested during <strong>the</strong>
negotiations for his appointment, Caught in this net was Richard Feynman, one <strong>of</strong> <strong>the</strong> great<br />
physicists <strong>of</strong> <strong>the</strong> modern era, who was forced to attend MIT due to Columbia University's<br />
quota system. We can infer that his acceptance at Princeton University for graduate<br />
studies was also subject to ethnic strictures: Philip Morse, Feynman's pr<strong>of</strong>essor at MIT, felt<br />
compelled to note in his recommendation that Feynman's "physiognomy <strong>and</strong> manner . . .<br />
show no trace <strong>of</strong> this characteristic [i.e., being Jewish] <strong>and</strong> I do not believe <strong>the</strong> matter<br />
will be any great h<strong>and</strong>icap." (Feynman would go on to<br />
become <strong>the</strong> third winner <strong>of</strong> <strong>the</strong> <strong>Einstein</strong> Prize, after Godel <strong>and</strong> Schwinger.) In this<br />
environment, as an eccentric Jewish pr<strong>of</strong>essor with a bohemian lifestyle <strong>and</strong> a heavy<br />
German accent, who moreover had devoted much <strong>of</strong> his life to a doomed search for an<br />
elusive <strong>the</strong>ory that would unite relativity with quantum mechanics, <strong>Einstein</strong> was<br />
unquestionably an outsider. Clearly, he would appreciate <strong>the</strong> appearance in Princeton <strong>of</strong><br />
ano<strong>the</strong>r outsider whose eccentricities, if not genius, surpassed his own.<br />
It was in 1933 that Godel first met <strong>Einstein</strong>, during a visit to <strong>the</strong> institute, when <strong>the</strong>y were<br />
introduced by ano<strong>the</strong>r emigre, Paul Oppen-heim. They became friends in 1942, after<br />
Godel, too, had joined <strong>the</strong> institute, <strong>and</strong> remained close until <strong>Einstein</strong>'s death in 1955.<br />
When Godel was ill in February 1951, he wrote to his mo<strong>the</strong>r, Marianne, that "during my<br />
sickness <strong>Einstein</strong> was <strong>of</strong> course extraordinarily nice to me <strong>and</strong> visited me many <strong>time</strong>s both<br />
in <strong>the</strong> hospital <strong>and</strong> at home." The exact circumstances surrounding <strong>the</strong> transition from<br />
acquaintance to friendship remain unknown, <strong>and</strong> not by accident. The two men cherished<br />
<strong>the</strong> privacy <strong>of</strong> <strong>the</strong>ir relationship. After his good friend died, Godel wrote to his mo<strong>the</strong>r that<br />
<strong>the</strong> fact "that people never mention me in connection with <strong>Einstein</strong> is very satisfactory to<br />
me (<strong>and</strong> would be to him, too, since he was <strong>of</strong> <strong>the</strong> opinion that even a famous man is<br />
entitled to a private life). After his death I have already been invited twice to say<br />
something about him, but naturally I declined."<br />
He was happy, however, to confide some details <strong>of</strong> <strong>the</strong>ir friendship to satisfy his mo<strong>the</strong>r's<br />
curiosity. Godel, it emerged, would meet <strong>Einstein</strong> at his home each day between ten <strong>and</strong><br />
eleven in <strong>the</strong> morning, <strong>and</strong> <strong>the</strong> two would walk to <strong>the</strong> institute, a journey that generally<br />
took half an hour. At one or two in <strong>the</strong> afternoon, <strong>the</strong>y would return home, discussing<br />
politics, philosophy <strong>and</strong> physics. This schedule gives new meaning to <strong>Einstein</strong>'s comment<br />
that he went to his <strong>of</strong>fice "just to have <strong>the</strong> privilege <strong>of</strong> walking home with Kurt Godel."<br />
These walks, in fact, consumed some thirty percent <strong>of</strong> his workday. They could also be<br />
dangerous. "I know <strong>of</strong> one occasion," wrote <strong>Einstein</strong>'s secretary, Helen Dukas, in 1946,<br />
"when a car hit a tree after its driver suddenly recog-<br />
nized <strong>the</strong> face <strong>of</strong> <strong>the</strong> beautiful old man walking along <strong>the</strong> street." Godel's more severe<br />
countenance, by contrast, was no threat to traffic. "I have so far," he wrote to his mo<strong>the</strong>r,<br />
"not found my 'fame' burdensome in any way. That begins only when one becomes so<br />
famous that one is known to every child in <strong>the</strong> street, as is <strong>the</strong> case <strong>of</strong> <strong>Einstein</strong>."<br />
Godel, by nature a pessimist about human affairsóthough an optimist about <strong>the</strong> power <strong>of</strong><br />
reasonówas balanced by <strong>the</strong> more optimistic <strong>Einstein</strong>. Yet <strong>Einstein</strong> too, as Godel wrote to<br />
his mo<strong>the</strong>r, "was in many respects a pessimist. In particular, he didn't have a very good<br />
opinion <strong>of</strong> humanity in general. Among o<strong>the</strong>r things, he based this on <strong>the</strong> fact that those
who wished to do some good, like Christ, Moses, Mohammed, etc., ei<strong>the</strong>r died a violent<br />
death or had to use violence against his followers."<br />
Both men were skeptical <strong>of</strong> Bohr <strong>and</strong> Heisenberg's Copenhagen interpretation <strong>of</strong> quantum<br />
mechanics, <strong>and</strong> Godel was skeptical as well <strong>of</strong> <strong>Einstein</strong>'s efforts to unify relativity with<br />
quantum physics. Each lived in a modest home, while colleagues like Von Neumann<br />
inhabited mansions. Their households mingled; <strong>the</strong>y exchanged housewarming gifts. Their<br />
lives became interconnected. It was a familiar sight in Princeton to see <strong>the</strong> two friends<br />
walking home from <strong>the</strong> institute, arguing in <strong>the</strong>ir mo<strong>the</strong>r tongue about politics <strong>and</strong> general<br />
relativity. <strong>Einstein</strong>, like many intellectuals, favored Adlai Stevenson for president. "Godel,"<br />
however, <strong>Einstein</strong> remarked, "has really gone completely crazy. He voted for Eisenhower."<br />
Whoever came to know <strong>the</strong> one, ipso facto became acquainted with <strong>the</strong> o<strong>the</strong>r.<br />
Submarines Again<br />
Their most famous discoveries behind <strong>the</strong>m, <strong>Einstein</strong> <strong>and</strong> Godel led increasingly quiet lives<br />
in <strong>the</strong> backwaters <strong>of</strong> <strong>the</strong>ir respective fields. In domains <strong>the</strong>y had once ruled as titans <strong>the</strong>y<br />
were now but part <strong>of</strong> <strong>the</strong> furniture, albeit, as <strong>Einstein</strong> cracked, "museum pieces." "In<br />
Prince-ton," he told friends, "I am known as <strong>the</strong> village idiot." As <strong>the</strong> war<br />
years dragged on <strong>and</strong> <strong>the</strong> light <strong>of</strong> science prepared to cast <strong>the</strong> shadow <strong>of</strong> <strong>the</strong> atomic bomb,<br />
<strong>the</strong> two brightest minds in <strong>the</strong> scientific firmament drifted ever far<strong>the</strong>r from <strong>the</strong> center. A<br />
few doors away from <strong>Einstein</strong>'s <strong>of</strong>fice in Fine Hall, Neils Bohr <strong>and</strong> John Wheeler were<br />
working out <strong>the</strong> details <strong>of</strong> nuclear fission for employment in a nuclear device. <strong>Einstein</strong>,<br />
meanwhile, alone <strong>and</strong> all but inaccessible, pursued his dream not <strong>of</strong> splitting <strong>the</strong> atom but<br />
<strong>of</strong> unifying physics. Visiting <strong>the</strong> institute a few years earlier, <strong>the</strong> fa<strong>the</strong>r <strong>of</strong> <strong>the</strong> atomic<br />
bomb, J. Robert Oppen-heimer, commented that "<strong>Einstein</strong> is completely cuckoo." The<br />
gods, as is well known, love irony. After <strong>the</strong> war, Oppenheimer found himself director <strong>of</strong><br />
<strong>the</strong> institute <strong>and</strong> thus, nominally at least, <strong>Einstein</strong>'s boss.<br />
In 1935, <strong>the</strong> year Oppenheimer paid his visit to <strong>the</strong> institute, <strong>the</strong> fa<strong>the</strong>r <strong>of</strong> relativity<br />
announced that in his view, <strong>the</strong> idea <strong>of</strong> constructing a bomb by splitting <strong>the</strong> atom was as<br />
promising as "firing at birds in <strong>the</strong> dark, in a neighborhood that has few birds." Yet <strong>Einstein</strong><br />
would eventually coauthor a letter to Roosevelt suggesting that <strong>the</strong> prospects for an<br />
atomic bomb be explored, becoming thus, in effect, <strong>the</strong> fa<strong>the</strong>r <strong>of</strong> <strong>the</strong> fa<strong>the</strong>r <strong>of</strong> <strong>the</strong> bomb.<br />
<strong>Einstein</strong>'s paternity, however, was at most symbolic. The causal efficacy <strong>of</strong> his letter to<br />
Roosevelt appears to have been minimal. Never<strong>the</strong>less, <strong>the</strong> some<strong>time</strong> pacifist put his<br />
fingerprints on <strong>the</strong> most lethal weapon ever devised by man, <strong>and</strong> he would spend <strong>the</strong> rest<br />
<strong>of</strong> his life preaching against <strong>the</strong> deployment <strong>of</strong> <strong>the</strong> weapon he had urged Roosevelt to<br />
develop.<br />
While <strong>the</strong> cream <strong>of</strong> <strong>the</strong> physicists <strong>and</strong> ma<strong>the</strong>maticians, many <strong>of</strong> <strong>the</strong>m close associates <strong>of</strong><br />
<strong>Einstein</strong> <strong>and</strong> Godel, including Von Neumannó whose brilliance stood out even among <strong>the</strong><br />
cluster <strong>of</strong> luminaries at Los Alamosóhad assembled for <strong>the</strong> bomb project in Oppenheimer's<br />
back yard in New Mexico, Godel <strong>and</strong> <strong>Einstein</strong> remained behind in Princeton, lost in clouds<br />
<strong>of</strong> abstruse ma<strong>the</strong>matics <strong>and</strong> philosophy. Godel had embarked on an intellectual task that
would prove as elusive as <strong>Einstein</strong>'s search for a unified field <strong>the</strong>ory. He took up <strong>the</strong> quest,<br />
inaugurated by Georg Cantor, <strong>of</strong> determining <strong>the</strong> cardinality <strong>of</strong> <strong>the</strong> continuumóin plain<br />
English, counting <strong>the</strong> number <strong>of</strong> points on a line. (It was this that had<br />
preoccupied Godel on his nocturnal sojourns during his summer break at Blue Hill in Maine,<br />
not, as nervous locals thought, assisting German submarines prowling <strong>of</strong>f <strong>the</strong> rugged<br />
coast.) Everyone knew, <strong>of</strong> course, that <strong>the</strong> number <strong>of</strong> points on <strong>the</strong> real number line is<br />
infinite, but after Cantor's epoch-making discovery that infinity itself came in different<br />
sizes, <strong>the</strong> hunt was on to discover <strong>the</strong> exact size <strong>of</strong> this infinity.<br />
Cantor had begun <strong>the</strong> quest in 1878 with his "continuum hypo<strong>the</strong>sis" in which he speculated<br />
that <strong>the</strong> number <strong>of</strong> points on a line, 2x∞, is <strong>the</strong> very next infinite number, X1? after <strong>the</strong><br />
smallest infinite number, X0, <strong>the</strong> cardinality <strong>of</strong> <strong>the</strong> set <strong>of</strong> all natural numbers. He thus<br />
hypo<strong>the</strong>sized that 2X∞ = X1.But he died defeated, having failed to prove his conjecture.<br />
Plagued by bouts <strong>of</strong> depression, he was several <strong>time</strong>s confined to a sanatorium, a sad<br />
precedent repeated by Godel. Godel, for his part, after much effort, was finally able to<br />
prove in <strong>the</strong> late 1930s <strong>and</strong> early 1940s that <strong>the</strong> continuum hypo<strong>the</strong>sis was consistent with<br />
Zermelo-Fraenkel set <strong>the</strong>oryóthat is, that it could not be disproved. Later, in 1963, a brash<br />
young ma<strong>the</strong>matician, Paul Cohen, <strong>of</strong> Stanford University, who had worked on <strong>the</strong> problem<br />
for two years at <strong>the</strong> institute, looking down his nose at <strong>the</strong> petty concerns <strong>of</strong> mere<br />
logicians, succeeded in proving that it could not be proved from <strong>the</strong>se same axioms.<br />
Cohen's result is known as <strong>the</strong> independence <strong>of</strong> <strong>the</strong> continuum hypo<strong>the</strong>sis, Godel's as <strong>the</strong><br />
consistency <strong>of</strong> <strong>the</strong> continuum hypo<strong>the</strong>sis.<br />
<strong>Einstein</strong>, embracing his doomed search for a unified field <strong>the</strong>ory <strong>and</strong> deemed a security<br />
risk for his sympathies for <strong>and</strong> contacts in <strong>the</strong> <strong>world</strong> <strong>of</strong> "socialism," discovered that he was<br />
not to be trusted with <strong>the</strong> defense <strong>of</strong> his adopted l<strong>and</strong>. His opportunity to serve (a lesser<br />
one) came not from <strong>the</strong> desert at Los Alamos but from <strong>the</strong> sea. The U.S. Department <strong>of</strong> <strong>the</strong><br />
Navy engaged him not to design new gyroscopes, as he had once done for his fa<strong>the</strong>rl<strong>and</strong>,<br />
but to calculate <strong>the</strong> explosive potentialities <strong>of</strong> torpedoes. Calculate he did, but though<br />
expensive experiments would later confirm that <strong>Einstein</strong>'s results had indeed been<br />
accurate, <strong>the</strong>re is no evidence that his contributions were put into practice during <strong>the</strong><br />
war.<br />
If <strong>Einstein</strong>'s ma<strong>the</strong>matics did little to advance <strong>the</strong> war effort, he had greater success in<br />
contributing financially. Not, to be sure, by emptying his pockets (which were already<br />
empty, as his wife made certain, distrusting his financial competence). Instead he was<br />
asked to donate <strong>the</strong> original manuscript <strong>of</strong> his 1905 treatise on special relativity for an<br />
auction to produce funds to support <strong>the</strong> war. But <strong>Einstein</strong>, untouched by manuscript<br />
fetishism, had discarded <strong>the</strong> original. Unfazed, <strong>the</strong> authorities asked him simply to rewrite<br />
it. The amazed pr<strong>of</strong>essor was happy to oblige, <strong>and</strong> in due course wrote again his famous<br />
paper, this <strong>time</strong> dictating it to his faithful assistant, Helen Dukas. "Did I really say that?" he<br />
interjected from <strong>time</strong> to <strong>time</strong> to <strong>the</strong> unflappable Ms. Dukas. Reality, here, clearly, has<br />
competed with fiction. It was Jorge Luis Borges who wrote "Pierre Menard, Author <strong>of</strong> <strong>the</strong><br />
Quixote," in which Menard "did not want to compose ano<strong>the</strong>r Quixoteówhich is easyóbut<br />
<strong>the</strong> Quixote itself." Did he succeed? "Cervantes' text <strong>and</strong> Menard's," Borges notes dryly, "are<br />
verbally identical, but <strong>the</strong> second is almost infinitely richer . . . more ambiguous ..."<br />
Whe<strong>the</strong>r or not <strong>Einstein</strong>'s second writing <strong>of</strong> his 1905 paper is more ambiguous than <strong>the</strong>
original, it is certainly richer: an insurance company bought <strong>the</strong> new manuscript for $6.5<br />
million.<br />
It is a sign <strong>of</strong> both her boredom <strong>and</strong> her skill as a seamstress that throughout <strong>the</strong> war years<br />
Adele is said to have contributed to <strong>the</strong> Austrian relief services a dress a day to be given to<br />
a young child, in recognition <strong>of</strong> which she received a bust <strong>of</strong> her late fa<strong>the</strong>r from <strong>the</strong><br />
Viennese authorities after <strong>the</strong> war. Godel is not known to have made any direct<br />
contribution to <strong>the</strong> Allies' war effort. He told his friend Atle Selberg, however, that he<br />
volunteered after <strong>the</strong> war ended to serve as a civil defense aircraft spotter. He also<br />
proceeded to formalize his commitment to <strong>the</strong> United States, becoming a citizen <strong>of</strong> his<br />
adopted country in 1947. As witnesses for <strong>the</strong> ceremony he brought along Morgenstern <strong>and</strong><br />
<strong>Einstein</strong>. He had already alarmed <strong>the</strong> former by confiding to him, in consternation, that he<br />
had discovered an "inconsistency" in <strong>the</strong> Constitution. Apprised by Morgenstern <strong>of</strong> <strong>the</strong><br />
danger ahead, <strong>Einstein</strong> took it upon himself to distract his friend on <strong>the</strong> way to <strong>the</strong><br />
swearing<br />
in, entertaining him with worn-out jokes <strong>and</strong> twice-told anecdotes. <strong>Einstein</strong> might have<br />
been even more concerned if he had known that for years <strong>the</strong> FBI had been intercepting<br />
<strong>and</strong> reading parts <strong>of</strong> Godel's correspondence with his mo<strong>the</strong>r, who was living in Vienna.<br />
The strategy proved unsuccessful. When judge Philip Forman, who only a few years earlier<br />
had ushered <strong>Einstein</strong> himself into <strong>the</strong> l<strong>and</strong> <strong>of</strong> liberty, asked Godel casually, "Do you think a<br />
dictatorship like that in Germany could ever arise in <strong>the</strong> United States?" he received a<br />
spirited reply in <strong>the</strong> affirmative. Godel launched into an account <strong>of</strong> how <strong>the</strong> United States<br />
Constitution formally permitted just such a regime to arise. Shrewdly, however, <strong>the</strong> judge<br />
cut <strong>of</strong>f <strong>the</strong> great logician before he could hit full stride, <strong>and</strong> <strong>the</strong> ceremony came to a<br />
peaceful conclusion, leaving Godel's new homel<strong>and</strong> to fend for itself against <strong>the</strong> opening<br />
he had discerned in its founding principles. Years later, asked for a legal analogy for his<br />
incompleteness <strong>the</strong>orem, he would comment that a country that depended entirely upon<br />
<strong>the</strong> formal letter <strong>of</strong> its laws might well find itself defenseless against a crisis that had not,<br />
<strong>and</strong> could not, have been foreseen in its legal code. The analogue <strong>of</strong> his incompleteness<br />
<strong>the</strong>orem, applied to <strong>the</strong> law, would guarantee that for any legal code, even if intended to<br />
be fully explicit <strong>and</strong> complete, <strong>the</strong>re would always be judgments "undecided" by <strong>the</strong> letter<br />
<strong>of</strong> <strong>the</strong> law.<br />
If Godel made no explicit contribution to <strong>the</strong> war, still his ma<strong>the</strong>matical work, in particular<br />
his foundational papers on recursive functionsówhich constitute <strong>the</strong> soul, if one can put it<br />
thus, <strong>of</strong> <strong>the</strong> computing machineówould contribute in a pr<strong>of</strong>ound way to <strong>the</strong> project <strong>of</strong><br />
building bigger <strong>and</strong> bigger <strong>the</strong>rmonuclear bombs, <strong>and</strong> more generally to <strong>the</strong> still ongoing<br />
program <strong>of</strong> computer-based military technology. Similarly, while <strong>Einstein</strong> took no part in<br />
<strong>the</strong> deliberations <strong>of</strong> Oppenheimer's bomb makers at Los Alamos, his foundational work in<br />
relativity formed part <strong>of</strong> <strong>the</strong> <strong>the</strong>oretical background <strong>of</strong> <strong>the</strong> very practical results reached<br />
in <strong>the</strong> heart <strong>of</strong> <strong>the</strong> New Mexican desert. Godel <strong>and</strong> <strong>the</strong> computer, <strong>Einstein</strong> <strong>and</strong> <strong>the</strong> bomb.<br />
Nei<strong>the</strong>r man contributed to <strong>the</strong> technology (or its ethos), but each one's research was<br />
essential background for those who did.<br />
Not only did <strong>Einstein</strong> <strong>and</strong> Godel refrain from contributing to <strong>the</strong> ethos <strong>of</strong> <strong>the</strong>se<br />
technologies, <strong>the</strong>y were completely against it. Their "unfashionable pursuits" gained <strong>the</strong>m
fame but not friends. <strong>Einstein</strong>, pursuing <strong>the</strong> elusive unified field <strong>the</strong>ory, was a lone figure<br />
unwilling to forgo determinism in physical <strong>the</strong>ory or realism in <strong>the</strong> quantum <strong>world</strong>. Godel<br />
was a rare spirit keeping <strong>the</strong> faith that in spite <strong>of</strong> his incompleteness <strong>the</strong>orem <strong>and</strong> <strong>the</strong><br />
independence result achieved by Paul Cohen, <strong>the</strong> ma<strong>the</strong>matical universe <strong>of</strong> sets <strong>and</strong><br />
numbers was a fully determined, complete reality. Where Cohen led <strong>the</strong> way for <strong>the</strong><br />
majority with his belief that <strong>the</strong>re was no more objective truth value attached to Cantor's<br />
continuum hypo<strong>the</strong>sis than to Euclid's parallel postulate, only a choice <strong>of</strong> which convention<br />
to follow, Godel never ceased believing that <strong>the</strong> comparison with Euclid was misguided.<br />
The true axioms that would settle <strong>the</strong> continuum hypo<strong>the</strong>sis, he felt, were out <strong>the</strong>re to be<br />
found. As he wrote to Alonzo Church, who shared Cohen's beliefs, "You know that I<br />
disagree about <strong>the</strong> philosophical consequences <strong>of</strong> Cohen's result. In particular, I don't think<br />
realists need expect any permanent ramifications as long as <strong>the</strong>y are guided, in <strong>the</strong> choice<br />
<strong>of</strong> <strong>the</strong> axioms, by ma<strong>the</strong>matical intuition <strong>and</strong> by o<strong>the</strong>r criteria <strong>of</strong> rationality."<br />
"<strong>Einstein</strong> <strong>and</strong> Me": Scientists as Philosophers<br />
The increasingly philosophical turn taken by <strong>the</strong> two thinkers was becoming more <strong>and</strong> more<br />
unfashionable. In 1935, <strong>Einstein</strong>, with <strong>the</strong> assistance <strong>of</strong> Boris Podolsky <strong>and</strong> Nathan Rosen,<br />
published a kind <strong>of</strong> philosophical manifesto (which contained what is known popularly as<br />
<strong>the</strong> EPR paradox), "Can Quantum-Mechanical Description <strong>of</strong> Physical Reality Be Considered<br />
Complete?" The authors suggested that <strong>the</strong> Copenhagen interpretation <strong>of</strong> quantum<br />
mechanics, pioneered by Bohr <strong>and</strong> Heisenberg, led to paradoxical results, including<br />
instantaneous, noncausal action between spatially separated events (which <strong>the</strong>y dubbed<br />
"spooky action at a distance"). Not only did this little essay<br />
arouse <strong>the</strong> ire <strong>of</strong> <strong>the</strong> quantum-mechanical establishment, including Bohr <strong>and</strong> Wolfgang<br />
Pauli, but on its own terms <strong>the</strong> document seemed as much philosophy as physics. <strong>Einstein</strong><br />
brought to bear what seemed to be a priori philosophical considerations concerning <strong>the</strong><br />
"completeness" <strong>of</strong> a physical <strong>the</strong>oryówhich, <strong>the</strong> reader was informed, requires <strong>the</strong><br />
correspondence <strong>of</strong> each significant element <strong>of</strong> <strong>the</strong> <strong>the</strong>ory with an "element <strong>of</strong> reality"óas<br />
well as what constitutes physical reality, "no reasonable definition <strong>of</strong> which," <strong>Einstein</strong><br />
insisted, permits what is real in one system to depend on <strong>the</strong> measurement <strong>of</strong> ano<strong>the</strong>r<br />
system.<br />
<strong>Einstein</strong>'s philosophical perspective was a form <strong>of</strong> realism. Godel too was committed to<br />
realism, in <strong>the</strong> physical as well as <strong>the</strong> ma<strong>the</strong>matical realm. He believed that ma<strong>the</strong>matical<br />
objects <strong>and</strong> properties exist objectively <strong>and</strong> independently <strong>of</strong> knowledge <strong>of</strong> <strong>the</strong>m by <strong>the</strong><br />
human mind. He was aware <strong>of</strong> <strong>the</strong> parallel between his <strong>and</strong> <strong>Einstein</strong>'s philosophies. "The<br />
heuristics <strong>of</strong> <strong>Einstein</strong> <strong>and</strong> Bohr," he told Hao Wang, "are stated in <strong>the</strong>ir correspondence.<br />
Cantor might also be classified with <strong>Einstein</strong> <strong>and</strong> me. Heisenberg <strong>and</strong> Bohr are on <strong>the</strong> o<strong>the</strong>r<br />
side." His own philosophical manifesto appeared in 1947, disguised as a popular survey <strong>of</strong><br />
<strong>the</strong> status <strong>of</strong> <strong>the</strong> continuum hypo<strong>the</strong>sis, entitled "What is Cantor's Continuum Problem?" He<br />
had been invited in I 946 by Lester Ford, editor <strong>of</strong> <strong>the</strong> American Ma<strong>the</strong>matical Monthly, to<br />
make a contribution to a "What is . . . ?" series whose purpose was to provide an<br />
introduction to "a small aspect <strong>of</strong> higher ma<strong>the</strong>matics" in "as simple, elementary <strong>and</strong><br />
popular a way" as possible. Godel, however, took this as an opportunity to formulate a<br />
manifesto declaring <strong>and</strong> defending his ma<strong>the</strong>matical Platonism. As an epistemological<br />
adjunct, he introduced <strong>the</strong> concept <strong>of</strong> ma<strong>the</strong>matical intuition, for which formal pro<strong>of</strong> was<br />
no substitute. "I don't see any reason," he wrote, "why we should have less confidence in
this kind <strong>of</strong> perception, i.e., in ma<strong>the</strong>matical intuition, than in sense perception, which<br />
induces us to build up physical <strong>the</strong>ories." According to Godel, since <strong>the</strong> continuum is a real<br />
object, it was only a matter <strong>of</strong> <strong>time</strong> before new axioms would be discovered that would<br />
settle <strong>the</strong> continuum hypo<strong>the</strong>sis, axioms that would "force <strong>the</strong>mselves upon us as being<br />
true."<br />
The manifesto, though shocking, was no surprise. The stage had been set by a previous<br />
essay, his contribution to <strong>the</strong> volume on Bertr<strong>and</strong> Russell for <strong>the</strong> "Library <strong>of</strong> Living<br />
Philosophers," edited by P.A. Schilpp. This series, inaugurated in 1938, was devoted to<br />
great living philosophers at <strong>the</strong> twilight <strong>of</strong> <strong>the</strong>ir careers. The person to whom a volume was<br />
dedicated would contribute an intellectual autobiography, followed by critical assessments<br />
<strong>of</strong> his philosophy provided by leading figures in <strong>the</strong> field, to be succeeded by <strong>the</strong> author's<br />
responses to those essays. In <strong>the</strong> volume on Russell, Godel cited with approval Russell's<br />
remark (written before his encounter with Wittgenstein) that "logic is concerned with <strong>the</strong><br />
real <strong>world</strong> just as truly as zoology, though with its more abstract <strong>and</strong> general features."<br />
This expression <strong>of</strong> logical realism Russell himself later rejected under Wittgenstein's<br />
influence, a rejection for which Godel took him to task. Much <strong>of</strong> Godel's contribution to<br />
<strong>the</strong> Schilpp volume consisted <strong>of</strong> a sustained critique <strong>of</strong> Russell's ma<strong>the</strong>matical philosophy,<br />
<strong>and</strong> he looked forward to Russell's response.<br />
He had reason to look forward to it. In his letter <strong>of</strong> November 1942 inviting Godel to<br />
contribute to <strong>the</strong> volume, Schilpp had said that "in talking <strong>the</strong> matter over last night with<br />
Lord Russell in person, 1 learned that he too would not only very greatly appreciate your<br />
participation in this project, but that he considers you <strong>the</strong> scholar par excellence in this<br />
field." In <strong>the</strong> event, however, Godel was late in submitting his essay, <strong>and</strong> Russell claimed<br />
that having already responded to <strong>the</strong> o<strong>the</strong>r contributors, he "lacked <strong>the</strong> leisure" to<br />
compose a proper reply to Godel. Deeply disappointed, Godel attempted unsuccessfully to<br />
change Russell's mind. The only response that appeared in <strong>the</strong> Schilpp volume was Russell's<br />
comment that since it was "eighteen years since [I had] last worked on ma<strong>the</strong>matical logic"<br />
it would have taken him "a long <strong>time</strong> to form a critical estimate <strong>of</strong> Dr. Godel's opinions."<br />
Behind this remark lurked an ambivalence on Russell's part toward his great colleague. The<br />
incompleteness <strong>the</strong>orems were not calculated to arouse warm feelings in Russell's breast.<br />
As we saw earlier, it is likely that Russell did not fully grasp <strong>the</strong>m. He also claimed that, in<br />
any case, he hail never subscribed to <strong>the</strong> dream <strong>of</strong> <strong>the</strong> Hilbert school<br />
that a consistency pro<strong>of</strong> could be found for his logical system. This would take some <strong>of</strong> <strong>the</strong><br />
sting out <strong>of</strong> Godel's discoveries, but it would apply only to <strong>the</strong> second incompleteness<br />
<strong>the</strong>orem. The first <strong>the</strong>orem, which nei<strong>the</strong>r Russell nor anyone else had expected,<br />
demonstrated <strong>the</strong> essential incompleteness <strong>of</strong> any formal logical system in capturing <strong>the</strong><br />
truths <strong>of</strong> ma<strong>the</strong>matics. Its effect, <strong>the</strong>refore, was to deal a mortal blow to <strong>the</strong> formalist<br />
project <strong>of</strong> which Russell's masterpiece, Principia Ma<strong>the</strong>matica, was <strong>the</strong> crowning<br />
achievement. Godel had added insult to injury by deciding, in <strong>the</strong> pro<strong>of</strong> <strong>of</strong> his <strong>the</strong>orem, to<br />
focus exclusively on Principia Ma<strong>the</strong>matica as an example <strong>of</strong> a formal system that is<br />
essentially incomplete. Not only had Russell been surpassed by Godel as <strong>the</strong> preeminent<br />
logician <strong>of</strong> <strong>the</strong>ir <strong>time</strong>, his magnum opus had been shown by Godel to be in an important<br />
respect a failure.
If Russell had been eclipsed in logic by Godel, he was overshadowed in philosophy by his<br />
former student Ludwig Wittgenstein. "It is not an altoge<strong>the</strong>r pleasant experience," wrote<br />
Russell late in life, "to find oneself regarded as antiquated after having been, for a <strong>time</strong>, in<br />
<strong>the</strong> fashion. It is difficult to accept this experience gracefully." He attributed this change<br />
<strong>of</strong> fashion explicitly to <strong>the</strong> rise <strong>of</strong> Wittgenstein, "by whom I was superseded in <strong>the</strong> opinion<br />
<strong>of</strong> many British philosophers." Not only <strong>the</strong> content but <strong>the</strong> very form <strong>of</strong> Russell's writings<br />
was rendered obsolete by Wittgenstein's works, which called to mind <strong>the</strong> Zeus-like<br />
pronouncements <strong>of</strong> pre-Socratic philosophers such as Par-menides <strong>and</strong> Heraclitus more<br />
than <strong>the</strong> didactic, quasi-scientific prose that Russell had made his trademark. And<br />
Wittgenstein was not <strong>the</strong> end <strong>of</strong> it. The final grain <strong>of</strong> salt in Russell's wounds came from<br />
<strong>Einstein</strong>, whose revolution in physics eclipsed in <strong>the</strong> popular imagination not only Russell's<br />
contributions to logic <strong>and</strong> philosophy but Godel's <strong>and</strong> Wittgenstein's as well. If he had it to<br />
do over again, said Russell wistfully at <strong>the</strong> end <strong>of</strong> his life, he would have become a<br />
physicist.<br />
Triply eclipsed, Russell came to <strong>the</strong> Institute for Advanced Study in <strong>the</strong> spring <strong>of</strong> 1943,<br />
while Godel was composing his critique <strong>of</strong> Russell's philosophy <strong>of</strong> ma<strong>the</strong>matics, with a large<br />
chip on his shoulder. He noticed Godel in <strong>the</strong> audience when he lectured but was so out <strong>of</strong><br />
touch<br />
with recent developments in logic <strong>and</strong> ma<strong>the</strong>matics that he failed to recognize Von<br />
Neumann. When he met for discussions at <strong>Einstein</strong>'s home with Godel, <strong>Einstein</strong> <strong>and</strong> Pauli, it<br />
was predictable that Russell would be unsatisfied, as revealed by <strong>the</strong> barbed comments he<br />
committed to his Autobiography about "<strong>the</strong> German bias for metaphysics" <strong>of</strong> his<br />
companions, "all three <strong>of</strong> <strong>the</strong>m Jews." That his application, several years earlier, for<br />
membership at <strong>the</strong> Institute for Advanced Study, though supported by <strong>Einstein</strong>, had been<br />
declined was <strong>the</strong> final insult. On <strong>the</strong> o<strong>the</strong>r side, Russell's dismissal <strong>of</strong> Godel as an<br />
"unadulterated Platonist" did not earn him a warm spot in Godel's heart.<br />
That <strong>the</strong> only pr<strong>of</strong>essional philosopher in <strong>the</strong> quartet <strong>of</strong> Russell, <strong>Einstein</strong>, Godel <strong>and</strong> Pauli<br />
was disappointed with <strong>the</strong> o<strong>the</strong>rs' attachment to <strong>the</strong> gr<strong>and</strong> old style <strong>of</strong> "German<br />
metaphysics" is a clear sign <strong>of</strong> <strong>the</strong> low estate to which philosophy had been reduced. As <strong>the</strong><br />
last <strong>of</strong> <strong>the</strong> great philosopher-scientists, <strong>Einstein</strong> <strong>and</strong> Godel were anomalies. But if <strong>the</strong><br />
physicist <strong>and</strong> <strong>the</strong> logician had both become increasingly philosophical since joining <strong>the</strong><br />
institute, <strong>the</strong>re remained a difference. For <strong>the</strong> most part, <strong>Einstein</strong>'s philosophy was<br />
immanent in his physics, just as <strong>the</strong> God <strong>of</strong> his beloved Spinoza, <strong>the</strong> great pan<strong>the</strong>ist, was<br />
contained in <strong>the</strong> <strong>world</strong>. Much <strong>of</strong> Godel's philosophy, too, was contained in his ma<strong>the</strong>matics,<br />
but he took pains to make some <strong>of</strong> it explicitóseparate, though closely related to his<br />
formal discoveriesójust as <strong>the</strong> God <strong>of</strong> his philosophical hero, Leibniz, was a being apart.<br />
Godel's desire was to become a great philosopher in <strong>the</strong> tradition <strong>of</strong> Plato, Leibniz, <strong>and</strong><br />
Kant, but he discovered that he had set this goal too late in his life, having devoted his<br />
best years to logic, ma<strong>the</strong>matics <strong>and</strong> physics. The scope <strong>of</strong> his ambitionsó<strong>and</strong> <strong>the</strong> degree<br />
to which he was at cross purposes with <strong>the</strong> Zeitgeistócan be seen in <strong>the</strong> fourteen<br />
philosophical <strong>the</strong>ses he committed to his notebooks in <strong>the</strong> 1960s under <strong>the</strong> title "My<br />
Philosophical Viewpoint." "Concepts," he states, "have an objective existence." Along <strong>the</strong><br />
same lines he writes that "materialism is false." Nei<strong>the</strong>r <strong>the</strong>sis is surprising or unexpected.<br />
But he goes fur<strong>the</strong>r: "The <strong>world</strong> is rational." This puts one in mind <strong>of</strong> philosophical <strong>the</strong>ism,<br />
according to which <strong>the</strong> order <strong>of</strong> <strong>the</strong> <strong>world</strong> reflects <strong>the</strong> order
<strong>of</strong> <strong>the</strong> supreme mind governing it. Plato, a philosopher Godel greatly admired, held<br />
similarly that all order is a reflection <strong>of</strong> rationality. Concerning religion Godel asserts,<br />
"Religions are, for <strong>the</strong> most part, bad, but religion [i.e., belief in God?] is not."<br />
He goes on: "There are o<strong>the</strong>r <strong>world</strong>s <strong>and</strong> rational beings <strong>of</strong> a different <strong>and</strong> higher kind....<br />
The[se] higher beings are connected to <strong>the</strong> o<strong>the</strong>rs by analogy, not by composition." And so<br />
on. Godel was striving for a new picture, a <strong>world</strong>view that would put <strong>the</strong> <strong>world</strong> into a<br />
better perspective than it is at present. From his discussions late in life with Hao Wang, it<br />
emerges that he believed that <strong>the</strong> proper philosophy should capture axiomaticallyóthough<br />
not purely formallyó<strong>the</strong> fundamental concepts that underlie reality, which he took to<br />
include "reason, cause, substance, accidens [a traditional Latin term], necessity, value,<br />
God, cognition, force, <strong>time</strong>, form, content, matter, life, truth, idea, reality, possibility."<br />
Godel <strong>the</strong> philosopher, according to Wang, said he wished to "do for metaphysics as much<br />
as Newton did for physics."<br />
<strong>Einstein</strong>, <strong>of</strong> course, had already done for physics what Newton did for physics. But if his<br />
philosophical aspirations did not match Godel's, <strong>the</strong> direction <strong>of</strong> his thought <strong>of</strong>ten did. Like<br />
Godel, he was opposed to positivism, but had, when necessary, exploited its weapons to<br />
fur<strong>the</strong>r his own scientific <strong>and</strong> philosophical objectives. The protopositivism <strong>of</strong> <strong>Einstein</strong>'s<br />
special <strong>the</strong>ory <strong>of</strong> relativity, which identified <strong>time</strong> with what could be measured by<br />
synchronized clocks, was not a philosophical calling cardóthough <strong>the</strong> Vienna Circle seized<br />
on it as suchóbut simply <strong>the</strong> right tool for <strong>the</strong> job. Minkowski's generalization <strong>of</strong> special<br />
relativity into <strong>the</strong> geometrical <strong>the</strong>ory <strong>of</strong> four-dimensional space-<strong>time</strong> was a different tool<br />
for a different job, which in turn opened <strong>the</strong> door for <strong>Einstein</strong>'s broader geometrization <strong>of</strong><br />
space-<strong>time</strong> in his general <strong>the</strong>ory <strong>of</strong> relativity, a development that is cold comfort to<br />
positivists hoping to adopt <strong>Einstein</strong> as a true believer. In like manner, with his<br />
incompleteness <strong>the</strong>orem, Godel exploited <strong>the</strong> favorite tool <strong>of</strong> <strong>the</strong> ma<strong>the</strong>matical positivist,<br />
formal systems <strong>of</strong> pro<strong>of</strong>, to construct a pro<strong>of</strong> that formal systems for number <strong>the</strong>ory will<br />
always be incomplete. In essence, <strong>the</strong> <strong>the</strong>orem was a mechanical, algorithmic<br />
demonstration <strong>of</strong> <strong>the</strong> limits <strong>of</strong><br />
mechanical, algorithmic methods, <strong>and</strong> as it turned out, <strong>of</strong> <strong>the</strong> inescapable limitations <strong>of</strong><br />
<strong>the</strong> computer. It was at once a jewel in <strong>the</strong> crown <strong>of</strong> formalism <strong>and</strong> a warning to those<br />
who would embrace it.<br />
At <strong>the</strong> heart <strong>of</strong> Godel <strong>and</strong> <strong>Einstein</strong>'s opposition to positivism was <strong>the</strong>ir unfashionable<br />
realism, <strong>the</strong>ir reluctance to make ontology, <strong>the</strong> <strong>the</strong>ory <strong>of</strong> what is, subservient to<br />
epistemology, <strong>the</strong> <strong>the</strong>ory <strong>of</strong> what can be known. At bottom, <strong>the</strong> positivist mentality<br />
consists in deriving ontology from epistemology. This was <strong>the</strong> source <strong>of</strong> Ernst Mach's positivistic<br />
objection to atomic <strong>the</strong>ory, since individual atoms will never be directly<br />
encountered by humans. But <strong>the</strong> springs <strong>of</strong> Mach's philosophy ran deeper. His rejection <strong>of</strong> a<br />
reality "beyond" what appears to human sensibility was a simplified version <strong>of</strong> Kant's<br />
philosophy. The "Copernican Revolution" in epistemology inaugurated by Kant consisted in<br />
<strong>the</strong> radical doctrine that <strong>the</strong> known must conform to <strong>the</strong> knower. The hard-nosed,<br />
ultraempiricist Mach had derived his positivism from Kant, who was not a realist but an<br />
idealist, albeit <strong>of</strong> <strong>the</strong> deep, German, transcendental variety, not (as Kant saw it) <strong>of</strong> <strong>the</strong><br />
shallow British strain <strong>of</strong> George Berkeley.
Still, idealism is idealism, whe<strong>the</strong>r British or German. Although Kant, unlike Mach,<br />
recognized <strong>the</strong> existence <strong>of</strong> a reality beyond what appears to us, he made it clear that <strong>the</strong><br />
objects <strong>of</strong> science are not <strong>the</strong> "things in <strong>the</strong>mselves" that lie behind "<strong>the</strong> appearances," but<br />
ra<strong>the</strong>r <strong>the</strong> appearances <strong>the</strong>mselves. This was a doctrine rejected by both <strong>Einstein</strong> <strong>and</strong><br />
Godel. Godel made his objections explicit: Whereas it was a "fruitful viewpoint [to make] a<br />
distinction between subjective <strong>and</strong> objective elements in our knowledge (which is so<br />
impressively suggested by Kant's comparison with <strong>the</strong> Copernican system), [when such a<br />
doctrine] appears in <strong>the</strong> history <strong>of</strong> science, <strong>the</strong>re is at once a tendency to exaggerate it<br />
into a boundless subjectivism. . . . Kant's doctrine <strong>of</strong> <strong>the</strong> unknowability <strong>of</strong> <strong>the</strong> things in<br />
<strong>the</strong>mselves is one example. ..."<br />
Godel, however, was not through with Kant. In an essay written in 1961 but never<br />
published, he noted that it was "a general feature <strong>of</strong> Kant's assertions that literally<br />
understood <strong>the</strong>y are false, but in a broader sense contain deeper truths." He had in mind<br />
Kant's doctrine<br />
that in proving geometrical <strong>the</strong>orems we always need new geometrical intuitions. This,<br />
Godel pointed out, is provably false. But if we substitute "ma<strong>the</strong>matical" for "geometrical,"<br />
<strong>the</strong> result is a truth that follows directly from Godel's incompleteness <strong>the</strong>orem. What was<br />
needed, <strong>the</strong>n, for <strong>the</strong> continual development <strong>of</strong> ma<strong>the</strong>matics (<strong>and</strong>, one might add,<br />
philosophy), was "a procedure that should produce in us a new state <strong>of</strong> consciousness in<br />
which we describe in detail <strong>the</strong> basic concepts we use in our thought, or grasp o<strong>the</strong>r basic<br />
concepts hi<strong>the</strong>rto unknown to us." This he claimed to have found in <strong>the</strong> later<br />
"transcendental phenomenology" <strong>of</strong> Edmund Husserl. "Transcendental phenomenology," he<br />
wrote in a draft <strong>of</strong> a letter to <strong>the</strong> ma<strong>the</strong>matician-philosopher Gian-Carlo Rota, "carried<br />
through, would be nothing more nor less than Kant's critique <strong>of</strong> pure reason transformed<br />
into an exact science," which "far from destroying traditional metaphysics . . . would<br />
ra<strong>the</strong>r prove a solid foundation for it." In Husserl, Godel thought he had found a form <strong>of</strong><br />
idealism that, though derived from Kant's, was not incompatible with realism. That Husserl<br />
shared Godel's disdain for unreconstructed Kantianism is apparent from a remark he made<br />
in 1915: "German idealism has always made me want to throw up."<br />
<strong>Einstein</strong>'s objections, in turn, to <strong>the</strong> new quantum mechanicsóin particular his formulation<br />
<strong>of</strong> <strong>the</strong> EPR paradoxóreflected a rejection <strong>of</strong> <strong>the</strong> Kantian turn in epistemology in its<br />
simplified reconstruction by pos-itivists like Mach. The uncertainty principle, after all, is<br />
an example <strong>of</strong> <strong>the</strong> same tendency to draw ontological conclusions from epistemologi-cal<br />
premises, in this instance, from our inability in principle to know simultaneously <strong>the</strong><br />
position <strong>and</strong> velocity <strong>of</strong> a subatomic particle, to <strong>the</strong> nonexistence <strong>of</strong> such a combined<br />
state. Not only did <strong>Einstein</strong> reject this reasoning, he resisted what he took to be<br />
Heisenberg's more fundamental belief that we should ab<strong>and</strong>on <strong>the</strong> very idea <strong>of</strong> "quantum<br />
reality." For <strong>Einstein</strong>, as for Godel, philosophy <strong>without</strong> ontology was an illusion, <strong>and</strong><br />
physics <strong>without</strong> philosophy reduced to engineering. (And for <strong>Einstein</strong>, engineering was a<br />
poor substitute for physics. When his eldest son, Hans, decided on an engineering career,<br />
his fa<strong>the</strong>r wrote that he was pleased that Hans had found a subject to concentrate on but<br />
also that
"what he is interested in isn't really important, even if it is, alas, engineering. One cannot<br />
expect one's children to inherit a mind.")<br />
An Offer That Couldn't Be Refused<br />
Before Princeton, <strong>Einstein</strong>'s <strong>and</strong> Godel's philosophical sen<strong>time</strong>nts had proceeded on parallel<br />
but independent lines. At <strong>the</strong> institute, however, <strong>the</strong> lines began to converge. In thought<br />
as in life, <strong>Einstein</strong> found himself increasingly entwined with Godel. In 1949 Godel received<br />
from his friend an unexpected housewarming gift, "a wonderful flower vase." At <strong>the</strong> <strong>time</strong><br />
<strong>Einstein</strong> was celebrating his seventieth birthday, <strong>and</strong> "after long searches" Godel finally<br />
settled on a birthday gift for his friend, an etching. (Adele had knit <strong>Einstein</strong> a sweater but<br />
decided against sending it.)<br />
There was ano<strong>the</strong>r gift as well. Godel had been invited by P.A. Schilpp to contribute to a<br />
new volume in his series, "The Library <strong>of</strong> Living Philosophers," a volume in honor <strong>of</strong><br />
<strong>Einstein</strong>'s seventieth birthday, to be entitled Albert <strong>Einstein</strong>: Philosopher-Scientist. It is <strong>the</strong><br />
only volume in <strong>the</strong> series to be devoted to a scientist. No doubt aware <strong>of</strong> Godel's friendship<br />
with <strong>Einstein</strong>, Schilpp must have assumed his invitation was an <strong>of</strong>fer Godel couldn't refuse.<br />
He didn't. Godel's essay would be his second exercise in this venue, following <strong>the</strong> 1944<br />
essay on Russell. (In <strong>time</strong> he would draft a third essay, this <strong>time</strong> on Rudolph Carnap,<br />
though he never submitted a final version, <strong>and</strong> he declined a fourth invitation to write on<br />
Popper.)<br />
Godel wasted no <strong>time</strong> setting to work. Schilpp suggested a title, "The Realistic St<strong>and</strong>point<br />
in Physics <strong>and</strong> Ma<strong>the</strong>matics," but Godel rejected it. He had in mind an ontological<br />
investigationó<strong>the</strong> gr<strong>and</strong> philosophical quest for <strong>the</strong> reality <strong>of</strong> <strong>time</strong>óreinterpreted as an<br />
examination <strong>of</strong> what relativity <strong>the</strong>ory has to teach us about this question, which has<br />
exercised <strong>the</strong> philosophical imagination from Parmenides <strong>and</strong> Plato to Kant. Godel<br />
informed Schilpp that he would submit a brief essay, <strong>of</strong> three to five pages, on <strong>the</strong> topic,<br />
"The Theory <strong>of</strong> Relativity <strong>and</strong> Kant." It is not known whe<strong>the</strong>r Schilpp appreciated <strong>the</strong> per-<br />
versify <strong>of</strong> Godel's proposal, in which he would put forward <strong>the</strong> <strong>the</strong>sis that Kant had<br />
anticipated <strong>Einstein</strong>. For <strong>Einstein</strong> was (<strong>and</strong> is) widely viewed not as having confirmed Kant<br />
but ra<strong>the</strong>r as refuting him. Kant dedicated a large part <strong>of</strong> his Critique <strong>of</strong> Pure Reason to<br />
<strong>the</strong> attempt to establish a priori that Newton was <strong>the</strong> final truth about physics <strong>and</strong> Euclid<br />
<strong>the</strong> last word on geometry, whereas <strong>Einstein</strong> demonstrated empirically that both Euclid<br />
<strong>and</strong> Newton were wrong. (Kant also claimed that logic would never take a step beyond<br />
Aristotle, a view made nonsense by Frege <strong>and</strong> Godel. The great philosopher, it seems, got<br />
it completely wrong about Newton, Euclid <strong>and</strong> Aristotle. No one's perfect.)<br />
What on earth was Godel thinking? One thing is clear. It was not simply <strong>the</strong> long walks with<br />
<strong>Einstein</strong> that had aroused Godel's interest in <strong>the</strong> problem <strong>of</strong> <strong>time</strong>. On his first visit to <strong>the</strong><br />
institute, he had been delighted to attend a seminar on quantum mechanics given by his<br />
friend Von Neumann. He had begun his studies at <strong>the</strong> University <strong>of</strong> Vienna in physics <strong>and</strong><br />
had maintained an active interest ever since. ("Active interest" in Godel's case means a<br />
level <strong>of</strong> competence that for any normal person would constitute a career.) Since <strong>the</strong>
question <strong>of</strong> <strong>time</strong> is at <strong>the</strong> center <strong>of</strong> special relativity, he could not have failed to attend to<br />
it. The primary source <strong>of</strong> his interest in <strong>time</strong>, however, was his preoccupation with<br />
"idealistic" philosophers, from Parmenides <strong>and</strong> Plato to Leibniz <strong>and</strong> Kant. Each <strong>of</strong> <strong>the</strong>se<br />
thinkers, in his own way, questioned <strong>the</strong> ultimate reality <strong>of</strong> <strong>time</strong>. For Plato, things in <strong>time</strong><br />
never really are but are always coming-to-be. Time itself, he said in his cosmological<br />
dialogue <strong>the</strong> Timaeus, is but a "moving image <strong>of</strong> eternity." Godel was deeply sympa<strong>the</strong>tic<br />
with Plato's philosophy, but his true hero was Leibniz. Why <strong>the</strong>n did he choose to discuss,<br />
for <strong>the</strong> Schilpp volume, <strong>the</strong> relationship <strong>of</strong> relativity <strong>the</strong>ory not to Leibniz but to Kant, for<br />
whom he had mixed feelings?<br />
A partial answer is that in composing an essay on <strong>Einstein</strong>, Godel was operating from<br />
within <strong>the</strong> modern scientific framework inaugurated by Newton. It is true that Godel<br />
believed that physics in <strong>the</strong> modern era should have followed Leibniz, a philosopher in <strong>the</strong><br />
tradition <strong>of</strong> German idealism, ra<strong>the</strong>r than his rival, Newton, but <strong>the</strong> fact remained that<br />
<strong>Einstein</strong>'s framework was Newtonian. Consistent, <strong>the</strong>n, with his<br />
methodology <strong>of</strong> undermining from within, Godel chose to demonstrate certain (radical)<br />
formal <strong>and</strong> philosophical consequences from within <strong>the</strong> "host framework," modern physics<br />
from Newton to <strong>Einstein</strong>. As with his incompleteness results, he believed that this strategy<br />
made his formal derivations inescapable <strong>and</strong> his philosophical conclusions hard to resist. A<br />
consequence <strong>of</strong> Godel's methodology is that <strong>the</strong> question <strong>of</strong> "realism" in his <strong>Einstein</strong> essay<br />
becomes delicate. In that essay, he defends <strong>the</strong> reality <strong>of</strong> relativistic space-<strong>time</strong>, much as<br />
he promoted, in <strong>the</strong> philosophical implications he drew from his incompleteness <strong>the</strong>orem,<br />
<strong>the</strong> reality <strong>of</strong> <strong>the</strong> natural numbers, <strong>and</strong>, in his consistency pro<strong>of</strong> for <strong>the</strong> continuum<br />
hypo<strong>the</strong>sis relative to set <strong>the</strong>ory, <strong>the</strong> reality <strong>of</strong> sets. But whereas he remained convinced<br />
<strong>of</strong> <strong>the</strong> fundamental reality <strong>of</strong> numbers from within his own philosophical system, not just<br />
that <strong>of</strong> its "host" framework, his preference for Leibniz leaves open <strong>the</strong> question <strong>of</strong><br />
whe<strong>the</strong>r, from within his own (never fully realized) philosophy <strong>of</strong> space <strong>and</strong> <strong>time</strong>, he<br />
believed that <strong>the</strong> physical <strong>world</strong> as such was founded on something yet more fundamental,<br />
like <strong>the</strong> "monads" introduced by Leibniz. In his discussion <strong>of</strong> <strong>Einstein</strong>, however, he is<br />
operating within <strong>the</strong> philosophical framework <strong>of</strong> relativity <strong>the</strong>ory, <strong>and</strong> so he adopts <strong>the</strong><br />
familiar st<strong>and</strong>point that space-<strong>time</strong> is part <strong>of</strong> <strong>the</strong> ultimate furniture <strong>of</strong> reality. The<br />
question he poses, <strong>the</strong>n, is precisely this: if we believe in <strong>the</strong> ultimate reality <strong>of</strong><br />
relativistic space-<strong>time</strong>, are we forced to be idealists about <strong>time</strong>?<br />
Ontology <strong>and</strong> Epistemology: The Two Axes <strong>of</strong> Philosophy<br />
. . . Almost all accounts <strong>of</strong> <strong>the</strong> concept <strong>of</strong> ma<strong>the</strong>matical truth can be identified with<br />
serving one or ano<strong>the</strong>r <strong>of</strong> <strong>the</strong>se masters [semantics <strong>and</strong> epistemology] at <strong>the</strong> expense <strong>of</strong><br />
<strong>the</strong> o<strong>the</strong>r.<br />
PAUL BENACERRAF<br />
The importance Godel attached to his investigation <strong>of</strong> this question in his contribution to<br />
<strong>the</strong> <strong>Einstein</strong> volume cannot be overstated. A month
after beginning his work in earnest, he wrote his mo<strong>the</strong>r that he was so preoccupied with<br />
it, he would have to give up writing letters until it was completed. Time, he told Wang<br />
years later, remains, even after <strong>Einstein</strong>, <strong>the</strong> philosophical question. Contrast this with<br />
what <strong>the</strong> influential ("analytical") philosopher <strong>of</strong> science Hilary Putnam has written: "I do<br />
not believe that <strong>the</strong>re are any longer any philosophical problems about Time; <strong>the</strong>re is only<br />
<strong>the</strong> physical problem <strong>of</strong> determining <strong>the</strong> exact physical geometry <strong>of</strong> <strong>the</strong> four dimensional<br />
continuum we inhabit." And Putnam is hardly alone. For Godel, however, <strong>time</strong> is "that<br />
mysterious <strong>and</strong> seemingly self-contradictory being which, on <strong>the</strong> o<strong>the</strong>r h<strong>and</strong>, seems to<br />
form <strong>the</strong> basis <strong>of</strong> <strong>the</strong> <strong>world</strong>'s <strong>and</strong> our own existence." Plato could not have put it better.<br />
Godel's preoccupation with <strong>the</strong> question <strong>of</strong> <strong>the</strong> reality <strong>of</strong> <strong>time</strong> went against <strong>the</strong> grain <strong>of</strong> <strong>the</strong><br />
philosophical tradition that dominated (<strong>and</strong> still dominates) his adopted country. His<br />
project more closely resembled that <strong>of</strong> "continental" philosophers like Martin Heidegger,<br />
whose magnum opus gives away <strong>the</strong> central problem in its title, Being <strong>and</strong> Time, <strong>and</strong>, more<br />
closely still, On <strong>the</strong> Phenomenology <strong>of</strong> <strong>the</strong> Consciousness <strong>of</strong> Internal Time, by Edmund<br />
Husserl, a work to which Godel would in <strong>the</strong> course <strong>of</strong> <strong>time</strong> devote a great deal <strong>of</strong><br />
attention, with special reference to Husserl's distinction between physical <strong>time</strong> <strong>and</strong><br />
"internal <strong>time</strong>-consciousness." Unlike Heidegger, however, as well as o<strong>the</strong>r philosophers <strong>of</strong><br />
<strong>time</strong> such as Henri Bergson <strong>and</strong> J.M.E. McTaggart, who, like Godel, took <strong>the</strong> question <strong>of</strong><br />
<strong>the</strong> ontological implications <strong>of</strong> <strong>the</strong> reality <strong>of</strong> <strong>time</strong> seriously, Godel had no interest in yet<br />
ano<strong>the</strong>r inconclusive metaphysical debate in which <strong>the</strong> elusiveness <strong>of</strong> <strong>the</strong> subject would be<br />
matched only by <strong>the</strong> density <strong>of</strong> <strong>the</strong> surrounding prose. He chose instead to examine <strong>the</strong><br />
traditional philosophical question from within an untraditional ma<strong>the</strong>matical context,<br />
<strong>Einstein</strong>'s <strong>the</strong>ory <strong>of</strong> relativity. The idea, as with his incompleteness <strong>the</strong>orem, was to<br />
establish formal results that would have deep philosophical implications. Godel would be a<br />
pathfinder, <strong>the</strong>n, along two directions: a ma<strong>the</strong>matical approach to <strong>the</strong> philosophy <strong>of</strong><br />
<strong>time</strong>, <strong>and</strong> a philosophical assessment <strong>of</strong> <strong>the</strong> ma<strong>the</strong>matics <strong>of</strong> relativity. <strong>Einstein</strong> himself had<br />
been reluctant to engage in <strong>the</strong> latter. The few<br />
remarks he did allow himself verged on inconsistency. He agreed with <strong>the</strong> philosopher<br />
Emile Meyerson that <strong>the</strong> temporal component <strong>of</strong> space-<strong>time</strong> was not a mere fourth<br />
"spatial" dimension, but at o<strong>the</strong>r <strong>time</strong>s he spoke <strong>of</strong> <strong>the</strong> "rigid four-dimensional space <strong>of</strong> <strong>the</strong><br />
special <strong>the</strong>ory <strong>of</strong> relativity." When his friend Michele Besso died, he wrote to his widow<br />
that "now he has preceded me a little by parting from this strange <strong>world</strong>. This means<br />
nothing. To us believing physicists <strong>the</strong> distinction between past, present, <strong>and</strong> future [i.e.,<br />
between what is now <strong>and</strong> what is not now] has only <strong>the</strong> significance <strong>of</strong> a stubborn illusion."<br />
Yet Carnap wrote that <strong>Einstein</strong> had told him that "<strong>the</strong> now means something special for<br />
man, something which physics cannot speak to." His physics was already deeply<br />
philosophical. He had no wish to take <strong>time</strong> away from scientific research to match <strong>the</strong><br />
philosophers in discussions about <strong>the</strong> nature <strong>of</strong> existence.<br />
Godel did. He would complete <strong>the</strong> philosophical journey <strong>Einstein</strong> had begun in <strong>the</strong> <strong>the</strong>ory<br />
<strong>of</strong> relativity. By stretching <strong>Einstein</strong>'s ma<strong>the</strong>matics to <strong>the</strong> limit, he would simplify <strong>and</strong><br />
clarify <strong>the</strong> philosophy, showing <strong>the</strong> <strong>world</strong> that one can plumb <strong>the</strong> depths <strong>of</strong> being <strong>and</strong> <strong>time</strong><br />
<strong>without</strong> disappearing into a Black Forest <strong>of</strong> Heideggerian prose. To appreciate how Godel<br />
was able to accomplish this it is necessary to underst<strong>and</strong> that <strong>the</strong> two fundamental axes<br />
along which <strong>the</strong> course <strong>of</strong> philosophy is plotted are ontology <strong>and</strong> epistemology.<br />
You can assess any position in philosophy by <strong>the</strong> relationship it proposes between being<br />
<strong>and</strong> knowing. Some traditions, like <strong>the</strong> Greek one <strong>of</strong> Plato <strong>and</strong> Aristotle, place ontology in
<strong>the</strong> center, while o<strong>the</strong>rs, like <strong>the</strong> modern period inaugurated by Descartes, put <strong>the</strong><br />
emphasis on epistemology. Clearly, however, a complete philosophy will have to do justice<br />
to both. Unfortunately, <strong>the</strong>re is a deep <strong>and</strong> irreducible tension between <strong>the</strong> two<br />
perspectives that makes a reconciliation difficult to achieve. The ontological perspective,<br />
"<strong>the</strong> view from nowhere" (to exploit Thomas Nagel's evocative phrase) seems to leave no<br />
room for "<strong>the</strong> <strong>world</strong> as I found it" (to borrow an expression from Wittgenstein). If we note<br />
fur<strong>the</strong>r that <strong>the</strong> <strong>world</strong> as we experience it, "our <strong>world</strong>," is essentially temporal, whereas<br />
<strong>the</strong> logical <strong>and</strong> empirical conditions for<br />
existence as such are more spatial, in <strong>the</strong> most general sense, than temporal, we begin to<br />
take a true measure <strong>of</strong> <strong>the</strong> problem Godel set for himself. We also begin to appreciate <strong>the</strong><br />
sympathy Godel would come to feel for Husserl's later phenomenology, <strong>the</strong> goal <strong>of</strong> which<br />
was to do justice to ontology <strong>without</strong> neglecting epistemology, a goal attempted by Kant,<br />
whose system, never<strong>the</strong>less, by GodePs lights, failed to do jus- tice to one element <strong>of</strong> <strong>the</strong><br />
dialectic: ontology.<br />
Now, Godel understood that <strong>the</strong> advent <strong>of</strong> relativity <strong>the</strong>ory enabled one for <strong>the</strong> first <strong>time</strong><br />
to cast <strong>the</strong> question <strong>of</strong> <strong>the</strong> reality <strong>of</strong> <strong>time</strong> into a <strong>the</strong>oretical context amenable to formal<br />
ma<strong>the</strong>matical methods. His approach to <strong>the</strong> philosophy <strong>of</strong> <strong>time</strong>, <strong>the</strong>n, would take <strong>the</strong> form<br />
<strong>of</strong> a frontal assault on <strong>the</strong> ontological implications <strong>of</strong> relativity <strong>the</strong>ory. Can one<br />
consistently maintain both <strong>the</strong> existence <strong>of</strong> <strong>time</strong>, intuitively understood, <strong>and</strong> <strong>the</strong> truth <strong>of</strong><br />
relativity <strong>the</strong>ory? This was a peculiar question, however, because special relativity was an<br />
epistemologically inspired <strong>the</strong>ory (which is what made it <strong>the</strong> apple <strong>of</strong> <strong>the</strong> positivists' eye)<br />
that sprang from such a priori assumptions as that <strong>time</strong> is determined by <strong>the</strong> measurement<br />
<strong>of</strong> simultaneity via synchronized clocks, <strong>and</strong> such a posteriori observations as that different<br />
clocks in different inertial frames deliver different simultaneity results <strong>and</strong> that <strong>the</strong>re is no<br />
objective method for privileging one inertial frame above all o<strong>the</strong>rs. <strong>Einstein</strong> himself,<br />
<strong>the</strong>n, had already drawn ontological conclusions from episte-mological premises. Godel<br />
continued drawing conclusions beyond <strong>the</strong> point where <strong>Einstein</strong> stopped. He would out-<br />
<strong>Einstein</strong> <strong>Einstein</strong> by taking <strong>the</strong> physicist's own ontological reasoning to its logical<br />
conclusion. And he would do this by answering a question that <strong>Einstein</strong>, though fully aware<br />
<strong>of</strong> it, wished to sidestep: is <strong>the</strong> temporal component t <strong>of</strong> four-dimensional relativistic<br />
space-<strong>time</strong>, i.e., what remains <strong>of</strong> "<strong>time</strong>" after relativity <strong>the</strong>ory, really <strong>time</strong>?<br />
In his response to Godel's paper in <strong>the</strong> Schilpp volume, <strong>Einstein</strong> acknowledged that "<strong>the</strong><br />
problem here involved disturbed me at <strong>the</strong> <strong>time</strong> <strong>of</strong> <strong>the</strong> building up <strong>of</strong> <strong>the</strong> general <strong>the</strong>ory <strong>of</strong><br />
relativity." This problem he described as follows: "Is what remains <strong>of</strong> temporal connection<br />
between <strong>world</strong>-points in <strong>the</strong> <strong>the</strong>ory <strong>of</strong> relativity an asymmetrical relation<br />
[like <strong>time</strong>, intuitively understood, <strong>and</strong> unlike space|, or would one be just as much<br />
justified ... to assert A is before P las to assert that A is after P] . . . ?" The issue could also<br />
be put this way: is relativistic space-<strong>time</strong> in essence a space or a <strong>time</strong>?<br />
An Atom Smasher for <strong>the</strong> Mind
Even to raise this question, one must first distinguish between <strong>time</strong> intuitively understood<br />
<strong>and</strong> t, <strong>the</strong> temporal component <strong>of</strong> relativistic space-<strong>time</strong>. While <strong>Einstein</strong> himself was<br />
attuned to this distinction, <strong>the</strong> very success <strong>of</strong> relativity <strong>the</strong>ory had encouraged most<br />
thinkers to conflate <strong>the</strong> two concepts. The situation was deeply reminiscent <strong>of</strong> what Godel<br />
had encountered when he constructed his incompleteness <strong>the</strong>orem in response to <strong>the</strong><br />
Hilbert program, where <strong>the</strong> issue was <strong>the</strong> relationship <strong>of</strong> formal demonstrability to<br />
intuitive ma<strong>the</strong>matical truth. Here too, many researchers conflated <strong>the</strong> two, making <strong>the</strong><br />
idea <strong>of</strong> a "relationship" seem moot. Let us recall Godel's comment to Wang: "... formalists<br />
considered formal demonstrability to be an analysis <strong>of</strong> <strong>the</strong> concept <strong>of</strong> ma<strong>the</strong>matical truth<br />
<strong>and</strong>, <strong>the</strong>refore, were <strong>of</strong> course not in a position to distinguish <strong>the</strong> two." With equal<br />
justification, he could have said that relativistic physicists <strong>and</strong> analytical philosophers <strong>of</strong><br />
science were not in a position to distinguish <strong>the</strong> temporal component <strong>of</strong> four-dimensional<br />
relativistic space-<strong>time</strong> from <strong>the</strong> intuitive concept <strong>of</strong> <strong>time</strong>. Godel's essay on <strong>Einstein</strong>, <strong>the</strong>n,<br />
was not an "excursion," as it is <strong>of</strong>ten taken to be, but ra<strong>the</strong>r a continuation <strong>of</strong> <strong>the</strong> "Godel<br />
program" <strong>of</strong> testing <strong>the</strong> limits <strong>of</strong> formal methods in capturing intuitive concepts. Having<br />
raised this issue, how does one <strong>the</strong>n devise a thought experiment to distinguish <strong>the</strong> two<br />
ma<strong>the</strong>matically (if indeed <strong>the</strong>y are distinguishable)? In subatomic physics, one can submit<br />
particles to <strong>the</strong> extreme forces <strong>of</strong> an accelerator, or "atom smasher," which results in<br />
particles that are indistinguishable under less-extreme conditions revealing <strong>the</strong>mselves as<br />
distinct. Godel would devise a method for sub-<br />
jecting <strong>the</strong> concept <strong>of</strong> space-<strong>time</strong> to similarly extreme conditionsóin this case,<br />
geometrically, not dynamically, extremeóso that invisible differences between <strong>the</strong> two<br />
concepts would become manifest. This too was a continuation <strong>of</strong> a methodology Godel had<br />
employed in his incompleteness <strong>the</strong>orem. The method consists in creating what can be<br />
called limit cases, formal constructions that by design are so extreme that <strong>the</strong>y limit,<br />
ma<strong>the</strong>matically, <strong>the</strong> possible intuitive interpretations <strong>the</strong>y will admit. For his<br />
incompleteness <strong>the</strong>orem, Godel devised a formal system toge<strong>the</strong>r with a series <strong>of</strong><br />
ingenious definitions <strong>and</strong> coordinations for which it could be demonstrated that <strong>the</strong><br />
concept <strong>of</strong> formal pro<strong>of</strong>, as it appeared in <strong>the</strong> system, could not, on pain <strong>of</strong> contradiction,<br />
be interpreted as representing intuitive ma<strong>the</strong>matical truth. He did this by constructing a<br />
formula that was provably unprovable, but intuitively true.<br />
In his contribution to <strong>the</strong> <strong>Einstein</strong> volume, Godel would construct a <strong>world</strong> model for <strong>the</strong><br />
equations <strong>of</strong> general relativity whose geometry was so extreme that <strong>the</strong> temporal<br />
component <strong>of</strong> <strong>the</strong> resulting space <strong>time</strong> structure could not reasonably be seen as<br />
representing intuitive lime. Kin stein had already succeeded, in <strong>the</strong> <strong>the</strong>ory <strong>of</strong> relativity, in<br />
bringing about <strong>the</strong> geometrization <strong>of</strong> physics. What Godel did was to construct a limit case<br />
for <strong>the</strong> relativistic geometrization <strong>of</strong> <strong>time</strong>. He would do this by bringing to <strong>the</strong> fore various<br />
properties that anything deserving <strong>the</strong> name <strong>of</strong> intuitive <strong>time</strong> would have to possess,<br />
including <strong>Einstein</strong>'s requirement that <strong>the</strong> series <strong>of</strong> events be asymmetrically ordered, so<br />
that if A is before B, it cannot also be after B. Godel would <strong>the</strong>n demonstrate<br />
ma<strong>the</strong>matically that in <strong>the</strong> <strong>world</strong> model he had constructed, <strong>the</strong>re were continuous<br />
<strong>time</strong>like <strong>world</strong> lines connecting any two events, so that even if B were observed occurring<br />
after A, one could undertake a journeyóin a very fast spaceshipóthat would take one to B<br />
before one reached A. From this, Godel would conclude that <strong>the</strong> space-<strong>time</strong> structure in<br />
such a <strong>world</strong> was clearly a space, not a <strong>time</strong>, <strong>and</strong> <strong>the</strong>refore that t, <strong>the</strong> temporal<br />
component <strong>of</strong> space-<strong>time</strong>, was in fact ano<strong>the</strong>r spatial dimensionónot <strong>time</strong> as we<br />
underst<strong>and</strong> it in ordinary experience.
A journey along <strong>the</strong> closed, continuous <strong>time</strong>like <strong>world</strong> lines Godel had discovered in (what<br />
came to be known as) <strong>the</strong> Godel universe could only be described as <strong>time</strong> travel. Godel<br />
had achieved an amazing demonstration that <strong>time</strong> travel, strictly understood, was<br />
consistent with <strong>the</strong> <strong>the</strong>ory <strong>of</strong> relativity. Enthusiasts <strong>of</strong> <strong>time</strong> travel would in due course<br />
become excited by this discovery, but <strong>the</strong>y would fail to see that <strong>the</strong> primary result was a<br />
powerful argument that if <strong>time</strong> travel is possible, <strong>time</strong> itself is not. If his results held up<br />
<strong>and</strong> his interpretation <strong>of</strong> <strong>the</strong>m survived scrutiny, Godel would have succeeded in<br />
demonstrating, ma<strong>the</strong>matically, a result about <strong>the</strong> reality (or unreality) <strong>of</strong> <strong>time</strong> that had<br />
eluded idealist philosophers for centuries, from Plato to Kant, <strong>and</strong> he would have done so,<br />
once again, as a spy, this <strong>time</strong> in <strong>the</strong> house <strong>of</strong> physics. Before <strong>Einstein</strong>'s very eyes, a<br />
metamorphosis had occurred. The <strong>the</strong>ory he had devised to capture <strong>time</strong>, to pin it down<br />
ma<strong>the</strong>matically <strong>and</strong> render it amenable to human underst<strong>and</strong>ing, had been transformed, in<br />
GodePs h<strong>and</strong>s, into a disappearing hat trick.<br />
<strong>Einstein</strong> was impressed. "Kurt GodePs essay," he wrote, "constitutes, in my opinion, an<br />
important contribution to <strong>the</strong> general <strong>the</strong>ory <strong>of</strong> relativity, especially to <strong>the</strong> analysis <strong>of</strong> <strong>the</strong><br />
concept <strong>of</strong> <strong>time</strong>." But he set <strong>the</strong> stage for how o<strong>the</strong>rs would respond when he stated that<br />
"it will be interesting to weigh whe<strong>the</strong>r <strong>the</strong>se [cosmological solutions] are not to be<br />
excluded on physical grounds." Most thinkers, once <strong>the</strong>y had recovered from <strong>the</strong> shock <strong>of</strong><br />
GodePs discovery, would restrict <strong>the</strong>ir response to enquiring whe<strong>the</strong>r <strong>the</strong> Godel universe<br />
was sufficiently realistic from a physical st<strong>and</strong>point to be taken seriously.<br />
<strong>Einstein</strong> himself, however, had a bad track record in acknowledging ma<strong>the</strong>matical<br />
consequences <strong>of</strong> general relativity as physically realistic. When Karl Schwarzschild, a<br />
German colleague, discovered in 1916 that if a star began an extreme gravitational<br />
collapse into itself, its mass would eventually reach a critical point after which space-<strong>time</strong><br />
would be so severely curved that nothing inside (what is now known as) <strong>the</strong> "event<br />
horizon," including light, would be able to escape, <strong>Einstein</strong> dismissed <strong>the</strong> "Schwarzschild<br />
singularity" as a ma<strong>the</strong>matical<br />
anomaly with no physical significance. He was wrong. We now call <strong>the</strong>se singularities<br />
"black holes," <strong>and</strong> astronomers find <strong>the</strong>m at <strong>the</strong> center <strong>of</strong> every galaxy.<br />
Later, in 1917, <strong>the</strong> Dutch astronomer Wellem de Sitter proposed a cosmological model for<br />
general relativity in which <strong>the</strong> universe was not static, as <strong>Einstein</strong> believed it to be, but<br />
ra<strong>the</strong>r exp<strong>and</strong>ing. Still later, in 1922, Aleks<strong>and</strong>r Friedmann, a Russian ma<strong>the</strong>matician <strong>and</strong><br />
physicist, argued that it was a consequence <strong>of</strong> general relativity that <strong>the</strong> universe must be<br />
ei<strong>the</strong>r contracting or exp<strong>and</strong>ing. <strong>Einstein</strong> rejected both de Sitter <strong>and</strong> Friedman as having<br />
produced unphysical ma<strong>the</strong>matical models <strong>of</strong> <strong>the</strong> universe. But since <strong>the</strong>ir ideas were<br />
consistent with general relativity, he proposed a new, ad hoc principle to be added to<br />
relativity, <strong>the</strong> "cosmological constant," whose sole purpose was to introduce an<br />
antigravitational law that would counteract <strong>the</strong> forces <strong>of</strong> gravity that de Sitter <strong>and</strong><br />
Friedman had deduced would o<strong>the</strong>rwise cause <strong>the</strong> universe to spiral inwards or exp<strong>and</strong><br />
outwards. Once again, <strong>Einstein</strong> turned out to be wrong. We now have overwhelming<br />
empirical evidence that <strong>the</strong> universe is indeed exp<strong>and</strong>ing, <strong>and</strong> <strong>Einstein</strong> would eventually<br />
call <strong>the</strong> introduction <strong>of</strong> <strong>the</strong> cosmological constant "my greatest blunder." Yet here too he<br />
would be wrong: <strong>the</strong>re now appears to be an antigravitational force that is accelerating<br />
<strong>the</strong> universe's expansion.
In all <strong>of</strong> <strong>the</strong>se cases, <strong>Einstein</strong> rejected <strong>world</strong> models for general relativity on <strong>the</strong> grounds<br />
that <strong>the</strong> extreme geometrical conditions <strong>the</strong>y represented were inconsistent not with <strong>the</strong><br />
letter <strong>of</strong> general relativity, but with his own intuitions about <strong>the</strong> shape <strong>of</strong> <strong>the</strong> universe.<br />
Each <strong>time</strong>, however, <strong>the</strong> real <strong>world</strong> refused to cooperate with <strong>the</strong> great physicist's a priori<br />
dem<strong>and</strong>s. It should not, <strong>the</strong>n, be taken as decisive that when Godel proposed a new,<br />
geometrically extreme <strong>world</strong> model for general relativity, <strong>Einstein</strong> was inclined to question<br />
its physical significance.<br />
Yet <strong>Einstein</strong> did take Godel to have made an important discovery about <strong>the</strong> nature <strong>of</strong> <strong>time</strong>.<br />
The question was what exactly this discovery meant. What had Godel really been up to in<br />
this beautiful <strong>and</strong> mysterious work? On this <strong>the</strong>re was a deafening silence. ("Like most<br />
o<strong>the</strong>rs," wrote a distinguished philosopher <strong>of</strong> physics decades after Godel's essay first<br />
appeared, "I avoid <strong>the</strong> puzzling issue <strong>of</strong> what Godel really thought he was showing about<br />
<strong>time</strong> <strong>and</strong> stick to <strong>the</strong> easier stuff on closed <strong>time</strong>like loops.") Unlike Godel's achievements in<br />
logic, which, after <strong>the</strong> initial shock wore <strong>of</strong>f, became gradually understood, his<br />
cosmological results on <strong>Einstein</strong>'s <strong>the</strong>ory <strong>of</strong> relativity remained an enigma, <strong>and</strong> all too<br />
soon, a distant memory. The question, however, cannot be avoided: What really happened<br />
when Godel became <strong>Einstein</strong>? How exactly did he achieve his shocking results on <strong>time</strong> <strong>and</strong><br />
relativity, <strong>and</strong> what should physicists <strong>and</strong> philosophers have made <strong>of</strong> <strong>the</strong>m? After all is said<br />
<strong>and</strong> done, can it really be true that we, who look forward to dessert while nibbling on our<br />
salads, are living in a <strong>world</strong> <strong>without</strong> <strong>time</strong>?<br />
7 The Sc<strong>and</strong>al <strong>of</strong><br />
Big "T" <strong>and</strong> Little "t"<br />
"Scientific people," proceeded <strong>the</strong> Time Traveler, "... know very well that <strong>time</strong> is only a<br />
kind <strong>of</strong> space. "<br />
H.G. WELLS, THE TIME MACHINE<br />
Albert <strong>Einstein</strong>: Philosopher-Scientist, dedicated by P.A. Schilpp to <strong>the</strong> physicist on <strong>the</strong><br />
occasion <strong>of</strong> his seventieth birthday, was a great success. It remains <strong>the</strong> most influential<br />
volume in <strong>the</strong> Library <strong>of</strong> Living Philosophers, not least because <strong>of</strong> its wonderful debate<br />
between Kin-stein <strong>and</strong> Niels Bohr, friends <strong>and</strong> adversaries, on <strong>the</strong> future <strong>of</strong> <strong>the</strong><br />
Copenhagen interpretation <strong>of</strong> quantum mechanics. <strong>Einstein</strong>'s walking companion, in turn,<br />
had worked intensely on his own contribution, writing to his mo<strong>the</strong>r that <strong>the</strong> work left him<br />
little leisure for correspondence. That summer, he canceled as well his accustomed<br />
vacation trip to <strong>the</strong> seashore. When <strong>the</strong> auspicious volume finally appeared, he cannot<br />
have failed to be disappointed by <strong>the</strong> near silence with which his essay was greeted.
To be sure, <strong>the</strong>re was a minor stir among astrophysicists <strong>and</strong> cos-mologists concerning <strong>the</strong><br />
validity <strong>of</strong> Godel's construction <strong>of</strong> new <strong>world</strong> models for general relativity. The initial<br />
response, however, was that he had simply made a mistake in his physics. No less a<br />
physicist than S. Ch<strong>and</strong>rasekhar, who had attended a talk Godel had given on his new<br />
models at Princeton, published an article with J.P. Wright in <strong>the</strong> Proceedings <strong>of</strong> <strong>the</strong><br />
National Academy <strong>of</strong> Sciences claiming that Godel had<br />
made an error when he described <strong>the</strong> possibility <strong>of</strong> <strong>time</strong> travel as along a geodesicó<strong>the</strong><br />
path <strong>of</strong> inertial motion, or free fallóin <strong>the</strong> Godel universe. This no doubt contributed to <strong>the</strong><br />
lack <strong>of</strong> interest in Godel's essay among philosophers. If <strong>the</strong> physical premise was faulty,<br />
why bo<strong>the</strong>r to examine <strong>the</strong> philosophy?<br />
But was Godel really in error? Amazingly, <strong>the</strong> editors <strong>of</strong> <strong>the</strong> Proceedings had not seen fit to<br />
consult <strong>the</strong> author himself before publishing a report <strong>of</strong> his alleged error concerning an<br />
elementary concept <strong>of</strong> relativity <strong>the</strong>ory. Might it not have been Ch<strong>and</strong>rasekhar <strong>and</strong> Wright,<br />
not Godel, who had made a mistake? This possibility seems not to have occurred to <strong>the</strong><br />
editors, yet it turned out to have been <strong>the</strong> case, a fact demonstrated not by a physicist but<br />
by a philosopher, Howard Stein, who showed clearly that <strong>time</strong> travel in <strong>the</strong> Godel universe<br />
could take place only under great acceleration, which could be provided by a spaceship,<br />
not along <strong>the</strong> free-fall path <strong>of</strong> a geodesic. More astonishing yet, however, Stein could not<br />
get <strong>the</strong> correction <strong>of</strong> Ch<strong>and</strong>rasekhar <strong>and</strong> Wright accepted for publication. Only when Godel<br />
himself intervened did <strong>the</strong> fact finally make it into print that his argument for <strong>the</strong><br />
possibility <strong>of</strong> <strong>time</strong> travel was relativistically valid.<br />
What had gone wrong? Clearly, regardless <strong>of</strong> Godel's reputation as a great logician, <strong>the</strong><br />
astrophysics community saw him as an outsider, <strong>and</strong> moreover as attempting to swim<br />
against <strong>the</strong> intellectual tide. But <strong>the</strong> sc<strong>and</strong>al <strong>of</strong> disregard extended to philosophy as well.<br />
Godel's contribution to <strong>the</strong> Schilpp volume had almost no impact on <strong>the</strong> community <strong>of</strong><br />
philosophers. Except for a few highly technical discussions <strong>of</strong> <strong>the</strong> physics, with some brief<br />
though poignant glances at Godel's philosophical goals, his argument that relativity <strong>the</strong>ory,<br />
correctly understood, provides strong support for <strong>the</strong> great philosophers throughout history<br />
who were skeptical <strong>of</strong> <strong>the</strong> objective reality <strong>of</strong> <strong>time</strong>, went unheeded. Naturally, <strong>the</strong>re was<br />
some interest in <strong>the</strong> question <strong>of</strong> <strong>time</strong> travel. There always is. It is a topic <strong>of</strong> perennial<br />
fascination among thoughtful <strong>and</strong> imaginative people, <strong>and</strong> <strong>the</strong> fact that Godel had derived<br />
such an exotic conclusion from <strong>the</strong> respectable equations <strong>of</strong> relativity inevitably raised a<br />
few eyebrows. But on <strong>the</strong> question <strong>of</strong> whe<strong>the</strong>r he had succeeded in showing that<br />
<strong>time</strong> is ideal <strong>the</strong>re was a pr<strong>of</strong>ound silence. If Godel had not been taken seriously as a<br />
philosopher before his contribution to <strong>the</strong> Schilpp volume, nothing changed after its<br />
appearance.<br />
Quite simply, he had never been a member <strong>of</strong> <strong>the</strong> club: he was out <strong>of</strong> touch <strong>and</strong> out <strong>of</strong><br />
step with <strong>the</strong> philosophical establishment, in Princeton as elsewhere, <strong>and</strong> <strong>the</strong> reason was<br />
not hard to fathom. Just as Wittgenstein's language-centered early work, <strong>the</strong> Tractatus,<br />
had helped set <strong>the</strong> philosophical agenda following World War I, not least in Godel's Vienna,<br />
in <strong>the</strong> aftermath <strong>of</strong> <strong>the</strong> Second World War, Wittgenstein's later, still linguistically oriented<br />
philosophy came to dominate again, this <strong>time</strong> in Godel's newly adopted country. To many<br />
philosophers, it must have seemed as if Godel had slept through not one but two
Wittgenstein revolutions. It added insult to injury that W.V.O. Quine, <strong>the</strong> dominant figure<br />
for years in American philosophy <strong>and</strong> <strong>the</strong> most analytic <strong>of</strong> analytical philosophers, was also<br />
absent from Godel's thinking. Godel himself was acutely aware <strong>of</strong> this alienation. When <strong>the</strong><br />
<strong>time</strong> came for his essay on Cantor's continuum problem to be reprinted in a now classic<br />
collection coedited by his Princeton colleague Paul Benacerraf <strong>and</strong> Harvard's Hilary<br />
Putnam, he would not agree to <strong>the</strong> republication until he had been convinced that <strong>the</strong><br />
editors would not deride his essay. Shameful it may have been that coming out <strong>of</strong> nowhere<br />
in every sense his highly compressed, paradoxical-sounding philosophical defense <strong>of</strong><br />
temporal idealism, based on an arcane new cosmological model for an abstruse physical<br />
<strong>the</strong>ory, failed to arouse more than a murmur. But it was not surprising. Yet why, more<br />
than half a century after it was proposed <strong>and</strong> ignored, is Godel's argument still worthy <strong>of</strong><br />
attention? What had Godel really accomplished?<br />
What Your Parents Never Told You About <strong>the</strong> Age <strong>of</strong> <strong>the</strong> Universe<br />
The problem Godel inherited from <strong>Einstein</strong> had been understood for centuries to concern<br />
<strong>the</strong> most fundamental aspect <strong>of</strong> human experience.<br />
For Kant, space <strong>and</strong> <strong>time</strong> are <strong>the</strong> two essential "forms <strong>of</strong> human sensibility," with <strong>time</strong>, as<br />
<strong>the</strong> form <strong>of</strong> both "inner" <strong>and</strong> "outer sense," being <strong>the</strong> more basic. Yet <strong>time</strong> is far more<br />
elusive than space. Capturing <strong>time</strong> through ma<strong>the</strong>maticsóa form <strong>of</strong> thought from which<br />
philosophers since Plato have taken pains to remove anything remotely temporaló is like<br />
trying to trap water with a net. With <strong>the</strong> advent <strong>of</strong> <strong>Einstein</strong>'s <strong>the</strong>ory <strong>of</strong> relativity, however,<br />
<strong>the</strong> mystery <strong>of</strong> this form <strong>of</strong> being was widely taken to have been resolved. Philosophers<br />
could finally relax. <strong>Einstein</strong> had taken care <strong>of</strong> business.<br />
Appearances, however, can be deceptive. The universe, for example, as everyone knows,<br />
is very old. Its exact age is a matter for debate, but <strong>the</strong>re is no disagreement that it runs<br />
to billions <strong>of</strong> years. We marvel that as frail <strong>and</strong> isolated a species as we are can have<br />
achieved such impressive wisdom about <strong>the</strong> origins <strong>of</strong> everything that is. In truth, however,<br />
it is more than marvelous to have discovered <strong>the</strong> age <strong>of</strong> <strong>the</strong> universe. It is impossible. For<br />
if <strong>the</strong> universe is n years old, its present state comes n years after <strong>the</strong> moment when it all<br />
began. In 1905, however, <strong>Einstein</strong> had demonstrated in <strong>the</strong> special <strong>the</strong>ory <strong>of</strong> relativity that<br />
<strong>the</strong>re is no such thing as "<strong>the</strong> present state <strong>of</strong> <strong>the</strong> universe," that is, what would be<br />
revealed by a snapshot <strong>of</strong> <strong>the</strong> universe as it exists at this very moment. The relativity <strong>of</strong><br />
simultaneity implies that what is taken to be "now" relative to one inertial frame will differ<br />
from what is "now" in ano<strong>the</strong>r frame if <strong>the</strong> second frame is in motion relative to <strong>the</strong> first. It<br />
follows immediately that if <strong>the</strong> <strong>the</strong>ory <strong>of</strong> relativity is correct, <strong>the</strong>re simply is no such thing<br />
as "<strong>the</strong> present state <strong>of</strong> <strong>the</strong> entire universe" <strong>of</strong> four-dimensional space-<strong>time</strong>. <strong>Einstein</strong><br />
himself said this quite clearly: "The four-dimensional continuum is now no longer<br />
resolvable objectively into sections, all <strong>of</strong> which contain simultaneous events; 'now' loses<br />
for <strong>the</strong> spatially extended <strong>world</strong> its objective meaning."<br />
None <strong>of</strong> this was lost on Godel. To him, <strong>the</strong>re was an inconsistency between <strong>Einstein</strong>'s<br />
<strong>the</strong>ory <strong>and</strong> <strong>the</strong> everyday belief that <strong>time</strong>, unlike space, "passes" or "flows." On this<br />
question, two assumptions dominated, <strong>the</strong>n as now, in <strong>the</strong> popular as well as <strong>the</strong> scientific<br />
consciousness. Both <strong>of</strong> <strong>the</strong>m arc faulty. The first is that special relativity is compatible
with <strong>the</strong> passing <strong>of</strong> <strong>time</strong>, as long as it is acknowledged that this flow has only local, as<br />
opposed to global, significance. The o<strong>the</strong>r is that <strong>the</strong> <strong>world</strong> according to special relativity<br />
is a fixed four-dimensional space-<strong>time</strong> "block," but that this does not conflict with <strong>the</strong><br />
deliverances <strong>of</strong> ordinary experience. The former fails because whatever <strong>the</strong> flow <strong>of</strong> <strong>time</strong><br />
is, it is not a merely geometrical fact <strong>and</strong> thus cannot enjoy only local existence. A river's<br />
course, for example, may curve locallyómay be serpentine near us but straight<br />
elsewhereóbut what would it mean to say that <strong>the</strong> river flowed only in our neighborhood?<br />
Concerning <strong>the</strong> second assumption, one need only recognize <strong>the</strong> befuddlement that would<br />
ensue if one were to try to act on <strong>the</strong> assumption that today's breakfast is no more actual<br />
than yesterday's or tomorrow's, that <strong>the</strong> future, like <strong>the</strong> present, has already arrived. ("The<br />
future is now," reads <strong>the</strong> logo <strong>of</strong> Hudsucker Enterprises in <strong>the</strong> film The Hudsucker Proxy.<br />
This makes for an entertaining story but an unconvincing metaphysics <strong>of</strong> everyday life.)<br />
Should I still be wondering what to order for breakfast yesterday, as I am for tomorrow, or<br />
should I cancel both orders because <strong>the</strong> meals have already arrived? And since <strong>the</strong> present<br />
is no more real than <strong>the</strong> past <strong>and</strong> I am still lying on <strong>the</strong> beach as I was last summer, why<br />
am I identifying only with <strong>the</strong> "I" that is presently shivering in <strong>the</strong> cold? Am I simply making<br />
a mistake? Or are <strong>the</strong>re as many "I's" as <strong>the</strong>re are moments in <strong>time</strong>, <strong>and</strong> if so, are <strong>the</strong>y all<br />
me, or only parts <strong>of</strong> me? (I have spatial parts, <strong>of</strong> course: head, h<strong>and</strong>s, feet; do I also have<br />
"temporal parts"?) The confidence <strong>of</strong> <strong>the</strong> popular (<strong>and</strong> not so popular) mind is misplaced<br />
when it clings to <strong>the</strong> belief that all is well, temporally speaking, between <strong>the</strong> universe <strong>and</strong><br />
Dr. <strong>Einstein</strong>. All is not well at all.<br />
But as indicated by its name, special relativity is not <strong>the</strong> full <strong>the</strong>ory <strong>of</strong> relativity. Its<br />
validity is restricted to so-called inertial reference frames, those that are unaccelerated<br />
<strong>and</strong> move in straight lines. The final, comprehensive <strong>the</strong>ory, general relativity, has no such<br />
restrictions. It includes an account <strong>of</strong> gravity, <strong>the</strong> first <strong>the</strong>ory <strong>of</strong> gravity to replace<br />
Newton's. Since it is gravity that governs <strong>the</strong> universe as a whole, general relativity is <strong>the</strong><br />
foundation <strong>of</strong> <strong>the</strong> modern science <strong>of</strong> cosmology. If<br />
special relativity, moreover, introduced <strong>the</strong> discovery that matter is equivalent to energy,<br />
<strong>the</strong> general <strong>the</strong>ory announced <strong>the</strong> identity <strong>of</strong> gravity with space-<strong>time</strong> curvature. Matter in<br />
motion determines <strong>the</strong> shape <strong>of</strong> space-<strong>time</strong>. The possibility arises that some reference<br />
frames might be privileged, namely those that follow, as Godel put it, <strong>the</strong> mean motion <strong>of</strong><br />
matter in <strong>the</strong> universe. Time relative to those frames <strong>of</strong> reference bears <strong>the</strong> designation<br />
"cosmic <strong>time</strong>," <strong>and</strong> this opens up <strong>the</strong> possibility that <strong>time</strong> in something like <strong>the</strong><br />
pre<strong>the</strong>oretical sense might after all be consistent with relativity, in particular with general<br />
relativity. It is <strong>time</strong> in this sense that is (or should be) invoked when cosmologists speak <strong>of</strong><br />
<strong>the</strong> age <strong>of</strong> <strong>the</strong> universe.<br />
The question remains, however, how closely this new concept <strong>of</strong> <strong>time</strong> resembles what <strong>time</strong><br />
was thought to be before <strong>Einstein</strong>. The astrophysicist James Jeans, whom Godel would cite<br />
by name when he came to discuss <strong>the</strong>se issues, thought <strong>the</strong> resemblance was very close<br />
indeed. With <strong>the</strong> advent <strong>of</strong> general relativity <strong>and</strong> cosmic <strong>time</strong>, "<strong>time</strong> regained a real<br />
objective existence, although only on <strong>the</strong> astronomical scale." Since, moreover, every<br />
known relativistically possible universe "makes [in this way] a real distinction between<br />
space <strong>and</strong> <strong>time</strong>," Jeans believed, "[we have] every justification for reverting to our old
intuitional belief that past, present, <strong>and</strong> future have real objective meaning." In short, "we<br />
are free to believe that <strong>time</strong> is real." Just this Godel would put to <strong>the</strong> test.<br />
What Godel Means by Time<br />
At issue is <strong>the</strong> leitmotif <strong>of</strong> Godel's lifework, <strong>the</strong> dialectic <strong>of</strong> <strong>the</strong> formal <strong>and</strong> <strong>the</strong> intuitive,<br />
here, <strong>of</strong> formal versus intuitive <strong>time</strong>, between what remains <strong>of</strong> <strong>time</strong> in <strong>the</strong> <strong>the</strong>ory <strong>of</strong><br />
relativity <strong>and</strong> <strong>the</strong> <strong>time</strong> <strong>of</strong> everyday life. The difference between <strong>the</strong>se two conceptions is<br />
crucial. It can be illuminated by considering what <strong>the</strong> early-twentieth-century philosopher<br />
J.M.E. McTaggart called <strong>the</strong> A-series <strong>and</strong> B-series. The B-series is founded on <strong>the</strong><br />
characterization <strong>of</strong> dates <strong>and</strong> <strong>time</strong>s in terms <strong>of</strong> <strong>the</strong><br />
fixed relationship <strong>of</strong> "before" <strong>and</strong> "after." It is a structurally or "geometrically" defined<br />
series, analogous to a space. It is <strong>the</strong> temporal series captured by calendars <strong>and</strong> by history<br />
books. The year 1865, for example, comesónow <strong>and</strong> foreveróbefore 1965 <strong>and</strong> after 1765,<br />
<strong>and</strong> <strong>the</strong>se structural, "geometric" facts are fixed <strong>and</strong> unchangeable. The A-series, in<br />
contrast, is essentially fluid or dynamic. It contains <strong>the</strong> "moving now," i.e., <strong>the</strong> present<br />
moment, which is always in flux. That your dentist appointment is at 3 p.m. on May 19 is a<br />
B-series fact that has been marked on your calendar for months. It will remain a fact after<br />
<strong>the</strong> appointment is long <strong>forgotten</strong>. That now, however, is <strong>the</strong> very date <strong>and</strong> <strong>time</strong> <strong>of</strong> <strong>the</strong><br />
appointment is a scary A-series fact that has not obtained until this very moment, <strong>and</strong> will<br />
happily no longer obtain tomorrow. (It is no accident that a famous philosophical essay on<br />
<strong>the</strong> A-series is entitled "Thank Goodness That's Over.")<br />
Though <strong>the</strong> A-series represents, intuitively, <strong>the</strong> most fundamental aspect <strong>of</strong> <strong>time</strong>óindeed,<br />
what distinguishes <strong>time</strong> from spaceóit is marked by several concomitants, each one<br />
difficult to capture in <strong>the</strong> formal language <strong>of</strong> ma<strong>the</strong>matics. First is <strong>the</strong> fact that one <strong>time</strong>ó<br />
nowóis privileged over all o<strong>the</strong>rs. This privilege passes from <strong>time</strong> to <strong>time</strong>. What is now will<br />
soon be <strong>the</strong>n. Second, according to this conception, <strong>time</strong> passes, or flows, or lapses, <strong>and</strong> in<br />
a certain "direction": what is future becomes present, <strong>the</strong>n past. Third, unlike both space<br />
<strong>and</strong> <strong>the</strong> B-series, "position" in <strong>the</strong> A-series is not ontologically neutral. Whereas to exist in<br />
New Jersey is to exist no less than in New York (protests by New Yorkers notwithst<strong>and</strong>ing),<br />
to "exist in <strong>the</strong> past" is no longer to exist at all. Socrates had his <strong>time</strong> on stage, but it<br />
passed, he died, <strong>and</strong> his name has been removed from <strong>the</strong> rolls. (It follows that <strong>the</strong>re is<br />
nothing subjective or mind-bound about <strong>the</strong> A-series, i.e., about what is happening now. If<br />
<strong>the</strong>re is such a thing as "inner rime"ó<strong>the</strong> subject, it would appear, <strong>of</strong> Husserl's<br />
investigationsó<strong>the</strong>n this must be distinguished from <strong>the</strong> A-series.) Fourth, while <strong>the</strong> past<br />
has passed <strong>and</strong> is now forever fixed <strong>and</strong> determinate, <strong>the</strong> future remains, as <strong>of</strong> now, open.<br />
Simultaneity, finally, since it determines what<br />
really exists at <strong>the</strong> same <strong>time</strong> as o<strong>the</strong>r things exist, is absolute <strong>and</strong> non-relative. We<br />
cannot, merely by choosing a frame <strong>of</strong> reference, determine what really exists at this<br />
moment. Ei<strong>the</strong>r my friend in Paris is speaking on <strong>the</strong> phone at very same <strong>time</strong> at which I<br />
am writing this, or she isn't, regardless <strong>of</strong> how I try to determine, via synchronized clocks,<br />
whe<strong>the</strong>r her speaking is occurring at <strong>the</strong> same moment as my writing.
Intuitively, <strong>time</strong> is characterized by both <strong>the</strong> A- <strong>and</strong> <strong>the</strong> B-series. If <strong>time</strong> as we experience<br />
it in everyday life, however, is to be identified with formal <strong>time</strong>ó<strong>time</strong> as it is studied in<br />
physicsóa problem arises. What we call "t," <strong>the</strong> temporal component <strong>of</strong> relativistic space<strong>time</strong>,<br />
can be consistently interpreted as representing <strong>the</strong> B-series. The problem lies with<br />
<strong>the</strong> A-series. Since, as <strong>Einstein</strong> put it, in special relativity "'now' loses for <strong>the</strong> spatially<br />
extended <strong>world</strong> its objective meaning"ó that is, <strong>the</strong>re is no objective, <strong>world</strong>wide "now"óit<br />
appears that "t" cannot represent <strong>the</strong> A-series, in which <strong>the</strong>re is a single <strong>world</strong>wide "now"<br />
whose "flux" constitutes <strong>the</strong> change in what exists that characterizes temporal, but not<br />
spatial, reality. This should come as no surprise. One <strong>of</strong> <strong>the</strong> most striking characteristics <strong>of</strong><br />
relativistic space-<strong>time</strong> is that space <strong>and</strong> <strong>time</strong> are no longer to be considered independent<br />
beings but ra<strong>the</strong>r two inextricably intertwined components <strong>of</strong> a single new kind <strong>of</strong> being,<br />
not space or <strong>time</strong> but ra<strong>the</strong>r space-<strong>time</strong>.<br />
The A-series cannot be made to resemble space. What keeps this seemingly obvious fact<br />
hidden from many formal thinkers, whe<strong>the</strong>r physicists or logicians, is that in special<br />
relativity, "t" is formally distinguished from <strong>the</strong> three spatial dimensions. In <strong>the</strong> definition,<br />
for example, <strong>of</strong> <strong>the</strong> space-<strong>time</strong> "interval"ó<strong>the</strong> unique relationship between any two space<strong>time</strong><br />
events that is frame-invariant, hence agreed upon by all observers, no matter <strong>the</strong>ir<br />
state <strong>of</strong> motionó<strong>the</strong> temporal variable, "t," is distinguished from <strong>the</strong> three spatial variables<br />
by being preceded by a negative sign. All this demonstrates, however, is that <strong>time</strong> in<br />
special relativity has a different "geometry" from <strong>the</strong> spatial dimensions, not that it is a<br />
qualitatively different kind <strong>of</strong> being, namely something that "flows." To be blind to this<br />
fact is to confuse <strong>the</strong> formal with <strong>the</strong> intuitive.<br />
It is not for nothing that with <strong>the</strong> <strong>the</strong>ory <strong>of</strong> relativity <strong>Einstein</strong> is said to have accomplished<br />
<strong>the</strong> geometrization <strong>of</strong> physics (an achievement for which, as we have seen, he owed a<br />
great debt to <strong>the</strong> ma<strong>the</strong>matician Minkowski, his long-suffering teacher at <strong>the</strong> Technical<br />
Institute in Zurich, who took <strong>the</strong> bold step <strong>of</strong> re-creating special relativity in a fourdimensional<br />
geometric framework). It is not just that <strong>Einstein</strong> reconceived <strong>the</strong> geometry <strong>of</strong><br />
<strong>the</strong> universe. Ra<strong>the</strong>r, in special relativity, he made <strong>the</strong> defining characteristic <strong>of</strong> <strong>time</strong> not<br />
its qualitative distinction from space, as Kant <strong>and</strong> Newton had done, but ra<strong>the</strong>r its<br />
contribution to <strong>the</strong> geometry <strong>of</strong> four-dimensional space-<strong>time</strong>. Similarly, in general<br />
relativity, he not only provided a new geometry for <strong>the</strong> laws <strong>of</strong> gravity, he defined gravity<br />
itself geometrically, as space-<strong>time</strong> curvature. One <strong>of</strong> <strong>Einstein</strong>'s claims to fame, after all, is<br />
his uncanny ability not only to provide new descriptions <strong>of</strong> old phenomena but new<br />
definitions as well. In this, as in many o<strong>the</strong>r aspects <strong>of</strong> his discoveries, he is as much<br />
philosopher as physicist. The coup de grace came when he replaced Newton's intuitively<br />
evident Euclidean ma<strong>the</strong>matics with unintuitive non-Euclidean geometry.<br />
Time as it appears in relativity <strong>the</strong>ory, <strong>the</strong>n, was ripe for consideration in <strong>the</strong> "Godel<br />
program" <strong>of</strong> assessing <strong>the</strong> extent to which intuitive ideas can be captured by formal<br />
concepts. This is what Godel had in mind when he titled his contribution to <strong>the</strong> Schilpp<br />
volume, "A Remark About <strong>the</strong> Relationship Between Relativity Theory <strong>and</strong> Idealistic<br />
Philosophy." The "idealistic philosophers" he was referring to were thinkers like<br />
Parmenides, Plato <strong>and</strong> Kant, who questioned whe<strong>the</strong>r our subjective experience <strong>of</strong> <strong>the</strong><br />
flow <strong>of</strong> <strong>time</strong> has an objective correlative. To such thinkers, <strong>time</strong> was always an ontological<br />
suspect. As before, when he examined <strong>the</strong> relationship <strong>of</strong> intuitive arithmetic truth, or big<br />
"T," to its representation as formal ma<strong>the</strong>matical pro<strong>of</strong> in Russell's Principia Ma<strong>the</strong>matical,<br />
Godel would begin by clarifying <strong>the</strong> distinction between intuitive <strong>time</strong> <strong>and</strong> little "t," its<br />
formal representation in <strong>Einstein</strong>'s <strong>the</strong>ory <strong>of</strong> relativity as <strong>the</strong> temporal component <strong>of</strong> four-
dimensional <strong>Einstein</strong>-Minkowski space-<strong>time</strong>. Drawing from his contribution to <strong>the</strong> Schilpp<br />
volume as well as <strong>the</strong> longer versions <strong>of</strong> this<br />
essay that have now been published, we can say that Godel characterized intuitive<br />
<strong>time</strong>ó"what everyone understood by <strong>time</strong> before relativity <strong>the</strong>ory"óas "Kantian," or<br />
"prerelativistic." Time in this intuitive sense, he said, is "a one-dimensional manifold that<br />
provides a complete linear ordering <strong>of</strong> all events in nature." This "objective lapse <strong>of</strong> <strong>time</strong>"<br />
is "directly experienced" <strong>and</strong> "involves a change in <strong>the</strong> existing [i.e., in what actually<br />
exists]." Time in <strong>the</strong> intuitive sense, for Godel, is something "whose essence is that only<br />
<strong>the</strong> present really exists." In particular, it "means (or is equivalent to <strong>the</strong> fact) that reality<br />
consists <strong>of</strong> an infinity <strong>of</strong> layers <strong>of</strong> 'now' which come into existence successively." These<br />
features Godel took to be essential properties <strong>of</strong> <strong>time</strong> in <strong>the</strong> intuitive sense, since<br />
"something <strong>without</strong> <strong>the</strong>se properties can hardly be called <strong>time</strong>." Clearly, <strong>time</strong> so<br />
characterized is reflected in <strong>the</strong> A-series, <strong>and</strong> indeed Godel refers to McTaggart by name<br />
in his essay. The question that remains is whe<strong>the</strong>r this intuitive concept can be captured<br />
by <strong>the</strong> formal methods <strong>of</strong> relativity.<br />
Godel's Dialectical Dance with Time<br />
As he had previously done in his incompleteness <strong>the</strong>orem, Godel demonstrated that those<br />
who fail to grasp <strong>the</strong> distinction between <strong>the</strong> intuitive <strong>and</strong> <strong>the</strong> formal concept are not in a<br />
position to make a proper assessment <strong>of</strong> <strong>the</strong>ir relationship. Having made that distinction<br />
with remarkable clarity, he was able to establish, by an ingenious <strong>and</strong> entirely unsuspected<br />
formal argumentówhich in itself, as <strong>Einstein</strong> pointed out, was a major contribution to<br />
relativity <strong>the</strong>oryó<strong>the</strong> inability <strong>of</strong> <strong>the</strong> formal representation to capture <strong>the</strong> intuitive<br />
concept. Godel's dialectical dance with intuitive <strong>and</strong> formal <strong>time</strong> in <strong>the</strong> <strong>the</strong>ory <strong>of</strong> relativity<br />
contained an intricate series <strong>of</strong> steps. We begin with a large-scale view <strong>of</strong> <strong>the</strong> structure <strong>of</strong><br />
Godel's argument, <strong>the</strong>n move on to a closer examination. First <strong>the</strong> forest, <strong>the</strong>n <strong>the</strong> trees.<br />
The opening move concerns <strong>the</strong> more limited special <strong>the</strong>ory <strong>of</strong> relativity. Given that <strong>the</strong> A-<br />
series contains <strong>the</strong> flux <strong>of</strong> "now," <strong>the</strong> absence<br />
<strong>of</strong> an objective, <strong>world</strong>wide "now" in special relativity rules out its existence. But absent<br />
<strong>the</strong> A-series <strong>the</strong>re is no intuitive <strong>time</strong>. What remains, formal <strong>time</strong> as represented by <strong>the</strong><br />
little "t" <strong>of</strong> <strong>Einstein</strong>-Minkowski space-<strong>time</strong>, cannot be identified with <strong>the</strong> intuitive <strong>time</strong> <strong>of</strong><br />
everyday experience. The conclusion, for Godel, is inescapable: if relativity <strong>the</strong>ory is valid,<br />
intuitive <strong>time</strong> disappears.<br />
Step two takes place when Godel reminds us that special relativity is "special" in that it<br />
recognizes only inertial frames in constant velocity relative to each o<strong>the</strong>r. It does not<br />
include an account <strong>of</strong> gravity. <strong>Einstein</strong>'s general <strong>the</strong>ory <strong>of</strong> relativity, in contrast, <strong>of</strong> which<br />
<strong>the</strong> special is a special case, does. In general relativity, as we have seen, gravity itself is<br />
defined as space-<strong>time</strong> curvature, determined, in turn, by <strong>the</strong> distribution <strong>of</strong> matter in<br />
motion. It follows that whereas in special relativity no frames <strong>of</strong> reference or systems in<br />
motion are privileged, in <strong>the</strong> general <strong>the</strong>ory some are distinguished, namely those that, in<br />
Godel's words, "follow <strong>the</strong> mean motion <strong>of</strong> matter" in <strong>the</strong> universe. In <strong>the</strong> actual <strong>world</strong>, it
turns out, <strong>the</strong>se privileged frames <strong>of</strong> reference can be coordinated so that <strong>the</strong>y determine<br />
an objective remnant <strong>of</strong> <strong>time</strong>: <strong>the</strong> "cosmic <strong>time</strong>" we encountered earlier. In general<br />
relativity, <strong>the</strong>n, <strong>time</strong> (<strong>of</strong> a sort) reappears.<br />
But no sooner has <strong>time</strong> reentered <strong>the</strong> scene than Godel proceeds to step three, where he<br />
exploits <strong>the</strong> fact that <strong>Einstein</strong> has fully geometrized space-<strong>time</strong>. The equations <strong>of</strong> general<br />
relativity permit alternative solutions, each <strong>of</strong> which determines a possible universe, a<br />
relativistically possible <strong>world</strong>. Solutions to <strong>the</strong>se complex equations are rare, but in no<br />
<strong>time</strong> at all Godel discovers a relativistically possible universe (actually, a set <strong>of</strong> <strong>the</strong>m)ónow<br />
known as <strong>the</strong> Godel universeó in which <strong>the</strong> geometry <strong>of</strong> <strong>the</strong> <strong>world</strong> is so extreme that it<br />
contains space-<strong>time</strong> paths unthinkable in more familiar universes like our own. In one such<br />
Godel universe, it is provable that <strong>the</strong>re exist closed <strong>time</strong>like curves such that if you travel<br />
fast enough, you can, though always heading toward your local future, arrive in <strong>the</strong> past.<br />
These closed loops or circular paths have a more familiar name: <strong>time</strong> travel. But if it is<br />
possible in such <strong>world</strong>s, Godel argues, to return to one's<br />
past, <strong>the</strong>n what was past never passed at all. But a <strong>time</strong> that never truly passes cannot<br />
pass for real, intuitive <strong>time</strong>. The reality <strong>of</strong> <strong>time</strong> travel in <strong>the</strong> Godel universe signals <strong>the</strong><br />
unreality <strong>of</strong> <strong>time</strong>. Once again, <strong>time</strong> disappears.<br />
But <strong>the</strong> dance is not over. For <strong>the</strong> Godel universe, after all, is not <strong>the</strong> actual <strong>world</strong>, only a<br />
possible one. Can we really infer <strong>the</strong> nonexistence <strong>of</strong> <strong>time</strong> in this <strong>world</strong> from its absence<br />
from a merely possible universe? In a word, yes. Or so Godel argues. Here he makes his<br />
final, his most subtle <strong>and</strong> elusive step, <strong>the</strong> one from <strong>the</strong> possible to <strong>the</strong> actual. This is a<br />
mode <strong>of</strong> reasoning close to Godel's heart. His ma<strong>the</strong>matical Platonism, which committed<br />
him to <strong>the</strong> existence <strong>of</strong> a realm <strong>of</strong> objects that are not accidental like you <strong>and</strong> meówho<br />
exist, but might not haveóbut necessary, implied immediately that if a ma<strong>the</strong>matical<br />
object is so much as possible, it is necessary, hence actual. This is so because what<br />
necessarily exists cannot exist at all unless it exists in all possible <strong>world</strong>s.<br />
This same mode <strong>of</strong> reasoning, from <strong>the</strong> possible to <strong>the</strong> actual, occurs in <strong>the</strong> "ontological<br />
argument" for <strong>the</strong> existence <strong>of</strong> God employed by Saint Anselm, Descartes <strong>and</strong> Leibniz.<br />
According to this argument, one cannot consider God to be an accidental beingóone that<br />
merely happens to existóbut ra<strong>the</strong>r a necessary one that, if it exists at all, exists in every<br />
possible <strong>world</strong>. It follows that if God is so much as possible, He is actual. This means that<br />
one cannot be an a<strong>the</strong>ist unless one is a "supera<strong>the</strong>ist," i.e., someone who denies not just<br />
that God exists but that He is possible. Experience teaches us that ordinary, garden-variety<br />
a<strong>the</strong>ists are not always willing to go fur<strong>the</strong>r <strong>and</strong> embrace su-pera<strong>the</strong>ism. Following in <strong>the</strong><br />
footsteps <strong>of</strong> Leibniz, Godel, too, constructed an ontological argument for God. Then,<br />
concerned that he would be taken for a <strong>the</strong>ist in an a<strong>the</strong>istic age, he never allowed it to<br />
be published.<br />
In arguing from <strong>the</strong> mere possibility <strong>of</strong> <strong>the</strong> Godel universe, in which <strong>time</strong> disappears, to <strong>the</strong><br />
nonexistence <strong>of</strong> <strong>time</strong> in <strong>the</strong> actual <strong>world</strong>, Godel was employing a mode <strong>of</strong> reasoning in<br />
which he had more confidence than most <strong>of</strong> his philosophical colleagues. In <strong>the</strong> case <strong>of</strong> <strong>the</strong>
Godel universe, he reasoned that since this possible <strong>world</strong> is governed by <strong>the</strong> same physical<br />
laws that obtain in <strong>the</strong> actual <strong>world</strong>ódiffering from our <strong>world</strong> only in <strong>the</strong> large-scale<br />
distribution <strong>of</strong> matter <strong>and</strong> motionóit cannot be that whereas <strong>time</strong> fails to exist in that<br />
possible <strong>world</strong>, it is present in our own. To deny this, Godel reasoned, would be to assert<br />
that "whe<strong>the</strong>r or not an objective lapse <strong>of</strong> <strong>time</strong> exists (i.e., whe<strong>the</strong>r or not a <strong>time</strong> in <strong>the</strong><br />
ordinary sense exists) depends on <strong>the</strong> particular way in which matter <strong>and</strong> its motion are<br />
arranged in this <strong>world</strong>." Even though this would not lead to an outright contradiction, he<br />
argued, "never<strong>the</strong>less, a philosophical view leading to such consequences can hardly be<br />
considered as satisfactory." But it is provable that <strong>time</strong> fails to exist in <strong>the</strong> Godel universe.<br />
It cannot, <strong>the</strong>refore, exist in our own. The final step is taken; <strong>the</strong> curtain comes down:<br />
<strong>time</strong> really does disappear.<br />
Into <strong>the</strong> Forest<br />
Such, in broad outlines, is <strong>the</strong> structure <strong>of</strong> Godel's argument. Even from this brief sketch,<br />
it should be apparent how complex <strong>and</strong> subtle was <strong>the</strong> case Godel made for <strong>the</strong> ideality <strong>of</strong><br />
<strong>time</strong>, a far cry from <strong>the</strong> amateurish philosophical fumblings with which he is frequently<br />
credited. To appreciate <strong>the</strong> full force <strong>of</strong> his reasoning, however, it is necessary to look<br />
more closely at <strong>the</strong> details <strong>of</strong> his argument, to get close to <strong>the</strong> trees in <strong>the</strong> forest. His very<br />
first step, from little "t" <strong>and</strong> special relativity to temporal idealism, went unappreciated,<br />
in part because, as he remarked about his incompleteness <strong>the</strong>orem <strong>and</strong> big "T,"<br />
ma<strong>the</strong>matical truth, <strong>the</strong>re was a widespread failure to appreciate <strong>the</strong> distinction between<br />
<strong>the</strong> formal <strong>and</strong> <strong>the</strong> intuitive. There still is. Even today, one can find distinguished<br />
proponents <strong>of</strong> <strong>the</strong> view that special relativity implies only that <strong>the</strong> flow <strong>of</strong> <strong>time</strong> must be<br />
tied to a frame <strong>of</strong> reference, <strong>and</strong> that <strong>the</strong> relativity <strong>of</strong> simultaneityócombined with <strong>the</strong><br />
fact that <strong>the</strong> progress <strong>of</strong> "now" represents <strong>the</strong> flux <strong>of</strong> realityósimply means that reality<br />
itself must be relativized to a frame <strong>of</strong> reference.<br />
The question not asked is this: does this conclusion make any sense? Fifty years ago Godel<br />
had <strong>the</strong> answer: "<strong>the</strong> concept <strong>of</strong> existence . . . cannot be relativized <strong>without</strong> destroying its<br />
meaning completely."<br />
How does Godel know this? Perhaps relativity has revised what we mean by existence? This<br />
Godel considered nonsense. Science, he maintained in his discussions with Hao Wang, does<br />
not analyze concepts, as does philosophy. It applies <strong>the</strong>m. "The notion <strong>of</strong> existence," in<br />
particular, "is one <strong>of</strong> <strong>the</strong> primitive concepts with which we must begin as given. It is <strong>the</strong><br />
clearest concept we have." To appreciate <strong>the</strong> force <strong>of</strong> Godel's re-ductio ad absurdum,<br />
<strong>the</strong>n, it is first necessary to recognize <strong>the</strong> absur-dum. Not everything can be relativized.<br />
You can relativize velocity to a frame <strong>of</strong> reference. You can recognize that what's on my<br />
left is on your right <strong>and</strong> that what is here for me is <strong>the</strong>re for you, that is, that when I say it<br />
is raining here, you agree that it is raining <strong>the</strong>re. But reality as such is absolute. One<br />
cannot speak coherently <strong>of</strong> "my reality" or "your reality," "reality here" versus "reality<br />
<strong>the</strong>re." When people say things like "my reality is a <strong>world</strong> in which people care for each<br />
o<strong>the</strong>r," <strong>the</strong>y meanó or should meanóthat this is <strong>the</strong>ir subjective view <strong>of</strong> <strong>the</strong> <strong>world</strong>, how it is<br />
or should be. But <strong>the</strong>re is still only one objective reality, which includes <strong>the</strong> fact that this<br />
is your view <strong>of</strong> <strong>the</strong> <strong>world</strong>. If a doctrine implies <strong>the</strong> opposite, it is that doctrine that has to<br />
go. We can have a <strong>world</strong> in which <strong>the</strong>re is <strong>time</strong> or a <strong>world</strong> in which <strong>the</strong>re is existence, but<br />
not both. Godel made <strong>the</strong> only rational choice: a <strong>world</strong> <strong>without</strong> <strong>time</strong>.
Since <strong>the</strong>re is no single objective <strong>world</strong>wide "now" in special relativity, <strong>and</strong> since <strong>the</strong>re<br />
cannot be multiple rivers <strong>of</strong> <strong>time</strong> each <strong>of</strong> which determines <strong>the</strong> advance <strong>of</strong> reality, it<br />
follows that <strong>the</strong>re simply is no such thing as <strong>the</strong> universal, <strong>world</strong>wide flux <strong>of</strong> "now" or lapse<br />
<strong>of</strong> <strong>time</strong> consistent with relativity. As Godel put it, "each observer has his own set <strong>of</strong> 'nows,'<br />
<strong>and</strong> none <strong>of</strong> <strong>the</strong>se various systems <strong>of</strong> layers can claim <strong>the</strong> prerogative <strong>of</strong> representing <strong>the</strong><br />
objective lapse <strong>of</strong> <strong>time</strong>." Special relativity, <strong>the</strong>n, is not simply "incomplete" with respect to<br />
intuitive <strong>time</strong>. <strong>Einstein</strong>'s <strong>the</strong>ory is inconsistent with <strong>the</strong> existence <strong>of</strong> <strong>the</strong> A-series, with <strong>the</strong><br />
reality <strong>of</strong> <strong>time</strong> in <strong>the</strong> intuitive sense. There is simply no way around it: if <strong>time</strong> as it is<br />
experienced in ordinary life is to be not ideal but fully<br />
real, <strong>Einstein</strong> must be wrong. And so he is. Or ra<strong>the</strong>r, <strong>the</strong> special <strong>the</strong>ory <strong>of</strong> relativity is to<br />
be replaced by <strong>the</strong> general <strong>the</strong>ory, which contains a universal <strong>the</strong>ory <strong>of</strong> motion, including<br />
acceleration due to gravity. Hope remains. But this hope too Godel will quash, beginning<br />
with <strong>the</strong> second step <strong>of</strong> his argument.<br />
No Time for Time Travel<br />
In general relativity, as we have seen, one can define, if not <strong>time</strong> itself, at least a kind <strong>of</strong><br />
simulacrum <strong>of</strong> <strong>the</strong> real thing, namely, "cosmic <strong>time</strong>," determined by those frames <strong>of</strong><br />
reference whose motion follows <strong>the</strong> mean motion <strong>of</strong> matter in <strong>the</strong> universe. This was a<br />
possibility opened up by <strong>Einstein</strong>'s geometrization <strong>of</strong> space <strong>and</strong> <strong>time</strong>. The only constraints<br />
placed on this geometrization are those determined by <strong>the</strong> laws <strong>of</strong> general relativity. Any<br />
possible universe that obeys <strong>the</strong>se rules must be, by <strong>the</strong> letter <strong>of</strong> relativity, physically<br />
possible. What Godel discoveredóby <strong>the</strong> judicious use <strong>of</strong> ingenious new geometrical<br />
methods that <strong>the</strong>mselves constituted an important advance in relativistic<br />
ma<strong>the</strong>maticsówas that <strong>the</strong>re are solutions to <strong>the</strong> equations <strong>of</strong> general relativity that<br />
provide <strong>world</strong> models in which all matter is rotating. Yet absent Newton's absolute space,<br />
with respect to what is <strong>the</strong> universe supposed to be rotating? "As a substitute for absolute<br />
space," said Godel, "we have a certain inertial field which determines <strong>the</strong> motion <strong>of</strong> bodies<br />
upon which no forces act.... This inertial field determines <strong>the</strong> behavior <strong>of</strong> <strong>the</strong> axis <strong>of</strong> a<br />
completely free gyroscope." This is what Godel used to define universal rotation: "It is with<br />
respect to <strong>the</strong> spatial directions defined in this way (by a free gyroscope ...) that matter<br />
will have to rotate." (Gyroscopes, it will be recalled, entered <strong>Einstein</strong>'s thought elsewhere,<br />
when he helped improve <strong>the</strong>ir design for use on U-boats during WWI.) In <strong>the</strong>se rotating or<br />
"Godel universes," Godel proved, no single objective cosmic <strong>time</strong> can be defined. The last<br />
remnant <strong>of</strong> something even approximating intuitive <strong>time</strong> cannot be introduced into <strong>the</strong>se<br />
Godel universes, on pain <strong>of</strong> contradiction.<br />
If one stake in <strong>the</strong> heart is good, two are better. Godel discovered that in a subclass <strong>of</strong> <strong>the</strong><br />
rotating universes, those that are not exp<strong>and</strong>ing, <strong>the</strong> large-scale geometry <strong>of</strong> <strong>the</strong> <strong>world</strong> is<br />
so warped that <strong>the</strong>re exist space-<strong>time</strong> curves that bend back on <strong>the</strong>mselves so far that<br />
<strong>the</strong>y close; that is, <strong>the</strong>y return to <strong>the</strong>ir starting point. A highly accelerated spaceship<br />
journey along such a closed path, or <strong>world</strong> line, could only be described as <strong>time</strong> travel.<br />
And it would be some spaceship. Godel worked out <strong>the</strong> length <strong>and</strong> <strong>time</strong> for <strong>the</strong> journey, as<br />
well as <strong>the</strong> exact speed <strong>and</strong> fuel requirements. The top speed would be a significant<br />
fraction <strong>of</strong> <strong>the</strong> velocity ot light, <strong>and</strong> <strong>the</strong> fuel requirements, too, would be enormous. (One<br />
<strong>the</strong>orist has calculated that even with a perfectly efficient rocket engine, <strong>the</strong> spaceship
would require 1012 grams <strong>of</strong> fuel for every 2 grams <strong>of</strong> payload.) Paradoxically, however,<br />
<strong>the</strong> very fact that this inconceivably fast spaceship would return its passengers to <strong>the</strong> past<br />
demonstrated, by Godel's lights, that <strong>time</strong> itselfóhence speed <strong>and</strong> motionóis but an<br />
illusion. For if we can revisit <strong>the</strong> past, it still exists. How else could it be revisited? You<br />
can't revisit New Jersey if New Jersey is no longer <strong>the</strong>re, <strong>and</strong> you can't return to <strong>time</strong> t if t<br />
has departed from <strong>the</strong> realm <strong>of</strong> existence. Thus temporal distanceópast <strong>and</strong> futureóturns<br />
out to be as onto-logically neutral as <strong>the</strong> measure <strong>of</strong> space. This is something that even <strong>the</strong><br />
"friends <strong>of</strong> Godel," who in recent years have stepped forward to defend his account <strong>of</strong> <strong>time</strong><br />
travel as logically <strong>and</strong> physically coherent, have failed to note. For Godel, if <strong>the</strong>re is <strong>time</strong><br />
travel, <strong>the</strong>re isn't <strong>time</strong>. The goal <strong>of</strong> <strong>the</strong> great logician was not to make room in physics for<br />
one's favorite episode <strong>of</strong> Star Trek, but ra<strong>the</strong>r to demonstrate that if one follows <strong>the</strong> logic<br />
<strong>of</strong> relativity fur<strong>the</strong>r even than its fa<strong>the</strong>r was willing to venture, <strong>the</strong> results will not just<br />
illuminate but eliminate <strong>the</strong> reality <strong>of</strong> <strong>time</strong>.<br />
Protecting Time from Godel<br />
Such, in essence, was <strong>the</strong> argument put forward by Godel in "A Remark About <strong>the</strong><br />
Relationship Between Relativity Theory <strong>and</strong> Idealistic<br />
Philosophy," a gift for his friend Albert <strong>Einstein</strong> in <strong>the</strong> Schilpp volume dedicated to <strong>the</strong><br />
great physicist on <strong>the</strong> occasion <strong>of</strong> his seventieth birthday. Six pages was all Godel needed<br />
to defeat <strong>time</strong>. Over fifty years later, however, what Godel really accomplished in this<br />
brief compass remains hidden. He had once again constructed a surprising "limit case," a<br />
formal structure whose "geometry" or "syntax" limited <strong>the</strong> possible interpretations it could<br />
be invested with. In <strong>the</strong> case <strong>of</strong> big "T," arithmetic truth, he was able to prove in his<br />
incompleteness <strong>the</strong>orem that <strong>the</strong> logical system he had constructed could successfully<br />
capture <strong>the</strong> concept <strong>of</strong> formal pro<strong>of</strong> but could not, on pain <strong>of</strong> contradiction, represent<br />
truth. Before <strong>the</strong> incompleteness <strong>the</strong>orem, it was possible to mistake pro<strong>of</strong> for truth.<br />
Afterward, with Godel's introduction <strong>of</strong> <strong>the</strong> "syntactically extreme" conditions <strong>of</strong> his formal<br />
systemó<strong>the</strong> conceptual analogue <strong>of</strong> an atom smasheróno reasonable person could fail ro<br />
see <strong>the</strong> distinction. In his contribution to relativity <strong>the</strong>ory, Godel, once again, constructed<br />
a limit case, this <strong>time</strong> for <strong>the</strong> relativistic geo-metrization <strong>of</strong> <strong>time</strong>. That is, he had<br />
demonstrated that in <strong>the</strong> ma<strong>the</strong>matical construction <strong>of</strong> <strong>the</strong> Godel universe, little "t," <strong>the</strong><br />
variable that represents <strong>the</strong> temporal component <strong>of</strong> four-dimensional space-<strong>time</strong>, cannot<br />
bear <strong>the</strong> st<strong>and</strong>ard interpretation <strong>of</strong> <strong>time</strong> in <strong>the</strong> intuitive sense. Indeed, he proved that it<br />
cannot even be interpreted as "cosmic <strong>time</strong>," itself at most a simulacrum <strong>of</strong> <strong>the</strong> real thing.<br />
Once again, he had been able to make a discovery because he had used his philosophical<br />
eye to isolate <strong>the</strong> essential properties that distinguish <strong>the</strong> intuitive from <strong>the</strong> formal<br />
concept, in this case, <strong>the</strong> properties that make intuitive <strong>time</strong> <strong>time</strong>, <strong>and</strong> was thus in a<br />
positionóas those who look little "t" to be an analysis <strong>of</strong> intuitive <strong>time</strong> were notóto prove<br />
that <strong>the</strong>se features were excluded by <strong>the</strong> very geometrical structure <strong>of</strong> <strong>the</strong> Godel<br />
universe. Whereas in our <strong>world</strong>, it was possibleóif you didn't look too closelyóto confuse<br />
formal, relativistic <strong>time</strong> with <strong>time</strong> as ordinarily conceived, this identification became<br />
patently unacceptable in <strong>the</strong> extreme geometrical environment represented by <strong>the</strong> Godel<br />
universe. What once was hidden was now revealed.
The similarities continue. Just as David Hilbert tried at first to avoid <strong>the</strong> consequences <strong>of</strong><br />
<strong>the</strong> incompleteness <strong>the</strong>orem by inventing a new rule <strong>of</strong> logical inference out <strong>of</strong> whole<br />
cloth, so too <strong>the</strong> relativistic establishment, in <strong>the</strong> person <strong>of</strong> Stephen Hawking, tried to get<br />
around <strong>the</strong> embarrassing consequences introduced by <strong>the</strong> Godel universe. If <strong>the</strong> annoying<br />
Godel universe was consistent with <strong>the</strong> laws <strong>of</strong> general relativity, why not change <strong>the</strong> laws?<br />
Hawking thus introduced what he called <strong>the</strong> "chronology protection conjecture" (though a<br />
better name would have been <strong>the</strong> "anti-Godel amendment"), which proposed a<br />
modification <strong>of</strong> general relativity whose primary goal was to rule out <strong>the</strong> possibility <strong>of</strong><br />
<strong>world</strong> models like GodePs, with <strong>the</strong>ir awkward chronologies permitting closed temporal<br />
loops <strong>and</strong> causal chains with no beginning. Despite having, as Russell noted in a different<br />
context, all <strong>the</strong> advantages <strong>of</strong> <strong>the</strong>ft over honest toil, Hawking's chronology protection<br />
conjecture has won few adherents, its ad hoc character betraying itself.<br />
Rarely Have So Many Understood So Little About So Much<br />
If it is shocking that such a pr<strong>of</strong>ound insight into <strong>the</strong> philosophical implications <strong>of</strong> <strong>the</strong><br />
<strong>the</strong>ory <strong>of</strong> relativity has had little impact on physicists, it is dismaying that Godel's ideas<br />
have failed to catch <strong>the</strong> attention <strong>of</strong> philosophers. In this atmosphere <strong>of</strong> neglect, it is<br />
hardly surprising that <strong>the</strong> striking dissimilarity between GodePs two great contributions to<br />
<strong>the</strong> dialectic <strong>of</strong> <strong>the</strong> formal <strong>and</strong> <strong>the</strong> intuitive has also gone unnoticed. Godel was at once a<br />
ma<strong>the</strong>matical realist <strong>and</strong> a temporal idealist. He concluded from <strong>the</strong> incompleteness <strong>of</strong><br />
Hilbert's pro<strong>of</strong>-<strong>the</strong>oretic system for arithmetic that <strong>the</strong> Platonic realm <strong>of</strong> numbers cannot<br />
be fully captured by <strong>the</strong> formal structures <strong>of</strong> logic. For Godel, <strong>the</strong> devices <strong>of</strong> formal pro<strong>of</strong><br />
are too weak to capture all that is true in <strong>the</strong> <strong>world</strong> ot numbers, not to say in ma<strong>the</strong>matics<br />
as a whole. When it came to relativistic cosmology, however, he took <strong>the</strong> opposite tack.<br />
The conse<br />
quence <strong>of</strong> his discoveries for <strong>Einstein</strong>'s realm was not that relativity was too weak to<br />
encompass all that is true about <strong>time</strong>, but ra<strong>the</strong>r that relativity is just fine, whereas <strong>time</strong><br />
in <strong>the</strong> intuitive sense is an illusion. Relativity, by Godel's lights, does not capture <strong>the</strong><br />
essence <strong>of</strong> intuitive <strong>time</strong>, because when it comes to <strong>time</strong>, our intuitions betray us. "As we<br />
present <strong>time</strong> to ourselves," he said, "it simply does not agree with fact. To call <strong>time</strong><br />
subjective is just a euphemism." This, for Godel, was <strong>the</strong> point <strong>of</strong> intersection between<br />
Kant's idealism <strong>and</strong> <strong>the</strong> temporal idealism implicit in <strong>Einstein</strong>'s physics.<br />
Having failed to notice <strong>the</strong> asymmetry between <strong>the</strong> two incompletenesses Godel<br />
discovered, his colleagues in <strong>the</strong> relativistic <strong>and</strong> philosophical establishments were <strong>of</strong><br />
course in no position to comprehend it. It remains one <strong>of</strong> <strong>the</strong> most important unanswered<br />
questions in our underst<strong>and</strong>ing <strong>of</strong> Godel's philosophy. A promising line might proceed as<br />
follows. In <strong>the</strong> case <strong>of</strong> his incompleteness <strong>the</strong>orem, Godel could compare <strong>the</strong> welldetermined<br />
set <strong>of</strong> <strong>the</strong>orems <strong>of</strong> formal arithmetic with <strong>the</strong> equally well founded<br />
deliverances <strong>of</strong> intuitive, i.e., unformalized, arithmetic, accumulated over millennia by<br />
<strong>the</strong> <strong>world</strong>'s great ma<strong>the</strong>maticians, from which no contradictions have been derived. Even<br />
<strong>the</strong> concept <strong>of</strong> set, as employed "naively" by ma<strong>the</strong>maticians, has not led to paradoxes.<br />
"This concept <strong>of</strong> set," Godel pointed out, "according to which a set is anything obtainable<br />
from <strong>the</strong> integers (or some o<strong>the</strong>r well-defined objects) by iterated application <strong>of</strong> <strong>the</strong><br />
operation <strong>of</strong> 'set <strong>of</strong>,' <strong>and</strong> not something obtained by dividing <strong>the</strong> totality <strong>of</strong> all existing<br />
things into two categories, has never led to any antinomy whatsoever." Russell's Paradox,<br />
in contrast, arose precisely from attempts like Frege's to formalize Cantor's intuitive <strong>the</strong>ory
<strong>of</strong> sets by "dividing <strong>the</strong> totality <strong>of</strong> all existing things into two categories," those that fall<br />
under a given concept <strong>and</strong> those that don't. "These contradictions," Godel reminded us,<br />
"did not appear within ma<strong>the</strong>matics but near its outermost boundary toward philosophy." It<br />
is formalisms like Hilbert's <strong>and</strong> Russell's that are problematic; everyday ma<strong>the</strong>matical<br />
practice is not founded on a mistake.<br />
Things st<strong>and</strong> o<strong>the</strong>rwise with <strong>time</strong>. Whereas special <strong>and</strong> general relativity are coherent,<br />
well-formulated, well-understood physical <strong>the</strong>ories that have enjoyed extensive empirical<br />
confirmation, our ordinary, pre<strong>the</strong>oretical, conceptions <strong>of</strong> <strong>time</strong>, i.e., <strong>of</strong> <strong>the</strong> A-series,<br />
cannot be trusted. The pro<strong>of</strong> <strong>of</strong> this comes from <strong>the</strong> fact that our own experience <strong>of</strong> <strong>time</strong><br />
in <strong>the</strong> actual <strong>world</strong> as something that lapses might well be indistinguishable from how one<br />
would perceive "<strong>time</strong>" in <strong>the</strong> Godel universe, in which intuitive <strong>time</strong>, which lapses, is<br />
provably absent. If a form <strong>of</strong> experience is compatible with both a <strong>the</strong>sis <strong>and</strong> its anti<strong>the</strong>sis,<br />
it cannot be taken as reliable testimony for ei<strong>the</strong>r. The fact, <strong>the</strong>n, that <strong>the</strong> <strong>the</strong>ory <strong>of</strong><br />
relativity fails to account for <strong>the</strong> deliverances <strong>of</strong> our everyday experience <strong>of</strong> <strong>time</strong><br />
suggested to Godel not that <strong>Einstein</strong>'s <strong>the</strong>ory is incomplete, but ra<strong>the</strong>r that our sense <strong>of</strong><br />
intuitive <strong>time</strong> is founded on a misunderst<strong>and</strong>ing or misapprehension. In <strong>the</strong> clash between<br />
<strong>Einstein</strong> <strong>and</strong> everyday experience, it is experience that has to yield.<br />
Such an answer to <strong>the</strong> fundamental question <strong>of</strong> Godel's asymmetrical responses to his two<br />
incompletenesses has not heret<strong>of</strong>ore been proposed, for <strong>the</strong> very simple reason that <strong>the</strong><br />
question itself has never been raised. The failure <strong>of</strong> his contemporariesó<strong>and</strong> oursóto<br />
appreciate what Godel has accomplished with his <strong>Einstein</strong>ian inheritance is a sad tale<br />
indeed. Rarely have so many understood so little about so much. Godel's "detour" into<br />
relativity has been dismissed as a bit <strong>of</strong> intellectual dabbling by someone outside his field<br />
<strong>and</strong> out <strong>of</strong> his depth. No one has seen this work for what it was: a continued development<br />
<strong>of</strong> Godel's program <strong>of</strong> probing <strong>the</strong> limits <strong>of</strong> formal methods in capturing intuitive concepts,<br />
a move from <strong>the</strong> big "T" <strong>of</strong> ma<strong>the</strong>matical truth to <strong>the</strong> little "t" <strong>of</strong> relativistic <strong>time</strong>. As a<br />
consequence <strong>of</strong> this failure, no one asked why Godel's responses to his incompleteness<br />
results in <strong>the</strong> two cases were diametrically opposed.<br />
The details <strong>of</strong> Godel's conclusions about little "t" were also neglected. Cosmologists<br />
questioned whe<strong>the</strong>r <strong>the</strong> possibility <strong>of</strong> <strong>time</strong> travel in <strong>the</strong> nonexp<strong>and</strong>ing Godel universe was<br />
consistent with relativity, but made little note <strong>of</strong> <strong>the</strong> primary purpose for which he had<br />
constructed<br />
<strong>the</strong>se <strong>world</strong> models, which was to show that since <strong>time</strong> travel was possible, <strong>time</strong> was not.<br />
And when it became clear that his new <strong>world</strong> models were indeed relativistically<br />
consistent, attention was diverted once more from <strong>the</strong> essential to <strong>the</strong> inessential: now<br />
cosmologists asked whe<strong>the</strong>r <strong>the</strong> actual <strong>world</strong> is an exp<strong>and</strong>ing Godel universe. The<br />
foundation <strong>of</strong> Godel's case for temporal idealism, his modal argument from <strong>the</strong> possibility<br />
<strong>of</strong> <strong>the</strong> Godel universe to <strong>the</strong> nonexistence <strong>of</strong> <strong>time</strong> in <strong>the</strong> actual <strong>world</strong>, disappeared from<br />
sight.<br />
Who's Kurt Godel?
Though misunderstood <strong>and</strong> underappreciated, Godel's birthday present for <strong>Einstein</strong> did<br />
attract some immediate attention, if not from philosophers <strong>the</strong>n at least from <strong>the</strong><br />
guardians <strong>of</strong> relativity. Of <strong>the</strong> two great <strong>the</strong>ories <strong>of</strong> modern physics, general relativity is<br />
clearly <strong>the</strong> philosophical cousin, leading naturally to speculation on <strong>the</strong> origin, shape <strong>and</strong><br />
fate <strong>of</strong> <strong>the</strong> universeóa highly <strong>the</strong>oretical, not to say metaphysical, preoccupation <strong>of</strong><br />
philosophers from <strong>time</strong> immemorialówhereas quantum mechanics has immediate<br />
implications for technology <strong>and</strong> practice, from lasers <strong>and</strong> microchips to <strong>the</strong> whole panoply<br />
<strong>of</strong> information-<strong>the</strong>oretic hardware. Within <strong>the</strong> confines <strong>of</strong> general relativity itself,<br />
moreover, <strong>the</strong> question <strong>of</strong> <strong>time</strong> represents an especially elusive philosophical corner. What<br />
to do with <strong>time</strong> in special relativity is easy (if you know what to look for); what to do with<br />
it in general relativity is something else entirely. Since Godel's discoveries concerned an<br />
even more isolated niche <strong>of</strong> this already remote cornerónamely, speculations about<br />
geometrically extreme cosmologies with bizarre chronological consequences, not to<br />
mention Godel's even more arcane philosophical reflections based on <strong>the</strong>se monstrous<br />
modelsóit was to be expected that <strong>the</strong> small ripple raised by his rarefied achievements<br />
would soon fade away.<br />
Into this quiet pond, one spring day decades later, stepped <strong>the</strong> physicist John Wheeler, a<br />
colleague <strong>of</strong> Godel's at Princeton. He was<br />
with his friends <strong>and</strong> fellow physicists Kip Thorne <strong>and</strong> Charles Misner, with whom he was<br />
completing what would become one <strong>of</strong> <strong>the</strong> great texts in general relativity, called simply<br />
Gravitation. The sunshine beckoned <strong>and</strong> <strong>the</strong> three betook <strong>the</strong>mselves across campus to <strong>the</strong><br />
grassy knolls <strong>of</strong> <strong>the</strong> institute, <strong>the</strong>re to meet "Wheeler's friend Kurt Godel. The warmth <strong>of</strong><br />
<strong>the</strong> day notwithst<strong>and</strong>ing, <strong>the</strong> old logician was found wrapped in his overcoat, <strong>the</strong> electric<br />
heater in his <strong>of</strong>fice turned on. Wheeler <strong>and</strong> friends had a question. Could Godel shed light<br />
on <strong>the</strong> relationship between his incompleteness <strong>the</strong>orem <strong>and</strong> Heisenberg's uncertainty<br />
principle? No. For Godel, it was bad taste even to pose such a question. Heisenberg's<br />
principle was <strong>the</strong> finest flower <strong>of</strong> <strong>the</strong> Copenhagen interpretation <strong>of</strong> quantum mechanics,<br />
itself <strong>the</strong> blue-eyed boy <strong>of</strong> positivism. It represented <strong>the</strong> high-water mark <strong>of</strong> indeterminism<br />
in physicsóin effect, a rejection <strong>of</strong> Leibniz's principle <strong>of</strong> sufficient reason, so beloved <strong>of</strong><br />
Godeló<strong>and</strong> <strong>the</strong> acme <strong>of</strong> irrealism in physical science. As such, it was <strong>the</strong> very thorn on <strong>the</strong><br />
rose for both <strong>Einstein</strong> <strong>and</strong> Godel. As Godel put it, "in physics . . . <strong>the</strong> possibility <strong>of</strong><br />
knowledge <strong>of</strong> objec-tivizable states <strong>of</strong> affairs is denied, <strong>and</strong> it is asserted that we must be<br />
content to predict <strong>the</strong> results <strong>of</strong> observations. This is really <strong>the</strong> end <strong>of</strong> all <strong>the</strong>oretical<br />
science in <strong>the</strong> usual sense." The incompleteness <strong>the</strong>orem, in contrast, was a definitive<br />
refutation <strong>of</strong> positivism. Its methods <strong>and</strong> formal conclusions, though positivistically<br />
acceptable, were <strong>of</strong> a piece with classical ma<strong>the</strong>matics. Moreover, <strong>the</strong> pro<strong>of</strong> itself, by<br />
Godel's lights, constituted strong evidence in favor <strong>of</strong> realism in ma<strong>the</strong>matics. To have<br />
suggested a connection or correlation between Heisenberg <strong>and</strong> Godel was a major faux<br />
pas.<br />
Godel notwithst<strong>and</strong>ing, however, Wheeler <strong>and</strong> his friends were not far <strong>of</strong>f <strong>the</strong> mark. It<br />
cannot be denied that <strong>the</strong>re are striking parallels between Godel's incompleteness <strong>and</strong><br />
Heisenberg's uncertainty (though tact should have counseled against pointing this out to<br />
Godel). For one thing, both thinkers were at pains to use methods that would be<br />
epistemologically acceptable to <strong>the</strong> most hard-headed positivist: formal systems in <strong>the</strong><br />
case <strong>of</strong> Godel's <strong>the</strong>orem, direct cm pirical observations in <strong>the</strong> case <strong>of</strong> Heisenberg's<br />
principle. Fur<strong>the</strong>r
more, each <strong>the</strong>orist drew ontological conclusions from epistemologi-cal premises,<br />
conclusions that established <strong>the</strong> intrinsic limitations <strong>of</strong> <strong>the</strong> epistemologically acceptable<br />
methods <strong>the</strong>y had employed. This form <strong>of</strong> argument is <strong>the</strong> very hallmark <strong>of</strong> positivism. It is<br />
also characterizes <strong>Einstein</strong>'s special <strong>the</strong>ory <strong>of</strong> relativity, a fact with which Heisenberg tried<br />
(unsuccessfully) to impress <strong>Einstein</strong>. That <strong>the</strong> conclusions Godel drew pointed to<br />
ma<strong>the</strong>matical realism, while Heisenberg made <strong>the</strong> case for physical irrealism, does not<br />
alter <strong>the</strong> fact that both thinkers blazed an ontological trail through <strong>the</strong> thickets <strong>of</strong><br />
epistemology, <strong>and</strong> that each inaugurated <strong>the</strong>reby an intellectual revolution whose full<br />
implications are yet to be realized. Not for nothing did Godel's colleague at <strong>the</strong> institute,<br />
Freeman Dyson, remark that "<strong>the</strong> two great conceptual revolutions <strong>of</strong> twentieth century<br />
science [are] <strong>the</strong> overturning <strong>of</strong> classical physics by Heisenberg <strong>and</strong> <strong>the</strong> overturning <strong>of</strong> <strong>the</strong><br />
foundations <strong>of</strong> ma<strong>the</strong>matics by Godel."<br />
Now Godel himself had a question. Would <strong>the</strong>re be a discussion in <strong>the</strong>ir new text <strong>of</strong> <strong>the</strong><br />
rotating universes he had discovered in relativity? No. Godel was disappointed. He was still<br />
seeking to discover whe<strong>the</strong>r <strong>the</strong> actual <strong>world</strong> is a (exp<strong>and</strong>ing) rotating Godel universe. The<br />
evidence for universal rotation, should it exist, would be found in <strong>the</strong> axes <strong>of</strong> rotation <strong>of</strong><br />
<strong>the</strong> surrounding galaxies. Wheeler was taken aback by <strong>the</strong> practical astronomical<br />
preoccupations <strong>of</strong> <strong>the</strong> great logician. Godel, he noted, "had taken down <strong>the</strong> great Hubble<br />
photographic atlas <strong>of</strong> <strong>the</strong> galaxies, lined up a ruler on each galactic image to estimate <strong>the</strong><br />
galaxy's axis <strong>of</strong> rotation, <strong>and</strong> compiled statistics <strong>of</strong> <strong>the</strong> orientation." The results, however,<br />
were negative.<br />
That Godel had made discoveries about rotating universes in general relativity had been<br />
known to Wheeler for many years. He was present in 1949 when Godel lectured on <strong>the</strong><br />
subject at <strong>Einstein</strong>'s seventieth birthday celebration. Yet he too, despite his impressive<br />
credentials, seems to have misunderstood what Godel was saying. "In a universe with an<br />
overall rotation," he wrote, attempting to summarize Godel's lecture, ". . . <strong>the</strong>re could<br />
exist <strong>world</strong> lines (space-<strong>time</strong> histories) that closed up in loops. In such a universe, one<br />
could, in principle, live one's life over<br />
<strong>and</strong> over again." Wheeler, unfortunately, has conflated a temporal circle with a cycle,<br />
precisely missing <strong>the</strong> force <strong>of</strong> Godel's conclusion that <strong>the</strong> possibility <strong>of</strong> closed, futuredirected,<br />
<strong>time</strong>like curves, i.e., <strong>time</strong> travel, proves that space-<strong>time</strong> is a space, not a <strong>time</strong><br />
in <strong>the</strong> intuitive sense. Whereas a circle is a figure in space, a cycle is a journey undertaken<br />
along a circular path, one that can be repeated, in Wheeler's words, "over <strong>and</strong> over again."<br />
Exactly how many <strong>time</strong>s, one wants to ask Wheeler, is <strong>the</strong> journey supposed to be<br />
repeated? The question clearly cannot be answered, since <strong>the</strong> <strong>time</strong> traveler's journey is not<br />
over <strong>time</strong>, along <strong>the</strong> closed <strong>time</strong>like curve: it is <strong>the</strong> curve itself. Just as one cannot ask <strong>of</strong><br />
a circle how many <strong>time</strong>s <strong>the</strong> points that constitute that figure have gone around, one<br />
cannot sensibly ask how <strong>of</strong>ten <strong>the</strong> <strong>time</strong> traveler in <strong>the</strong> Godel universe has made his or her<br />
trip.<br />
Wheeler should have known better. As he himself pointed out, an "unsettling consequence<br />
<strong>of</strong> <strong>Einstein</strong>'s 1905 special <strong>the</strong>ory <strong>of</strong> relativity is that <strong>time</strong> is relative." And not just relative,<br />
but "static," for "<strong>the</strong> o<strong>the</strong>r thing that special relativity did for <strong>time</strong> is join it with space into<br />
<strong>the</strong> four-dimensional entity space-<strong>time</strong> . .. [<strong>and</strong>] a consequence <strong>of</strong> this new space-<strong>time</strong><br />
view is that motion through <strong>time</strong>, or motion <strong>of</strong> <strong>time</strong> ... is replaced by static <strong>time</strong>." But, as<br />
Godel showed, a <strong>time</strong> that is relative or static is no <strong>time</strong> at all. Wheeler seems reluctant to
call a spade a spade. Yet he does entitle his chapter "The End <strong>of</strong> Time," so perhaps he<br />
does, after all, recognize this. Not at all. What Wheeler means by "<strong>the</strong> end <strong>of</strong> <strong>time</strong>" is not<br />
that it disappears in <strong>Einstein</strong>'s <strong>the</strong>ory as a consequence <strong>of</strong> being relative <strong>and</strong> static, but<br />
ra<strong>the</strong>r that, as he sees it, when <strong>the</strong> "Big Crunch" comes, after <strong>the</strong> "Big Bang," <strong>time</strong> will<br />
come to an end. "There was no 'before' <strong>the</strong> Big Bang," he writes, "<strong>and</strong> <strong>the</strong>re will be no<br />
'after' after <strong>the</strong> Big Crunch." Moreover, "every black hole brings an end to <strong>time</strong> <strong>and</strong> space<br />
... as surely as <strong>the</strong> Big Crunch will bring an end to <strong>the</strong> universe as a whole." What Godel<br />
has seen, it seems, Wheeler has not.<br />
A year after he introduced Misner <strong>and</strong> Thorne to Godel, Wheeler found himself in <strong>the</strong><br />
<strong>of</strong>fice <strong>of</strong> a colleague, <strong>the</strong> cosmologist James Peebles. In walked Peebles' student Dan<br />
Hawley, announcing that he had<br />
just completed his dissertation on <strong>the</strong> question <strong>of</strong> a preferred rotation among <strong>the</strong> galaxies.<br />
Godel, Wheeler commented, would be pleased. "Who's Godel?" asked Hawley. "The greatest<br />
logician since Aristotle," Wheeler replied. And much more. A phone call to Godel allowed<br />
Wheeler to apprise <strong>the</strong> greatest logician since Aristotle <strong>of</strong> <strong>the</strong> new work being done in<br />
Princeton on <strong>the</strong> rotation <strong>of</strong> <strong>the</strong> galaxies. Godel's queries, however, were soon too<br />
dem<strong>and</strong>ing for <strong>the</strong> physicist, so Wheeler h<strong>and</strong>ed him over to <strong>the</strong> student <strong>of</strong> cosmology. The<br />
questions quickly exhausted him, too, so <strong>the</strong> phone was passed yet again, this rime to<br />
Peebles. When <strong>the</strong> conversation finally concluded, <strong>the</strong>re was just one thing Peebles had to<br />
say: "My, I wish we had talked to him before we started this work."<br />
Though <strong>the</strong> <strong>world</strong> at large had not yet taken note <strong>of</strong> what Godel had accomplished in<br />
<strong>Einstein</strong>'s backyard, <strong>the</strong>re were rumblings among <strong>the</strong> cosmologists that something new was<br />
brewing. Just what this was, however, would remain hidden for years to come. That a<br />
noted cos-mologist was moved as recently as <strong>the</strong> 1990s to protect chronology from <strong>the</strong><br />
Godel universe suggests that <strong>the</strong> <strong>world</strong> is still not ready for Godel. Yet <strong>the</strong> mere fact that<br />
as distinguished a <strong>the</strong>orist as Stephen Hawking believed protection was needed, combined<br />
with <strong>the</strong> fact that his chronology protection conjecture has so far failed to attract a<br />
significant number <strong>of</strong> adherents, suggests that readiness may be near. The Zeitgeist, as<br />
Godel noted, has its own <strong>time</strong> <strong>and</strong> agenda.<br />
8 Twilight <strong>of</strong> <strong>the</strong> Gods<br />
We live in a <strong>world</strong> in which ninety-nine per cent <strong>of</strong> all beautiful things are destroyed in <strong>the</strong><br />
hud.<br />
KURT GODEL<br />
It all began with geometry. "Those ignorant <strong>of</strong> geometry shall not enter," Plato had<br />
inscribed over <strong>the</strong> entrance to his academy. In a passage admired by Godel, he says in<br />
Book 6 <strong>of</strong> The Republic that when students <strong>of</strong> geometry "make use <strong>of</strong> <strong>the</strong> visible forms [<strong>of</strong><br />
geometric figures] <strong>and</strong> reason about <strong>the</strong>m, <strong>the</strong>y are in fact thinking not <strong>of</strong> <strong>the</strong>se but <strong>of</strong> <strong>the</strong>
ideals [i.e., "ideas" or forms] <strong>the</strong>y resemble." Thus was <strong>the</strong> path cleared for Euclid, who<br />
succeededónot perfectly, as Kant thought, but to a considerable extentóin capturing those<br />
geometric forms in a system <strong>of</strong> axioms that remains <strong>the</strong> paradigm <strong>of</strong> <strong>the</strong>oretical<br />
knowledge, in ma<strong>the</strong>matics <strong>and</strong> logic no less than in physics.<br />
<strong>Einstein</strong>, who had <strong>the</strong> courage to employ an alternative to Euclid's system to describe <strong>the</strong><br />
actual <strong>world</strong>, was one <strong>of</strong> <strong>the</strong> first to grasp <strong>the</strong> difference between geometry as a formal<br />
science <strong>of</strong> deduction <strong>and</strong> geometry as an empirical account <strong>of</strong> physical space, a distinction<br />
he elaborated with gusto in his essay "Geometry <strong>and</strong> Experience." He had begun to follow<br />
in <strong>the</strong> footsteps <strong>of</strong> <strong>the</strong> Greek philosophers early in life, when his youthful imagination was<br />
captured by his "holy geometry booklet." Geometry, too, was <strong>the</strong> secret password for<br />
entrance to <strong>the</strong> Godel universe, a password <strong>Einstein</strong> himself was hesitant to invoke, yet it<br />
was also <strong>the</strong> key <strong>Einstein</strong> himself would employ<br />
to unlock <strong>the</strong> secrets to his unified field <strong>the</strong>ory, a key no one but he cared to turn.<br />
"<strong>Einstein</strong>'s now ab<strong>and</strong>oned dream <strong>of</strong> a geometrical unification <strong>of</strong> <strong>the</strong> forces <strong>of</strong> nature" is<br />
how John Wheeler described it decades later, in 1980.<br />
Everything Is Something Else<br />
Turning to geometry one more <strong>time</strong> for <strong>the</strong> solution to his final problem was for <strong>Einstein</strong> a<br />
case <strong>of</strong> going home with <strong>the</strong> girl he had brought to <strong>the</strong> dance. In creating relativity <strong>the</strong>ory,<br />
he had inaugurated <strong>the</strong> geometriza-tion <strong>of</strong> physics. The mysterious limit velocity <strong>of</strong> light<br />
was to be accounted for not by ad hoc mechanical devices like <strong>the</strong> strange shrinking<br />
behavior <strong>of</strong> measuring apparatuses, but ra<strong>the</strong>r by <strong>the</strong> geometrical structure <strong>of</strong> space-<strong>time</strong><br />
itself, a structure that has <strong>the</strong> limiting velocity <strong>of</strong> electromagnetic signals built into its<br />
very definition. Similarly, <strong>the</strong> force <strong>of</strong> gravity was explained not, as with Newton, as a<br />
mysterious instantaneous action at a distance that moves through an even more mysterious<br />
<strong>world</strong>-filling yet invisible substance known as e<strong>the</strong>r, but ra<strong>the</strong>r by <strong>the</strong> geometrical device<br />
<strong>of</strong> <strong>the</strong> curvature <strong>of</strong> space-<strong>time</strong>. Time itself had been tamedóor so it seemedóby its<br />
transformation into space, into <strong>the</strong> temporal component <strong>of</strong> four-dimensional space-<strong>time</strong>.<br />
In his way, <strong>Einstein</strong> turned out to be no less an ironist than Godel. Everything is really<br />
something else: <strong>time</strong> is really space, gravity is really geometrical curvature, energy is<br />
really mass. How can one not love such a thinker?<br />
That <strong>Einstein</strong> <strong>and</strong> Godel would meet on <strong>the</strong> field <strong>of</strong> geometry was altoge<strong>the</strong>r fitting. The<br />
parallel lines <strong>of</strong> <strong>the</strong>ir careers converged on <strong>the</strong> Godel universe, at once Godel's birthday<br />
present for his best friend <strong>and</strong> his entrance into <strong>Einstein</strong>'s arena <strong>of</strong> battle. Godel had<br />
carried <strong>Einstein</strong>'s geometrization <strong>of</strong> <strong>time</strong> to a surprising conclusion, forcing us to ques tion<br />
not just <strong>the</strong> truth but <strong>the</strong> very meaning <strong>of</strong> <strong>the</strong> <strong>Einstein</strong>ian starting point. The two walking<br />
companions had marched so far ahead <strong>of</strong> <strong>the</strong> rest <strong>of</strong> us that no one could tell whe<strong>the</strong>r it<br />
was <strong>the</strong>y or we who were<br />
147
lost. Godel's writings on <strong>Einstein</strong> did, however, provide cover for <strong>the</strong> attempt by Godel's<br />
friends to cheer <strong>the</strong> great logician, whose health, shortly after he completed his<br />
contribution to <strong>the</strong> Schilpp volume, became imperiled by a bleeding duodenal ulcer.<br />
Promotion to full pr<strong>of</strong>essor would not come until later, when <strong>the</strong> objections <strong>of</strong> some <strong>of</strong> his<br />
colleagues could be overcome. ("One crazy man [himself] on <strong>the</strong> faculty" is enough, said<br />
<strong>the</strong> ma<strong>the</strong>matician C.L. Siegel.) A solution was found when institute director<br />
Oppenheimer, who was on <strong>the</strong> selection committee for <strong>the</strong> first <strong>Einstein</strong> Awardóto be<br />
presented every three years, on <strong>the</strong> physicist's birthdayósuggested that it would be fitting<br />
to divide <strong>the</strong> honor between Godel <strong>and</strong> Julian Schwinger, <strong>the</strong> physicist from Harvard who<br />
would soon earn a Nobel Prize for his work in quantum electrodynamics. Thus on March 14,<br />
1951, <strong>Einstein</strong>'s birthday, after Von Neumann had delivered a brief speech in which he<br />
described Godel's work as "a l<strong>and</strong>mark which will remain visible far in space <strong>and</strong> <strong>time</strong>," <strong>the</strong><br />
aging physicist was able to return <strong>the</strong> favor <strong>of</strong> Godel's birthday gift <strong>of</strong> 1949 by personally<br />
h<strong>and</strong>ing his good friend <strong>the</strong> first <strong>Einstein</strong> Award. ("You deserve it," he said to Schwinger;<br />
"you don't need it," he remarked to Godel, who needed it most <strong>of</strong> all.)<br />
This was <strong>the</strong> first formal academic honor Godel had ever received. In due course he would<br />
get o<strong>the</strong>rs, including honorary degrees from Yale <strong>and</strong> Harvard (though not from Princeton;<br />
<strong>the</strong> invitation came too late) <strong>and</strong> <strong>the</strong> National Medal <strong>of</strong> Science, but he had already begun<br />
to withdraw from academic <strong>and</strong> social life. When he delivered <strong>the</strong> prestigious Gibbs<br />
Lecture that same year, 1951, before <strong>the</strong> American Ma<strong>the</strong>matical Society, <strong>the</strong> only logician<br />
ever to do so, it was <strong>the</strong> last talk he would ever deliver to a ma<strong>the</strong>matical audience, <strong>and</strong><br />
one <strong>of</strong> <strong>the</strong> last he would even attend. He had never, in any case, found much pr<strong>of</strong>it in<br />
attending formal talks. "I never go to lectures," he said, "because I have difficulty in<br />
following <strong>the</strong>m, even if I am well acquainted with <strong>the</strong> subject matter." For <strong>the</strong> remainder<br />
<strong>of</strong> his life, he would publish no new essays, nor even <strong>the</strong> Gibbs Lecture, which appeared<br />
only posthumously. His sun rose <strong>and</strong> set at <strong>the</strong> same moment.<br />
The Need for Roots<br />
For <strong>Einstein</strong>, <strong>the</strong> darkness had long since descended. After presenting his award to Godel,<br />
he had few years left to live. No new conquests graced his final decades. His last gr<strong>and</strong><br />
geometrical move, like Godel's, had been followed by no one. He became preoccupied with<br />
<strong>the</strong> attempt to mitigate <strong>the</strong> very forces he had helped set into motion: <strong>the</strong> positivism,<br />
inspired by Mach, that had set <strong>the</strong> stage for special relativity; <strong>the</strong> bold decision to take<br />
Planck's quantum as a genuine aspect <strong>of</strong> reality ra<strong>the</strong>r than a mere calculating device; <strong>the</strong><br />
courageous proposal, following Boltzmann, that probability be taken seriously in physics;<br />
<strong>the</strong> seminal work in forging a new quantum mechanics. Politically, he provided bookends<br />
to <strong>the</strong> sudden intrusion <strong>of</strong> physics into global politics. On one side was his recommendation<br />
to FDR that nuclear power, in <strong>the</strong> form <strong>of</strong> a bomb, be exploited to defeat his former<br />
homel<strong>and</strong>. On <strong>the</strong> o<strong>the</strong>róhis final act on <strong>the</strong> public stageówas his signature on a manifesto<br />
written by Bertr<strong>and</strong> Russell dem<strong>and</strong>ing <strong>world</strong>wide nuclear disarmament. His white whale,<br />
however, remained his never-ending, never-succeeding search for a unified field <strong>the</strong>ory,<br />
toge<strong>the</strong>r with his attempt to find a philosophical flaw in <strong>the</strong> Copenhagen interpretation <strong>of</strong><br />
quantum mechanics. Like Ahab, he took <strong>the</strong> hunt personally <strong>and</strong> was fully prepared to go<br />
down with <strong>the</strong> ship.<br />
Increasingly, he withdrew from <strong>the</strong> physics establishment to pursue <strong>the</strong> great beast in<br />
isolation. After he arrived at <strong>the</strong> institute, he never visited Europe again. He never drove a
car <strong>and</strong> never flew. His circle <strong>of</strong> friends diminished, with Godel <strong>the</strong> brightest star in his<br />
shrinking firmament. Never again would he enjoy <strong>the</strong> intellectual camaraderie that had<br />
formed a cloak against all <strong>the</strong> ugliness that beset his years in Berlin. Toward <strong>the</strong> end <strong>of</strong> his<br />
life he confessed that his strongest personal ties, including those to his wife <strong>and</strong> children,<br />
had all been failures. When his wife Elsa's daughter from a previous mar-riage, Useówhom<br />
he had once thought <strong>of</strong> marryingólay dying <strong>of</strong> cancer in Paris in 1934 at <strong>the</strong> tender age <strong>of</strong><br />
thirty-seven, he declined to accompany his wife ro attend to her. His first wife, Mileva<br />
Marie<br />
<strong>Einstein</strong>, died alone in Zurich, desperately unhappy, unreconciled with <strong>the</strong> man who had<br />
left her. His daughter with her, Lieserl, born out <strong>of</strong> wedlock, disappeared into <strong>the</strong> mists <strong>of</strong><br />
<strong>time</strong>. His gifted first son, Eduard, became schizophrenic <strong>and</strong> was deposited in a psychiatric<br />
clinic, where he remained for <strong>the</strong> rest <strong>of</strong> his life, unvisited by his fa<strong>the</strong>r. His second son,<br />
Hans Albert, always distant, remained so after he too emigrated to America. And <strong>the</strong><br />
second marriage, finally, <strong>of</strong> <strong>Einstein</strong>, like <strong>the</strong> first, was no success, though it did at least<br />
provide a slender root in an o<strong>the</strong>rwise rootless existence. Its removal in 1936, yet ano<strong>the</strong>r<br />
deracination, deeply affected him, surprising Elsa. "I never thought he loved me so much,"<br />
she told her friend Antonina Vallentin, "<strong>and</strong> that comforts me." Sad words, indeed, from a<br />
dying spouse. With Elsa's death his personal universe collapsed in on itself.<br />
For <strong>the</strong> remaining years <strong>of</strong> his life his most visceral human connection was to what he<br />
called his "tribe," his fellow Jews, <strong>the</strong> deepest root <strong>of</strong> this rootless man (though he<br />
somehow never managed to plant himself in <strong>the</strong> l<strong>and</strong> <strong>of</strong> Zion). "My relationship to <strong>the</strong><br />
Jewish people," he wrote, "has become my strongest human bond." Why he chose this for<br />
his fundamental human tie remains to be explained. One suspects that <strong>the</strong> French<br />
philosopher Simone Weil's dark study, The Need for Roots, contains greater hints than are<br />
found in <strong>the</strong> st<strong>and</strong>ard literature, which has difficulty acknowledging <strong>the</strong> degree <strong>of</strong> discord<br />
between <strong>Einstein</strong>'s self-confessed "tribalism" <strong>and</strong> his lifelong commitment to rationality <strong>and</strong><br />
internationalism.<br />
When <strong>the</strong> end finally came at <strong>the</strong> age <strong>of</strong> seventy-six, <strong>Einstein</strong> could not help feeling<br />
embarrassed at <strong>the</strong> larger-than-life icon he had become. In March 1955, a month before he<br />
took his final voyage, he confided to his long-<strong>time</strong> friend, Queen Elisabeth <strong>of</strong> Belgium, that<br />
"<strong>the</strong> exaggerated esteem in which my lifework is held makes me very ill at ease. I feel<br />
compelled to think <strong>of</strong> myself as an involuntary swindler." The day before he died, he<br />
requested his latest version <strong>of</strong> unification <strong>the</strong>ory <strong>and</strong> proceeded to make some<br />
calculations. He did not put up a light to remain living. "It is tasteless to prolong life<br />
artificially," he told Helen Dukas; "I have done my share, it is <strong>time</strong> to go."<br />
The Absence <strong>of</strong> <strong>the</strong> Muses<br />
According to <strong>the</strong> ma<strong>the</strong>matician Stanislaw Ulam, Godel too, toward <strong>the</strong> end <strong>of</strong> his life,<br />
nurtured fears that his contribution had been overestimated. Godel, Ulam said, had "a<br />
gnawing uncertainty that maybe all he had discovered was ano<strong>the</strong>r paradox a la Burali-<br />
Forte or Russell." Godel himself, however, denied this. "Ulam wrote a book . . . [in which<br />
he] says that perhaps I was never sure whe<strong>the</strong>r I had merely detected ano<strong>the</strong>r paradox like<br />
Burali-Forte's. This is absolutely false. Ulam doesn't underst<strong>and</strong> my result." Since Ulam is<br />
not here to defend himself, we cannot determine whe<strong>the</strong>r it is his memory or Godel's that
is at fault. O<strong>the</strong>rs confirm, however, that Godel, like o<strong>the</strong>r distinguished thinkers who<br />
joined <strong>the</strong> institute, occasionally wondered whe<strong>the</strong>r he had done enough to justify his<br />
appointment.<br />
Like <strong>Einstein</strong>, Godel had led a life <strong>of</strong> increasing isolation <strong>and</strong> reclu-siveness since coming to<br />
<strong>the</strong> institute, a tendency that only increased after he received <strong>the</strong> <strong>Einstein</strong> Award <strong>and</strong><br />
delivered <strong>the</strong> Gibbs Lecture. He too spent his final years in a lost cause, part formal <strong>and</strong><br />
part philosophical, searching for new axioms to decide <strong>the</strong> continuum hypo<strong>the</strong>sis (<strong>and</strong> thus<br />
settle <strong>the</strong> question <strong>of</strong> whe<strong>the</strong>r <strong>the</strong>re is an infinity between <strong>the</strong> number <strong>of</strong> points on a line<br />
<strong>and</strong> <strong>the</strong> cardinality <strong>of</strong> <strong>the</strong> natural numbers), <strong>and</strong> seeking a definitive refutation <strong>of</strong> <strong>the</strong><br />
<strong>the</strong>sisóbolstered by Cohen's independence result for <strong>the</strong> continuum hypo<strong>the</strong>sisóthat <strong>the</strong><br />
results <strong>of</strong> ma<strong>the</strong>matics are in some sense only <strong>the</strong> reflection <strong>of</strong> human convention. This<br />
was a <strong>the</strong>me he pursued in his Gibbs Lectureóin which he invoked his own incompleteness<br />
<strong>the</strong>orem as evidence for his Platonismóas well as in his contribution to yet ano<strong>the</strong>r Schilpp<br />
volume, devoted this <strong>time</strong> to Rudolf Carnap, his old friend <strong>and</strong> foe from <strong>the</strong> Vienna Circle.<br />
But he was never satisfied with this essay <strong>and</strong> did not allow it to be published.<br />
Godel <strong>and</strong> <strong>Einstein</strong>, two great thinkers each <strong>of</strong> whose earlier years had been marked by a<br />
string <strong>of</strong> successes that left <strong>the</strong>ir contemporaries breathless, spent <strong>the</strong>ir final decades in a<br />
doomed commitment to lost<br />
causes. What happened? Why were <strong>the</strong> muses absent during <strong>the</strong> twilight <strong>of</strong> <strong>the</strong> gods?<br />
No one, <strong>of</strong> course, has ever divined <strong>the</strong> secret <strong>of</strong> <strong>the</strong> muse (else we would all become<br />
Mozart), but we can never<strong>the</strong>less note certain salient factors in <strong>the</strong> striking lack <strong>of</strong> success<br />
<strong>Einstein</strong> <strong>and</strong> Godel enjoyed in <strong>the</strong>ir final years. What does it take to make a great<br />
scientific discovery? Two elements are crucial. One must have an insight into which<br />
problems are ripe for resolution, <strong>and</strong> one must <strong>the</strong>n have <strong>the</strong> craftóor invent itóto solve<br />
<strong>the</strong> problem one has had <strong>the</strong> audacity to recognize as solvable. Both elements, clearly,<br />
were present in <strong>Einstein</strong>'s success with relativity <strong>the</strong>ory <strong>and</strong> his early work in quantum<br />
mechanics. With regard to <strong>the</strong> first element, his biographer has pointed out that "in 1905,<br />
work on spectral lines could not have gone beyond an attempt at a phenomenological<br />
interpretation, even for <strong>Einstein</strong>. The fact that he did not attempt it show[ed] him to be a<br />
master <strong>of</strong> <strong>the</strong> art <strong>of</strong> <strong>the</strong> soluble." Both were prominent in his discovery <strong>of</strong> general<br />
relativity. No less a figure than Planck himself warned <strong>Einstein</strong> that it was hubris to<br />
attempt to rethink gravity after three hundred years <strong>of</strong> Newton. And <strong>the</strong> technique<br />
required to forge <strong>the</strong> new <strong>the</strong>ory turned out, unlike special relativity, to require highly<br />
nontrivial ma<strong>the</strong>matics that strained <strong>Einstein</strong>'s formal capacities almost to <strong>the</strong> breaking<br />
point.<br />
For Godel, too, each element had been present. In <strong>the</strong> incompleteness <strong>the</strong>orem, he<br />
understood that it was possible to test <strong>the</strong> limitations <strong>of</strong> formal systems, undermining <strong>the</strong><br />
confidence in purely deductive methods, inaugurated by Euclid, that had held sway for<br />
more than two thous<strong>and</strong> years. Finding, or ra<strong>the</strong>r creating, <strong>the</strong> methods needed to resolve<br />
this now solvable problemóincluding Godel numbering <strong>and</strong> <strong>the</strong> arithmetization <strong>of</strong><br />
metama<strong>the</strong>maticsówas perhaps GodePs chief boast. The continuum hypo<strong>the</strong>sis, too, which<br />
had defeated its inventor, Cantor, yielded (at least in part) to Godel's realization that its
consistency with <strong>the</strong> axioms <strong>of</strong> set <strong>the</strong>ory could now be settled, if one had <strong>the</strong> ingenuity to<br />
employ <strong>the</strong> new set-<strong>the</strong>oretical techniques Godel had managed to cook up.<br />
For o<strong>the</strong>r distinguished scientists, too, <strong>the</strong> same pattern for scientific discovery holds. The<br />
secret to James Watson <strong>and</strong> Francis Crick's discovery <strong>of</strong> <strong>the</strong> structure <strong>of</strong> DNA (leaving aside<br />
<strong>the</strong> small matter <strong>of</strong> Rosalind Franklin's desk drawer) was as much <strong>the</strong>ir realizationóalmost<br />
unique at <strong>the</strong> <strong>time</strong>óthat <strong>the</strong> problem was now solvable, as it was <strong>the</strong>ir technical<br />
competence in fitting toge<strong>the</strong>r all <strong>the</strong> pieces that lay scattered about, unconnected, in<br />
various workshops. In <strong>the</strong>ir case, <strong>the</strong> first step was probably <strong>the</strong> crucial one. They were not<br />
<strong>the</strong> only scientists equipped to solve <strong>the</strong> riddle once it was understood to be solvable; <strong>the</strong><br />
great Linus Pauling was pursuing <strong>the</strong> problem with equal fervor but, unfortunately for him,<br />
<strong>the</strong> wrong idea. This partly explains Watson <strong>and</strong> Crick's frantic anxiety to find a solution at<br />
breakneck speed <strong>and</strong> <strong>the</strong>ir obliviousness to <strong>the</strong> niceties <strong>of</strong> pr<strong>of</strong>essional ethics.<br />
More recently still, Andrew Wiles's dramatic solution to <strong>the</strong> problem <strong>of</strong> Fermat's last<br />
<strong>the</strong>orem also conforms to <strong>the</strong> pattern. Wiles himself has written <strong>of</strong> <strong>the</strong> moment he<br />
realized that <strong>the</strong> <strong>the</strong>orem could now be proved, that we could really get <strong>the</strong>re from here,<br />
<strong>and</strong> that apart from <strong>the</strong> usual cranks, he was alone in knowing this. Secreting himself away<br />
in his attic for years, he was able finally to bring forth <strong>the</strong> second element, <strong>the</strong> virtuoso<br />
technical methods with which all <strong>the</strong> pieces <strong>of</strong> <strong>the</strong> puzzle uncovered by his great<br />
predecessors could finally be sewn toge<strong>the</strong>r. (Even Wiles, however, it turned out, could<br />
not put all <strong>the</strong> pieces toge<strong>the</strong>r <strong>without</strong> help, after his initial pro<strong>of</strong> turned out to contain a<br />
flaw.)<br />
In <strong>the</strong> case <strong>of</strong> Godel <strong>and</strong> <strong>Einstein</strong>, <strong>the</strong>re is no indication that <strong>the</strong>ir ability to cook up <strong>the</strong><br />
second half <strong>of</strong> <strong>the</strong> recipe for scientific success had dwindled. Both men remained<br />
ma<strong>the</strong>matically nimble to <strong>the</strong> very end. It was with <strong>the</strong> first element that problems arose.<br />
They had simply bitten <strong>of</strong>f more than <strong>the</strong>y, or anyone else, could chew. To this day, half a<br />
century after <strong>Einstein</strong>'s failed efforts, we still do not have a clear path to <strong>the</strong> unification <strong>of</strong><br />
<strong>the</strong> very small with <strong>the</strong> very large, quantum mechanics with relativity. If, as some suspect,<br />
<strong>the</strong> most promising avenue lies in string <strong>the</strong>ory, with its exotic ma<strong>the</strong>matics <strong>of</strong> ten<br />
dimensions, <strong>the</strong>n <strong>Einstein</strong> clearly never bad a chance. There is simply no<br />
way even <strong>Einstein</strong> in his day could have dreamed up string <strong>the</strong>ory. As for quantum<br />
mechanics, while <strong>Einstein</strong>'s philosophical objections retain <strong>the</strong>ir power to haunt<br />
physicistsó"I cannot define <strong>the</strong> real problem," said Feynman in 1982, about <strong>the</strong> EPR<br />
paradox, "<strong>the</strong>refore I suspect <strong>the</strong>re's no real problem, but I'm not sure <strong>the</strong>re's no real<br />
problem"ó<strong>the</strong> final philosophical account <strong>of</strong> <strong>the</strong> nature <strong>of</strong> quantum reality for unreality)<br />
has yet to be written. Not only are we not <strong>the</strong>re yet; no one seems to know where we're<br />
going or how we will know when we get <strong>the</strong>re.<br />
Godel's lost causeófinding new axioms that will settle, in a convincing, non ad hoc,<br />
manner, <strong>the</strong> continuum hypo<strong>the</strong>sisóhas likewise seen little progress, ei<strong>the</strong>r to suggest<br />
wherein <strong>the</strong> answer lies or even to indicate whe<strong>the</strong>r <strong>the</strong>re will ever be a definitive answer.<br />
Godel had finally tackled a problem that was anything but ripe for resolution. Nor has<br />
<strong>the</strong>re been any breakthrough on settling <strong>the</strong> companion philosophical problem <strong>of</strong> <strong>the</strong><br />
extent to which ma<strong>the</strong>matics represents a reality independent <strong>of</strong> human convention. The<br />
appearance, posthumously <strong>and</strong> in different versions, <strong>of</strong> Godel's contribution to <strong>the</strong> Schilpp
volume on Carnap, entitled "Is Ma<strong>the</strong>matics Syntax <strong>of</strong> Language?", has revealed <strong>the</strong><br />
intricacies <strong>of</strong> Godel's attempts to settle accounts with his old friend from Vienna, but it<br />
has not adduced arguments that comm<strong>and</strong> universal assent. The essay represents a<br />
struggle more than a consummation.<br />
Sowing <strong>without</strong> Reaping<br />
Godel had received <strong>the</strong> invitation to contribute to <strong>the</strong> Carnap volume in 1953, just two<br />
years after he delivered his Gibbs Lecture, "Some <strong>Basic</strong> Theorems on <strong>the</strong> Foundations <strong>of</strong><br />
Ma<strong>the</strong>matics, <strong>and</strong> Their Implications." The common <strong>the</strong>me was to find a convincing<br />
argument in favor <strong>of</strong> Platonism <strong>and</strong> against conventionalism or formalism, using his<br />
incompleteness <strong>the</strong>orem as a powerful new weapon in <strong>the</strong> war. But Godel soon found that<br />
not even his superweapon could blast a shortcut<br />
through <strong>the</strong> tangled thickets <strong>of</strong> ma<strong>the</strong>matical ontology, an insight Wittgensteinómuch as<br />
his aims diverged from Godel'sóhad reached on his own years earlier, in his attempt to<br />
remove <strong>the</strong> spell that Godel's <strong>the</strong>orem had cast over philosophers. One might caricature<br />
Wittgenstein's conclusion in this way: whereas (parts <strong>of</strong>) ma<strong>the</strong>matics possess a beguiling<br />
symmetry, philosophy will always be to some extent messy <strong>and</strong> ugly. In his youth, he had<br />
gushed over <strong>the</strong> beauty <strong>of</strong> Russell's Prin-cipia Ma<strong>the</strong>matica, comparing it to music. In his<br />
later years, he took <strong>the</strong> o<strong>the</strong>r side, drawing attention to what he called <strong>the</strong> "motley" <strong>of</strong><br />
ma<strong>the</strong>matics. Godel, in contrast, kept faith that <strong>the</strong> beauty <strong>of</strong> ma<strong>the</strong>matics could be<br />
matched by philosophy. His contribution to <strong>the</strong> philosophy <strong>of</strong> <strong>time</strong> gave support for this<br />
belief, but he was well aware that his drafts <strong>of</strong> <strong>the</strong> Carnap paper, with <strong>the</strong>ir never-ending<br />
revisions, were anything but pretty. None, in his estimation, was worthy <strong>of</strong> publication. "In<br />
view <strong>of</strong> widely held prejudices," he finally wrote to Schilpp in 1959, "it may do more harm<br />
than good to publish half done work."<br />
The ontological project contra Carnap was never finished. Nor was his attempt to complete<br />
<strong>the</strong> o<strong>the</strong>r half <strong>of</strong> <strong>the</strong> philosophical coin, epistemology. Unlike o<strong>the</strong>r ma<strong>the</strong>matical<br />
Platonists such as Frege <strong>and</strong> Church, Godel understood <strong>the</strong> need to supply his<br />
ma<strong>the</strong>matical ontology with a convincing epistemology. Here he turned for help to Frege's<br />
contemporary <strong>and</strong> fellow philosopher, Edmund Husserl. He became a pr<strong>of</strong>ound (if unhappy)<br />
student <strong>of</strong> Husserl's recalcitrant texts, a project that consumed ever-greater amounts <strong>of</strong><br />
his <strong>time</strong> <strong>and</strong> energy. He counseled his surprised colleagues to do likewise, with what<br />
success, one can imagine. But here too, by life's end, though he had made considerable<br />
progress, nothing definitive emerged.<br />
Nor, for <strong>the</strong> remainder <strong>of</strong> his life, was Godel able to convince <strong>the</strong> physics or philosophy<br />
community that he had achieved a breakthrough on relativity <strong>the</strong>ory's philosophical<br />
consequences regarding <strong>the</strong> existence <strong>of</strong> <strong>time</strong>. His contribution to <strong>the</strong> Schilpp volume on<br />
<strong>Einstein</strong> may have provided cover for him to be <strong>of</strong>fered <strong>the</strong> <strong>Einstein</strong> Award, but it had<br />
signally failed to establish his bona fides as a<br />
philosopher, <strong>and</strong> it did nothing to turn philosophers' attention to <strong>the</strong> burning question <strong>of</strong><br />
<strong>the</strong> reality <strong>of</strong> <strong>time</strong>.
Suspicions <strong>of</strong> Piety<br />
Godel's attempt to discover <strong>the</strong> truth about <strong>the</strong> abstract universe <strong>of</strong> sets <strong>and</strong> numbers had<br />
stalled, as had his assault on <strong>the</strong> starry heavens. Undeterred, he aimed still higher. He<br />
tried to pin down God Himself, developing his own version <strong>of</strong> <strong>the</strong> Anselm-Descartes-Leibniz<br />
"ontolog-ical argument" for <strong>the</strong> existence <strong>of</strong> God, a being, by hypo<strong>the</strong>sis, so perfect, if His<br />
existence is possible at all, He must exist not just in <strong>the</strong> actual but in every possible <strong>world</strong>.<br />
The step from God's possibility to His actuality was relatively straightforward, given a<br />
suitable choice <strong>of</strong> axioms for one's "modal logic" (i.e., <strong>the</strong> logic <strong>of</strong> <strong>the</strong> modes <strong>of</strong> possibility<br />
<strong>and</strong> actuality). The hard part, Godel realizedóas had his hero Leibniz before himówas<br />
proving that a divine being was so much as possible. This Godel attempted to do via a<br />
highly compressed formal argument, which, once again, he declined to publish. He feared,<br />
he told his friends, that its publication might suggest to his skeptical philosophical<br />
colleagues that he actually believed in God, whereas (he claimed) in fact it was a mere<br />
formal exercise.<br />
His assessment <strong>of</strong> <strong>the</strong> religious inclinations <strong>of</strong> <strong>the</strong> philosophical community was probably<br />
accurate: "Ninety per cent <strong>of</strong> contemporary philosophers," he wrote to his mo<strong>the</strong>r in 1961,<br />
"see <strong>the</strong>ir principal task to be that <strong>of</strong> beating religion out <strong>of</strong> men's heads." Charles Parsons,<br />
a philosopher <strong>and</strong> logician at Harvard, tells a story that speaks to Godel's concerns. During<br />
an interview in 1955 for membership in <strong>the</strong> prestigious Society <strong>of</strong> Fellows at Harvard,<br />
where he was a first-year graduate student, he let it slip that he had done readings in<br />
<strong>the</strong>ology <strong>and</strong> found Pascal interesting. Even though, he insists, "I was not <strong>the</strong>n <strong>and</strong> never<br />
have been a Christian," he had <strong>forgotten</strong> that one <strong>of</strong> his examiners, <strong>the</strong> dean <strong>of</strong> American<br />
philosophers, W.V.O. Quine, "was a<br />
firm opponent <strong>of</strong> religion." Legend has it that when <strong>the</strong> meeting concluded, Quine was<br />
heard to say, "Good grief, Parsons is pious." Needless to say, he was not elected. (He was,<br />
however, invited to reapply.)<br />
Godel was no more successful in preventing himself from being considered pious. Word<br />
leaked out about his pro<strong>of</strong>, <strong>and</strong> no one, <strong>the</strong>n or now, was fooled into thinking it was a<br />
mere "formal exercise." When <strong>the</strong> pro<strong>of</strong> itself finally materialized, posthumously, problems<br />
were found in <strong>the</strong> details. Whe<strong>the</strong>r repair is possible is an open question, as is <strong>the</strong> problem<br />
<strong>of</strong> whe<strong>the</strong>r an amended pro<strong>of</strong>, with its revised premises, would be convincing. What is<br />
beyond dispute, however, is that <strong>the</strong> appearance <strong>of</strong> Godel's version <strong>of</strong> <strong>the</strong> ontological<br />
argument has had little effect on <strong>the</strong> confidence <strong>of</strong> philosophers that a formal<br />
demonstration <strong>of</strong> God's existence is impossible. Knocking religion out <strong>of</strong> people's heads<br />
continues to be a favorite philosophical pas<strong>time</strong>.<br />
Preoccupied with not appearing eccentric or out <strong>of</strong> fashion to his colleagues, Godel was<br />
never<strong>the</strong>less universally perceived as both. What <strong>the</strong> logician Solomon Feferman has<br />
characterized as Godel's "special caution" had one especially unfortunate effect: it kept<br />
him from contributing to important branches <strong>of</strong> logic <strong>and</strong> ma<strong>the</strong>matics that he himself had<br />
been instrumental in inaugurating. As Feferman points out, Godelóperhaps exaggerating<br />
<strong>the</strong> continued influence <strong>of</strong> <strong>the</strong> Hilbert school <strong>and</strong> <strong>the</strong> dominance <strong>of</strong> positivismóhaving
demonstrated <strong>the</strong> indefinability <strong>of</strong> arithmetic truth within formal arithmetic, declined to<br />
go on to provide a formal account <strong>of</strong> <strong>the</strong> concept <strong>of</strong> truth itself. That notable task was left<br />
to his colleague Alfred Tarski, with whose name <strong>the</strong> subject is now identified. Similarly,<br />
having laid <strong>the</strong> groundwork for much <strong>of</strong> <strong>the</strong> <strong>the</strong>ory <strong>of</strong> effective computability in his seminal<br />
discussions <strong>of</strong> recursive functions in his incompleteness <strong>the</strong>orem, Godel declined to provide<br />
a definitive account <strong>of</strong> effective computability, a central concept <strong>of</strong> today's <strong>the</strong>oretical<br />
computer science. That task was accomplished by Alan Turing, with contributions from<br />
Church, Stephen S. Kleene <strong>and</strong> J. Barkley Rosser. "One may wonder," says Feferman, "how<br />
logic might have been different had Godel been<br />
bolder in bringing his philosophical views into play in relation to his logical work."<br />
"The World Tends to Deteriorate"<br />
Godel's paranoia, <strong>the</strong>n, undoubtedly deprived <strong>the</strong> <strong>world</strong> <strong>of</strong> his contributions to important<br />
areas <strong>of</strong> modern thought. In <strong>time</strong> it would deprive him <strong>of</strong> life itself. Though he recuperated<br />
well enough from <strong>the</strong> bleeding ulcer he suffered in 1951, cheered by <strong>the</strong> <strong>Einstein</strong> Award<br />
<strong>and</strong> by <strong>the</strong> invitation to deliver <strong>the</strong> Gibbs Lecture, his mental <strong>and</strong> physical state would<br />
soon enter a downward journey from which he would never recover. In <strong>the</strong> coming decade<br />
<strong>the</strong> colleagues who had been closest to him, who had shepherded him since he first arrived<br />
in Princeton, would all die: <strong>Einstein</strong> in 1955, Von Neumann in 1957, Veblen in 1960.<br />
<strong>Einstein</strong> had kept it secret that he had a life-threatening heart conditionóan aneurysm, a<br />
weakening in <strong>the</strong> abdominal aorta, diagnosed in 1948óthat could take him at any moment,<br />
<strong>and</strong> when that moment finally arrived, Godel was stunned. His thoughts, always gloomy,<br />
took on a darker hue. "We live in a <strong>world</strong>," he-wrote, "in which ninety-nine per cent <strong>of</strong> all<br />
beautiful things are destroyed in <strong>the</strong> bud."<br />
He had studied Hegel <strong>and</strong> developed his own philosophy <strong>of</strong> history, according to which <strong>the</strong><br />
<strong>world</strong> is subject to large-scale "noncausal" laws: "There are structural laws in <strong>the</strong> <strong>world</strong><br />
which can't be explained causally." These did not, in his view, justify <strong>the</strong> post-<br />
Enlightenment, Christian belief in human progress. "The <strong>world</strong> tends to deteriorate," he<br />
wrote. "Good things appear from <strong>time</strong> to <strong>time</strong> in single persons <strong>and</strong> events . . . but <strong>the</strong><br />
general development tends to be negative." Christianity, with which he was generally<br />
sympa<strong>the</strong>tic, was no exception. It "was best at <strong>the</strong> beginning. Saints slow down <strong>the</strong><br />
downward movement." As Simone Weil put it, although "since [Christ's] day <strong>the</strong>re have<br />
been no very noticeable changes in men's behavior," "drops <strong>of</strong> purity"<br />
appear from <strong>time</strong> to <strong>time</strong>. Philosophy suffered a similar fate: "Philosophy tends to go<br />
down." Indeed, "it is, at best, at <strong>the</strong> point where Babylonian ma<strong>the</strong>matics was."<br />
Unsurprisingly, Godel had few interactions with <strong>the</strong> Babylonian philosophical establishment<br />
in Princeton or elsewhere. With most <strong>of</strong> his closest friends dead <strong>and</strong> Adele suffering from<br />
increasingly debilitating ailments (variously characterized as hypertension, arthritis,<br />
bursitis <strong>and</strong> gall bladder disease), he turned in his final decade to his old acquaintance<br />
Oskar Morgenstern, who was granted <strong>the</strong> dubious privilege <strong>of</strong> witnessing Godel's gradual
descent into full-blown paranoia <strong>and</strong> hypochondria. Where once Adele had been <strong>the</strong>re to<br />
tend to his needs, it was left henceforth to Godel to care for his wife.<br />
By 1968, Morgenstern had grown alarmed at how gaunt his friend had become. It became<br />
nearly impossible to persuade him to eat, with predictable consequences. Hypochondria<br />
joined forces with paranoia, <strong>and</strong> soon Godel was claiming that his doctors were lying to<br />
him, <strong>the</strong>ir medications misidentified, <strong>the</strong>ir textbooks ill-written. His distrust <strong>of</strong> doctors<br />
was combined with an all too realistic fear that <strong>the</strong>y would commit him to a psychiatric<br />
hospital. Soon he failed even to recognize <strong>the</strong>m, <strong>and</strong> Morgenstern in turn could barely<br />
recognize his friend, who was now hallucinating <strong>and</strong> looked like a "living corpse."<br />
Princeton too, according to Godel, was against him, <strong>and</strong> Morgenstern, who had thrice<br />
failed to get <strong>the</strong> university to award his friend an honorary degree, had a hard <strong>time</strong><br />
convincing him o<strong>the</strong>rwise. When he tried to calm Godel by assuring him that at least he<br />
was a true friend, Godel replied, sadly, that a real friend would have brought him cyanide.<br />
(Alan Turing, it should be recalled, succeeded in doing away with himself by this means,<br />
using a syringe to squirt <strong>the</strong> poison into an apple, imitating Godel's favorite fairy tale.) The<br />
hallucinations continued, as did <strong>the</strong> appeal for assistance in committing suicide.<br />
The imaginary health problems were made worse by real ones. In 1974, Godel's enlarged<br />
prostate blocked his urinary tract, a serious problem exacerbated by bis refusal to seek<br />
treatment. Only when <strong>the</strong> pain became unbearable did he finally check into a hospital,<br />
where he<br />
was ca<strong>the</strong>terized. But although surgery was recommended, he refused, rejecting <strong>the</strong><br />
diagnosis. He removed <strong>the</strong> ca<strong>the</strong>ter himself, which had to be forcibly reinserted. He never<br />
did relinquish his opposition to surgery, <strong>and</strong> in <strong>the</strong> end remained permanently<br />
ca<strong>the</strong>terized, leaving him in a condition <strong>of</strong> constant discomfort that could only intensify his<br />
depression. Into this fragile, pain-suffused body, Godel proceeded to insert a whole<br />
cabinet <strong>of</strong> medications, which Morgenstern was amazed his friend could survive. Godel<br />
ingested milk <strong>of</strong> magnesia, Metamucil, Keflex, M<strong>and</strong>elamine, Macrodantin, Gantanol,<br />
Achromycin, Ter-ramycin, Lanoxin, Quinidine, Imbricol, <strong>and</strong> Pericolase. The only thing he<br />
was reluctant to admit into his starved body was food.<br />
Despite bouts <strong>of</strong> hallucinations <strong>and</strong> a constant fear <strong>of</strong> people, he had moments <strong>of</strong> great<br />
lucidity <strong>and</strong> even charm, <strong>and</strong> managed a few extended human contacts, <strong>the</strong> most<br />
important being his association with <strong>the</strong> logician Hao Wang. Fortunately for posterity,<br />
Wang took it on himself to coax out <strong>of</strong> Godel his unwritten philosophy. From 1971 to 1972,<br />
<strong>the</strong> two met at Godel's <strong>of</strong>fice every o<strong>the</strong>r Wednesday for two hours. If <strong>the</strong>y could not meet<br />
in person, <strong>the</strong> discussion was carried out by phone. The accounts <strong>of</strong> <strong>the</strong>se exchanges <strong>and</strong><br />
Wang's observations on <strong>the</strong>m, published after GodePs death, are valuable documents,<br />
though it is unclear to what extent Godel was trying out ideas, or musings, sufficiently farfetched<br />
that he would have been reluctant to own up to <strong>the</strong>m in print. What <strong>the</strong><br />
philosopher Richard Rorty has aptly said <strong>of</strong> Plato also applies to Godel: we are still<br />
struggling to separate out <strong>the</strong> straight lines from <strong>the</strong> jokes.
When none <strong>of</strong> his remaining friends could persuade Godel to eat, his demise was<br />
guaranteed. In late December 1977, weighing sixty-five pounds, he was finally admitted to<br />
Princeton Hospital. He died on January 11, 1978, from what was diagnosed as "malnutrition<br />
<strong>and</strong> inanition" due to "personality disturbance." They buried him in Princeton Gemetery.<br />
Adele, whose capacity to save her husb<strong>and</strong> from himself had long since been exhausted,<br />
lived on until 1981.<br />
A friend for life, <strong>Einstein</strong> did not lie beside him in death. Fearing that his grave would<br />
become "a place <strong>of</strong> pilgrimage, where pilgrims<br />
would come to view <strong>the</strong> bones <strong>of</strong> a saint," he had asked to be cremated, his ashes<br />
scattered. Godel had no such fears. There was a memorial service at <strong>the</strong> institute, where<br />
Wang paid tribute to his colleague, as did <strong>the</strong> ma<strong>the</strong>matician Andre Weil, bro<strong>the</strong>r <strong>of</strong><br />
Simone Weil, whose dark assessment <strong>of</strong> human history so strongly resembled Godel's. His<br />
passing, unlike <strong>Einstein</strong>'s, attracted little attention. The disappearance <strong>of</strong> this giant planet<br />
disturbed no o<strong>the</strong>r orbit. To philosophers, <strong>the</strong>n as now, he was a simply a logician trying to<br />
pass as a philosopher.<br />
9 In What Sense Is Godel<br />
(or Anyone Else) a Philosopher?<br />
Engaging in philosophy is salutary, even when no positive results emerge. . . . The color is<br />
brighter, that is, reality appears more clearly as such.<br />
KURT GODEL<br />
It is difficult to protect our interests while we are alive. Much more so when we're dead.<br />
Godel, throughout his academic life, was exceptionally anxious to avoid being considered a<br />
philosophical dilettante or crank, <strong>and</strong> a "pious" one at that. In this, he failed completely.<br />
His "special caution" (as Feferman described it) succeeded only in keeping him from<br />
contributing to important fields <strong>of</strong> research. His refusal to publish his ontological argument<br />
for <strong>the</strong> existence <strong>of</strong> God fooled no one. He could not hide <strong>the</strong> fact that he was a kind <strong>of</strong><br />
believer <strong>and</strong> that his argument, like Leibniz's before him, was anything but a mere<br />
intellectual exercise. What Russell called Godel's "unadulterated Platonism" marked him<br />
for some as an intellectual throwback to "precritical" <strong>time</strong>s, before Kant launched his<br />
"critique" <strong>of</strong> pure reason, a police operation by which Kant intended to curb <strong>the</strong><br />
pretensions to <strong>the</strong>oretical knowledge <strong>of</strong> most, if not all, philosophers who preceded him.<br />
Even Godel's writings on <strong>Einstein</strong> succeeded only in convincing cosmologists that strange<br />
things happen when a logician pays too much attention to <strong>the</strong> equations <strong>of</strong> relativity, while<br />
forgetting <strong>the</strong>ir physical meaning. His absence, in turn,
from <strong>the</strong> Wittgenstein revolutionówhich turned philosophy toward <strong>the</strong> human construction<br />
<strong>of</strong> language <strong>and</strong> <strong>the</strong> "games" people play with itócombined with his refusal to pay homage<br />
to <strong>the</strong> preeminent figure in contemporary analytical philosophy, W.V.O. Quine, marked<br />
him as a philosophical castaway.<br />
After GodePs death, his colleague <strong>and</strong> amanuensis Hao Wang published excerpts <strong>of</strong> <strong>the</strong>ir<br />
philosophical discussions, in an attempt to give a more rounded <strong>and</strong> positive cast to Godel's<br />
image as a thinker. He too failed. Too many philosophers had already argued to <strong>the</strong><br />
contrary. The full extent <strong>of</strong> <strong>the</strong> damage became clear during a symposium held nearly two<br />
decades after Godel's death to celebrate his contributions to philosophy. The celebration<br />
resembled a wake.<br />
Who Buried Kurt Godel?<br />
On a cold February day in 1995, a distinguished group <strong>of</strong> philosophers, ma<strong>the</strong>maticians <strong>and</strong><br />
logicians assembled to honor Godel in a symposium hosted by Boston University entitled<br />
"Godel's General Philosophical Significance." The speakers included those who had been<br />
assigned <strong>the</strong> task <strong>of</strong> guarding Godel's <strong>legacy</strong> by ga<strong>the</strong>ring his published <strong>and</strong> unpublished<br />
essays, with editorial introductions, in a definitive edition <strong>of</strong> <strong>the</strong> Collected Works. Plato's<br />
question from The Republic, however, hung in <strong>the</strong> air: "Who will guard <strong>the</strong> guardians?" It<br />
was clear from <strong>the</strong> start that <strong>the</strong> speakers had come not to praise Godel but to bury him.<br />
John Dawson, however, one <strong>of</strong> <strong>the</strong> guardians, was different. He alluded to <strong>the</strong> neglect <strong>of</strong><br />
Godel as a philosopher <strong>of</strong> <strong>time</strong>, but drew attention to <strong>the</strong> fact that <strong>the</strong>re were exceptions.<br />
As he put it in his soon to be published biography <strong>of</strong> Godel, "To date, only a single volume<br />
(<strong>Yourgrau</strong> 1991) [The Disappearance <strong>of</strong> Time] has examined in any detail . . . <strong>the</strong><br />
ramifications <strong>of</strong> [GodePs] cosmological work for <strong>the</strong> philosophy <strong>of</strong> <strong>time</strong>." Yet nei<strong>the</strong>r he nor<br />
any o<strong>the</strong>r speaker attempted to remedy <strong>the</strong> lack <strong>of</strong> attention that had been paid to Godel's<br />
writings devoted to <strong>the</strong> subject.<br />
Some recounted personal anecdotes about <strong>the</strong>ir encounters with Godel, while o<strong>the</strong>rs spoke<br />
briefly about his Platonism. Two speakers, however, stood out. Warren Goldfarb, from<br />
Harvard, ano<strong>the</strong>r guardian, addressed <strong>the</strong> question <strong>of</strong> what Godel had succeeded in<br />
accomplishing in philosophy. The answer, according to Goldfarb, was nothing. The<br />
audience, which included Hao Wang as well as <strong>the</strong> author <strong>of</strong> <strong>the</strong> only book devoted to <strong>the</strong><br />
ramifications <strong>of</strong> Godel's cosmological work for <strong>the</strong> philosophy <strong>of</strong> <strong>time</strong> (myself), was<br />
stunned. A h<strong>and</strong> was raised during <strong>the</strong> question period. "Do I underst<strong>and</strong> you correctly,<br />
Pr<strong>of</strong>essor Goldfarb, that in your judgment Godel, though a great logician, was a<br />
philosophical fool?" A polite smile was Goldfarb's only answer.<br />
But ano<strong>the</strong>r scheduled speaker, Burton Dreben, also from Harvard, could not restrain<br />
himself: "Wait until you hear my talk!" And in due course, Dreben delivered. He made<br />
explicit in his presentation (entitled, simply, "Godel") what had been implicit in his<br />
colleague's talk, that Godel was a logician trying to pass as a philosopher. For Dreben, it<br />
seemed, this was a kind <strong>of</strong> sc<strong>and</strong>al. He was moved to deliver a sermon on <strong>the</strong> harm that is<br />
done when people who are good at purely formal thinking get <strong>the</strong> idea that <strong>the</strong>y are also<br />
qualified to contribute to philosophy. That Dreben's own position in <strong>the</strong> philosophical <strong>world</strong><br />
owed much to his reputation in formal logic was an irony that seemed lost on him. Being<br />
good at one task, he stressed, says nothing <strong>of</strong> your ability to succeed at <strong>the</strong> o<strong>the</strong>r. Fascists,
he said, were some<strong>time</strong>s good at science, but that doesn't mean we should take <strong>the</strong>m<br />
seriously when <strong>the</strong>y try <strong>the</strong>ir h<strong>and</strong> at philosophy.<br />
Godel, by Dreben's lights, was a throwback to a benighted, pre-Kantian era <strong>of</strong> philosophy,<br />
a vagabond in <strong>the</strong> modern vineyards painstakingly planted by <strong>the</strong> likes <strong>of</strong> Kant,<br />
Wittgenstein <strong>and</strong> Quine. To Dreben, <strong>the</strong>se thinkers represented <strong>the</strong> future. Godel was <strong>the</strong><br />
past. What especially irked him was that Godel had <strong>the</strong> audacity "in this day <strong>and</strong> age" to<br />
engage in rational <strong>the</strong>ology, a reference, no doubt, to Godel's recasting <strong>of</strong> <strong>the</strong> ontological<br />
argument. He seemed almost driven to despair by what Godel had done. If Goldfarb had<br />
quietly begun <strong>the</strong> delicate task <strong>of</strong> burial, Dreben was in haste to get <strong>the</strong> body<br />
into <strong>the</strong> ground. But would <strong>the</strong> ground receive <strong>the</strong> man being celebrated? By what right<br />
had Godel, <strong>the</strong> logician, ventured to teach philosophers about <strong>the</strong>ir own discipline? What<br />
kind <strong>of</strong> discipline is philosophy, anyway?<br />
The Philosopher on <strong>the</strong> Train<br />
If you meet a philosopher on a train <strong>and</strong> ask him his pr<strong>of</strong>ession, he is likely to lie. It is not<br />
that philosophers are especially prone to lying, but ra<strong>the</strong>r that philosophy is a peculiar<br />
pr<strong>of</strong>ession. To tell your fellow passenger that you are a philosopher opens up an awkward<br />
line <strong>of</strong> questioning. To begin with, describing yourself as a philosopher is like calling<br />
yourself a sage, a seeker <strong>of</strong> wisdom. We all seek wisdom, after all, but that won't feed <strong>the</strong><br />
bulldog. A safer response is to account oneself a philosophy pr<strong>of</strong>essor. This is fine, unless<br />
you happen to be an actual philosopher, in which case it is just ano<strong>the</strong>r lie. As <strong>the</strong><br />
philosopher Leo Strauss once said, you are as likely to find a real philosopher in a<br />
philosophy department as you are to discover a Picasso in <strong>the</strong> department <strong>of</strong> fine arts.<br />
(Wittgenstein, though he taught for years at Cambridge University, is correctly described<br />
as a philosopher, not a pr<strong>of</strong>essor.) If you take <strong>the</strong> plunge, however, <strong>and</strong> accept <strong>the</strong> label <strong>of</strong><br />
philosopher, you must be prepared for <strong>the</strong> disappointment when your listener hears that<br />
you don't live in a hut on a mountaintop, haven't uncovered <strong>the</strong> secret <strong>of</strong> life, <strong>and</strong> cannot<br />
explain why <strong>the</strong> <strong>world</strong> exists. If you are foolish enough to go fur<strong>the</strong>r <strong>and</strong> attempt to<br />
describe your lifelong attempt to reconcile <strong>the</strong> epistemology <strong>of</strong> ma<strong>the</strong>matics with its<br />
ontology, be prepared to encounter a look in which boredom <strong>and</strong> horror are blended<br />
equally. Best, <strong>the</strong>refore, to say simply that you are an architect, <strong>and</strong> leave it at that.<br />
Godel, one can be certain, was well aware <strong>of</strong> how <strong>the</strong> <strong>world</strong> regards those who dare to call<br />
<strong>the</strong>mselves philosophers; he would have been reluctant to describe himself that way to his<br />
fellow traveler. Being <strong>of</strong>ficially a logician gave him adequate cover. Though Dreben <strong>and</strong> a<br />
host <strong>of</strong> o<strong>the</strong>rs took him to he a logician trying to pass as a philosopher, he is more<br />
accurately described as a philosopher posing as a logician. More than most academic<br />
philosophers, he engaged in philosophy in a manner <strong>of</strong> which Parmenides <strong>and</strong> Plato would<br />
have been proud: asking fundamental questions about <strong>the</strong> nature <strong>of</strong> <strong>time</strong>, being, death,<br />
God <strong>and</strong> <strong>the</strong> <strong>world</strong> <strong>of</strong> transcendent forms, or "ideas."
He shared, too, Socrates' <strong>and</strong> Wittgenstein's mistrust <strong>of</strong> philosophy as just ano<strong>the</strong>r paid<br />
pr<strong>of</strong>ession. "To do philosophy is a special vocation," he wrote. "We do see <strong>the</strong> truth, yet<br />
error would reign." The special vocation carried with it duties <strong>and</strong> dangers. "Philosophy,"<br />
he said, "is a persecuted science." He was not thinking <strong>of</strong> <strong>the</strong> danger <strong>of</strong> not acquiring<br />
tenure, but ra<strong>the</strong>r <strong>of</strong> <strong>the</strong> risk you take if you question <strong>the</strong> ruling paradigm. Whereas "moral<br />
relativity," for example, is a widespread catchphrase that encourages us to attribute <strong>the</strong><br />
lack <strong>of</strong> progress in ethics to <strong>the</strong> fact that philosophers have a constitutional predilection to<br />
objectify what is merely subjective, Godel made a different, shrewder assessment:<br />
"Actually, it would be easy to get a strict ethicsóat least no harder than o<strong>the</strong>r basic<br />
scientific problems. Only <strong>the</strong> result would be unpleasant, <strong>and</strong> one does not want to see it<br />
<strong>and</strong> avoids facing itó to some extent even consciously."<br />
The more pressing question is whe<strong>the</strong>r Godel <strong>the</strong> philosopher deserved <strong>the</strong> respect<br />
accorded full-fledged members <strong>of</strong> <strong>the</strong> academy. The answer, one feels, ought to be clear,<br />
yet academic philosophy is a most peculiar discipline. Despite tracing its lineage to such<br />
thinkers as Parmenides, Heraclitus, Socrates <strong>and</strong> Plato, philosophy's right to exist is<br />
constantly called into question, not least by its leading practitioners, from Kant to Carnap,<br />
from Wittgenstein to Quine. For philosophy as such pretends to a kind <strong>of</strong> knowledge<br />
greater than anything mere mortals seem equipped to discover. Kant put it thus in a<br />
beautiful passage in <strong>the</strong> Critique <strong>of</strong> Pure Reason:<br />
The light dove, cleaving <strong>the</strong> air in her free flight, <strong>and</strong> feeling its resistance, might imagine<br />
that its flight would be still easier in empty space. It was thus that Plato left <strong>the</strong> <strong>world</strong> <strong>of</strong><br />
<strong>the</strong> senses, as<br />
setting too narrow limits to <strong>the</strong> underst<strong>and</strong>ing, <strong>and</strong> ventured out beyond it on <strong>the</strong> wings <strong>of</strong><br />
<strong>the</strong> Ideas, in <strong>the</strong> empty space <strong>of</strong> <strong>the</strong> pure underst<strong>and</strong>ing.<br />
Sweeping aside traditional metaphysics from Plato to Leibniz as insufficiently grounded in<br />
<strong>the</strong> bedrock <strong>of</strong> sense experience, Kant <strong>of</strong>fered a comprehensive account <strong>of</strong> what man can<br />
<strong>and</strong> cannot know. His critique changed <strong>the</strong> course <strong>of</strong> modern philosophy, but unfortunately<br />
it was never clear whe<strong>the</strong>r his account <strong>of</strong> <strong>the</strong> nature <strong>of</strong> things violated his own precepts.<br />
Can sensory experience, for example, on its own teach us what sensory experience can <strong>and</strong><br />
cannot accomplish?<br />
This problem seems endemic to <strong>the</strong> enterprise <strong>of</strong> constructing a philosophical system. All<br />
too <strong>of</strong>ten, a philosopher finds himself painted into an epistemic corner <strong>of</strong> his own making.<br />
The famous motto <strong>of</strong> <strong>the</strong> Vienna Circle, for example, that <strong>the</strong> meaning <strong>of</strong> a proposition is<br />
its method <strong>of</strong> verification, was demolished by one <strong>of</strong> <strong>the</strong>ir own, an astute philosopher who<br />
had once belonged to <strong>the</strong> circle. In a classic essay, Carl Hempel pointed out that since <strong>the</strong><br />
"verifiability [or empiricistl criterion <strong>of</strong> meaning" cannot itself be verified by experience,<br />
by its own account it lacks cognitive significance. In itself, it is nei<strong>the</strong>r true nor false. The<br />
circle's criterion for meaning turned out to be meaningless.<br />
A still more dramatic example comes from <strong>the</strong> first (<strong>and</strong> only) book Wittgenstein published<br />
during his life<strong>time</strong>, <strong>the</strong> Tractatus Logico-Pbilosopbicus, in which, like Kant before him, he
attempted to set limits to what can be known, or ra<strong>the</strong>r, to what can be said, thus helping<br />
to inaugurate <strong>the</strong> famous "linguistic turn" in philosophy. For Wittgenstein, nothing that was<br />
<strong>of</strong> genuine valueósuch as <strong>the</strong> beautiful, <strong>the</strong> good or <strong>the</strong> meaning <strong>of</strong> lifeócould actually be<br />
stated (as opposed to "shown"), <strong>and</strong> everything that could be said, which amounted to <strong>the</strong><br />
substance <strong>of</strong> physical science, was absent <strong>of</strong> value. Displaying a greater sense <strong>of</strong> irony than<br />
had Kant, he acknowledged that <strong>the</strong> book itself attempted to say what could not be said.<br />
Strictly speaking, he admitted (or boasted), it was all nonsense. But deep non-<br />
sense. One must, he added poetically, "throw away <strong>the</strong> ladder after |one| has climbed up<br />
it." F.P. Ramsey, a brilliant young ma<strong>the</strong>matician who wrote an astute review <strong>of</strong> <strong>the</strong><br />
Tractatus at <strong>the</strong> tender age <strong>of</strong> twenty, was unimpressed. Though he would go on to<br />
become a close associate <strong>of</strong> Wittgenstein's, he never forgot or forgave <strong>the</strong> nonsense <strong>of</strong> <strong>the</strong><br />
Tractatus. "Philosophy must be <strong>of</strong> some use," he wrote, "<strong>and</strong> we must take it seriously." If,<br />
however, he added, "<strong>the</strong> chief proposition <strong>of</strong> philosophy is that philosophy is nonsense,"<br />
<strong>the</strong>n we must take this seriously "<strong>and</strong> not pretend, as Wittgenstein does, that it is<br />
important nonsense." More succinctly: "What we can't say we can't say, <strong>and</strong> we can't<br />
whistle it ei<strong>the</strong>r."<br />
Wittgenstein would in <strong>time</strong> renounce much <strong>of</strong> <strong>the</strong> Tractatus, but he never recanted his<br />
skepticism about <strong>the</strong> existence <strong>of</strong> philosophy as a kind <strong>of</strong> "superscience," first among<br />
equals. The temptation to engage in deep philosophical pronouncements, he said,<br />
amounted to a kind <strong>of</strong> psychological disorder or mental cramp. The jobóindeed <strong>the</strong> dutyó<strong>of</strong><br />
a genuine philosopher was to enlighten <strong>the</strong> patient by showing him that <strong>the</strong> illusion <strong>of</strong><br />
depth was <strong>the</strong> result merely <strong>of</strong> his skating on <strong>the</strong> thin ice <strong>of</strong> confused linguistic practice.<br />
The tantalizing figures that appeared to linger deep within <strong>the</strong> ice were only reflections <strong>of</strong><br />
<strong>the</strong> subject looking at himself. Upon awakening from his philosophical slumbers, <strong>the</strong><br />
patient, like one <strong>of</strong> Freud's subjects, would arise from <strong>the</strong> couch, shake <strong>of</strong>f his dream, <strong>and</strong><br />
return to <strong>the</strong> sober dreariness <strong>of</strong> everyday life. Language would, in his memorable phrase,<br />
no longer be on holiday. It was hardly surprising, <strong>the</strong>n, that when Wittgenstein's fellow<br />
Austrian Karl Popper addressed <strong>the</strong> Cambridge Metaphysical Society on <strong>the</strong> topic, "Are<br />
There Philosophical Problems?", his positive response roused considerable ire. Sparks flew,<br />
a poker was grabbed by Wittgenstein from a smoldering fireplace, <strong>and</strong> material was<br />
provided for a lively book eventually to be written about <strong>the</strong> incident.<br />
Skepticism about <strong>the</strong> existence <strong>of</strong> philosophy, however, did not keep Wittgenstein away<br />
from <strong>the</strong> university. He fled once, only to return <strong>and</strong> take up residence at his old haunt.<br />
Trinity College at Cambridge University, <strong>and</strong> become for decades <strong>the</strong> dominant figure in<br />
<strong>the</strong><br />
academic philosophical <strong>world</strong>. Unlike Socrates, Plato, Frege <strong>and</strong> Russell, but like Marx <strong>and</strong><br />
Freud (<strong>and</strong> Jesus), he surrounded himself with a group <strong>of</strong> disciples who spread <strong>the</strong> gospel<br />
far <strong>and</strong> wide. It is an irony that precisely those philosophers who have called into question<br />
<strong>the</strong> existence <strong>of</strong> philosophy in <strong>the</strong> traditional sense have been, <strong>and</strong> continue to be, <strong>the</strong><br />
most influential in <strong>the</strong> halls <strong>of</strong> academe. Kant <strong>the</strong> protopos-itivist, Carnap <strong>the</strong> actual<br />
positivist, Wittgenstein <strong>the</strong> patron saint <strong>of</strong> positivism, Quine <strong>the</strong> preeminent opponent <strong>of</strong><br />
"first philosophy," <strong>the</strong>se are <strong>the</strong> movers <strong>and</strong> shakers <strong>of</strong> modern academic philosophy in <strong>the</strong><br />
Anglo-American <strong>world</strong>. Godel, a disciple <strong>of</strong> none, had sealed his fate. How could he hope<br />
to be taken seriously by an academy whose founding fa<strong>the</strong>rs he refused to embrace?
"Precritical"<br />
Goldfarb <strong>and</strong> Dreben, in contrast, did embrace <strong>the</strong>m. And <strong>the</strong>y were far from alone. "After<br />
<strong>the</strong> devastating attacks by Wittgenstein <strong>and</strong> Quine," wrote <strong>the</strong> philosopher Paul Horwich in<br />
1990, "it is now widely believed that <strong>the</strong> sciences exhaust what can be known <strong>and</strong> that <strong>the</strong><br />
promise <strong>of</strong> metaphysics was an intellectually dangerous illusion." At <strong>the</strong> Boston University<br />
conference, when Goldfarb rose to deliver his assessment, he did so as a colleague <strong>of</strong><br />
Quine <strong>and</strong> a follower <strong>of</strong> Wittgenstein who clearly knew his duty regarding this<br />
"intellectually dangerous illusion." Though Goldfarb is a logician <strong>of</strong> distinction <strong>and</strong> an<br />
important commentator on Wittgenstein, <strong>the</strong> soundness <strong>of</strong> his philosophical judgment<br />
seems inversely proportional to his proximity to Godel. Here too he is not alone.<br />
Since Godel is not here to defend himself, <strong>the</strong> task falls to o<strong>the</strong>rs, but to assess <strong>the</strong> text <strong>of</strong><br />
Goldfarb's presentation, entitled "On Godel's General Philosophical Outlook," is difficult,<br />
since although some speakers submitted <strong>the</strong>ir talks for publication, Goldfarb <strong>and</strong> Dreben<br />
did not. Fortunately, however, a transcript does exist <strong>of</strong> a closely related presen-<br />
ration, entitled "Godel's Philosophy," which he gave in July 1990 before a meeting <strong>of</strong> <strong>the</strong><br />
Association for Symbolic Logic in Helsinki. Opening this talk with Godel's view that<br />
metaphysics should ideally be presented by a small set <strong>of</strong> primitive axioms, Goldfarb<br />
commented that for Godel, "all <strong>the</strong> important content <strong>of</strong> <strong>the</strong> primitive concepts can by<br />
exhibited in precise axiomatic relations to o<strong>the</strong>r concepts." Though this description is not<br />
as clear as it should be, it puts one in mind <strong>of</strong> <strong>the</strong> doctrine <strong>of</strong> "implicit definition" adopted<br />
by Hilbert, for whom <strong>the</strong> meanings <strong>of</strong> <strong>the</strong> primitive concepts in an axiomatic system are<br />
exhausted by <strong>the</strong>ir relationships to <strong>the</strong> o<strong>the</strong>r concepts. In geometry, for example, <strong>the</strong>re is<br />
no more to being a line than its relationship to points <strong>and</strong> planes. Godel, however,<br />
explicitly rejected this doctrine, as did his predecessor Frege. That is why he insisted that<br />
to know <strong>the</strong> primitive concepts, one must not only underst<strong>and</strong> <strong>the</strong>ir relationships to <strong>the</strong><br />
o<strong>the</strong>r primitives but must grasp <strong>the</strong>m on <strong>the</strong>ir own, by a kind <strong>of</strong> "intuition."<br />
Goldfarb continued: "There is no hint" in what Godel wrote, he said, "that <strong>the</strong> truths <strong>of</strong><br />
metaphysics are problematic in any special way, or pose special problems <strong>of</strong> our access to<br />
<strong>the</strong>m." Thus Godel's view, according to Goldfarb, is "precritical, in Kant's sense <strong>of</strong> 'critical.'"<br />
In fact, however, Godel rejected Kant's critical assessment <strong>of</strong> <strong>the</strong> possibilities for<br />
systematic metaphysics, a rejection founded not on ignorance but ra<strong>the</strong>r on a deep<br />
underst<strong>and</strong>ing <strong>of</strong> Kant. Yet Goldfarb chose to describe Godel not as "post-Kantian" but<br />
"precritical," i.e., as a philosophical naif, not up to speed on Kant, ra<strong>the</strong>r than as someone<br />
steeped in Kant who never<strong>the</strong>less rejected much (though not all) <strong>of</strong> Kant's "critique."<br />
Indeed, <strong>the</strong> longer, original drafts <strong>of</strong> Godel's contribution to <strong>the</strong> Schilpp volume on<br />
<strong>Einstein</strong>, entitled "Some Observations about <strong>the</strong> Relationship <strong>of</strong> Theory <strong>of</strong> Relativity to<br />
Kantian Philosophy," leave no room for doubt that Godel had a pr<strong>of</strong>ound underst<strong>and</strong>ing <strong>of</strong><br />
Kant, which enabled him to demonstrate a striking <strong>and</strong> previously unsuspected connection<br />
between Kantian idealism <strong>and</strong> Ein-steinian relativity. This newly published essay makes<br />
clear that Godel, though he accepted certain elements <strong>of</strong> Kant's philosophy, systematically
ejected its main thrust, which assimilated knowledge to <strong>the</strong> knower, not <strong>the</strong> known, <strong>and</strong><br />
thus gave Kant's philosophy a subjectivist cast. To characterize <strong>the</strong> author <strong>of</strong> this essay as<br />
"precritical" is perverse.<br />
Also <strong>of</strong>f <strong>the</strong> mark was Goldfarb's assessment that Godel, in his naivete, failed somehow to<br />
appreciate <strong>the</strong> difficulty <strong>of</strong> finding <strong>the</strong> right concepts <strong>and</strong> axioms for metaphysics. Nothing<br />
could be fur<strong>the</strong>r from <strong>the</strong> truth. Time, for example, in relation to being, Godel considered<br />
one <strong>of</strong> <strong>the</strong> basic concepts, but he believed that <strong>the</strong> attempt to discover what is<br />
fundamental about our thinking about <strong>time</strong> can receive no assistance from physics, which,<br />
he argued, combines concepts <strong>without</strong> analyzing <strong>the</strong>m. Instead, we must reconstruct <strong>the</strong><br />
original nature <strong>of</strong> our thinking, a project fraught with difficulty. For assistance he turned<br />
not to <strong>Einstein</strong> but to Husserl <strong>and</strong> phenomenology, but he acknowledged that "<strong>the</strong> problem<br />
<strong>of</strong> <strong>time</strong> is important <strong>and</strong> difficult. For twenty-five years Husserl worked on just this one<br />
problem: <strong>the</strong> concept <strong>of</strong> <strong>time</strong>." The situation in ma<strong>the</strong>matics was no different. "The way . .<br />
. we form ma<strong>the</strong>matical objects," he said, "from what is givenó<strong>the</strong> question <strong>of</strong><br />
constitutionórequires a phenomenological analysis. But <strong>the</strong> constitution <strong>of</strong> <strong>time</strong> <strong>and</strong> <strong>of</strong><br />
ma<strong>the</strong>matical objects is difficult."<br />
Since <strong>the</strong> fundamental concepts are primitive <strong>and</strong> <strong>the</strong>ir meaning is not exhausted by <strong>the</strong>ir<br />
relationships to o<strong>the</strong>r concepts, how can we manage to gain some insight into <strong>the</strong>m? What<br />
is required, said Godel, is "a clarification <strong>of</strong> meaning that does not consist in defining." His<br />
fellow Platonists, Plato <strong>and</strong> Frege, had little to say on how to accomplish this. Husserl,<br />
however, who early in his career had debated Frege on <strong>the</strong> foundations <strong>of</strong> arithmetic,<br />
devoted himself to just this task, <strong>and</strong> Godel, in turn, devoted himself to <strong>the</strong> task <strong>of</strong><br />
discovering what Husserl had found. Husserl called his new way "phenomenology," which<br />
Godel described as a method by which we can "focus more sharply on <strong>the</strong> concepts<br />
concerned by directing our attention in a certain way, namely, onto our own acts in <strong>the</strong><br />
use <strong>of</strong> <strong>the</strong>se concepts." If we are successful, said Godel, we achieve "a new state <strong>of</strong><br />
consciousness in which we describe in detail <strong>the</strong> basic concepts we use in our thought."<br />
Exactly what this new method came down to is not easy to fathom. Godel struggled for<br />
years to pierce <strong>the</strong> veil <strong>of</strong> Husserl's re-barbative prose, to follow him on his long, winding<br />
way. "I don't particularly like Husserl's way," he told Wang, "long <strong>and</strong> difficult." Indeed, "I<br />
love everything brief," he wrote to his mo<strong>the</strong>r, "<strong>and</strong> find that in general <strong>the</strong> longer a work<br />
is, <strong>the</strong> less <strong>the</strong>re is in it." Yet Goldfarb appeared to have no difficulty underst<strong>and</strong>ing<br />
Husserl, <strong>and</strong> no qualms about dismissing out <strong>of</strong> h<strong>and</strong> what Godel hoped to find by studying<br />
his writings. According to Goldfarb, Godel gives phenomenology "a highly subjectivist cast"<br />
<strong>and</strong> "provides no evidence that observation <strong>of</strong> one's stream <strong>of</strong> consciousness" will give<br />
insight into <strong>the</strong> concepts one is employing. The "passing show," Goldfarb assured his<br />
audience, will never assist us in grasping concepts or finding new axioms. The great Frege<br />
had demonstrated this. "The muddled results <strong>of</strong> bare phenomenological examination," said<br />
Goldfarb, "were pointedly <strong>and</strong> effectively criticized by Frege in his review <strong>of</strong> Husserl's<br />
Philosopbie der Arithmetik."<br />
Nei<strong>the</strong>r Husserl nor Godel, however, thought <strong>of</strong> phenomenology as merely taking note <strong>of</strong><br />
one's stream <strong>of</strong> consciousness or gazing at <strong>the</strong> "passing show." That is <strong>the</strong> method <strong>of</strong> an<br />
empiricist like Hume, not a rationalist like Husserl. Godel saw phenomenology as an<br />
attempt to reconstruct our original use <strong>of</strong> basic ideas, to focus not on ways to employ or<br />
combine concepts, as we do in science or everyday life, but ra<strong>the</strong>r on recovering what we
meant in <strong>the</strong> first place by our most fundamental acts <strong>of</strong> thought. This is a difficult,<br />
painful process that involves a redirection <strong>of</strong> our thinking toward self-reflection. Since<br />
both Godel <strong>and</strong> Husserl (in his later period) were conceptual realists, <strong>the</strong> self-reflection at<br />
issue concerns underst<strong>and</strong>ing how we grasp real, objective concepts; so <strong>the</strong> subjectivism<br />
Goldfarb feared was an illusion. Subjectivism, as Goldfarb employed this term, is in<br />
opposition to objectivism. What Godel found valuable in Husserl, however, was a turn to<br />
<strong>the</strong> thinking subject, <strong>the</strong> source <strong>of</strong> cognition, which was meant not as an alternative to<br />
objectivism, but ra<strong>the</strong>r as an account <strong>of</strong> how what is objective is given to us. Indeed, "in<br />
<strong>the</strong> last analysis," wrote Godel,<br />
"<strong>the</strong> Kantian philosophy rests on <strong>the</strong> idea <strong>of</strong> phenomenology, albeit in a not entirely clear<br />
way." And it is phenomenology, according to Godel, which "entirely as intended by Kant,<br />
avoids both <strong>the</strong> death-defying leaps <strong>of</strong> idealism into a new metaphysics as well as <strong>the</strong><br />
positivis-tic rejection <strong>of</strong> all metaphysics."<br />
From Frege to Godel<br />
Goldfarb's invocation <strong>of</strong> Frege was especially misplaced. To begin with, <strong>the</strong> review by<br />
Frege he cited concerned Husserl's early, non-Platonist, psychologistic account <strong>of</strong><br />
arithmetic, whereas <strong>the</strong> phenomenology that caught Godel's eye was an attempt by Husserl<br />
late in his career to reconcile ma<strong>the</strong>matical Platonism <strong>and</strong> conceptual realism with <strong>the</strong><br />
capacities <strong>of</strong> <strong>the</strong> human intellect. Frege's early critique does not speak to this project.<br />
Indeed, one can ask how exactly Frege came up with his own analysis <strong>of</strong> <strong>the</strong> concept <strong>of</strong><br />
number in his path-breaking study The Foundations <strong>of</strong> Arithmetic. A natural number, Frege<br />
begins, is what answers <strong>the</strong> question "How many?" ("How much?" is answered by a real<br />
number.) But how many what} First we must determine what it is that is really numbered.<br />
His answer: not <strong>the</strong> nine planets <strong>the</strong>mselves (each by itself is one, <strong>and</strong> <strong>the</strong> group as such is<br />
also one), but ra<strong>the</strong>r <strong>the</strong> concept is a planet. To assign a number to a concept, <strong>the</strong>n, is to<br />
determine how many instances it has. The number nine numbers <strong>the</strong> planets because <strong>the</strong><br />
concept is a planet has nine instances. Such are <strong>the</strong> opening moves in Frege's deep <strong>and</strong><br />
beautiful analysis <strong>of</strong> <strong>the</strong> concept <strong>of</strong> number, at <strong>the</strong> conclusion <strong>of</strong> which he <strong>of</strong>fers an<br />
explicit definition: <strong>the</strong> number assigned to <strong>the</strong> concept F is <strong>the</strong> extension <strong>of</strong> <strong>the</strong> concept<br />
equinumerous with F. (Concepts are equinumerous when <strong>the</strong>re is a one-to-one correlation<br />
between <strong>the</strong>ir extensions, i.e., to <strong>the</strong> set <strong>of</strong> objects falling under each. In effect, <strong>the</strong>n, for<br />
Frege, <strong>the</strong> number three is <strong>the</strong> set <strong>of</strong> all trios, i.e., <strong>of</strong> all things that arc three in number.)<br />
Nothing<br />
like this had ever been proposed before. As Frege himself put it, his discussion raised <strong>the</strong><br />
subject to an entirely new level.<br />
But where did this pr<strong>of</strong>ound analysis come from? Was it just a lucky guess? Was it <strong>the</strong> result<br />
<strong>of</strong> an empirical survey Frege had conducted among <strong>the</strong> citizens <strong>of</strong> Jena? Or perhaps <strong>the</strong><br />
great logician was simply grinding out <strong>the</strong>orems from some preexisting set <strong>of</strong> axioms? None<br />
<strong>of</strong> <strong>the</strong>se "explanations," <strong>of</strong> course, holds water, nor was Frege simply sitting back <strong>and</strong><br />
regarding <strong>the</strong> "passing show." He had turned his attention, as no one had before him, to
what we are really doing when we employ <strong>the</strong> concept <strong>of</strong> number, when we use numbers,<br />
that is, both to count things <strong>and</strong> to do arithmetic. But this redirection <strong>of</strong> attention is just<br />
what Godel, following Husserl, called phenomenology. And Wittgenstein, too, it would<br />
appear. At one point in his notebooks, he remarked that one could with justice call his<br />
method <strong>of</strong> investigation <strong>of</strong> <strong>the</strong> correct use <strong>of</strong> words, phenomenology. Where Wittgenstein<br />
speaks <strong>of</strong> words, Godel refers to concepts. "'Trying to see (i.e., underst<strong>and</strong>) a concept<br />
more clearly,'" says Godel, "is <strong>the</strong> correct way <strong>of</strong> expressing <strong>the</strong> phenomenon vaguely<br />
described as 'examining what we mean by a word.'"<br />
The goal <strong>of</strong> this method is to enable us to discern <strong>the</strong> concept itself, free from <strong>the</strong><br />
encrustations <strong>of</strong> historical practice <strong>and</strong> psychological necessity. This was as much Frege's<br />
goal as it was Godel's. As Frege put it, "Often it is only after immense intellectual effort,<br />
which may have been continued over centuries, that humanity at last succeeds in<br />
achieving knowledge <strong>of</strong> a concept in its pure form, in stripping <strong>of</strong>f <strong>the</strong> irrelevant accretions<br />
which veil it from <strong>the</strong> eyes <strong>of</strong> <strong>the</strong> mind." But concepts, <strong>of</strong> course, aren't physical objects.<br />
We can't literally see <strong>the</strong>m. Ra<strong>the</strong>r, says Godel, "we perceive objects <strong>and</strong> underst<strong>and</strong><br />
concepts. Underst<strong>and</strong>ing is a different kind <strong>of</strong> perception." He adopted <strong>the</strong> Kantian term<br />
"intuition" for <strong>the</strong> quasi-perceptive grasping by <strong>the</strong> mind's eye <strong>of</strong> concepts <strong>and</strong> o<strong>the</strong>r "ideal<br />
objects." Goldfarb, once again, is <strong>of</strong>fended: "Kantian 'intuition' is a matter <strong>of</strong> <strong>the</strong><br />
presentation <strong>of</strong> objects (in sensibility) in space <strong>and</strong> <strong>time</strong>; none <strong>of</strong> this is in<br />
Godel's notion." Why <strong>the</strong>n did Godel continue to employ this term for his own purposes?<br />
Goldfarb makes no effort to resolve this question. Why not? For <strong>the</strong> question does have an<br />
answer, which lies in <strong>the</strong> fact that for a Pla-tonist or conceptual realist, <strong>the</strong> mind<br />
encounters an ideal entity in a manner parallel to <strong>the</strong> way <strong>the</strong> eye tracks a physical<br />
object. In both cases, we are confronted with something real that we have not ourselves<br />
created. We grasp it, <strong>the</strong>refore, only partially <strong>and</strong> in stages, gaining new insights as we<br />
shift our perspective. "We begin with vague perceptions <strong>of</strong> a concept," says Godel, "as we<br />
see an animal from far away or take two stars for one before using <strong>the</strong> telescope." Since<br />
<strong>the</strong> entity is not a child <strong>of</strong> our own imagination, we will never exhaust <strong>the</strong> information to<br />
be gained by different ways <strong>of</strong> approaching it, but we may reach a limit after which we no<br />
longer find ourselves bumping up against surprises or running into mysteries. If that<br />
threshold is achievedóas Godel thought it was by Turing's analysis <strong>of</strong> <strong>the</strong> concept <strong>of</strong><br />
effective computabilityówe are satisfied that our perception, or quasi-perception, is now<br />
adequate to our purpose <strong>and</strong> that we have an acceptable (though not perfect) intuition <strong>of</strong><br />
<strong>the</strong> concept or object. This explains why Godel rejected Kant's more limited conception <strong>of</strong><br />
intuition, as well as why he chose to retain <strong>the</strong> Kantian term while giving it a more<br />
generous interpretation.<br />
Goldfarb's attempt to enlist Frege in his critique <strong>of</strong> Godel was worse than misguided. It<br />
kept him from noting <strong>the</strong> deep philosophical affinities between <strong>the</strong>se two giants <strong>of</strong> modern<br />
logic. They swam in a sea <strong>of</strong> ultraempiricism, but managed somehow not to get wet. Their<br />
philosophy was an uncompromising Platonism, reminiscent <strong>of</strong> Plato himself. "In arithmetic,"<br />
wrote Frege, "we are not concerned with objects which we come to know as something<br />
alien from <strong>without</strong> through <strong>the</strong> medium <strong>of</strong> <strong>the</strong> senses, but with objects given directly to<br />
our reason, <strong>and</strong>, as its nearest kin, utterly transparent to it." Plato could not have put it<br />
better. As he wrote in <strong>the</strong> Phaedo, "When <strong>the</strong> soul investigates by itself it passes into <strong>the</strong><br />
realm <strong>of</strong> what is pure, ever existing . . . <strong>and</strong> unchanging . . . [<strong>and</strong> is] akin to it."
Frege <strong>and</strong> Godel were "logicists" who believed that ma<strong>the</strong>matics, with <strong>the</strong> exception <strong>of</strong><br />
geometry, could be derived from logic, which <strong>the</strong>y took to include set <strong>the</strong>ory, toge<strong>the</strong>r<br />
with <strong>the</strong> correct definitions <strong>of</strong> <strong>the</strong> fundamental concepts. Nei<strong>the</strong>r believed, as did <strong>the</strong><br />
positivists, that <strong>the</strong> derivability from definitions, or analyticity, <strong>of</strong> arithmetic meant that<br />
its formulas lacked cognitive content. Godel made it clear that what is analytic is true in<br />
virtue <strong>of</strong> <strong>the</strong> meanings <strong>of</strong> <strong>the</strong> concepts involved. Each held that geometrical knowledge<br />
was different, founded on a kind <strong>of</strong> a priori physical intuition, as intimated by Kant, a<br />
philosopher both had studied deeply, though <strong>the</strong>y rejected large parts <strong>of</strong> his thinking. Each<br />
believed that <strong>the</strong> fundamental logical <strong>and</strong> metaphysical relationship is that <strong>of</strong><br />
predicationó<strong>the</strong> <strong>the</strong>ory <strong>of</strong> which, predicate logic, Frege had invented from whole<br />
clothówhose metaphysical correlative is falling under a concept (<strong>the</strong> central idea, not<br />
accidentally, <strong>of</strong> Frege's philosophy <strong>of</strong> arithmetic). Both held that <strong>the</strong> only means <strong>of</strong><br />
escaping from <strong>the</strong> personal, solipsistic <strong>world</strong> <strong>of</strong> <strong>the</strong> ideas in your head was to grasp<br />
abstract concepts, which exist not in our minds but ra<strong>the</strong>r in an objectively real "concept<br />
space," <strong>and</strong> that by this means we are granted access both to <strong>the</strong> physical <strong>world</strong> (by<br />
employing <strong>the</strong> concept <strong>of</strong> physical object) <strong>and</strong> to what Frege called <strong>the</strong> "third realm" <strong>of</strong><br />
ideal entities. "Concepts are <strong>the</strong>re," wrote Godel, "but not in any definite place. They . . .<br />
form <strong>the</strong> 'conceptual space,'" while for Frege, "in <strong>the</strong> external <strong>world</strong>, in <strong>the</strong> whole <strong>of</strong> space<br />
<strong>and</strong> all that <strong>the</strong>rein is, <strong>the</strong>re are no concepts ... no numbers."<br />
One place where Godel went beyond Frege was in attempting to forge a systematic<br />
epistemology that would account for our ability to make contact with "conceptual space"<br />
or <strong>the</strong> "third realm." Here Godel turned to Husserl, whom he saw as improving on <strong>the</strong><br />
Kantian philosophy, which is strong in epistemology but weak in ontology (weak, that is,<br />
for realists like Godel, Frege <strong>and</strong> Husserl). Yet perversely, it is just here, where a serious<br />
Platonist has finally decided to settle his epistemological accounts, that Goldfarb chooses<br />
to take Godel to taskóattempting to enlist Frege against Godel just when Godel was<br />
engaged in repairing a hole in Frege's philosophyóby citing<br />
Frege's irrelevant critique <strong>of</strong> Husserl's earlier, non-Platonist work. And this from a fellow<br />
logician, indeed, one <strong>of</strong> <strong>the</strong> guardians <strong>of</strong> Godel's <strong>legacy</strong>, an editor <strong>and</strong> contributor to <strong>the</strong><br />
definitive Collected Works <strong>of</strong> Kurt Godel. How is this to be explained?<br />
Goldfarb <strong>and</strong> Dreben's performances at <strong>the</strong> Boston University conference (<strong>and</strong> in Goldfarb's<br />
case, also at Helsinki) were no aberration. In <strong>the</strong>ir steadfast refusal to find anything <strong>of</strong><br />
value in Godel's contributions to philosophy <strong>and</strong> <strong>the</strong>ir unembarrassed display <strong>of</strong><br />
condescension, <strong>the</strong>y were simply carrying on a great <strong>and</strong> noble tradition among<br />
pr<strong>of</strong>essional philosophers. Long before <strong>the</strong> two spoke in Boston, for example, Charles<br />
Chihara, a philosopher <strong>of</strong> ma<strong>the</strong>matics at <strong>the</strong> University <strong>of</strong> California at Berkeley, had<br />
written a series <strong>of</strong> essays in which he took pains not just to criticize but to ridicule Godel's<br />
ma<strong>the</strong>matical episte-mology. He continued this project in a book published in 1990, Constructibility<br />
<strong>and</strong> Ma<strong>the</strong>matical Existence, which proved too much for at least one reviewer.<br />
"Unfortunately," wrote E.P. James, "Chihara's account <strong>of</strong> Godelian Platonism follows <strong>the</strong><br />
common line <strong>of</strong> regarding Godel as a logician par excellence but a philosophical fool."<br />
James went on to demonstrate that Chihara had simply gotten Godel wrong, concluding<br />
that "having failed to appreciate <strong>the</strong> complexity <strong>of</strong> Godel's beliefs about physical<br />
perception, it is easy for Chihara to argue against a caricature <strong>of</strong> his analogous beliefs<br />
about ma<strong>the</strong>matical 'perception.'" Once again, Godel's invocation <strong>of</strong> ma<strong>the</strong>matical intuition<br />
had succeeded in arousing <strong>the</strong> ire <strong>of</strong> a pr<strong>of</strong>essional philosopher. "At present," Godel wrote,
"ma<strong>the</strong>maticians are prejudiced against intuition." Philosophers, too. For Godel's<br />
philosophy <strong>of</strong> <strong>time</strong>, on this point as on o<strong>the</strong>rs, has been accorded a similarly cold<br />
reception.<br />
"The correct response to Godel"<br />
In his 1987 book Asymmetries in Time, Paul Horwich, <strong>the</strong>n at MIT, <strong>of</strong>fered fresh<br />
reexaminations <strong>of</strong> such staples <strong>of</strong> <strong>the</strong> space-<strong>time</strong> literature as <strong>the</strong> "direction <strong>of</strong> <strong>time</strong>" <strong>and</strong><br />
<strong>the</strong> difference between past <strong>and</strong> fu-<br />
ture. One aspect that was new was his decision to revisit Godel's argument for <strong>the</strong><br />
possibility <strong>of</strong> <strong>time</strong> travel, <strong>and</strong> newer still was <strong>the</strong> conclusion he reached, that <strong>the</strong><br />
argument was actually valid. What was not new was his neglect <strong>of</strong> <strong>the</strong> philosophical<br />
motiveó<strong>the</strong> demonstration <strong>of</strong> <strong>the</strong> ideality <strong>of</strong> <strong>time</strong>óbehind Godel's development <strong>of</strong> his new<br />
<strong>world</strong> models. In contrast, Milic Capek, ano<strong>the</strong>r philosopher <strong>of</strong> science, had argued<br />
decades before Horwich against Godel's case for <strong>time</strong> travel precisely on <strong>the</strong> grounds that<br />
if it held water, it followed that <strong>time</strong> was not real, a conclusion he found unacceptable.<br />
(Karl Popper had argued similarly.) Thus, an ironic situation: Capek (but not Horwich)<br />
agreed with Godel that <strong>the</strong> possibility <strong>of</strong> <strong>time</strong> travel signaled <strong>the</strong> ideality <strong>of</strong> <strong>time</strong>, while<br />
Horwich (but not Capek) took sides with Godel on <strong>the</strong> genuine possibility <strong>of</strong> <strong>time</strong> travel.<br />
Nei<strong>the</strong>r sided with Godel in maintaining both that <strong>time</strong> travel is possible <strong>and</strong> that <strong>the</strong>refore<br />
<strong>time</strong> is merely ideal.<br />
Attention, however, was paid to this dual aspect <strong>of</strong> Godel's reasoning in a small book <strong>of</strong><br />
mine published in 1991, The Disappearance <strong>of</strong> Time: Kurt Godel <strong>and</strong> <strong>the</strong> Idealistic Tradition<br />
in Philosophy. When an exp<strong>and</strong>ed edition appeared in 1999, some notice was taken, <strong>and</strong> a<br />
small but growing cottage industry developed <strong>of</strong> attempts to assess what Godel was really<br />
up to in his writings on <strong>Einstein</strong>. The Zeitgeist, however, is not so easily deflected. On <strong>the</strong><br />
question <strong>of</strong> whe<strong>the</strong>r Godel really had something important to contribute to <strong>the</strong> philosophy<br />
<strong>of</strong> <strong>time</strong>, as well as to <strong>the</strong> philosophy <strong>of</strong> ma<strong>the</strong>matics <strong>and</strong> <strong>the</strong> ontological argument for God,<br />
an impolite skepticism remains dominant.<br />
That <strong>the</strong> spirit <strong>of</strong> <strong>the</strong> <strong>time</strong> is still unmoved became most apparent in 1995, <strong>the</strong> year <strong>of</strong> <strong>the</strong><br />
Godel conference in Boston, when a distinguished philosopher <strong>of</strong> space <strong>and</strong> <strong>time</strong>, John<br />
Earman, responded to me <strong>and</strong> o<strong>the</strong>rs by devoting an appendix <strong>of</strong> his book Bangs, Crunches,<br />
Whimpers, <strong>and</strong> Shrieks to Godel's argument for <strong>the</strong> ideality <strong>of</strong> <strong>time</strong>. He began auspiciously<br />
by noting "<strong>the</strong> relative neglect <strong>of</strong> <strong>the</strong> philosophical moral Godel himself wanted to draw<br />
from his solutions to EFE [<strong>Einstein</strong>'s Held equations!." He continued in like manner by<br />
observing, with becoming piety, that "a deeply held conviction <strong>of</strong> someone<br />
<strong>of</strong> Godel's stature deserves serious consideration." At long last, it appeared, <strong>the</strong> space-<strong>time</strong><br />
establishment was giving Godel his due. But <strong>the</strong> appearance, like <strong>time</strong> itself, was an<br />
illusion. Earman's assessment <strong>of</strong> what Godel had to teach us was entirely negative. He<br />
couldn't resist adding, moreover, that <strong>the</strong> neglect <strong>of</strong> Godel's philosophical moral had after<br />
all been "benign." Then, just in case his message wasn't sufficiently clear, he concluded by<br />
observing that <strong>Einstein</strong>, in his reply to Godel in <strong>the</strong> Schilpp volume, had "brushed aside"
Godel's discussion <strong>of</strong> idealism, adding that "this seems to me to be <strong>the</strong> correct response to<br />
Godel." So much for <strong>the</strong> attention due someone <strong>of</strong> Godel's stature.<br />
That Earman took issue with Godel is not <strong>the</strong> point. The point is that once again a noted<br />
philosopher felt no embarrassment in dismissing Godel with maximum condescension.<br />
Moreover, just as Goldfarb had attempted to enlist someone Godel admired, namely Frege,<br />
to attack him, Earman employed <strong>Einstein</strong> to back up his claim that Godel's attempt at<br />
philosophy should be "brushed aside." Once again, however, <strong>the</strong> enlistment was premature.<br />
It is true that <strong>Einstein</strong>, in his response to Godel, sidestepped <strong>the</strong> issue <strong>of</strong> idealism, but he<br />
was speaking as a physicist whose interest was in <strong>the</strong> implications <strong>of</strong> Godel's important new<br />
discovery (as <strong>Einstein</strong> described it) for <strong>the</strong> <strong>the</strong>ory <strong>of</strong> relativity. He was not, like Goldfarb,<br />
participating in a conference devoted to Godel's contributions to philosophy, nor, like<br />
Earman, discoursing on <strong>the</strong> philosophy <strong>of</strong> <strong>time</strong> for an audience <strong>of</strong> philosophers. On his own,<br />
<strong>Einstein</strong> shared Godel's appreciation <strong>of</strong> Kant, <strong>and</strong> also his reservations, <strong>and</strong> he too had<br />
occasionally speculated on <strong>the</strong> relationship between relativity <strong>the</strong>ory <strong>and</strong> our everyday<br />
experience <strong>of</strong> <strong>time</strong>. Unlike Godel, however, he had no intention <strong>of</strong> engaging directly with<br />
<strong>the</strong> philosophical literature, <strong>and</strong> had no plans to follow up his philosophical speculations in<br />
any depth. He was not, in short, an appropriate ally in a campaign to put Godel in his<br />
philosophical place.<br />
<strong>Einstein</strong> aside, Earman made it clear that nothing in Godel's argument impressed him. He<br />
looked askance at everything, even something as straightforward as Godel's clear<br />
appreciation that if <strong>time</strong> flows, this flux represents <strong>the</strong> coming into existence <strong>of</strong> new<br />
states <strong>of</strong> affairs, <strong>and</strong><br />
that <strong>the</strong> relativity <strong>of</strong> simultaneity could not sensibly be taken to imply <strong>the</strong> relativity <strong>of</strong><br />
existence. Without disturbing himself to dispute GodePs conclusion, Earman commented<br />
only that "this is a pretty piece <strong>of</strong> ordinary language philosophizing . . . but like most <strong>of</strong> its<br />
ilk, it leaves one up in <strong>the</strong> air . . . one can wonder how such intuitions can support such<br />
weighty philosophical morals." Once again GodePs invocation <strong>of</strong> intuition had gotten him<br />
into trouble. But Earman had added a new twist. Somehow, Godel, with his distrust <strong>of</strong><br />
"ordinary language philosophizing" exemplified by Wittgenstein <strong>and</strong> his school, had been<br />
lumped toge<strong>the</strong>r with this movement. The intuitions Godel had in mind, however, were <strong>the</strong><br />
result <strong>of</strong> <strong>the</strong> highly rational exercise <strong>of</strong> turning one's attention to <strong>the</strong> nature <strong>of</strong> <strong>the</strong><br />
concepts <strong>the</strong>mselves, not <strong>of</strong> tuning one's ear to <strong>the</strong> marketplace <strong>of</strong> "ordinary language" <strong>and</strong><br />
everyday conversation. Language as such, for Godel, had nothing to do with it. The crux <strong>of</strong><br />
Earman's critique concerned <strong>the</strong> relevance <strong>of</strong> <strong>time</strong> travel in a merely possible Godel<br />
universeówhere one might grant that <strong>time</strong> is an illusionóto <strong>the</strong> existence <strong>of</strong> <strong>time</strong> in our<br />
own <strong>world</strong>. Since <strong>the</strong> extreme geometry <strong>of</strong> <strong>the</strong> Godel universe, which allows for <strong>the</strong><br />
possibility <strong>of</strong> closed, future-directed, <strong>time</strong>like curves, is not a feature <strong>of</strong> <strong>the</strong> actual <strong>world</strong>,<br />
Earman argued, Godel had failed to show <strong>time</strong>'s absence in <strong>the</strong> <strong>world</strong> in which we live. Yet<br />
Earman did not deny that <strong>the</strong> experience <strong>of</strong> <strong>the</strong> lapse <strong>of</strong> <strong>time</strong> does not decide <strong>the</strong> issue,<br />
since it is by hypo<strong>the</strong>sis something we might well have in common with <strong>the</strong> denizens <strong>of</strong> <strong>the</strong><br />
Godel universe. That <strong>the</strong> geometry <strong>of</strong> our <strong>world</strong>, moreover, does not in itself exclude <strong>the</strong><br />
possibility <strong>of</strong> <strong>the</strong> flow <strong>of</strong> <strong>time</strong> is also not decisive, since that shows at most that <strong>the</strong><br />
conditions necessary for <strong>the</strong> existence <strong>of</strong> <strong>time</strong> are present in our <strong>world</strong>. But necessary<br />
conditions are not <strong>the</strong> same as sufficient conditions, <strong>and</strong> absent <strong>the</strong> testimony <strong>of</strong> <strong>the</strong><br />
experience <strong>of</strong> <strong>time</strong> (which cuts both ways), <strong>and</strong> <strong>of</strong> <strong>the</strong> laws <strong>of</strong> nature (which are <strong>the</strong> same<br />
in both <strong>world</strong>s), it is hard to see that anything at all could decide in favor <strong>of</strong> <strong>the</strong> existence<br />
<strong>of</strong> <strong>time</strong> in <strong>the</strong> actual <strong>world</strong>. In short, our <strong>world</strong> would appear, by Godel's lights, to be in<br />
principle indistinguishable from a universe in which <strong>time</strong> is demonstrably absent.
How could one miss <strong>the</strong> force <strong>of</strong> this argument? In company with o<strong>the</strong>r philosophers <strong>of</strong><br />
physics, Earman verges on conflating <strong>the</strong> geometrical prerequisites for <strong>the</strong> lapse <strong>of</strong> <strong>time</strong><br />
with <strong>the</strong> actual lapse itself. Indeed, <strong>the</strong> latter is something Earman clearly has trouble<br />
taking seriously. But if we don't acknowledge <strong>the</strong> flow <strong>of</strong> <strong>time</strong>, one can hear Godel saying,<br />
<strong>the</strong>n <strong>the</strong> game is already over: <strong>the</strong>re simply is no such thing as <strong>time</strong> in <strong>the</strong> intuitive sense,<br />
<strong>and</strong> an argument for <strong>the</strong> ideality <strong>of</strong> <strong>time</strong> is not needed. In fact, Godel actually did say just<br />
this: "One may take <strong>the</strong> st<strong>and</strong>point," he wrote in <strong>the</strong> Schilpp volume on <strong>Einstein</strong>, "that <strong>the</strong><br />
idea <strong>of</strong> an objective lapse <strong>of</strong> <strong>time</strong> (whose essence is that only <strong>the</strong> present really exists) is<br />
meaningless. But this is no way out <strong>of</strong> <strong>the</strong> dilemma; for by this very opinion one would take<br />
<strong>the</strong> idealistic viewpoint."<br />
One can, <strong>of</strong> course, in good conscience, disagree with Godel's conclusions. He did not<br />
suffer from <strong>the</strong> illusion that he had said <strong>the</strong> last word on <strong>the</strong> reality <strong>of</strong> <strong>time</strong>, which, as we<br />
saw earlier, he described to Hao Wang as perhaps <strong>the</strong> philosophical question. But it is<br />
difficult to underst<strong>and</strong> how one can disagree with <strong>the</strong> judgment that, whe<strong>the</strong>r he is<br />
ultimately right or wrong, Godel has made a pr<strong>of</strong>ound contribution to philosophy. In<br />
addressing <strong>the</strong> question <strong>of</strong> <strong>time</strong>'s existence, a question that has haunted philosophers from<br />
Parmenides <strong>and</strong> Plato to Kant, he brought to bear <strong>the</strong> most powerful, most fully developed<br />
formal account <strong>of</strong> <strong>time</strong> ever constructed, <strong>the</strong> <strong>the</strong>ory <strong>of</strong> relativity, <strong>and</strong> proceeded to<br />
distinguish, with great precision, <strong>the</strong> formal from <strong>the</strong> intuitive concept, rel-ativistic from<br />
intuitive <strong>time</strong>. He <strong>the</strong>n demonstrated, simply <strong>and</strong> elegantly, that <strong>the</strong> existence <strong>of</strong> intuitive<br />
<strong>time</strong>, which lapses, is inconsistent with <strong>the</strong> truth <strong>of</strong> special relativity, in particular, with<br />
<strong>the</strong> relativity <strong>of</strong> simultaneity <strong>and</strong> <strong>the</strong> equivalence <strong>of</strong> all inertial frames. He proceeded,<br />
next, to remind us that in general relativity certain reference frames can be seen as<br />
privileged, <strong>and</strong> thatóif <strong>the</strong> distribution <strong>of</strong> matter <strong>and</strong> motion is accommodatingówe can<br />
construct a simulacrum <strong>of</strong> intuitive <strong>time</strong>, namely cosmic <strong>time</strong>. He <strong>the</strong>n made a deep <strong>and</strong><br />
surprising contribution to relativistic physics by discovering new <strong>world</strong> models in which,<br />
due to <strong>the</strong> unhappy distribution <strong>of</strong> matter <strong>and</strong> motion, even cosmic <strong>time</strong><br />
cannot be constructed. In addition, he proved that in some <strong>of</strong> <strong>the</strong>se new <strong>world</strong> models, or<br />
Godel universes, <strong>time</strong> travel is possible, which once again proves that in such <strong>world</strong>s <strong>the</strong>re<br />
is no such thing as cosmic or intuitive <strong>time</strong>. To clinch <strong>the</strong> deal, he brought to bear a modal<br />
argument, from possibility to actuality, making <strong>the</strong> case that if <strong>time</strong> was ideal in <strong>the</strong> Godel<br />
universe, which contains <strong>the</strong> same physical laws as ours, <strong>and</strong> whose inhabitants might well<br />
experience <strong>time</strong> in <strong>the</strong> same way we do, <strong>the</strong>n it cannot be that <strong>the</strong> mere difference in <strong>the</strong><br />
distribution <strong>of</strong> matter <strong>and</strong> motion in <strong>the</strong> two <strong>world</strong>s accounts for <strong>the</strong> fact that in one <strong>time</strong><br />
exists, whereas in <strong>the</strong> o<strong>the</strong>r it does not. Finally, in a deep <strong>and</strong> original interpretation <strong>of</strong><br />
Kant, who is usually taken to have been refuted by <strong>Einstein</strong>, he argued that, correctly<br />
understood, Kant's doctrine <strong>of</strong> <strong>the</strong> ideality <strong>of</strong> <strong>time</strong> bears striking affinities with <strong>the</strong><br />
temporal idealism implicit in <strong>the</strong> <strong>the</strong>ory <strong>of</strong> relativity. If this is not <strong>the</strong> right way to do<br />
philosophy, what is?<br />
Godel as Philosopher<br />
The case for Godel as philosopher is unassailable. Though he published few essays that<br />
could be considered explicit contributions to philosophy, <strong>the</strong>y suffice to establish him as an<br />
important philosopher <strong>of</strong> ma<strong>the</strong>matics <strong>and</strong> <strong>of</strong> space <strong>and</strong> <strong>time</strong>. The posthumous publication
<strong>of</strong> several more <strong>of</strong> his philosophical studies, including <strong>the</strong> Gibbs Lecture, his contribution<br />
to <strong>the</strong> Schilpp volume on Carnap, <strong>and</strong> <strong>the</strong> longer version <strong>of</strong> his contribution to <strong>the</strong> Schilpp<br />
volume on <strong>Einstein</strong>, confirm this assessment. The essay he wrote for <strong>the</strong> Schilpp volume on<br />
Russell, which contained new <strong>and</strong> insightful discussions <strong>of</strong> Frege as well as Russell on <strong>the</strong><br />
question <strong>of</strong> meaning, including an illuminating <strong>and</strong> prescient comparison <strong>of</strong> Russell on<br />
"denoting" with Frege on "sense <strong>and</strong> reference," leaves little room for doubt that in <strong>the</strong><br />
philosophy <strong>of</strong> language, too, his abilities were striking. He had clearly mastered <strong>the</strong><br />
writings <strong>of</strong> most <strong>of</strong> <strong>the</strong> seminal figures in twentieth-century analytical philosophy,<br />
including Frege, Russell <strong>and</strong> Carnap.<br />
He knew Wittgenstein's Tractatus, too, though it is not known how well, <strong>and</strong> was a student<br />
<strong>of</strong> <strong>the</strong> writings <strong>of</strong> <strong>the</strong> founding fa<strong>the</strong>r <strong>of</strong> continental philosophy, Husserl. The split between<br />
<strong>the</strong> analytical <strong>and</strong> continental schools, which, sadly, holds sway to this day, did not<br />
intimidate him. Here as elsewhere he proved himself free from <strong>the</strong> philosophical<br />
prejudices around him. He attended seriously, as well, to <strong>the</strong> history <strong>of</strong> philosophy,<br />
devoting endless hours to <strong>the</strong> study <strong>of</strong> Leibniz <strong>and</strong> acquiring a pr<strong>of</strong>ound underst<strong>and</strong>ing <strong>of</strong><br />
Kant. His grasp <strong>of</strong> Hegel astonished <strong>the</strong> logician <strong>and</strong> philosopher Georg Kreisel, a man not<br />
easy to impress. Looking over <strong>the</strong> set <strong>of</strong> quotations from Hegel that Godel had assembled,<br />
Kreisel remarked that "<strong>the</strong> publication <strong>of</strong> such an anthology is likely to produce a minor<br />
revolution in philosophy." He also studied Plato <strong>and</strong> Aristotle, as well as <strong>the</strong> medieval<br />
philosophers, but we do not know <strong>the</strong> extent <strong>of</strong> his familiarity with <strong>the</strong>se figures. We do<br />
know that he put his grasp <strong>of</strong> <strong>the</strong> history <strong>of</strong> philosophy to creative use, enlisting his<br />
knowledge <strong>of</strong> Kant to help him comprehend <strong>the</strong> philosophical significance <strong>of</strong> <strong>the</strong> <strong>the</strong>ory <strong>of</strong><br />
relativity, <strong>and</strong> turning to Husserl's phenomenology for assistance in developing an<br />
epistemology adequate to <strong>the</strong> Platonist ontology he espoused for ma<strong>the</strong>matics. He<br />
believed that <strong>the</strong> history <strong>of</strong> philosophy could help free us from prejudice. "Even science,"<br />
he said, "is very heavily prejudiced in one direction. Knowledge in everyday life is also<br />
prejudiced. Two methods to transcend such prejudices are: (1) phenomenology; (2) going<br />
back to o<strong>the</strong>r ages."<br />
Overarching much <strong>of</strong> his research in philosophy <strong>and</strong> logic was <strong>the</strong> "Godel program," <strong>the</strong><br />
investigation <strong>of</strong> <strong>the</strong> limits <strong>of</strong> formal methods in capturing intuitive concepts. This was<br />
clearly a philosophical enterprise, though he carried it out using both formal <strong>and</strong><br />
philosophical tools. If <strong>the</strong> leitmotif <strong>of</strong> <strong>the</strong> twentieth century was formalism, in <strong>the</strong> most<br />
general sense, his incompleteness <strong>the</strong>orem was unquestionably <strong>the</strong> single most important<br />
contribution to this subject. As it had been for Plato, ma<strong>the</strong>matics was for him a deep<br />
source <strong>of</strong> philosophical inspiration, in itself, in its relationship to logic, <strong>and</strong> in its ability to<br />
describe <strong>the</strong> physical <strong>world</strong>. A lifelong opponent <strong>of</strong> positivism, both in<br />
<strong>the</strong> narrow, technical sense <strong>of</strong> <strong>the</strong> program embraced by <strong>the</strong> Vienna Circle <strong>and</strong> in <strong>the</strong><br />
broader sense <strong>of</strong> a philosophical tendency endemic to every age, he attempted to<br />
reappropriate ma<strong>the</strong>matics <strong>and</strong> logic for <strong>the</strong> o<strong>the</strong>r side. "One bad effect <strong>of</strong> logical<br />
positivism," he said in his conversations with Wang, "is its claim <strong>of</strong> being intimately<br />
associated with ma<strong>the</strong>matical logic. . . . Ma<strong>the</strong>matical logic should be used more by<br />
nonpositivistic philosophers. The positivists have a tendency to represent <strong>the</strong>ir philosophy<br />
as a consequence <strong>of</strong> logic, to give it scientific dignity."<br />
If in his published writings he aimed for maximum precision <strong>and</strong> minimum controversy,<br />
stripping down his contributions until only <strong>the</strong> bones were leftóthat is, until all that
emained was what was most amenable to rigorous demonstration <strong>and</strong> unavoidable<br />
philosophical interpretationóin his notebooks <strong>and</strong> in his conversations with Wang, he felt<br />
free to engage in <strong>the</strong> most wide-ranging, fundamental speculations, flying through thin air<br />
as high as pure thought could take him, with no fear <strong>of</strong> crashing since he had no intention<br />
<strong>of</strong> l<strong>and</strong>ing. There is a fair amount <strong>of</strong> rough coal in <strong>the</strong>se extravagant musings, but <strong>the</strong>re<br />
are also diamonds, sparks <strong>of</strong> insight into noncausal "laws" <strong>of</strong> historical development, <strong>the</strong><br />
limitations <strong>of</strong> mechanical biology, a concept <strong>of</strong> "absolute" pro<strong>of</strong>, <strong>and</strong> <strong>the</strong> possibility <strong>of</strong> an<br />
afterlife. If <strong>Einstein</strong> accepted his end with equanimity, underst<strong>and</strong>ing that he had long<br />
since fired <strong>of</strong>f his best shots, it is underst<strong>and</strong>able that Godel resented his. In many ways,<br />
he had just begun to fight.<br />
An Ugly Body in Beautiful Clothing<br />
Even on his deathbed, <strong>Einstein</strong> had been engaged in a desperate effort to find a unified<br />
field <strong>the</strong>ory. When his eyes finally closed, it fell to Bruria Kaufman, his last collaborator,<br />
<strong>and</strong> his friend Godel to attend to <strong>the</strong> papers that remained in <strong>the</strong> great physicist's <strong>of</strong>fice.<br />
Toge<strong>the</strong>r, <strong>the</strong>y found a blackboard filled with sad equations that led nowhere. The fruits <strong>of</strong><br />
<strong>Einstein</strong>'s earlier endeavors, however, remain <strong>the</strong> jewels in<br />
<strong>the</strong> crown <strong>of</strong> physics in <strong>the</strong> twentieth century. His <strong>the</strong>ories uncovered <strong>the</strong> ma<strong>the</strong>matical<br />
symmetries hidden in <strong>the</strong> noise <strong>and</strong> confusion <strong>of</strong> <strong>the</strong> physical <strong>world</strong>.<br />
This fact deeply impressed Godel. Ma<strong>the</strong>matics itself, which he had encountered more<br />
directly than <strong>Einstein</strong> had, was for him a source <strong>of</strong> wonder <strong>and</strong> admiration. "It is given to<br />
us in its entirety <strong>and</strong> does not change," he said, "unlike <strong>the</strong> Milky Way. That part <strong>of</strong> it <strong>of</strong><br />
which we have a perfect view seems beautiful, suggesting harmony." More than this,<br />
however, he recognized that <strong>the</strong>se symmetries are not exclusive to <strong>the</strong> separate <strong>world</strong> <strong>of</strong><br />
pure, transcendent "forms." "Ma<strong>the</strong>matics is applied to <strong>the</strong> real <strong>world</strong>," he wrote, "<strong>and</strong> has<br />
proved fruitful. This suggests that <strong>the</strong> ma<strong>the</strong>matical <strong>and</strong> empirical parts are in harmony<br />
<strong>and</strong> that <strong>the</strong> real <strong>world</strong> is also beautiful." Like Plato in <strong>the</strong> Timaeus, he believed that<br />
however <strong>the</strong> "real" <strong>world</strong> had come about, it was based on a divine model. "O<strong>the</strong>rwise," he<br />
said, "ma<strong>the</strong>matics would be just an ornament <strong>and</strong> <strong>the</strong> real <strong>world</strong> would be like an ugly<br />
body in beautiful clothing."<br />
On his departure from this <strong>world</strong>, he left boxes <strong>of</strong> papers at <strong>the</strong> institute, begging for<br />
discovery. The first to do so, his biographer John Dawson, found two bound notebooks<br />
containing calculations that had nothing to do with logic or pure ma<strong>the</strong>matics. It turned<br />
out <strong>the</strong>y were Godel's recordings <strong>of</strong> <strong>the</strong> angular orientations <strong>of</strong> galaxies. The purist <strong>of</strong> pure<br />
logicians had never ceased trying to discover whe<strong>the</strong>r <strong>the</strong> actual <strong>world</strong> we inhabit is a<br />
Godel universe. His efforts, however, were unnecessary. In a deep sense, it is clear enough<br />
that we all do live in Godel's universe.<br />
Notes
Chapter 1<br />
2 key to his trunk: See Peter Suber, 1992, "Fifty Years Later, <strong>the</strong> Questions Remain:<br />
Kurt Godel at Blue Hill," Ellsworth American, August 27. 2<br />
Folsing, 1998, Albert <strong>Einstein</strong>, trans. Ewald Osers<br />
"From a distance": Albrecht<br />
(New York: Penguin <strong>Books</strong>), 689.<br />
2 "just to have <strong>the</strong> privilege": Oskar Morgenstern, in a letter to <strong>the</strong> Austrian<br />
government in 1965, cited in Hao Wang, 1996, A Logical journey: From Godel to Philosophy<br />
(Cambridge, Mass.: MIT Press), 57.<br />
3 "<strong>the</strong> most significant": W.V. Quine, 1952, text <strong>of</strong> citation for Godel's honorary<br />
doctorate from Harvard University, June, cited in Wang 1996, 2.<br />
4 "Here you see that portion": Folsing 1998, 283.<br />
4 "When I objected": Werner Heisenberg, 1983, Encounters with <strong>Einstein</strong> <strong>and</strong> O<strong>the</strong>r Essays<br />
on People, Places, <strong>and</strong> Particles (Princeton: Princeton University Press), 114.<br />
4 "on equal terms with <strong>Einstein</strong>": Freeman Dyson, 1993, From Eros to Gaia (New York:<br />
Penguin <strong>Books</strong>), 161.<br />
5 "Only fables": Godel, letter to his mo<strong>the</strong>r, April 21, 1965, cited in Wang 1996, 45.<br />
5 "You know, once you start calculating": Folsing 1998, 311.<br />
6 "Every boy in <strong>the</strong> streets": Constance Reid, 1986, Hilbert-Courant (New York:<br />
Springer-Verlag), 142.<br />
7 Time, "that mysterious...": Kurt Godel, "A Remark About <strong>the</strong> Relationship Between<br />
Relativity Theory <strong>and</strong> Idealistic Philosophy," in Paul Arthur Schilpp, ed., 1949, Albert<br />
<strong>Einstein</strong>: Philosopher-Scientist (LaSalle, 111.: Open Court), 557.
8 "chronology protection conjecture": Stephen Hawking, 1992, "Chronology Protection<br />
Conjecture," Physical Review D46, no. 2 (July 15).<br />
Chapter 2<br />
9 The German man <strong>of</strong> science: Arthur Miller, 1986, Imagery in Scientific Thought<br />
(Cambridge, Mass.: MIT Press), 108.<br />
9 It is a remarkable fact: Kurt Godel, 1946-1949, "Some Observations About <strong>the</strong><br />
Relationship Between Theory <strong>of</strong> Relativity <strong>and</strong> Kantian Philosophy," in Solomon<br />
Feferman et al., eds., 1995, Kurt Godel: Collected Works, vol. 3, Unpublished Essays <strong>and</strong><br />
Lectures (New York: Oxford University Press), 230.<br />
10 "I have firmly resolved": Folsing 1998, 333.<br />
10 "Of course he has no children": Suber 1992.<br />
10 "There is no doubt": Paul Weingartner <strong>and</strong> Leopold Schmetterer, 1987, eds., Godel<br />
Remembered (Naples: Bibliopolis), 32.<br />
10 "Albert himself": Dennis Overbye, 1999, "<strong>Einstein</strong>, Confused in Love <strong>and</strong>, Some<strong>time</strong>s,<br />
Physics," New York Times, August 31.<br />
11 Finding himself trapped: Paul Halmos, 1988, / Want to Be a Ma<strong>the</strong>matician: An<br />
Automatkography in Three Parts (Washington, D.C., Ma<strong>the</strong>matical Association <strong>of</strong> America).<br />
11 "to make <strong>the</strong> serious things in <strong>the</strong> <strong>world</strong> tolerable": Philipp Frank, 1989, <strong>Einstein</strong>: His<br />
Life <strong>and</strong> Times (New York, Da Capo Press), 281-282.<br />
12 <strong>Einstein</strong> "was an experienced sight reader": Gerald Holton <strong>and</strong> Yehuda Elkana, 1982,<br />
eds., Albert <strong>Einstein</strong>: Historical <strong>and</strong> Cultural Perspectives (Mineola, N.Y.: Dover), 410.<br />
12 Begriffsscbrift: Gottlob Frege, 1967, Begriffsschrift, A Formula Language, Modeled upon<br />
That <strong>of</strong> Arithmetic, for Pure Thought," in Jean van Heijenoort, ed., From Frege to Godel: A<br />
Source Book in Ma<strong>the</strong>matical Logic, 1879-1931, 182 (Cambridge, Mass.: Harvard University<br />
Press).
12 "deeply religious unbeliever": Gerald Holton, 1998, "<strong>Einstein</strong> <strong>and</strong> <strong>the</strong> Cultural Roots<br />
<strong>of</strong> Modern Science," Daedalus, Winter, 16.<br />
13 Godel was not a pan<strong>the</strong>ist: Hao Wang, 1987, Reflections on Kurt Godel (Cambridge,<br />
Mass.: MIT Press), 18.<br />
13 Spinoza's God: Wang 1996, 152. 13 Whereas "ninety per cent <strong>of</strong> philosophers":<br />
Ibid., 107.<br />
13 "All three <strong>of</strong> <strong>the</strong> o<strong>the</strong>rs": Bertr<strong>and</strong> Russell, 2000, Autobiography (London: Routledge),<br />
466. 13 "I am not a Jew": Wang 1996, 112.<br />
13 "Kurt had a friendly attitude": Weingartner <strong>and</strong> Schmetterer 1987, 33.<br />
14 "I frequently held an opinion": John Dawson, 1997, Logical Dilemmas (Wellesley,<br />
Mass.: A.K. Peters).<br />
15 "You became a ma<strong>the</strong>matician": Wang 1996, 87.<br />
15 "You have a vicarious": Karl Menger, 1994, Reminiscences <strong>of</strong> <strong>the</strong> Vienna Circle <strong>and</strong> <strong>the</strong><br />
Ma<strong>the</strong>matical Colloquium (Dordrecht, Ne<strong>the</strong>rl<strong>and</strong>s: Kluwer Academic Publishers), 222.<br />
15 "Who ever became more intelligent": Ibid., 223<br />
15 "If you had a particular problem": Weingartner <strong>and</strong> Schmetterer 1987, 33.<br />
16 "Kant is a sort <strong>of</strong> highway": Ibid., 74.<br />
16 "At <strong>the</strong> Institute in Princeton": Holton 1998, 18.<br />
18 "holy geometry booklet": Albert <strong>Einstein</strong>, "Autobiographical Notes," in Schilpp
1949, 11. 18 "Time," Kant himself said: Norman Kemp Smith, 1965, trans., Critique <strong>of</strong><br />
Pure<br />
Reason, by Immanuel Kant (New York: St. Martin's Press), B 50. 18 "We cannot obtain for<br />
ourselves": Ibid., B 156. 18 "<strong>the</strong> primary object <strong>of</strong> <strong>Einstein</strong>'s": G.J. Whitrow, 1980, The<br />
Natural Philosophy<br />
nf Time (Oxford- ('l:ircn
27 by constructing "far-fetched grounds for jealousy": Ibid, 154.<br />
28 Tractatus: Ludwig Wittgenstein, 1961, Tractatus Logico-Philosophicus, trans. D.F.<br />
Pears <strong>and</strong> B.F. McGuinness (London: Routledge £>C Kegan Paul).<br />
28 "what we cannot speak about": Wittgenstein 1961, 7<br />
28 "Today we . . . out-Wittgensteined <strong>the</strong>se Wittgensteinians": Menger 1994, 210.<br />
30 "I. .. believe myself to have found": Wittgenstein 1961, 5.<br />
30 "I can testify to this": Weingartner <strong>and</strong> Schmetterer 1987, 40.<br />
32 "I put my faith in organization": Holton 1998, 16.<br />
32 "Despite <strong>the</strong>ir remoteness": Kurt Godel, 1964, "What Is Cantor's Continuum Problem?"<br />
(1964 supplement), in Feferman et al. 1990, 11/268.<br />
33 "Mach's way": Holton 1998, 9.<br />
34 "since <strong>the</strong> ma<strong>the</strong>maticians pounced on <strong>the</strong> relativity <strong>the</strong>ory": Folsing 1998, 245.<br />
35 "The antipathy <strong>of</strong> <strong>the</strong>se scholars": Albert <strong>Einstein</strong>, "Autobiographical Notes," in<br />
Schilpp 1949, 49.<br />
35 "We really experience only": Gottlob Frege, "Thought," in Michael Beaney, ed.,<br />
1997, The Frege Reader (Oxford: Blackwell), 339. 35 Lenin's critique <strong>of</strong> Mach: V.I. Lenin,<br />
1950, Materialism <strong>and</strong> Empirio-Criticism,<br />
trans. A. Fineberg (London: Lawrence & Wisehart). 37<br />
David Lindley, 2001, Boltzmann's Atom (New<br />
"I don't believe that atoms exist!":<br />
York: Free Press), vii, xi.
37 <strong>Einstein</strong>'s Annalen derPhysik publications: Albert <strong>Einstein</strong>, 1905a, "On <strong>the</strong><br />
Electrodynamics <strong>of</strong> Moving Bodies," Annalen der Physik 17:891-921; Albert <strong>Einstein</strong>, 1905b,<br />
"On <strong>the</strong> Movement <strong>of</strong> Particles Suspended in Fluids at Rest, as Postulated by <strong>the</strong> Molecular<br />
Theory <strong>of</strong> Heat," Annalen der Physik 17:549-60.<br />
38 "<strong>the</strong> scientist must appear": Schilpp 1949, 684.<br />
38 Wittgenstein's view <strong>of</strong> "logical space": Wittgenstein 1961, 1.13 <strong>and</strong> 2.11.<br />
39 His most influential work: Moritz, Schlick, 1918, Allgemeine Erkenntnislehre (IWrliii:<br />
Spritigrr Verlag).<br />
I'1<br />
"Sad, Inn true": Mcncci ll>'M, Sh.<br />
40 "Now you damned bastard": See David Edmonds <strong>and</strong> John Eidinow, 2001,<br />
Wittgenstein's Poker (New York: HarperCollins), 142-148.<br />
41 "It is to be hoped": Ibid., 146.<br />
41 Nelbock after <strong>the</strong> Anschluss: Ibid., 147.<br />
47 "Dear Colleague": Bertr<strong>and</strong> Russell, 1902, "Letter to Frege," in van Heijenoort 1967,<br />
124.<br />
48 "The sole possible foundations <strong>of</strong> arithmetic": Frege, 1902, "Letter to Russell," in van<br />
Heijenoort 1967, 128.<br />
48 "The development <strong>of</strong> philosophy since <strong>the</strong> Renaissance": Kurt Godel, 1961, "The<br />
Modern Development <strong>of</strong> <strong>the</strong> Foundations <strong>of</strong> Ma<strong>the</strong>matics in <strong>the</strong> Light <strong>of</strong> Philosophy," in<br />
Feferman et al. 1995, III/375-376.<br />
49 "Thus came into being": Ibid., 379. 49 in Godel's words: Ibid.<br />
49 "an a priorism with <strong>the</strong> sign reversed": Ibid., 383.
49 "[A] system <strong>of</strong> truths": Warren Goldfarb, "The Philosophy <strong>of</strong> Ma<strong>the</strong>matics in Early<br />
Positivism," in R.N. Giere, ed., 1996, Origins <strong>of</strong> Logical Empiricism, Minnesota Studies in<br />
<strong>the</strong> Philosophy <strong>of</strong> Science, vol. 16, 213-230 (University <strong>of</strong> Minnesota Press), 214.<br />
49 "<strong>the</strong> unreasonable effectiveness <strong>of</strong> ma<strong>the</strong>matics": Eugene Wigner, 1979, "The<br />
Unreasonable Effectiveness <strong>of</strong> Ma<strong>the</strong>matics in <strong>the</strong> Natural Sciences," in Symmetries <strong>and</strong><br />
Reflections: Scientific Essays, 222-237 (Woodbridge, Conn.: Ox Bow Press).<br />
50 "It is not true, as Kant urged": Hans Hahn, "The Crisis in Intuition," in Brian<br />
McGuinness, ed., 1980, Empiricism, Logic, <strong>and</strong> Ma<strong>the</strong>matics: Philosophical Papers, 731-02<br />
(Dordrecht, Ne<strong>the</strong>rl<strong>and</strong>s: D. Reidel), 101.<br />
50 "The essence <strong>of</strong> this view": Kurt Godel, 1951, "Some <strong>Basic</strong> Theorems on <strong>the</strong><br />
Foundations <strong>of</strong> Ma<strong>the</strong>matics <strong>and</strong> Their Implications" (Gibbs Lecture, 1951), in Feferman et<br />
al. 1995, III/319.<br />
Chapter 4<br />
51 Every spy's life: Anais Nin, 1974, A Spy in <strong>the</strong> House <strong>of</strong> Love, (A<strong>the</strong>ns, Ohio: Swallow<br />
Press), 73.<br />
51 "[Russell] brought to light": Kurt Godel, 1944, "Russell's Ma<strong>the</strong>matical Logic," in<br />
Feferman et al. 1990, 11/124.<br />
52 "Where else would reliability <strong>and</strong> truth be found": David Hilbert, 1925, "On <strong>the</strong><br />
Infinite," in van Heijenoort 1967, 375.<br />
52 "The definitive clarification": Ibid. 370-371.<br />
52 "As concerns <strong>the</strong> <strong>world</strong>": John Dawson, 1985, "Discussion on <strong>the</strong> Foundations <strong>of</strong><br />
Ma<strong>the</strong>matics," History <strong>and</strong> Philosophy <strong>of</strong> Logic 5:116.<br />
52 "A very simple fact now seems": Ibid.<br />
53 Who pays any attention: e. e. cummings, "since feeling is first," 1926, in Complete'<br />
Poems: 1904-1962 (New York: Liveright, 1994).
54 "I can begin to hear <strong>the</strong> sound <strong>of</strong> machinery": Ray Monk, 1990, Ludivig Witlgeii stein:<br />
The Duty <strong>of</strong> Genius (New York: Free Press), 13.<br />
55 Frege's rejection <strong>of</strong> implicit definition: Gottlob Krege, 1980, The foundations <strong>of</strong><br />
Arithmetic: A I .ogico- Arithmetic Enquiry into <strong>the</strong> Concept <strong>of</strong> Number, trans. J.I,. Austin<br />
(Evanston. III.: Northwestern University I'ress). 68.<br />
57 "Continued appeals to ma<strong>the</strong>matical intuition": Kurt Godel, "What Is Cantor's<br />
Continuum Problem?" (1964 supplement), in Feferman et al. 1990, 11/269.<br />
58 "I would be very much interested": Dawson 1997, 70.<br />
58 "Von Neumann, from <strong>the</strong> beginning": Dawson 1984, 122.<br />
58 Von Neumann's striking prescience: S.M. Ulam, 1976, Adventures <strong>of</strong> a Ma<strong>the</strong>matician<br />
(Berkeley: University <strong>of</strong> California).<br />
62 "I was ra<strong>the</strong>r depressed": Benjamin Y<strong>and</strong>ell, 2002, The Honors Class: Hilbert's Problems<br />
<strong>and</strong> Their Solvers (Natick, Mass.: A.K. Peters), 75.<br />
68 Similarly, as a direct consequence: William F. Dowling, 1989, "There are no safe<br />
virus tests," American Ma<strong>the</strong>matical Monthly 96, 835-836.<br />
69 God's mercy preserves ma<strong>the</strong>matics: Simone Weil, 1987, "The Pythagorean Doctrine,"<br />
in Intimations <strong>of</strong> Christianity Among <strong>the</strong> Ancient Greeks (London: Ark Paperbacks), 165.<br />
71 Hilbert <strong>and</strong> Grommer: Y<strong>and</strong>ell 2002, 18.<br />
72 Hilbert <strong>and</strong> Max Born: Reid 1986, 105.<br />
72 "Formalists considered": Kurt Godel, letter to Hao Wang, 1968, in Feferman et al.<br />
2003, V/404.<br />
73 Godel <strong>and</strong> Zermelo at Bad Elster: Dawson 1997, 76. 73 Russell's comments: Ibid.,<br />
77.
73 "Russell evidently misinterprets my result": Kurt Godel, 1973, letter to Abraham<br />
Robinson, in Feferman et al. 2003, V/201.<br />
74 Post noted: Emil Post, "Absolutely Unsolvable Problems ami Relatively llndeud able<br />
Propositions: Account <strong>of</strong> an Anticipation," in Martin Davis, cd., I9(, S, ih
84 <strong>Einstein</strong> in Berlin: See Thomas levenson, 2003, FJnstein in Berlin (New York: Bantam<br />
Hooks).<br />
84 "li is not entirely clear": Folsiiu; 1998, 330.<br />
87 "I am told in all steamship bureau*": Dawson 1997, 149.<br />
88 "das Reich der Zwei": Kurt Vonnegut, 1999, Mo<strong>the</strong>r Night (New York: Delta <strong>Books</strong>).<br />
Chapter 6<br />
89 When pygmies cast such long shadows: Gian-Carlo Rota, 1997, Indiscrete Thoughts<br />
(Boston: Birkhaiiser), 257.<br />
89 "as like Oxford as monkeys can make it": Ronald W. Clark, 1976, The Life <strong>of</strong><br />
Bertr<strong>and</strong> Russell, Alfred A. Knopf, New York, 233. 89 "Princeton is a wonderful piece <strong>of</strong><br />
earth": Folsing 1998, 679. 89 "ten <strong>time</strong>s more congenial": Dawson 1997, 168.<br />
91 "<strong>the</strong>re is any danger": Ibid., 158.<br />
92 "Is his [mental] illness": Y<strong>and</strong>ell 2002, 52.<br />
93 "in today's terms, <strong>Einstein</strong>": Peter Bucky, 1992, The Private Albert <strong>Einstein</strong> (Kansas<br />
City: Andrews <strong>and</strong> McMeel), 102.<br />
93 Feynman's "physiognomy <strong>and</strong> manner": James Gleick, 1992, Genius: The Life <strong>and</strong><br />
Science <strong>of</strong> Richard Feynman (New York: Pan<strong>the</strong>on <strong>Books</strong>), 84.<br />
94 "during my sickness <strong>Einstein</strong>": Hao Wang, 1987, Reflections on Kurt Godel<br />
(Cambridge, Mass.: MIT Press), 37.<br />
94 Godel wrote to his mo<strong>the</strong>r: Wang 1996, 39.
94 "I know <strong>of</strong> one occasion": Wang 1987, 32.<br />
95 "That begins only": Wang 1996, 56.<br />
95 <strong>Einstein</strong> . . . "was in many respects a pessimist": Wang 1987, 39-40.<br />
95 "Godel," however: Ibid., 32.<br />
95 "museum pieces": Holton 1998, 4.<br />
96 "<strong>Einstein</strong> is completely cuckoo": Folsing 1998, 693. 96 "firing at birds in <strong>the</strong> dark":<br />
Ibid., 709.<br />
98 "Did I really say that?": Ibid., 716.<br />
98 "Cervantes' text": Jorge Luis Borges, 1964, "Pierre Menard, Author <strong>of</strong> <strong>the</strong> Quixote," in<br />
Labyrinths: Selected Stories <strong>and</strong> O<strong>the</strong>r 'Writings (New York: New Directions), 42.<br />
99 "Do you think a dictatorship": Dawson 1997, 180.<br />
100 "You know that I disagree": Mark van Atten <strong>and</strong> Juliette Kennedy, 2003, "Godel's<br />
Philosophical Development," The Bulletin <strong>of</strong> Symbolic Logic, vol. 9, no. 4, December,<br />
425^476, 470.<br />
100 In 1935, <strong>Einstein</strong>: Albert <strong>Einstein</strong>, Boris Podolsky, <strong>and</strong> Nathan Rosen, 1935, "Can<br />
Quantum-Mechanical Description <strong>of</strong> Physical Reality Be Considered Complete?" Physical<br />
Review 47 (May 15), 777-780.<br />
101 "The heuristics <strong>of</strong> <strong>Einstein</strong> <strong>and</strong> Bohr": Wang 1996, 175.<br />
101 His own philosophical manifesto: Feferman 1990, 11/176-187.<br />
101 "I don't see any reason": "What Is Cantor's Continuum Problem?" (1964 supplement),<br />
in Feferman 1990, 11/254-270, 268.
102 a previous essay: Paul Arthur Schilpp, ed., 1944, The Philosophy <strong>of</strong> Bertr<strong>and</strong> Russell<br />
(La Salle, 111.: Open Court).<br />
102 "in talking <strong>the</strong> matter over": Charles Parson, 1944, "Introductory Note to [Godel] 1944,"<br />
in Feferman 1990, 11/102.<br />
102 Russell's comment: Ibid., 102.<br />
103 "It is not an altoge<strong>the</strong>r pleasant experience": IWrtratuI Russell, 1 c>Sl>, My I'hiln-<br />
Sti/ihli til l)fl'flt)l>Hlfll\ (I .union<br />
(III will l'.l|iclli.n !
110 Almost all accounts: Paul Benacerraf, "Ma<strong>the</strong>matical Truth," in Paul Benacerraf <strong>and</strong><br />
Hilary Putnam, eds., 1983, Philosophy <strong>of</strong> Ma<strong>the</strong>matics: Selected Readings (Cambridge:<br />
Cambridge University Press), 403-420, 403.<br />
111 "I do not believe": Hilary Putnam, 1967, "Time <strong>and</strong> Physical Geometry," journal <strong>of</strong><br />
Philosophy 64, reprinted in Hilary Putnam 1979, Ma<strong>the</strong>matics, Matter, <strong>and</strong> Method:<br />
Philosophical Papers, vol. I (Cambridge: Cambridge University Press), 198-205, 205.<br />
Ill <strong>time</strong> is "that mysterious": Kurt Godel, A Remark About <strong>the</strong> Relationship between<br />
Relativity Theory <strong>and</strong> Idealistic Philosophy," in Paul Arthur Schilpp, ed., 1949, Al bert<br />
<strong>Einstein</strong>: Philosopher-Scientist (LaSalle, 111.: Open Court), 557.<br />
111 "continental" philosophers: Martin Heidegger, 1962, Being <strong>and</strong> 'lime, transl. J.<br />
Macquarrie <strong>and</strong> E. Robinson, (New York: Harper & Row); F.dmund 1 lusserl, 1990, On <strong>the</strong><br />
Phenomenology <strong>of</strong> <strong>the</strong> Consciousness <strong>of</strong> internal lime, trans. J. 11. Brough (Dordrecht,<br />
Ne<strong>the</strong>rl<strong>and</strong>s: Kluwer).<br />
112 "now he has preceded me a little": Folsing 1998, 741.<br />
112 "<strong>the</strong> now means something special": Rudolf Carnap, "Intellectual Autobiography," in<br />
P.A. Schilpp, ed., The Philosophy <strong>of</strong> Rudolf Carnap (La Salle, 111.: Open Court), 1963: 37-<br />
63, 37-38.<br />
113 "Is what remains <strong>of</strong> temporal connection": Albert <strong>Einstein</strong>, "Reply to Criticisms," in<br />
Schilpp 1949, 687.<br />
116 "Kurt Godel's essay": Ibid., 687. 117-18 "Like most o<strong>the</strong>rs": Lawrence Sklar, quoted in<br />
Palle <strong>Yourgrau</strong>, 1999, Godel Meets <strong>Einstein</strong>: Time Travel in <strong>the</strong> Godel Universe (Chicago:<br />
Open Court), xiii, note 14.<br />
Chapter 7<br />
120 Only when Godel himself intervened: Dawson 1997, 184-185. 122 "The fourdimensional<br />
continuum is now": Albert <strong>Einstein</strong>,1961, Relativity: The Special <strong>and</strong> General<br />
Theory (New York: Corn Publishers), 149.<br />
124 "<strong>time</strong> regained a real objective existence": James Jeans, "Man <strong>and</strong> <strong>the</strong> Universe," Sir<br />
I lalley Stewart Lecture, 1936, in James Jeans et al., eds., 1936, Scientific Progress,<br />
(London: Allen & Unwin), I 1-38, 22-23.
125 a famous philosophical essay on <strong>the</strong> A-series: A.N. Prior, 1959, "Thank Goodness I<br />
hat's Over," Philosophy ª4, 12-17.<br />
131 "Whe<strong>the</strong>r or not an objective lapse": Kurt Godel, "A Remark about <strong>the</strong> Relationship<br />
Between Relativity Theory <strong>and</strong> Idealistic Philosophy," in Schilpp 1949, 562.<br />
132 "The concept <strong>of</strong> existence": Ibid., 558, note 5.<br />
132 "The notion <strong>of</strong> existence": Wang 1996, 150.<br />
00 : Kurt Godel, "A Remark about <strong>the</strong> Relationship Between Relativity Theory <strong>and</strong> Idealistic<br />
Philosophy," in Schilpp 1949, 558.<br />
133 "As a substitute for absolute space": Kurt Godel, 1949, "Lecture on Rotating<br />
Universes," in Feferman et al. 1995, III/271.<br />
136 "chronology protection conjecture": Hawking 1992.<br />
137 "As we present <strong>time</strong> to ourselves": Wang 1996, 320. 137 "This concept <strong>of</strong> set":<br />
Godel 1964, 258-259.<br />
137 "These contradictions": Godel 1961, 377.<br />
140 one <strong>of</strong> <strong>the</strong> great texts in general relativity: Charles Misner, Kip Thorne, John Wheeler,<br />
Gravitation (New York: Worth), 1973.<br />
140 "In physics ... <strong>the</strong> possibility <strong>of</strong> knowledge": Godel 1961, 377.<br />
141 "<strong>the</strong> two great conceptual revolutions": Dyson 1993, 263.<br />
141 "Godel, he noted, "had taken down": J.A. Wheeler, Geons, Black Holes, <strong>and</strong> Quantum<br />
Foam: A Life in Physics (New York: W.W. Norton & Co.), 1998, 309-310.
141 "In a universe with an overall rotation": Wheeler 1998, 310.<br />
142 an "unsettling consequence": Ibid., 344.<br />
142 "<strong>the</strong> o<strong>the</strong>r thing that special relativity did": Ibid., 345.<br />
142 Moreover, "every black hole": Ibid., 350.<br />
143 "My, I wish we had talked to him": Ibid., 310.<br />
Chapter 8<br />
145 We live in a <strong>world</strong> in which ninety-nine per cent: Dawson 1997, 208. 145 Plato<br />
says: Wang 1996, 300.<br />
145 "Geometry <strong>and</strong> Experience": Albert <strong>Einstein</strong>, 1921 "Geometry <strong>and</strong> Experience," in<br />
Albert <strong>Einstein</strong>, 1954, Ideas <strong>and</strong> Opinions (New York: Crown), 232-246.<br />
146 "<strong>Einstein</strong>'s now ab<strong>and</strong>oned dream": John Wheeler, 1980, "Beyond <strong>the</strong> Black Hole," in<br />
Harry Woolf, ed., Some Strangeness in <strong>the</strong> Proportion (Reading, Mass.: Addison-Wesley),<br />
344.<br />
147 "One crazy man on <strong>the</strong> faculty": Dawson 1997, 194. 147 "You don't need it": Ibid.,<br />
195.<br />
147 "I never go to lectures": Ibid., 203, note 5.<br />
149 "I never thought he loved me so much": Levenson 2003, 430.<br />
149 "My relationship to <strong>the</strong> Jewish people": Ibid., 428.<br />
149 Simone Weil's dark study: Simone Weil, 1987, transl. A.F. Wills, The Need for<br />
Roots (London: Ark Paperbacks), 1987.<br />
',
149 "<strong>the</strong> exaggerated esteem": Levenson 2003, 431.<br />
149 "It is tasteless": Ibid., 432.<br />
150 "Godel's gnawing uncertainty": Ulam 1976, 80-81.<br />
150 "Ulam doesn't underst<strong>and</strong> my result": Wang 1996, 84.<br />
151 "in 1905, work on spectral lines": Fcilsing 1998, 224. 153 "I cannot define (he real<br />
problem": Regis 1987, 33.<br />
153 Godel's contribution (o <strong>the</strong> Schilpp volume: Godel l''53/9. III & V, in Feferman Vol. Ill,<br />
)'>1>S; \.H-\h2.<br />
15} "Some <strong>Basic</strong> Theorems on <strong>the</strong> Foundations <strong>of</strong> Ma<strong>the</strong>matics, <strong>and</strong> Their Implications":<br />
Godel, 1951, in Feferman et al. 1995, 111/304-323.<br />
154 "In view <strong>of</strong> widely held prejudices": Warren Goldfarb, "Introductory Note" to 1953/9,<br />
in Feferman et al. 1995, III/324.<br />
155 "Ninety per cent <strong>of</strong> contemporary philosophers": Kurt Godel, letter to Marianne<br />
Godel, in Feferman et al. 2003, IV/437.<br />
156 Willard V. Quine: Charles Parsons, 2002, "W.V. Quine: A Student's Eye View,"<br />
Harvard Review <strong>of</strong> Philosophy, Spring, 2002, X:6-10, 8.<br />
156-57 "One may wonder": Solomon Feferman, 1984, "Kurt Godel: Conviction <strong>and</strong> Caution,"<br />
Philosophia Naturalis 21, 1984: 546-562, 560.<br />
157 "There are structural laws": Wang 1996, 151-152. 157 Christianity "best at <strong>the</strong><br />
beginning": Ibid., 150.<br />
157 "Since [Christ's] day <strong>the</strong>re": John Hellman, 1982, Simone Weil: An Introduction to<br />
Her Thought (Wilfrid Laurier University Press), 60, 65.
158 "Philosophy tends to go down": Wang 1996, 150.<br />
158 "Babylonian ma<strong>the</strong>matics": Dawson 1997, 241, note 6.<br />
158 "living corpse": Ibid., 234.<br />
159 association with <strong>the</strong> logician Hao Wang: Wang 1996.<br />
160 <strong>Einstein</strong>'s ashes: Levenson 2003, 431.<br />
Chapter 9<br />
161 Engaging in philosophy is salutary: Wang 1996, 119.<br />
162 "To date, only a single volume": Dawson 1997, 269.<br />
165 "To do philosophy is a special vocation": Wang 1996, 308.<br />
165 "Philosophy," he said: Ibid., 166.<br />
165 "Actually, it would be easy": Ibid., 307.<br />
165 The light dove: Kant 1965, A5,B9.<br />
166 In a classic essay: Carl Hempel, "The Empiricist Criterion <strong>of</strong> Meaning," in A.J. Ayer,<br />
ed., 1959, Logical Positivism (New York: The Free Press), 108-129.<br />
166 <strong>the</strong> first (<strong>and</strong> only) book: Wittgenstein, 1961.<br />
167 "Philosophy must be <strong>of</strong> some use": F.P. Ramsey, 1931, ed. R.B. Braithwaite, The<br />
Foundations <strong>of</strong> Ma<strong>the</strong>matics <strong>and</strong> O<strong>the</strong>r Logical Essays (New York: Harcourt, Brace),<br />
1931,263.
167 "What we can't say we can't say": Ibid, 238.<br />
167 material was provided for a lively book: David Edmonds <strong>and</strong> John Eidinow 2001,<br />
Wittgenstein's Poker, HarperCollins.<br />
168 "After <strong>the</strong> devastating attacks": Paul Horwich, 1990, "The Growth <strong>of</strong> Now," Review <strong>of</strong><br />
J.R. Lucas, The Future, Times Literary Supplement, June 22-28, 672.<br />
168 entitled "Godel's Philosophy": Warren Goldfarb, 1990, "On Godel's Philosophy,"<br />
address to <strong>the</strong> Association <strong>of</strong> Symbolic Logic, Helsinki, July 20, typescript.<br />
169 entitled "Some Observations": Godel 1946/49, B2, CI.<br />
170 "<strong>the</strong> problem <strong>of</strong> <strong>time</strong> is important <strong>and</strong> difficult": Wang 1996, 319. 170 "The way...<br />
we form ma<strong>the</strong>matical objects: Ibid., 301.<br />
170 "a clarification <strong>of</strong> meaning": Godel 1961, 383.<br />
170 "focus more sharply on <strong>the</strong> concepts": Ibid.<br />
170 "a new state ol consciousness": Ibid.<br />
171 "1 don't piuticuhirly like HiisseiTs way": Wau|; I'Wf,, 168.<br />
171 "I love everything brief": Ibid., 43.<br />
172 "entirely as intended by Kant": Ibid., 387.<br />
172 in his path-breaking study The Foundations <strong>of</strong> Arithmetic: Frege 1980.<br />
173 "Trying to see (i.e., underst<strong>and</strong>) a concept": Wang 1996, 233.
173 "Often it is only after immense intellectual effort": Frege 1980, Introduction, vii.<br />
173 "we perceive objects <strong>and</strong> underst<strong>and</strong> concepts": Wang 1996, 235.<br />
174 "We begin with vague perceptions": Ibid., 235. 174 "In arithmetic," wrote Frege:<br />
Frege 1980, 115.<br />
174 "When <strong>the</strong> soul investigates by itself": Plato, 1981, transl. G.M.A. Grube, Five<br />
Dialogues (Indianapolis:, Hackett Publishing Company), Phaedo, 79d.<br />
175 Both held that <strong>the</strong> only means <strong>of</strong> escaping: Palle <strong>Yourgrau</strong>, 1989, review essay,<br />
Reflections on Kurt Godel, Philosophy <strong>and</strong> Phenomenological Research L, no. 2, 391^108,<br />
402-403.<br />
175 "Concepts are <strong>the</strong>re," wrote Godel: Wang 1996, 149.<br />
175 "in <strong>the</strong> external <strong>world</strong>, in <strong>the</strong> whole <strong>of</strong> space": Frege 1980, 99.<br />
176 Constructibility <strong>and</strong> Ma<strong>the</strong>matical Existence: Charles Chihara, 1990, Con-structibility<br />
<strong>and</strong> Ma<strong>the</strong>matical Existence (Oxford: Clarendon Press).<br />
176 "Chihara's account <strong>of</strong> Godelian Platonism": E.P. James, 1992, "The Problem <strong>of</strong><br />
Ma<strong>the</strong>matical Existence," Philosophical <strong>Books</strong> XXXIII, no. 3, July, 129-138. 176<br />
"ma<strong>the</strong>maticians are prejudiced against intuition": Wang 1996, 169.<br />
176 Asymmetries in Time: Paul Horwich, 1987, Asymmetries in Time: Problems in <strong>the</strong><br />
Philosophy <strong>of</strong> Science (Cambridge, Mass.: The MIT Press).<br />
177 Milic Capek, ano<strong>the</strong>r distinguished philosopher <strong>of</strong> science: Milic Capek, 1966, "The<br />
Inclusion <strong>of</strong> Becoming in <strong>the</strong> Physical World," in Milic Capek, ed., 1976, The Concepts <strong>of</strong><br />
Space <strong>and</strong> Time (Dordrecht: Reidel).<br />
177 Karl Popper had argued similarly: Karl Popper, 1982, The Open Universe: An Argument<br />
for Indeterminism (Totowa, New Jersey: Rowman <strong>and</strong> Littlefied), 2-3, note 2.
177 The Disappearance <strong>of</strong> Time: Palle <strong>Yourgrau</strong>, 1991, The Disappearance <strong>of</strong> Time: Kurt<br />
Godel <strong>and</strong> <strong>the</strong> Idealistic Tradition in Philosophy (New York Cambridge University Press).<br />
177 When an exp<strong>and</strong>ed edition appeared: Palle <strong>Yourgrau</strong>, 1999, Godel Meets <strong>Einstein</strong>: Time<br />
Travel in <strong>the</strong> Godel Universe (Chicago: Open Court).<br />
177 Bangs, Crunches, Whimpers, <strong>and</strong> Shrieks: John Earman, 1995, Bangs, Crunches,<br />
Whimpers, <strong>and</strong> Shrieks (New York: Oxford University Press).<br />
177 "<strong>the</strong> relative neglect <strong>of</strong> <strong>the</strong> philosophical moral": Ibid., 194-195.<br />
177 "a deeply held conviction": Ibid, 195.<br />
178 "this seems to me to be <strong>the</strong> correct response to Godel": Ibid., 200.<br />
179 "this is a pretty piece <strong>of</strong> ordinary language philosophizing": Ibid., 195.<br />
180 "One may take <strong>the</strong> st<strong>and</strong>point": in Schilpp 1949, 558, note 4. 182 "<strong>the</strong> publication<br />
<strong>of</strong> such an anthology": Wang 1987, 254.<br />
182 "Even science," he said, "is very heavily prejudiced": Wang 1987, 254.<br />
183 "One bad effect <strong>of</strong> logical positivism": Wang 1987, 308.<br />
184 "It is given to us in its entirety": Wang 1996, 151. 184 "Ma<strong>the</strong>matics is applied to<br />
<strong>the</strong> real <strong>world</strong>": Ibid., 151.<br />
184 "O<strong>the</strong>rwise," he said, "ma<strong>the</strong>matics would be just an ornament": Ibid.<br />
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Index<br />
Absolute space, 42, 133 Berlin, 84-85<br />
Aleph null, 45 Berlin Philharmonic, 82<br />
American Ma<strong>the</strong>matical Monthly, 101 Bernays, Paul, 71<br />
American Ma<strong>the</strong>matical Society, 147 Besso, Michele, 112<br />
Analytic geometry, Cartesian coordinates Black holes, 117<br />
<strong>and</strong>, 62 Bohr, Niels, 3, 17, 43, 96, 100, 101<br />
Annalen der Physik, 37 Boltzmann, Ludwig, 21, 32, 35-37, 148;<br />
Anselm, Saint, 130, 155 suicide, 38<br />
Anti-Semitism, 41. See also Jews Born, Max, 72<br />
"Are There Philosophical Problems?" Borowicka, Sylvia, 40<br />
(Popper), 167 Boston University, 162, 176
Aristotle, 182 Brahms, Johannes, 54<br />
Arithmetic: formal (FA), 57, 60-67, 69; Brno, 21<br />
intuitive (IA), 60-67; Peano Brouwer, I.. E. [., 24<br />
postulates <strong>and</strong>, 56-57; philosophy Brownian motion, 37-38<br />
<strong>and</strong>, 174; unprovability <strong>of</strong> Browns, Bobbie <strong>and</strong> George, 90<br />
consistency <strong>and</strong>, 58<br />
Logic, 169<br />
Bucky, Peter, 93 Association for Symbolic<br />
Asymmetries in Time (Horwich), 176 Calculation: infinitesimal calculus, 47; Atom<br />
bomb, 96, 99 recursive function <strong>and</strong>, 75<br />
Atomic <strong>the</strong>ory, 34-35, 37-38, 106 Cambridge University, 167<br />
Austria-Hungary, 80<br />
"Can Quantum-Mechanical Description <strong>of</strong><br />
Physical Be Considered Complete?" BadElster, 73<br />
(Podolsky/Rosen), 100<br />
Bahcall, John, 15 Cantor, Georg, 1, 51, 72, 121, 137, 151;<br />
Bamburger, Louis, 77 continuum hypo<strong>the</strong>sis <strong>and</strong>, 96-97,<br />
Bangs, Crunches, Whimpers, <strong>and</strong> Shrieks 100, 101; set <strong>the</strong>ory <strong>and</strong>, 44-47<br />
(Earman), 177 Capek, Milic, 177<br />
Befreiung, 81 Carnap, Rudolf, 25, 26, 27, 29, 86, 154,<br />
Begriffssckrift (Frege), 24<br />
168, 181; incompleteness <strong>the</strong>orem
Being <strong>and</strong> Time (Heidegger), 111 <strong>and</strong>, 70<br />
Benacerraf, Paul, 121<br />
Cartesian coordinates, Godel numbering<br />
Bcrgson, 1 Icnri, 1 I I <strong>and</strong>, 62<br />
Cezanne, Paul, 54<br />
motion <strong>and</strong>, 37-38; calculates<br />
Ch<strong>and</strong>rasekhar, S., 119-20<br />
explosive potentialities <strong>of</strong> torpedoes<br />
Chihara, Charles, 176<br />
for U.S. Navy, 97; as colleague <strong>of</strong><br />
Chomsky, Noam, 55<br />
Godel, 4, 14; common sense vs.<br />
Christianity, Godel <strong>and</strong>, 157<br />
ma<strong>the</strong>matics, 15-16; early residence<br />
Chronology protection conjecture, 8, 143 in Berlin, 84-85; Earman <strong>and</strong>, 178;<br />
Church, Alonzo, 68-69, 75, 76, 79, 156<br />
eccentricities, 92-93; EPR paradox,<br />
Cohen, Paul, 61-62, 97, 100 100, 153; family <strong>and</strong>, 148-49;<br />
Collected Works (Godel omnibus), 176 friendship with Godel, 88, 94-95;<br />
Computers: arithmetic <strong>and</strong>, 57; Godel <strong>and</strong>,<br />
geometry <strong>and</strong>, 17-18, 145-46; Godel<br />
99; Hilbert <strong>and</strong>, 54; incompleteness <strong>and</strong> Leibniz, 109-10; intuition <strong>and</strong>,<br />
<strong>the</strong>orem <strong>and</strong>, 68<br />
32; invited to work at <strong>the</strong> Princeton<br />
Concept: formal vs. intuitive <strong>and</strong> special Institute for Advanced Study, 78;<br />
relativity, 128-29; implicit definitions Kant <strong>and</strong>, 108-09; Mach <strong>and</strong>, 33-34;
<strong>and</strong>, 55, 169; metaphysics <strong>and</strong>, 169; ma<strong>the</strong>matics <strong>and</strong>, 5, 34, 53;<br />
natural numbers <strong>and</strong>, 172; perception<br />
Minkowski <strong>and</strong>, 34; music <strong>and</strong>,<br />
<strong>and</strong>, 174; predication <strong>and</strong>, 175. See 11-12; philosophy <strong>and</strong>, 101, 104,<br />
also Perception 107, 111-12; pipe <strong>and</strong>, 9-10, 93;<br />
Conceptual space, predication <strong>and</strong>, 175<br />
positivism <strong>and</strong>, 105-06, 107; unified<br />
"Confession to <strong>the</strong> Fuhrer" (Schrodinger),<br />
field <strong>the</strong>ory <strong>and</strong>, 100; Vienna Circle<br />
77, 80 <strong>and</strong>, 39, 40. See also General<br />
Continuum hypo<strong>the</strong>sis, 1, 96; Godel <strong>and</strong>,<br />
relativity <strong>the</strong>ory; Relativity <strong>the</strong>ory;<br />
97, 100, 101, 110, 150, 151, 153; set Special relativity <strong>the</strong>ory<br />
<strong>the</strong>ory <strong>and</strong>, 97, 151 <strong>Einstein</strong>, Eduard (son <strong>of</strong> AE), 149<br />
Cosmic <strong>time</strong>, 124, 129, 133, 135, 180-81<br />
<strong>Einstein</strong>, Elsa (Lowenthal, second wife <strong>of</strong><br />
Cosmology, 8, 139; intuitive <strong>time</strong> <strong>and</strong>, AE), 10, 84, 148, 149<br />
137; physical significance <strong>of</strong> <strong>Einstein</strong>, Hans Albert (son <strong>of</strong> AE), 93, 107,<br />
ma<strong>the</strong>matical models consequential to 149<br />
general relativity, 116-17 <strong>Einstein</strong>, Lieserl (daughter <strong>of</strong> AE), 149
Critique <strong>of</strong> Pure Reason, The (Kant), 16, Empiricism, positivism <strong>and</strong>, 52<br />
29, 109, 165 Entropy, probability <strong>and</strong>, 36<br />
cummings, e. e., 53<br />
Epistemology: incompleteness <strong>the</strong>orem<br />
<strong>and</strong>, 75; ma<strong>the</strong>matical intuition <strong>and</strong>,<br />
Dawson, John, 162, 184 101; ontology <strong>and</strong>, 3, 106, 107,<br />
Dedekind, Richard, 62 112-13, 141, 154, 175, 182;<br />
Definition, 55, 169. See also Properties positivism <strong>and</strong>, 31; Schlick <strong>and</strong>, 41<br />
Descartes, Rene, 62, 65, 130, 155<br />
EPR (<strong>Einstein</strong>-Podolsky-Rosen) paradox,<br />
Disappearance <strong>of</strong> Time, The (<strong>Yourgrau</strong>), 100, 153<br />
162, 177 Erkenntis, 70, 71<br />
DNA (deoxyribonucleic acid), 152 Euclid, 145<br />
Dreben, Burton, 163, 164, 168, 176 Euclidean geometry, 17-18; vs. Hilbert, 55;<br />
Dukas, Helen, 94-95 vs. logic, 51-52<br />
Dyson, Freeman, 1, 4, 141 Existence: accidental vs. necessary, 130;<br />
meaning <strong>and</strong>, 132<br />
Earman, John, 177-80<br />
Experience: intuitive <strong>time</strong> <strong>and</strong>, 138; Kant
<strong>Einstein</strong>, Albert, 83, 91, 147, 157, 183; <strong>and</strong>, 166; Mach <strong>and</strong>, 33, 34;<br />
atom bomb <strong>and</strong>, 96, 99; Brownian positivism <strong>and</strong>, 29-32, 38<br />
Feferman, Solomon, 156-57 <strong>time</strong> <strong>and</strong>, 105; <strong>time</strong> travel <strong>and</strong>, 129.<br />
Feigl, Herbert, 27, 29<br />
See also Euclidean geometry<br />
Feynman, Richard, 93, 153<br />
"Geometry <strong>and</strong> Experience" (<strong>Einstein</strong>),<br />
Flexner, Abraham, 77-78 145<br />
Ford, Lester, 101 Germany, 80<br />
Formal arithmetic (FA), 57, 60-67, 69; Gibbs Lecture, 147, 150, 181<br />
meta<strong>the</strong>ory <strong>of</strong> (MFA), 61-67. See also God: actuality vs. possibility <strong>and</strong>, 130;<br />
Ma<strong>the</strong>matical formalism<br />
<strong>Einstein</strong> <strong>and</strong>, 12, 104; Godel <strong>and</strong>,<br />
Formal <strong>time</strong>, temporal component (t) <strong>of</strong><br />
12-13, 23, 130, 161, 165. See also<br />
relativity <strong>the</strong>ory, 113-14, 115, 127,<br />
Religion<br />
129, 131, 135, 138 Godel, Adele (Porkert, wife <strong>of</strong> KG), 10, 26,<br />
Foundations <strong>of</strong> Arithmetic, The (Frege), 172<br />
87, 98, 158; death, 159; home <strong>and</strong>,<br />
Fraenkel, Abraham, 25, 72 91; new to Princeton, 89-90
Frank, Philipp, 11<br />
Godel, Kurt<br />
Frege, Gottlob, 29, 48, 51, 56, 62, 70,<br />
Career: first formal academic honor,<br />
137, 181; formal arithmetic <strong>and</strong>, 60; 147; invitations to work at <strong>the</strong><br />
Goldfarb <strong>and</strong>, 171, 172, 174-76,<br />
Princeton Institute for Advanced Study,<br />
178; Mach <strong>and</strong>, 35; ma<strong>the</strong>matical 79, 85; Schlick circle <strong>and</strong>, 27-29; as a<br />
logic <strong>and</strong>, 24-25; set <strong>the</strong>ory <strong>and</strong>, 45, student <strong>of</strong> number <strong>the</strong>ory, 22-23<br />
47; symbolic logic <strong>and</strong>, 23, 24 Incompleteness <strong>the</strong>orem, 3, 8, 50,<br />
Friedmann, Aleks<strong>and</strong>r, 117<br />
<strong>and</strong>,<br />
151, 182; effective calculability<br />
Fuld, Mrs. Felix, 78<br />
75, 76; Husserlian phenomenology<br />
Furtwangler, Philip, 22-23, 82, 92<br />
<strong>and</strong>, 107; initial response <strong>of</strong> <strong>the</strong><br />
Furtwangler, Wilhem, 82-83 ma<strong>the</strong>matical community, 70-71;<br />
legal code <strong>and</strong>, 99; ma<strong>the</strong>matical<br />
General relativity <strong>the</strong>ory, 18, 105, 140, formalism <strong>and</strong>, 53, 74-76, 105-06;<br />
151; cosmic <strong>time</strong> <strong>and</strong>, 124, 129; <strong>the</strong> methodology <strong>of</strong>, 115; vs. positivism,<br />
exp<strong>and</strong>ing universe <strong>and</strong>, 19;<br />
140; pro<strong>of</strong> <strong>of</strong>, 60-69; pro<strong>of</strong><br />
geometrization <strong>of</strong> <strong>time</strong> <strong>and</strong>, 115;<br />
procedures <strong>and</strong>, 24; pro<strong>of</strong> vs. truth
Godel rotating universes <strong>and</strong>, <strong>and</strong>, 135; response <strong>of</strong> Hilbert to, 71,<br />
133-34, 141; Godel universes <strong>and</strong>, 6, 72; response <strong>of</strong> Russell to, 73;<br />
116; Godel <strong>world</strong> models for, unprovability <strong>of</strong> consistency <strong>and</strong>, 58<br />
112-16, 117-18, 135; gravity <strong>and</strong>, Ma<strong>the</strong>matics, 22, 32; beauty <strong>and</strong>, 184;<br />
127, 129; Hawking chronology vs. Carnap, 30; cognitive content <strong>and</strong>,<br />
protection conjecture <strong>and</strong>, 136; 175; continuum hypo<strong>the</strong>sis <strong>and</strong>, 97,<br />
intuitive <strong>time</strong> <strong>and</strong>, 138; intuitive <strong>time</strong><br />
100, 101, 150, 151; vs. formalism,<br />
<strong>and</strong> inertial frames, 180; 52-53, 59; intuition <strong>and</strong>, 57-58;<br />
ma<strong>the</strong>matical formalism <strong>and</strong>, 55;<br />
Kant <strong>and</strong>, 175; Peano postulates <strong>and</strong>,<br />
physical significance <strong>of</strong> ma<strong>the</strong>matical 56-57; philosophy <strong>and</strong>, 112;<br />
models for, 116-17; response to<br />
philosophy <strong>and</strong> set <strong>the</strong>ory, 137; vs.<br />
Godel's <strong>world</strong> models for, 119-21;<br />
positivism, 52, 105-06, 182; Post on,<br />
<strong>time</strong> <strong>and</strong>, 138, 139 74; pro<strong>of</strong> vs. truth <strong>and</strong>, 136;<br />
Geometry, 175; analytic, 62;<br />
58;<br />
unprovability <strong>of</strong> consistency <strong>and</strong>,<br />
geometrization <strong>of</strong> <strong>time</strong>, 115; Godel vs. Wittgenstein, 50
universes <strong>and</strong>, 135-36, 179; Hilbert Personal like: <strong>the</strong> arts <strong>and</strong>, 4-5, 11;<br />
<strong>and</strong>, 55; painting <strong>and</strong>, 54; Plato <strong>and</strong>,<br />
becomes citizen <strong>of</strong> United States,<br />
I-tS; relativity iluoiy <strong>and</strong>, 127; sp.uo- l'K 99; ns colleague <strong>of</strong> I'.instein, 4, 14;<br />
death, 159; depression, 86; driving arithmetic truth (T) <strong>and</strong>, 127, 138;<br />
<strong>and</strong>, 25; early years in Princeton, intuitive <strong>time</strong> <strong>and</strong>, 137, 138, 142;<br />
89-90; eating disorder, 86, 159; intuitive <strong>time</strong> vs. temporal component<br />
eccentricities, 11, 91-92; friendship (t) <strong>of</strong> relativistic space-<strong>time</strong>, 114, 115,<br />
with <strong>Einstein</strong>, 88, 94-95; 135; McTaggart series <strong>and</strong>, 124-26,<br />
hypochondria, 9, 92, 158; leaves 128-29, 132; nonexistence <strong>of</strong>, 131;<br />
Austria for America, 87; paranoia, 5,<br />
ontological investigation <strong>of</strong> relativity<br />
9, 15, 92, 158; pessimism, 5; <strong>and</strong>, 108, 111, 112-13; Plato <strong>and</strong>,<br />
purchase <strong>of</strong> a home, 91; sanatoria 109; <strong>and</strong> reality <strong>of</strong>, 111, 113, 120,<br />
<strong>and</strong>, 9, 86; women <strong>and</strong>, 10, 26 130; special relativity <strong>and</strong>, 122-24,<br />
Philosophy, 22, 104, 158, 164, 165;<br />
126-27, 128-29, 131-33, 139; <strong>time</strong><br />
Christianity <strong>and</strong>, 157; vs. Earman, travel <strong>and</strong>, 181<br />
177-80; <strong>Einstein</strong> <strong>and</strong>, 108-09, 161; Time travel, 6, 142; cosmic <strong>time</strong> <strong>and</strong>,
vs. Goldfarb, 163, 168-72, 173-74,<br />
181; geometry <strong>and</strong>, 129; Godel<br />
175-76; history <strong>of</strong> philosophy <strong>and</strong>, universes <strong>and</strong>, 116, 120, 129-30, 134,<br />
182; Husserl <strong>and</strong>, 70, 154, 170-71, 138-39; nonexistence <strong>of</strong> <strong>time</strong> <strong>and</strong>,<br />
173, 175-76, 182; Kant <strong>and</strong>, 16, 130; relativity <strong>the</strong>ory <strong>and</strong>, 116, 120<br />
108-09, 161, 172, 182; Kantian Godel, Marianne (mo<strong>the</strong>r <strong>of</strong> KG), 25<br />
intuition <strong>and</strong>, 17, 18, 173-74; Godel, Rudolf (bro<strong>the</strong>r <strong>of</strong> KG), 21, 25, 27<br />
ma<strong>the</strong>matics <strong>and</strong>, 48; methodology, Godel formula, <strong>the</strong> (G), 65-67<br />
109-10, 180-81; neglect <strong>of</strong> <strong>the</strong> Godel numbering, natural numbers <strong>and</strong>,<br />
philosophical community, 162; 62-67<br />
ontology <strong>and</strong>, 107, 108, 111, 154, "Godel's Philosophy" (Goldfarb), 168<br />
161; vs. positivism, 32, 48; as Godel universes, 138-39, 184; vs. <strong>the</strong><br />
precritical to Kant, 169-70; reality <strong>of</strong><br />
actual <strong>world</strong>, 179; distribution <strong>of</strong><br />
<strong>time</strong> <strong>and</strong>, 180; religion <strong>and</strong>, 12-13, matter <strong>and</strong>, 6; geometry <strong>and</strong>, 135-36;<br />
105, 155-56; response to Godel's Hawking <strong>and</strong>, 7-8, 143; nonexistence<br />
general relativity <strong>world</strong> models, <strong>of</strong> <strong>time</strong> <strong>and</strong>, 131; rotating, 133-34;
119-21; Russell <strong>and</strong>, 102, 161, 181; <strong>time</strong> travel <strong>and</strong>, 116, 120, 129-30,<br />
self-reflection <strong>and</strong>, 171; set <strong>the</strong>ory 134, 181<br />
<strong>and</strong>, 137 Goebbels, Josef, 83<br />
Publications: Collected Works Goldfarb, Warren, 163, 168-72, 173-74,<br />
(posthumous omnibus), 176; "Is 175-76<br />
Ma<strong>the</strong>matics Syntax <strong>of</strong> Language?," Gomperz, Heinrich, 22<br />
153; "My Philosophical Viewpoint," Gould, Glenn, 54-55<br />
104; "Remark About <strong>the</strong> Gravitation (Misner/Thorne), 140<br />
Relationship Between Relativity<br />
Gravity, 17, 146, 151; cosmological<br />
Theory <strong>and</strong> Idealistic Philosophy, A,"<br />
constant <strong>and</strong>, 117; geometry <strong>and</strong>,<br />
127, 134-35; "Theory <strong>of</strong> Relativity 127; motion <strong>and</strong>, 123-24;<br />
<strong>and</strong> Kant, The," 108; "What is Schwarzchild singularity <strong>and</strong>, 116;<br />
Cantor's Continuum Problem?," 101 space-<strong>time</strong> <strong>and</strong>, 129<br />
Time: actual <strong>world</strong> vs. Godel universes, Graz, 80<br />
179; general relativity <strong>and</strong>, 115, 129; Grommer, Jakob, 71-72<br />
general relativity <strong>and</strong> rotating
universes, 133-34; general relativity<br />
llahn, Hans, 25, 27, 29, 52, 77, 86; on<br />
<strong>world</strong> models <strong>and</strong>, 112-16, 117-18, intuition, 50<br />
135; <strong>and</strong> geonietri/.ilion ot, I 1 *>; <strong>and</strong> I lawking, Stephen, 7 K, I \(,, 14 t<br />
ideality <strong>of</strong>, 177, ISO, I H I; intuitive<br />
I l.iwley, Dan, 14.' -It<br />
Hegel, Georg, 157, 182<br />
positivism, .32; primitive concepts<br />
Heidegger, Martin, 11 1 <strong>and</strong>, 169; <strong>of</strong> space <strong>and</strong> <strong>time</strong>, 17,<br />
18<br />
Heisenberg, Werner, 2, 4, 17, 43, 83, 100;<br />
uncertainty principle <strong>and</strong>, 3, 140<br />
Intuitionist ma<strong>the</strong>matics, 75. See also<br />
Ma<strong>the</strong>matical intuition<br />
Hempel, Carl, 165 Intuitive arithmetic, 60-67<br />
Herbr<strong>and</strong>, Jacques, 64, 68, 75, 86 Intuitive arithmetic truth (T), 127, 131,<br />
Hilbert, David, 8, 29, 74, 137, 169; 135, 138<br />
develops metama<strong>the</strong>matics, 24; Intuitive ma<strong>the</strong>matical truth, 114<br />
ma<strong>the</strong>matical formalism <strong>and</strong>, 52, Intuitive <strong>time</strong>, 137, 138; geometrical 53-57, 59;<br />
ma<strong>the</strong>matics <strong>and</strong>, 43-44,<br />
structure <strong>of</strong> <strong>the</strong> Godel universe <strong>and</strong>,<br />
46, 50; vs. recursive functions, 68, 135; Godel rotating universes <strong>and</strong>,<br />
69; response to incompleteness 133; special relativity <strong>and</strong>, 128-29,<br />
<strong>the</strong>orem, 71, 72<br />
132, 180; temporal component (t) <strong>of</strong>
Hitler, Adolf, 22, 83 relativity <strong>the</strong>ory <strong>and</strong>, 114, 115,<br />
127,<br />
Holton, Gerald, 16 129, 135<br />
Horwich, Paul, 168, 176<br />
Intuitive <strong>time</strong>, formal representation (t) in<br />
Husserl, Edmond, 70, 111, 113, 154, 170,<br />
<strong>Einstein</strong>-Minkowski space-<strong>time</strong>,<br />
182; Kantian idealism <strong>and</strong>, 107 113-14, 115, 127, 129, 131, 135,<br />
138<br />
Idealism: Husserlian phenomenology <strong>and</strong>, Irrational numbers, 44, 47, 71-72<br />
107; knowledge <strong>and</strong>, 106; Leibniz "Is Ma<strong>the</strong>matics Syntax <strong>of</strong> Language?"<br />
<strong>and</strong>, 109; Mach <strong>and</strong>, 35; (Godel), 153<br />
ma<strong>the</strong>matizing <strong>of</strong> physical <strong>the</strong>ory<br />
<strong>and</strong>, 37; relativity <strong>the</strong>ory <strong>and</strong>, 16; James, E. P., 176<br />
temporal, 131, 137, 181. See also Jeans, James, 124<br />
Kant, Immanuel Jews: <strong>Einstein</strong> as, 12, 85, 93-94;<br />
Ideality <strong>of</strong> <strong>time</strong>, 177, 180, 181 Furtwangler <strong>and</strong>, 83; Godel <strong>and</strong>, 13,<br />
Incompleteness <strong>the</strong>orem, 3, 8, 50, 151,<br />
86-87; reclassification as Mischlinge
182; effective calculability <strong>and</strong>, 75, <strong>and</strong>, 81-82; Schlick <strong>and</strong>, 41<br />
76; Husserlian phenomenology <strong>and</strong>,<br />
107; initial response <strong>of</strong> <strong>the</strong> Kant, Immanuel, 113, 161, 168, 175, 182;<br />
ma<strong>the</strong>matical community, 70-71;<br />
<strong>Einstein</strong> <strong>and</strong>, 16, 108-09, 178; vs.<br />
legal code <strong>and</strong>, 99; ma<strong>the</strong>matical Hahn, 50; intuition <strong>and</strong>, 173-74;<br />
formalism <strong>and</strong>, 53, 74-76, 105-06;<br />
phenomenology <strong>and</strong>, 172; Plato <strong>and</strong>,<br />
methodology <strong>of</strong>, 115; vs. positivism,<br />
165-66; vs. positivism, 29; reality<br />
140; pro<strong>of</strong> <strong>of</strong>, 60-69; pro<strong>of</strong> <strong>and</strong> idealism <strong>of</strong>, 106; relativity <strong>the</strong>ory<br />
procedures <strong>and</strong>, 24; pro<strong>of</strong> vs. truth <strong>and</strong>, 16, 108-09, 122, 169-70, 181;<br />
<strong>and</strong>, 135; response <strong>of</strong> Hilbert to, 71, space <strong>and</strong>, 17, 122; <strong>time</strong> <strong>and</strong>, 18,<br />
72; response <strong>of</strong> Russell to, 73; 122<br />
unprovability <strong>of</strong> consistency <strong>and</strong>, Karl Franzen University, 80<br />
58 Kleene, Stephen S., 156<br />
Infinitesimal numbers, 47<br />
Knowledge, 140; history <strong>of</strong> philosophy vs.<br />
Infinity, 54; continuum hypo<strong>the</strong>sis <strong>and</strong>, 97, prejudice, 182; idealism <strong>and</strong>, 106;<br />
150; set <strong>the</strong>ory <strong>and</strong>, 44-45, 48 Kant vs. positivism, 29; ma<strong>the</strong>matics<br />
Introduction to Ma<strong>the</strong>nidtiail Philosophy<br />
<strong>and</strong>, 30, 52; realism <strong>and</strong>, 101; reality
(Russell), 2S<br />
<strong>and</strong>, i<br />
Intuition, 179; I lalin on, SO; Katu mil, KoniKsberg, 70<br />
17.1ó74; mt-.iiiiiik .ind, S7; vs Krciscl, (icorg, 182<br />
Lambda calculus, 79<br />
Language: ma<strong>the</strong>matics as, 30; philosophy<br />
Ma<strong>the</strong>matics, 6, 50, 51, 122; computers<br />
<strong>and</strong>, 54, 57; continuum hypo<strong>the</strong>sis<br />
<strong>and</strong>, 167, 179; positivism <strong>and</strong>, 28; <strong>and</strong>, 1; <strong>Einstein</strong> <strong>and</strong>, 5, 15-16;<br />
symbolic, 23 Hilbert <strong>and</strong>, 24, 43-44, 46, 52,<br />
Leibniz, Gottfried, 15, 130, 140, 155, 182; 53-54; historical crises <strong>of</strong>, 47-48;<br />
<strong>Einstein</strong> <strong>and</strong>, 109-10; existence <strong>of</strong> incompleteness <strong>the</strong>orem <strong>and</strong>, 3, 8;<br />
God <strong>and</strong>, 13; founds infinitesimal logic <strong>and</strong>, 23-24; positivism <strong>and</strong>, 30,<br />
calculus, 47 32, 49-50, 52; reality <strong>and</strong>, 17, 100;<br />
Lenin, V. I., 35<br />
vs. sensory experience in physical<br />
"Library <strong>of</strong> Living Philosophers," 102,<br />
science, 30. See also Arithmetic<br />
108, 119 Ma<strong>the</strong>matics, Godel, 22, 32, 74; beauty Light,<br />
speed <strong>of</strong>, 2; frames <strong>of</strong> reference <strong>and</strong>, <strong>and</strong>, 184; vs. Carnap, 30; cognitive<br />
42-43 content <strong>and</strong>, 175; continuum<br />
Linguistics, structuralism <strong>and</strong>, 55 hypo<strong>the</strong>sis <strong>and</strong>, 97, 100, 101, 150,
Logic, 56, 175; vs. Euclidean geometry, 151; vs. formalism, 52-53, 59;<br />
51-52; Frege <strong>and</strong>, 23-24, 29, 30; intuition <strong>and</strong>, 57-58; Kant <strong>and</strong>, 175;<br />
incompleteness <strong>the</strong>orem <strong>and</strong>, 103; Peano postulates <strong>and</strong>, 56-57;<br />
lambda calculus, 79; numbers <strong>and</strong>,<br />
philosophy <strong>and</strong>, 112; philosophy <strong>and</strong><br />
136; positivism <strong>and</strong>, 29-30, 52, set <strong>the</strong>ory, 137; vs. positivism, 52,<br />
182-83; realism <strong>and</strong>, 102 105-06, 182; pro<strong>of</strong> vs. truth <strong>and</strong>,<br />
Logical space, physical systems <strong>and</strong>,<br />
136; unprovability <strong>of</strong> consistency<br />
38-39 <strong>and</strong>, 58; vs. Wittgenstein, 50. See alsc<br />
Loos, Adolf, 21-22<br />
Incompleteness <strong>the</strong>orem<br />
Lorentz, Hendrik Antoon, 42, 43, 55 Matter, 133; Godel universes <strong>and</strong>, 6; Lorentz<br />
transformations, inertial frames gravity <strong>and</strong>, 129; motion <strong>and</strong>, 124<br />
<strong>and</strong>, 42 Maxwell, James Clerk, 41-42<br />
Lowenthal, Ilsa, 148<br />
McTaggart, J. M. E., Ill<br />
McTaggart A <strong>and</strong> B series, 124-26, Mach, Ernst, 21, 32-37, 106, 148<br />
128-29, 132<br />
Mahler, Gustav, 16, 21<br />
Meaning: existence <strong>and</strong>, 132; intuition<br />
Marie, Mileva (first wife <strong>of</strong> AE), 148 <strong>and</strong>, 57-58<br />
Materialism <strong>and</strong> Empirio-Criticism (Lenin), Mechanics, probability <strong>and</strong>, 36
35 Menger, Karl, 15, 21, 26, 27, 29, 39, 77, 86<br />
Ma<strong>the</strong>matical formalism, 49, 137; Metama<strong>the</strong>matics, 24<br />
consistency <strong>and</strong>, 56, 58; Hilbert <strong>and</strong>, Metaphysics: Godel <strong>and</strong>, 169, 170;<br />
52, 53-57, 59; incompleteness Husserlian phenomenology <strong>and</strong><br />
<strong>the</strong>orem <strong>and</strong>, 53, 76, 105-06; pro<strong>of</strong><br />
Kantian idealism, 107; positivism<br />
<strong>and</strong>, 57, 59-60, 72, 74-75. See also <strong>and</strong>, 43; primitive concepts <strong>and</strong>, 169<br />
Formal arithmetic (FA) Meta<strong>the</strong>ory <strong>of</strong> formal arithmetic, 61-67<br />
Ma<strong>the</strong>matical intuition, 57, 101, 176. See Minimalism, 21; positivism <strong>and</strong>, 28<br />
also Intuitionist ma<strong>the</strong>matics Minkowski, Hermann, 5, 34<br />
Ma<strong>the</strong>matical logic, 23; set <strong>the</strong>ory <strong>and</strong>, Misner, Charles, 140<br />
24-25 Morgenstern, Oskar, 91, 98, 158, 159<br />
Ma<strong>the</strong>matical realism, 140-41 Morse, Philip, 93<br />
Ma<strong>the</strong>matical truth: formalism <strong>and</strong>, 49, Mo<strong>the</strong>r Night (Vonnegut), 88<br />
57, 72, 75; intuitive <strong>time</strong> <strong>and</strong>, 114; Motion, 133; Brownian, 37-38; Godel<br />
semantics vs. syntax <strong>and</strong>, 72, 74<br />
rotating universes, 134; gravity <strong>and</strong>.
12.1-24, 129; Maxwell vs. Newton, Painting, 21, 54<br />
41-42; relativity <strong>and</strong> frames <strong>of</strong> Parsons, Charles, 155-56<br />
reference, 43 Pascal, Blaise, 155<br />
Mozart, Wolfgang Amadeus, 11-12 Pauli, Wolfgang, 13, 101, 104<br />
Music, 21, 54; Berlin Philharmonic <strong>and</strong> Peano postulates, 56-57, 62<br />
Jews, 82; <strong>Einstein</strong> <strong>and</strong>, 11-12 Peebles, James, 142-43<br />
Myhill, John, 57<br />
Perception: intuition <strong>and</strong>, 32; Kantian<br />
"My Philosophical Viewpoint" (Godel), 104<br />
intuition <strong>and</strong>, 173-74; ma<strong>the</strong>matical<br />
intuition <strong>and</strong>, 101, 176. See also<br />
Nagel, Thomas, 112<br />
Concept<br />
Natkin, Marcel, 27 Phaedo (Plato), 174<br />
Natural numbers, 110; concepts <strong>and</strong>, 172; Phenomenology: Godel <strong>and</strong>, 107, 170-71,<br />
continuum hypo<strong>the</strong>sis <strong>and</strong>, 150; 173; Kantian idealism <strong>and</strong>, 107, 172<br />
definitions <strong>and</strong>, 55; as ideal forms, Philosophic der Arithmetik (Husserl), 171<br />
23; incompleteness <strong>the</strong>orem <strong>and</strong>, 61, Philosophy, 6; <strong>Einstein</strong> <strong>and</strong>, 101, 104, 107,<br />
62-69; infinity <strong>and</strong>, 44^46, 47; 111-12; Mach <strong>and</strong>, 32, 33;
ma<strong>the</strong>matical formalism <strong>and</strong>, 74;<br />
positivism <strong>and</strong>, 29-30, 31; quantum<br />
Peano postulates <strong>and</strong>, 56-57 mechanics <strong>and</strong>, 100-101;<br />
Natural science, 14-15; ma<strong>the</strong>matics <strong>and</strong>, Wittgenstein <strong>and</strong>, 166-68<br />
48 Philosophy, Godel, 22, 48, 104, 158, 164,<br />
Nazis, 86; Anschluss <strong>of</strong> Austria to<br />
165; Christianity <strong>and</strong>, 157; vs.<br />
Germany <strong>and</strong>, 80-82; vs.<br />
Earman, 177-80; <strong>Einstein</strong> <strong>and</strong>,<br />
Furtwangler, 83 108-09, 161; vs. Goldfarb, 163,<br />
Nelbock, Hans, 40<br />
168-72, 173-74, 175-76; history <strong>of</strong><br />
Nernst, Wal<strong>the</strong>r, 84<br />
70,<br />
philosophy <strong>and</strong>, 182; Husserl <strong>and</strong>,<br />
Newton, Sir Isaac, 33, 109-10, 133, 146,<br />
154, 170-71, 173, 175-76, 182; Kant<br />
151; founds infinitesimal calculus, 47; <strong>and</strong>, 16, 108-09, 161, 172, 182;<br />
vs. Maxwell, 41-42; vs. special Kantian intuition <strong>and</strong>, 17, 18,<br />
relativity, 19, 31 173-74; methodology, 109-10,<br />
Nin, Anais, 51<br />
philosophical<br />
180-81; neglect <strong>of</strong> <strong>the</strong><br />
Numbers, formal pro<strong>of</strong> <strong>and</strong>, 136 community, 162; ontology <strong>and</strong>, 107,
Number <strong>the</strong>ory: Godel <strong>and</strong>, 22-23, 56-57,<br />
108, 111, 154, 161; vs. positivism,<br />
62; Peano postulates <strong>and</strong>, 56-57, 62; 32, 48; as precritical to Kant, 169-70;<br />
symbolic logic <strong>and</strong>, 23<br />
reality <strong>of</strong> <strong>time</strong> <strong>and</strong>, 180; religion <strong>and</strong>,<br />
12-13, 105, 155-56; response to<br />
"On Godel's General Philosophical<br />
Godel's general relativity <strong>world</strong><br />
Outlook" (Goldfarb), 168 models, 119-21; Russell <strong>and</strong>, 102,<br />
On <strong>the</strong> Phenomenology <strong>of</strong> <strong>the</strong> 161, 181; self-reflection <strong>and</strong>, 171;<br />
set<br />
Consciousness <strong>of</strong> Internal Time<br />
<strong>the</strong>ory <strong>and</strong>, 137. See also Time, Godel<br />
(Husserl), 111<br />
Physical science, 140; ma<strong>the</strong>matizing <strong>of</strong><br />
Ontology: epistemology <strong>and</strong>, 3, 106, 107,<br />
physical <strong>the</strong>ory <strong>and</strong>, 37; positivism<br />
112-13, 141, 154, 175, 182; <strong>and</strong>, 29-30; positivism <strong>and</strong><br />
existence <strong>of</strong> God <strong>and</strong>, 13, 130, 155,<br />
ma<strong>the</strong>matical truth, 49; sensory<br />
156, 161; relativity <strong>the</strong>ory <strong>and</strong>, 108; experience <strong>and</strong>, 33; sensory<br />
temporal distance <strong>and</strong>, 134 experience vs. ma<strong>the</strong>matics <strong>and</strong>, 30<br />
Oppeiilienncr, |. Kobirt, V 1, 96, 147 Physical <strong>the</strong>ory: ma<strong>the</strong>niatizing <strong>of</strong>, 37;
Oswald, Williclin, \h physical reality <strong>and</strong>, (4, 101<br />
Physics, 140; geometrization <strong>of</strong>, 127; <strong>and</strong>, 100, 153; positivism <strong>and</strong>, 43,<br />
Godel <strong>and</strong>, 22; ma<strong>the</strong>matics <strong>and</strong>, 50; 48, 140<br />
ma<strong>the</strong>matizing <strong>of</strong> physical <strong>the</strong>ory Quantum physics, positivism <strong>and</strong>, 3<br />
<strong>and</strong>, 37; philosophy <strong>and</strong>, 112; Quantum reality, EPR paradox <strong>and</strong>, 153<br />
probability <strong>and</strong>, 36; <strong>without</strong> Quantum <strong>the</strong>ory, <strong>Einstein</strong> <strong>and</strong>, 4<br />
philosophy, 107 Quine, W. V. O., 121, 156, 168<br />
Planck, Max, 84, 148, 151<br />
Plato, 23, 162, 174, 182, 184; geometry Rational numbers, 44<br />
<strong>and</strong>, 145; Kant on, 165-66; <strong>time</strong> <strong>and</strong>, Realism, 38; Husserlian phenomenology 109<br />
<strong>and</strong> Kantian idealism, 107;<br />
Platonism, 44, 50, 101, 130, 174 knowledge <strong>and</strong>, 101; logic <strong>and</strong>, 102;<br />
Podolsky, Boris, 100 vs. positivism, 106, 140-41<br />
Popper, Karl, 21, 29, 167 Reality: frames <strong>of</strong> reference <strong>and</strong>, 131-32;<br />
Porkert, Ilsa, 10 Godel <strong>and</strong>, 105; idealism <strong>and</strong>, 106;<br />
Positivism: atomic <strong>the</strong>ory <strong>and</strong>, 35; <strong>Einstein</strong> knowledge <strong>and</strong>, 3; Mach <strong>and</strong>, 34-35,<br />
<strong>and</strong>, 105-06, 107, 148; vs. Godel, 37; ma<strong>the</strong>matical truth <strong>and</strong>, 49;
32, 48, 52, 105-06; Hilbert <strong>and</strong>, 48, ma<strong>the</strong>matics <strong>and</strong>, 17, 52, 100;<br />
49; vs. ma<strong>the</strong>matical realism, 140-41; physical <strong>the</strong>ory <strong>and</strong>, 34, 101;<br />
ma<strong>the</strong>matics <strong>and</strong>, 43^44, 49-50, 52;<br />
positivism <strong>and</strong>, 31; relativistic space-<br />
quantum physics <strong>and</strong>, 3; reality <strong>and</strong>, <strong>time</strong> <strong>and</strong>, 110; <strong>of</strong> <strong>time</strong>, 111, 120,<br />
31; sensory experience <strong>and</strong>, 38; 130, 180<br />
Wittgenstein <strong>and</strong>, 28, 29, 30 Real numbers, 44, 45^16, 172<br />
Possibility: existence <strong>of</strong> God <strong>and</strong>, 130; Recursive definitions, Godel numbering<br />
nonexistence <strong>of</strong> <strong>time</strong> <strong>and</strong>, 131 <strong>and</strong>, 63-64<br />
Post, Emil, 74 Recursive functions, 63-64, 66, 68-69, 76,<br />
Princeton, 89; abodes <strong>of</strong> <strong>the</strong> Godels, 90-91 99<br />
Princeton Institute for Advanced Study Reichenbach, Hans, 70<br />
(IAS), 2, 10, 79, 85; founded, 78; Relativistic <strong>time</strong>, temporal component (t) Russell<br />
<strong>and</strong>, 103-04 <strong>of</strong> relativity <strong>the</strong>ory, 113-14, 115,<br />
Principia Ma<strong>the</strong>matica 127, 131, 135, 138<br />
(Russell/Whitehead), 25, 73, 79, 103, Relativity <strong>the</strong>ory, 2, 118, 137, 182; formal 154<br />
vs. intuitive concept <strong>and</strong>, 128-29;<br />
Probability, 36, 148 geometrization <strong>of</strong> physics <strong>and</strong>, 127,<br />
Proceedings <strong>of</strong> <strong>the</strong> National Academy <strong>of</strong><br />
146; idealism <strong>and</strong>, 16; inertial frames
Sciences, 119 <strong>and</strong>, 180; Kant <strong>and</strong>, 18, 108-09, 122,<br />
Pro<strong>of</strong>, 54; consistency <strong>of</strong> arithmetic axioms 169-70, 181; Mach <strong>and</strong>, 33;<br />
<strong>and</strong>, 58; implicit definitions <strong>and</strong>, 55; Minkowski <strong>and</strong>, 34; reality <strong>and</strong>, 110;<br />
incompleteness <strong>the</strong>orem <strong>and</strong>, 24, 58,<br />
speed <strong>of</strong> light <strong>and</strong>, 43; temporal<br />
135; ma<strong>the</strong>matical formalism <strong>and</strong>, component (t) <strong>of</strong> relativistic space-<br />
57, 59-60, 105; vs. truth, 49, 57, 72, <strong>time</strong> <strong>and</strong>, 112-15; <strong>time</strong> <strong>and</strong>, 5, 7;<br />
74, 75, 135, 136 <strong>time</strong> travel <strong>and</strong>, 116, 120; verification<br />
Properties: Russell set <strong>the</strong>ory paradox <strong>and</strong>,<br />
<strong>and</strong>, 39-40. See also General relativity<br />
46-47, 51. See also Definition <strong>the</strong>ory; Special relativity <strong>the</strong>ory<br />
Putnam, Hilary, 111, 121<br />
Religion: <strong>Einstein</strong> <strong>and</strong>, 12; Godel <strong>and</strong>,<br />
Pythagorean <strong>the</strong>orem, 47<br />
12-13, 105, 155-56. Sec also God<br />
"Remark About <strong>the</strong> Relationship Between<br />
Quantum mechanics, 3, 83, 100; <strong>Einstein</strong><br />
Relativity Theory <strong>and</strong> Idealistic<br />
<strong>and</strong>, 4, I4S, 152-53; EPR paradox Philosophy, A" (Godel), 127, 134 !S<br />
Republic; The (Plato), 162 Space-<strong>time</strong>, 6, 105, 110, 112, 146;
Rockefeller, John D., 32<br />
124,<br />
geometry <strong>and</strong>, 127; gravity <strong>and</strong>,<br />
Rosen, Nathan, 100 129; vs. intuitive <strong>time</strong>, 114, 115;<br />
Rosser, J. Barkley, 156 ma<strong>the</strong>matical formalism <strong>and</strong>, 55;<br />
Rota, Gian-Carlo, 89 space vs. <strong>time</strong> <strong>and</strong>, 125-26<br />
Rotating universes, 6, 7, 133-34, 141, Special relativity <strong>the</strong>ory, 109, 112, 127, 184<br />
133, 148; epistemology vs. ontology,<br />
Russell, Bertr<strong>and</strong>, 13, 24, 25, 39, 148, 113; flow <strong>of</strong> <strong>time</strong> <strong>and</strong>, 122-23;<br />
154, 161, 181; formal arithmetic <strong>and</strong>, frames <strong>of</strong> reference <strong>and</strong>, 131-32;<br />
60; incompleteness <strong>the</strong>orem <strong>and</strong>, 73, geometrization <strong>of</strong> <strong>time</strong> <strong>and</strong>, 18;<br />
102-03; positivism <strong>and</strong>, 29, 30; on intuitive <strong>time</strong> <strong>and</strong>, 180; ma<strong>the</strong>matical<br />
Princeton, 89; response to formalism <strong>and</strong>, 55; matter <strong>and</strong>, 124;<br />
incompleteness <strong>the</strong>orem, 73; set<br />
McTaggart series <strong>and</strong>, 128-29; vs.<br />
<strong>the</strong>ory paradox <strong>and</strong>, 46^8, 72, 137<br />
Newton, 19, 31; philosophy <strong>and</strong>,<br />
112; positivism <strong>and</strong>, 3;<br />
Schilpp, P. A., 102, 108, 119, 181<br />
protopositivism <strong>of</strong>, 105; static <strong>time</strong><br />
Schlick, Moritz, 25, 29, 32, 39, 49, 86; <strong>and</strong>, 142; <strong>time</strong> <strong>and</strong>, 138, 139
murder <strong>of</strong>, 40-41<br />
Speed <strong>of</strong> light, 2; frames <strong>of</strong> reference <strong>and</strong>,<br />
Schlick circle, 27. See also Vienna Circle 42-43<br />
Schoenberg, Arnold, 21, 54-55 Spinoza, Benedict de, 12-13<br />
Schrodinger, Erwin, 77, 80-81, 83 Static <strong>time</strong>, nonexistence <strong>of</strong> <strong>time</strong> <strong>and</strong>, 142<br />
Schwarzchild, Karl, 116 Stein, Howard, 120<br />
Schwarzchild singularity, 116 Strauss, Leo, 164<br />
Schwatz, Boris, 12 Strauss, Richard, 21<br />
Schwinger, Julian, 147<br />
String <strong>the</strong>ory, quantum mechanics <strong>and</strong>,<br />
Science: au<strong>the</strong>nticity <strong>and</strong>, 56; positivism 152-53<br />
<strong>and</strong>, 31; sensory experience <strong>and</strong>, 33.<br />
Symbolic language, number <strong>the</strong>ory <strong>and</strong>, l.i<br />
See also Natural science; Physical Syntax: ma<strong>the</strong>matical formalism <strong>and</strong>, 53,<br />
science<br />
55-57; ma<strong>the</strong>matics <strong>of</strong> language <strong>and</strong>,<br />
Scientific discovery, 151-52 30; vs. semantics, 55, 57, 60, 74<br />
Semantics vs. syntax, 55, 57, 60, 74<br />
Sensory experience, 54; Kant <strong>and</strong>, 166; Tarski, Alfred, 156<br />
Mach <strong>and</strong>, 33, 34; positivism <strong>and</strong>, Taussky-Todd, Olga, 10, 13, 26, 30-31<br />
29-32, 38 Temporal idealism, 121, 131, 137, 138-39,
Set <strong>the</strong>ory, 32, 44, 48, 72, 110, 137, 175; 181<br />
continuum hypo<strong>the</strong>sis <strong>and</strong>, 97, 151; Temporality, 122, 123, 146; component (t) Frege<br />
<strong>and</strong>, 24-25, 45, 47; Russell in relativity <strong>the</strong>ory <strong>and</strong>, 112-15, 127,<br />
paradox <strong>and</strong>, 46-47, 51<br />
129, 131, 135, 138; McTaggart A<br />
Seyss-Inquart, Dr. Arthur, 82 <strong>and</strong> B series, 124-26, 128-29, 132;<br />
Siegel, C. L., 147 <strong>time</strong> travel <strong>and</strong>, 134<br />
Sitter, Wellem de, 1 17<br />
"Theory <strong>of</strong> Relativity <strong>and</strong> Kant, The"<br />
Snow White <strong>and</strong> <strong>the</strong> Seven Dwarfs, 4-5 (Godel), 108<br />
Space: absolute, 42; conceptual, 175; Thermodynamics, probability <strong>and</strong>, 36<br />
geometrization <strong>of</strong> rime <strong>and</strong>, I 15; Thorne, Kip, 140<br />
gyroscopic rotation <strong>and</strong> inertial fields, "Thought" (Frege), 35<br />
I 13; Jeans <strong>and</strong>, 124; Kant <strong>and</strong>, 17, 18, Dimwits (Plato), 184<br />
111; leal numbers ,\ih\, 44; relativity Time, i; entropy mi^. '6; geometri/.ntion<br />
<strong>the</strong>ory .mil, 11 vs. <strong>time</strong>, 125-26<br />
"I, I II, M; jeans <strong>and</strong>, 124; Kant<br />
<strong>and</strong>, 17, 18, 122; real numbers <strong>and</strong>, 44; relativity <strong>the</strong>ory <strong>and</strong>, 5, 7, 18-19, 31; vs. space,<br />
125-26; special <strong>the</strong>ory <strong>of</strong> relativity <strong>and</strong>, 105; static, 142. See also Space-<strong>time</strong><br />
Time, Godel: actual <strong>world</strong> vs. Godel
universes, 179; general relativity <strong>and</strong>, 115, 129; general relativity <strong>and</strong> rotating universes,<br />
133-34; general relativity <strong>world</strong> models <strong>and</strong>, 112-16, 117-18, 135; <strong>and</strong> geometrization <strong>of</strong>,<br />
115; <strong>and</strong> ideality <strong>of</strong>, 177, 180, 181; intuitive arithmetic truth (T) <strong>and</strong>, 127, 138; intuitive<br />
<strong>time</strong> <strong>and</strong>, 137, 138, 142; intuitive <strong>time</strong> vs. temporal component (t) <strong>of</strong> relativistic space<strong>time</strong>,<br />
114, 115, 135; McTaggart series <strong>and</strong>, 124-26, 128-29, 132; nonexistence <strong>of</strong>, 131;<br />
ontological investigation <strong>of</strong> relativity <strong>and</strong>, 108, 111, 112-13; Plato <strong>and</strong>, 109; <strong>and</strong> reality <strong>of</strong>,<br />
111, 113, 120, 130; special relativity <strong>and</strong>, 122-24, 126-27, 128-29, 131-33, 139; <strong>time</strong> travel<br />
<strong>and</strong>, 181<br />
Time travel, 6, 142; cosmic <strong>time</strong> <strong>and</strong>, 181; geometry <strong>and</strong>, 129; Godel universes <strong>and</strong>, 116,<br />
120, 129-30, 134, 138-39; nonexistence <strong>of</strong> <strong>time</strong> <strong>and</strong>, 130; relativity <strong>the</strong>ory <strong>and</strong>, 116, 120<br />
Tractatus Logico-Philosophicus<br />
(Wittgenstein), 82, 121, 182; logical space, 38-39; as nonsense, 166-67; positivism <strong>and</strong>, 28,<br />
30-31, 32, 50<br />
Transcendental phenomenology, idealism <strong>and</strong>, 107<br />
Transfinite numbers, 46, 51; aleph null, 45<br />
Trinity College, 167<br />
Truth: ma<strong>the</strong>matical, 49; vs. pro<strong>of</strong>, 49, 57, 72, 74, 75, 135, 136. See also Ma<strong>the</strong>matical<br />
truth<br />
Turing, Alan, 5, 156, 158, 174; computers <strong>and</strong>, 68<br />
Ulam, Stanislaw, 150 Uncertainty principle, 3, 140 Unified field <strong>the</strong>ory, 100, 146, 148, 183;<br />
string <strong>the</strong>ory <strong>and</strong>, 152-53<br />
U.S. Constitution, 98-99<br />
U.S. Navy, 97<br />
Universal rotation, 6, 7, 133-34, 141,
184 Universe, <strong>the</strong>: cosmic <strong>time</strong> <strong>and</strong>, 129;<br />
cosmological constant <strong>and</strong>, 117;<br />
special relativity <strong>and</strong> age <strong>of</strong>, 122-24.<br />
See also Godel universes University <strong>of</strong> Vienna, 21<br />
Veblen, Oswald, 77, 78-79, 91, 157 Vienna, 21-22; Anschluss <strong>of</strong> Austria <strong>and</strong>,<br />
80-82 Vienna Circle, 25, 28-29, 30-31; <strong>Einstein</strong><br />
<strong>and</strong>, 39, 40 Vonnegut, Kurt, 88 Von Neumann, John, 58, 70, 96, 104, 147,<br />
157<br />
Wang, Hao, 27, 105, 159, 160, 163<br />
Weil, Andre, 160<br />
Weil, Simone, 69, 157<br />
Wells, H. GÑ 119<br />
Weyl, Hermann, 58<br />
"What is Cantor's Continuum Problem?" (Godel), 101<br />
Wheeler, John, 96, 139-40, 141-43, 146
Whitehead, Albert North, 25<br />
Wittgenstein, Helene (sister <strong>of</strong> LW), 81, 82<br />
Wittgenstein, Hermine (sister <strong>of</strong> LW), 81, 82<br />
Wittgenstein, Ludwig, 29, 74, 112, 154, 164, 179, 182; Anschluss <strong>of</strong> Austria <strong>and</strong> siblings <strong>of</strong>,<br />
81-82; Brahms <strong>and</strong>, 54; logical space <strong>and</strong>, 38-39; Loos <strong>and</strong>, 21-22; ma<strong>the</strong>matics vs. sensory<br />
experience, 30; philosophy <strong>and</strong>, 166-68; positivism <strong>and</strong>, 28; Russell <strong>and</strong>, 103<br />
Wittgenstein, Margarete (sister <strong>of</strong> LW), 81<br />
Wittgenstein, Paul (bro<strong>the</strong>r <strong>of</strong> LW), 81-82<br />
"World Structure in <strong>the</strong> Large <strong>and</strong> in <strong>the</strong> Small" (Schrodinger), 80<br />
Wright, J. P., 119-20<br />
Zermelo, Ernst, 24, 72-73