Einstein and Gödel

Oh, to be a fly on a textbook when the century's greatest physicist walked home from work with its most influential mathematician

By David Berlinski|Friday, March 01, 2002
RELATED TAGS: COSMOLOGY

A picture taken in Princeton, New Jersey, in August 1950 shows Albert Einstein standing next to the Austrian logician Kurt Gödel. Both men are looking at the camera. Einstein is wearing a rumpled shirt and baggy slacks held up by suspenders. His body sags. Gödel, dressed in a white linen suit and wearing owlish spectacles, looks lean and almost elegant, the austerity of his expression softened by an odd sensuality that plays over the lower half of his face. The men are at ease; they are indulging the photographer. Clearly, they are friends. It is hardly surprising that they should have come to know each other. They were members of the Institute for Advanced Study at Princeton, and their offices were close. As refugees from the Third Reich, they had both felt the harsh breath of history and had in common the rich, throaty German language, a world of words in which the pivot of memory turns on Goethe, not Shakespeare. Although Einstein was a physicist and Gödel a mathematician, they shared an intellectual daring that transcended their disciplines.

Gödel's incompleteness theorems, published in 1931 when he was only 25, had rewritten the ground rules of modern science much as Einstein's theory of relativity had done 15 years before. Elementary arithmetic, Gödel demonstrated, is incomplete and will remain so. Whatever axiomatic system you base your calculations on, there are true statements that lie beyond the system's reach. Adding such statements to the system as further axioms does no good. The enriched system is also incomplete, the infection moving upward by degrees.

Einstein once remarked to Oskar Morgenstern, one of the cofounders of game theory, that he went to the Institute chiefly to walk home with Gödel. ("Um das Privileg zu haben, mit Gödel zu Fuss nach Hause gehen zu dürfen." There is in the original German a note of gentle deference that cannot quite be translated.) They did so often until Einstein's death in 1955. Yet their scientific affinities grew out of profound personal differences. Einstein was a man of unshakable self-confidence. Gödel retreated before controversy and twice suffered nervous collapses; he was, under the best of circumstances, a valetudinarian, and under the worst, a hypochondriac. When the two men met, in 1933, word of young Gödel's genius had yet to leave the academic cloister, where it was conveyed in whispers. Einstein, on the other hand, was 54, nearing the end of his productive career. Although he retained a sense of impudent playfulness, he had also acquired a marmoreal aspect, transcending fame itself to become one of the century's mythic figures, his plump, sad face known throughout the world.

These differences were inevitably reflected in the nature of the friendship. In a letter written to the biographer Carl Seelig, Einstein's secretary remarked on the "awed hush" that greeted Einstein whenever he appeared at seminars or conferences. Not even the sharp-tongued Wolfgang Pauli, a fellow Nobel Prize winner in physics, could bring himself to treat the great man as if he were mortal. Gödel seemed to share something of this attitude. In letters to his mother, he appeared to take pleasure in affirming that through his friendship with Einstein, he was basking in reflected glory. "I have so far been to his house two or three times," he wrote in 1946. "I believe it rarely happens that he invites anybody to his house."

Still, in the grandeur of their scientific achievements, Einstein and Gödel both stood alone and so must have turned to each another in part because they could turn to no one else. Although the content of their conversations has been lost, we can imagine at least one topic they must have discussed on those long evening walks. In 1948 Gödel turned his attention to Einstein's supreme creation, the general theory of relativity, and succeeded in coaxing a new and flamboyant universe from the alembic of its symbols. He did so by providing an exact solution to the heart of the theory—a field equation that allows one to calculate the force of a gravitational field—and his analysis reflects the distinctive characteristics of all his work. It is original and logically coherent, the argument set out simply but with complete and convincing authority. A sense of superb taste prevails throughout. There is no show.

And it is odd. It is distinctly odd.

The leading idea of general relativity—the fusion of space and time—is not hard to grasp. After all, space and time are fused in ordinary life as well. We locate an event (the assassination of JFK, for example) both in terms of where it took place (Dallas, Texas) and when it took place (roughly 1:30 EST on the afternoon of November 22, 1963). Three numbers suffice to mark the space of Dallas, Texas, on a three-dimensional map: longitude, latitude, and altitude. The place is pinpointed as an event in space-time if another number, the time, is added. And if an event can be defined by four numbers, then a series of events can be defined by a series of such numbers, trailing one another like elephants marching trunk to tail. In general relativity such series are called world lines.

General relativity then forges a far-flung connection between the geometry of space and time and the behavior of objects in motion within space and time. Imagine a marble placed on a mattress. Given a tap, the marble will move in a straight line. But place a bowling ball on the mattress, too, and the marble, given precisely the same tap, will roll down the sagging surface, its path changing from a straight to a curved line. The bowling ball's weight deforms the medium of the mattress, and the deformed medium influences the marble's movement.

Replace the bowling ball and marble with planets, stars, or wheeling galaxies, and the mattress with space-time itself, and a homely metaphor is transformed into the leading principle of a great physical theory. In a universe with no massive objects, there is no deformation of space and time, and the shortest route between two points is a straight line. When matter makes its fateful appearance, the shortest routes will curve. The first and most celebrated confirmation of this theory came in 1919, when astronomers established that the mass of the sun causes a beam of light to curve, as Einstein had predicted.

"For us believing physicists," Einstein once wrote, "the distinction between the past, the present, and the future is only an illusion." It was a melancholy remark, made as Einstein faced death, but it flowed directly from Einstein's special theory of relativity. Imagine a group of observers scattered carelessly throughout the cosmos. Each is able to organize the events of his life into a linear order—a world line of the kind just described. Each is convinced that his life consists of a series of nows, moving moments passing from the past to the present to the future. Special relativity urges a contrary claim. The observers scattered throughout space and time are all convinced their sense of now is universal. Now is, after all, now, is it not? Apparently not. Time passes at a different rate depending on how fast a person is moving: While one hour passes on Earth, only a few seconds might pass on a rocket ship hurtling away from Earth at nearly the speed of light. It is entirely possible that one man's now might be another man's past or future.

Gödel's solution to the field equation vindicated the deepest insight of Einstein's theory, namely that time is relative. But Einstein's theory of relativity suggests only that time does not exist in the conventional sense, not that time exists in no sense whatsoever. Einstein's claim is more subtle. He suggests that change is an illusion. Things do not become, they have not been, and they will not be: They simply are. Time is like space; it is precisely like space. In traveling to Singapore, I do not bring Singapore into existence. I reach Singapore, but the city has been there all along. So, too, I reach events in the future by displacing myself in time. I do not bring them into being. And if nothing is brought into being, there is no change.

Most cosmologists now agree that the universe expanded from a primordial explosion we call the Big Bang. Physicists talk, after all, about the first three minutes. But if this makes sense, it makes sense as well to talk of times after the first three minutes. And if time has an origin, and a uniform measure, then we are again within the bounds of Newton's universal clock, marking time throughout the cosmos. It is everywhere approximately 14 billion years after the Big Bang, and it is that time now.

But a universe proceeding from nothing to nowhere by means of an enthusiastic expansion is only one possibility. There are others. Some interpretations of the field equation are realized in a static but unstable universe, one that simply hangs around for all eternity if it manages to hang around at all. Then again the universe might be rotating in a void, turning serenely like a gigantic pinwheel. In a universe of this sort, each observer sees things as if he were at the center of the spinning, with the galaxies—indeed, the whole universe—rotating about him. And this strange assumption, Gödel demonstrated, satisfies the field equation of general relativity exactly.

A rotating universe is an idea reminiscent of ancient astrologers, who imagined observers clustered on the earth and the celestial sphere turning around them. But in Gödel's conception, the galaxies aren't the only things rotating. Everything else goes along for the ride. The galaxies rotate, and as they do, they drag space and time with them. Just as an expanding universe blows up space and time, a rotating universe turns space and time around in spirals. The same idea is at work but with a different effect. In a rotating universe, for instance, time travel becomes possible. By moving in a large enough circle around an axis, at something approaching the speed of light, an observer might catch his own temporal tail, returning to his starting point at some time earlier than his departure. The requisite paths are known as closed timelike curves.

When Gödel first published his work, the general reaction from other scientists was one of polite curiosity. Einstein was respectful but cautious, suggesting that perhaps Gödel's conclusions would simply be rejected on "physical grounds." (Gödel's solution rules out an expanding universe, which Einstein had by then grudgingly accepted.) Gödel himself never succeeded in making sense of time travel, whatever his solution suggested. Aside from the sorts of paradoxes beloved by science fiction writers—a time traveler who accidentally kills his own grandparents, say—time travel has more subtle, theoretical problems. There is no suggestion in Gödel's work, for instance, that time itself can come to a stop and then reverse course. Yet time travel would entail a journey, and in general relativity, as in real life, every journey takes time. From the traveler's perspective time would move forward minute by minute, even as it leaps backward when he arrives.

There are still deeper points at issue. If time moves in circles, and an observer can return to his own past, it seems to follow that effects might be their own causes. It is one thing to give up on time; it is quite another to give up on cause as a fundamental physical property.

And finally, there is the philosophical point, the one at the heart of Gödel's concerns. Granted, rotating universes may be physically unrealistic. But they are possible, and once seen as possibilities, they cannot be unseen. Within these strange contraptions, time is an illusion. But if time is an illusion in some universes, then the features of time that we take for granted in this particular universe must be accidents of creation, a matter of how matter and its motions are arranged in the world. But a philosophical view leading to this conclusion, Gödel remarked dryly, "can hardly be considered satisfactory." Time is far too deep a concept to arise accidentally.

Gödel spent the second half of his life absorbed by philosophy. Despite his experiences in Europe, he believed that "the world is rational." He was an optimist and a theist; and although he thought that "religions are for the most part bad," he insisted that "religion [itself] is not." A deity was at the center of his metaphysics. He entertained speculations about the afterlife, arguing that "the world in which we live is not the only one in which we shall live or have lived." He dismissed the Darwinian theory of evolution and declared flatly that "materialism was false." He was a mathematical Platonist, arguing with boldness that the human intellect is capable of perceiving pure mathematical abstractions, just as the human senses are capable of grasping material objects.

Still, in the end, Gödel concluded that his efforts had been unavailing. "By his own account," the late logician Hao Wang wrote in Reflections on Kurt Gödel, he did not "attain what he was looking for in philosophy." Much the same is true for Einstein. The great unified theory for which he had searched for more than 30 years ultimately eluded him. For much of that time, he had worked in isolation, a younger generation of physicists regarding his obsession as just that, treating him with reverence, of course, but a form of reverence in which the first faint curl of contempt might be seen.

Did Einstein and Gödel discuss such issues? Gödel's journals and biographers do not say. But the depth of their friendship made what diplomats often call frank discussions unnecessary. Gödel was skeptical of Einstein's quest for a unified theory, and Einstein must have regarded Gödel's philosophical investigations with a certain amused detachment. The deep and ineradicable melancholy in Einstein's personality made it impossible for him to regard optimism or theism with anything more than a sense of tolerant skepticism.

Their lives, too, revealed countervailing currents. Einstein sought solace not only in solitude but in a deliberate, carefully contrived release from the ordinary human bonds of family and friends. He divorced his first wife and never once saw his daughter, who was probably put up for adoption. His second marriage, to his cousin Elsa, was hardly an affair of passion.

Like Einstein, Gödel found ordinary social intercourse an immense chore. For most of his adult life, he was happily married to a former dancer in Viennese cabarets. Yet he was notoriously reclusive, working at the Institute for Advanced Study in a darkened room, never attending other scientists' lectures, solitary, obsessed, half-mad, consumed from within by the fires of an intellectual passion so powerful that by the end of his life they seemed quite literally to have consumed his frail flesh entirely. Gödel died, in 1978, of "inanition," in the lapidary words of his death certificate. He had refused to eat.

Long before then, Einstein's theory of general relativity had undergone an efflorescence, its dark, difficult interior yielding many interesting mathematical secrets. A wealth of observational evidence made it possible to test the theory against a cosmic instead of a local background. Most of all, a new generation of physicists fell under Einstein's sway and attended to the strange, unearthly music of his dream of a unified theory. Einstein's vision has proved too powerful to be dismissed. As for Gödel's, it awaits another time to be fully understood. Or perhaps another universe.


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