A World Without Time

ACKNOWLEDGMENTS

Having already written a book intended primarily for philosophers about Kurt Gödel’s attempt to make sense of Einstein’s theory of relativity, I was intrigued when William Frucht of Basic Books suggested I write another, this one accessible to normal readers. Such a book would focus on the sheer intellectual drama of the companionship of Gödel and Einstein—a relationship sorely neglected in the literature— and would place Gödel’s and Einstein’s epoch-making discoveries in the context of the great intellectual movements of the twentieth century, some of which, having helped to father, they tried, belatedly, to abandon. It was an offer too good to refuse, and I didn’t. The task, however, turned out to be far from easy, and Frucht had to endure not only the late delivery of the final manuscript, but the drumbeat of my complaints about his editorial adventures at the expense of my beloved prose; it may not have been much, but it was all mine. I am grateful to Frucht both for the initial invitation and for (what turned out to be) his wise editorial advice, at every stage, on how to improve the manuscript.

I have greatly benefited from discussing the book with Mary Sullivan and Ben Callard. Sullivan, attentive as ever to the task of trying to keep me honest, read large parts of the book and offered acute, sobering advice, which I took to heart. Callard’s critical remarks on every chapter were of great value and are reflected in the final draft. His strange affection for the work, moreover, kept my spirits up during the many dark moments when the project seemed to me ill-advised and misbegotten. Mark van Atten also read the entire manuscript. His extraordinarily detailed remarks, on both substance and style, were a great boon. Robert Tragesser shared with me his deep understanding of Gödel’s theorem, and Eli Hirsch helped make certain that my discussions of logic and mathematics were clear, accessible, and to the point. To each of these I owe a serious debt of gratitude, but especially to Callard and van Atten, for their extensive and thoughtful comments.

I would also like to express my appreciation for the conversations I had over a period of years with the late Hao Wang, of Rockefeller University, whom I came to know when I taught at Barnard College in New York. With Wang, who was one of Gödel’s closest associates in his final years, I spent endless hours discussing Gödel’s ideas, published and unpublished. The question of time was of particular interest to him. He confessed that, finding the topic uncongenial, he had resisted Gödel’s efforts to discuss his thoughts on this theme, and regretted it. I regretted it, too. It did not diminish, however, the fruitfulness of our conversations, nor the enjoyment we took in contemplating what Gödel has said and written.

In addition to help from my friends, I have also depended on the kindness of strangers. My copy editor, David Kramer, in addition to improving the clarity and flow of the text, made a number of comments and suggestions of a more substantial nature on a wide variety of topics, from music to mathematics, which were an unexpected gift. AnnaLee Pauls, photoduplication coordinator at the Rare Books and Special Collections department of Princeton University’s Firestone Library, provided much needed assistance in the delicate task of choosing and reproducing the right photos from the Gödel archives, housed in Firestone Library. I am grateful both for her assistance and for the spirit with which she provided it. I would also, finally, like to thank Arudra Burra, who is not a stranger, for his generous help, combined with fine judgment, in the search for the right photos to include in the book.

1 | A Conspiracy of Silence

In the summer of 1942, while German U-boats roamed in wolf packs off the coast of Maine, residents in the small coastal town of Blue Hill were alarmed by the sight of a solitary figure, hands clasped behind his back, hunched over like a comma with his eyes fixed on the ground, making his way along the shore in a seemingly endless midnight stroll. Those who encountered the man were struck by his deep scowl and thick German accent. Speculation mounted that he was a German spy giving secret signals to enemy warships. The dark stranger, however, was no German spy. He was Kurt Gödel, the greatest logician of all time, a beacon in the intellectual landscape of the last thousand years, and the prey he sought was not American ships bound for Britain but rather the so-called continuum hypothesis, a conjecture made by the mathematician Georg Cantor about the number of points on a line. Gödel was spending the summer vacationing at the Blue Hill Inn with his wife, Adele, although fellow visitors at the inn rarely saw either of them. They materialized for dinner, but were never observed actually eating. To the locals, Gödel’s scowl betrayed a dark disposition, but the innkeeper saw things differently. For her it was the expression of a man lost in thought. His last word to Blue Hill would not decide the issue. He sent a letter accusing the innkeeper of stealing the key to his trunk.

The place Gödel would return to in the fall was a long way from Blue Hill—the prestigious Institute for Advanced Study in Princeton, New Jersey. There he would no longer have to walk alone, arousing the suspicions of neighbors. He had a walking companion, a colleague at the institute and his best friend. There was no danger that his reputation would intimidate his companion. For his friend, another German-speaking refugee with a mathematical bent, was the most famous scientist of all time, Albert Einstein, whose own meditative strolls already irritated the residents of Princeton.

“From a distance,” a biographer wrote, “the [residents of Princeton] chuckled discreetly over [Einstein’s] habit of licking an ice cream on Nassau Street on his way home from Fine Hall and were astonished by his utterly un-American long walks through the streets of Princeton.” Indeed, toward the end of his career, when he was more or less retired, Einstein commented that his own work no longer meant much to him and that he now went to his office “just to have the privilege of walking home with Kurt Gödel.” Ironically, it was not the scowling Gödel but his smiling companion who had once given indirect aid to the German U-boats, when, during World War I, although a courageous and committed pacifist, Einstein had helped improve the gyroscopes used by the German navy. Gödel’s research would also, in the end, relate to gyroscopes, but these spun at the center of the universe, not in the dank bowels of submarines.

Washed up onto America’s shores by the storm of Nazism that raged in Europe in the 1930s, the two men had awakened to find themselves stranded in the same hushed academic retreat, Princeton’s Institute for Advanced Study, an exclusive intellectual club, whose members had only one assigned duty: to think. But Gödel and Einstein already belonged to an even more exclusive club. Together with another German-speaking theorist, Werner Heisenberg, they were the authors of the three most fundamental scientific results of the century. Each man’s discovery, moreover, established a profound and disturbing limitation. Einstein’s theory of relativity set a limit—the speed of light—to the flow of any information-bearing signal. And by defining time in terms of its measurement with clocks, he set a limit to time itself. It was no longer absolute but henceforth limited or relative to a frame of measurement. Heisenberg’s uncertainty principle in quantum mechanics set a limit on our simultaneous knowledge of the position and momentum of the fundamental particles of matter. This was not just a restriction on what we can know: for Heisenberg it signified a limit to reality. Finally, Gödel’s incompleteness theorem—“the most significant mathematical truth of the century,” as it would soon be described in a ceremony at Harvard University—set a permanent limit on our knowledge of the basic truths of mathematics: The complete set of mathematical truths will never be captured by any finite or recursive list of axioms that is fully formal. Thus, no mechanical device, no computer, will ever be able to exhaust the truths of mathematics. It follows immediately, as Gödel was quick to point out, that if we are able somehow to grasp the complete truth in this domain, then we, or our minds, are not machines or computers. (Enthusiasts of artificial intelligence were not amused.)

Einstein, Gödel, Heisenberg: three men whose fundamental scientific results opened up new horizons, paradoxically, by setting limits to thought or reality. Together they embodied the zeitgeist, the spirit of the age. Mysteriously, each had reached an ontological conclusion about reality through the employment of an epistemic principle concerning knowledge. The dance or dialectic of knowledge and reality— of limit and limitlessness—would become a dominant theme of the twentieth century. Yet Gödel’s and Einstein’s relations to their century were more uneasy than Heisenberg’s.

The zeitgeist took root most famously in quantum mechanics. Here Gödel and Einstein would find themselves in lonely opposition to Heisenberg, who, on the wrong side in the war of nations, chose the winning team in the wars of physics. Heisenberg was a champion of the school of positivism, in quantum physics known as the Copenhagen interpretation in deference to Heisenberg’s mentor, the Danish physicist Niels Bohr. What had been a mere heuristic principle in Einstein’s special relativity—deducing the nature of reality from limitations on what can be known—became for Heisenberg a kind of religion, a religion that Gödel and Einstein had no wish to join. Some, however, claimed to see in Gödel’s theorem itself an echo of Heisenberg’s uncertainty principle. That group did not include Gödel.

Einstein, himself one of the great pioneers of quantum mechanics, had known and inspired Heisenberg in Germany. In 1911, in Prague, years before Heisenberg came on the scene, Einstein once pointed out to his colleague Philipp Frank the insane asylum in the park below his study and remarked, “Here you see that portion of lunatics who do not concern themselves with quantum theory.” By Einstein’s lights, a bad situation had become even worse after Heisenberg. In an early encounter, Heisenberg, on the defensive against Einstein’s harangue against quantum mechanics, fought back: “When I objected that in [my approach] I had merely been applying the type of philosophy that he, too, had made the basis of his special theory of relativity, [Einstein] answered simply, ‘Perhaps I did use such philosophy earlier, and also wrote it, but it is nonsense all the same.’”

The two parted before the war, Einstein emigrating to the United States, Heisenberg remaining in Germany, to which he would remain loyal to the end. In Princeton, Einstein—pacifist, bohemian, socialist and Jew—was a man apart. To be sure, he found Gödel, but together they remained isolated and alone, not least because of their opposition to Heisenberg’s positivist worldview, which ruled the intellectual scene even as Heisenberg’s fatherland was attempting to dominate the world. Gödel and Einstein were not merely intellectual engineers, as so many of their brethren, inspired by positivism, had become, but philosopher-scientists. Ironically, while their stars had begun to wane, the sheer size of their reputations had made them unapproachable. Not to each other, however. “Gödel,” wrote their colleague Freeman Dyson, “was the only one of our colleagues who walked and talked on equal terms with Einstein.”

Their tastes, however, remained distinct. Einstein, a violinist, could never bring his friend to subject himself to the likes of Beethoven and Mozart. Gödel, in turn, had no more success, surely, in dragging Einstein to Snow White and the Seven Dwarfs, his favorite movie. History, sadly, does not record which of the seven dwarfs was Gödel’s favorite, but we do know why he favored fairy tales: “Only fables,” he said, “present the world as it should be and as if it had meaning.” That meaning, of course, may be dark. It is not known whether Alan Turing acquired an affection for Snow White from Gödel when he visited the institute in the 1930s, but some have heard an echo of the dark side of that tale in Turing’s decision to end his life by eating a poisoned apple when, as a reward for his having broken the Enigma code of the German navy, the British government ordered him to receive hormone injections as a “cure” for his homosexuality.

Einstein, before fleeing Germany, had already become a refugee from mathematics. He later said that he could not find, in that garden of many paths, the one to what is fundamental. He turned to the more earthly domain of physics, where the way to the essential was, he thought, clearer. His disdain for mathematics earned him the nickname “lazy dog” from his teacher Hermann Minkowski (who would soon recast the lazy dog’s special relativity into its characteristic four-dimensional form). “You know, once you start calculating,” Einstein would quip, “you shit yourself up before you know it.” Gödel’s journey, in contrast, was in the opposite direction. Having befriended Gödel, Einstein commented that he knew now, at last, that in mathematics too one could find a path to the fundamental. In befriending Einstein, Gödel was reawakened to his early interest in physics. On their long walks home from the office, Einstein, forever cheerful, would attempt to raise the spirits of the gloomy and pessimistic Gödel by recounting his latest insights on general relativity. Sadly, however, pessimism blossomed into paranoia. The economist Oskar Morgenstern, calling one day on his good friend, was shocked to find the great Gödel hiding in the cellar behind the furnace.

From those long walks that Einstein and Gödel shared, from their endless discussions, something beautiful would soon be born. The scene was pregnant with possibility. Time, which has taunted thinkers from Plato to Saint Augustine to Kant, had finally met its match in Einstein. While the U-boats of his former fatherland were stalking the Allied fleet, this most un-German of Germans was hunting a more elusive prey. He had amazed the world decades earlier when he alone succeeded in capturing time itself in the equations of relativity. “Every boy in the streets of Göttingen,” his countryman David Hilbert wrote, “understands more about four-dimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians.” Relativity had rendered time, the most elusive of beings, manageable and docile by transforming it into a fourth dimension of space, or rather, of relativistic space-time. Sharing with Gödel his latest thoughts on the four-dimensional universe of space-time that he himself had conjured into being, Einstein was sowing the seeds of relativity in the mind of a thinker who would later be described as a combination of Einstein and Kafka.

If Einstein had succeeded in transforming time into space, Gödel would perform a trick yet more magical: He would make time disappear. Having already rocked the mathematical world to its foundations with his incompleteness theorem, Gödel now took aim at Einstein and relativity. Wasting no time, he announced in short order his discovery of new and unsuspected cosmological solutions to the field equations of general relativity, solutions in which time would undergo a shocking transformation. The mathematics, the physics and the philosophy of Gödel’s results were all new. In the possible worlds governed by these new cosmological solutions, the so-called rotating or Gödel universes, it turned out that the space-time structure is so greatly warped or curved by the distribution of matter that there exist timelike future-directed paths by which a spaceship, if it travels fast enough—and Gödel worked out the precise speed and fuel requirements, omitting only the lunch menu—can penetrate into any region of the past, present or future.

Gödel, the union of Einstein and Kafka, had for the first time in human history proved, from the equations of relativity, that time travel was not a philosopher’s fantasy but a scientific possibility. Yet again he had somehow contrived, from within the very heart of mathematics, to drop a bomb into the laps of the philosophers. The fallout, however, from this mathematical bomb was even more perilous than that from the incompleteness theorem. Gödel was quick to point out that if we can revisit the past, then it never really “passed.” But a time that fails to pass is no time at all. Einstein saw at once that if Gödel was right, he had not merely domesticated time: he had killed it. Time, “that mysterious and seemingly self-contradictory being,” as Gödel put it, “which, on the other hand, seems to form the basis of the world’s and our own existence,” turned out in the end to be the world’s greatest illusion. In a word, if Einstein’s relativity was real, time itself was merely ideal. The father of relativity was shocked. Though he praised Gödel for his great contribution to the theory of relativity, he was fully aware that time, that elusive prey, had once again slipped his net.

But now something truly amazing took place: nothing. Although in the immediate aftermath of Gödel’s discoveries a few physicists bestirred themselves to refute him and, when this failed, tried to generalize and explore his results, this brief flurry of interest soon died down. Within a few years the deep footprints in intellectual history traced by Gödel and Einstein in their long walks home had disappeared, dispersed by the harsh winds of fashion and philosophical prejudice. A conspiracy of silence descended on the Einstein-Gödel friendship and its scientific consequences.

An association no less remarkable than the friendship between Michelangelo and Leonardo—if such had occurred—has simply vanished from sight. To this day, not only is the man on the street unaware of the intimate relationship between these two giants of the twentieth century, even the most exhaustive intellectual biographies of Einstein either omit all mention of this friendship or at best begrudge a sentence or two. Whereas a whole industry has grown up in search of Lieserl, the “love child” of Einstein’s first marriage, the child of the imagination that was born of the friendship of Einstein and Gödel has been abandoned.

Only in the last few years has this child, the Gödel universe, received any glimmer of recognition. This comes from the redoubtable Stephen Hawking. Revisiting the rotating Gödel universe, Hawking was moved to deliver the highest of compliments. So threatening did he find results like Gödel’s demonstrating the consistency of time travel with the laws of relativity, that he put forward what amounts to an anti-Gödel postulate. If accepted, Hawking’s famous “chronology protection conjecture” would precisely negate Gödel’s contribution to relativity. So physically unacceptable did Hawking find conclusions like Gödel’s that he felt compelled to propose what looks like an ad hoc modification of the laws of nature that would have the effect of ruling out the Gödel universe as a genuine physical possibility.

Hawking’s attempt to neutralize the Gödel universe shows how dangerous it is to break the conspiracy of silence that has shrouded the Gödel-Einstein connection. Not only does this mysterious silence hide from the world one of the most moving and consequential friendships in the history of science, it also keeps the world from realizing the full implications of the Einstein revolution. It is one thing to overturn, as Einstein did, Newton’s centuries-old conception of the absoluteness and independence of space and time. It is quite another to demonstrate that time is not just relative but ideal. Unlike Einstein, a classicist who forever sought continuity with the past, Gödel was at heart an ironist, a truly subversive thinker. With his incompleteness theorem he had shaken the foundations of mathematics, prompting the great mathematician David Hilbert to propose a new law of logic just to refute Gödel’s results. The Gödel universe, correctly understood, shares with the incompleteness theorem an underlying methodology and purpose. It is a bomb, built from cosmology’s most cherished materials, lobbed into the foundations of physics.

In the footsteps of Gödel and Einstein, then, can be heard an echo of the zeitgeist, a clue to the secret of the great and terrible twentieth century, a century that, like the seventeenth, will go down in history as one of genius. The residents of Blue Hill, preoccupied with war and the enemy out at sea, had failed to take the full measure of their man.

2 | A German Bias for Metaphysics

Physically they were opposites. Gödel, thin to the point of emaciation, hid his spectral body even in the heat of summer in overcoat and scarf. Gaunt, harrowed, and haunted, peering through thick glasses like an owl from another dimension, he could not fail to arouse suspicion. Early in life he had come to the conclusion that the less food one ate the better. This dubious insight he carried out with ruthless consistency, unencumbered by the excess baggage of common sense, a faculty he approached life without. His preconception, fueled by hypochondria that grew out of childhood rheumatic fever and by paranoia about the intentions of doctors, developed into a neurosis that would eventually take his life. During several periods of extreme stress he was confined to sanatoria, from one of which, by some accounts, he enlisted the services of his wife to escape. At his death he weighed a mere sixty-five pounds.

Einstein, in contrast, whose sanity was never in question, was as satisfied by a good sausage as by a good theorem. He had a taste for solid German cooking, which he consumed with relish, topped off by his omnipresent pipe. Friends and wives would be swept aside in the current of his turbulent life, but his pipe never left him. Late in life he was the proud owner of a respectable professorial paunch. “I have firmly resolved,” he wrote his wife Elsa, “to bite the dust, when my time comes, with the minimum of medical assistance, and until then to sin cheerfully . . . smoke like a chimney, work like a beaver, eat without thought or choice, and walk only in agreeable company, in other words, rarely.”

With brown hair and blue eyes, Gödel measured barely five feet six. This number came as a surprise to his colleagues. His intellectual presence was so great that his modest height often went unnoticed. His frailty, however, was obvious. “Of course he has no children,” the proprietor of the Blue Hill Inn said of Gödel; “he hasn’t the strength to make babies.” He did, however, have in his youth the strength to pursue women. “There is no doubt,” wrote a college friend, Olga Taussky-Todd, “about the fact that Gödel had a liking for members of the opposite sex, and he made no secret about this fact.” Gödel, she went on, was not beyond showing off his acquaintance with a pretty face. Taussky-Todd herself, to her dismay, was once enlisted to come to the mathematical aid of one such young woman who in turn was trying to make an impression on Gödel. Was this interest in women confined to Gödel’s youth? Not if his wife, Adele, is to believed. Teasing her husband, she quipped that the Institute for Advanced Study—which she liked to call an Altersversorgungsheim