Einstein and Gödel

A picture taken in Princeton, New Jersey in August of 1950 shows Albert Einstein standing next to the Austrian logician, Kurt Gödel. Einstein is wearing baggy slacks and a rumpled shirt. His body sags. Dressed in a white linen suit, and wearing owlish spectacles, Gödel looks lean and almost elegant in comparison, the austerity of his expression softened by a certain odd sensuality that plays over the lower half of his face. Plainly at ease, the men are indulging the photographer.

When their friendship began in 1941, Kurt Gödel was thirty-five. Ten years earlier, in a relatively short but symphonic paper of some thirty-five pages, he had created one of the monuments of modern thought. Elementary arithmetic, he had demonstrated, is incomplete and incompleteable. Whatever the axiomatic system by which arithmetic is expressed, there are true statements that lie beyond the system’s reach. They cannot be demonstrated. Adding such statements to the system as further axioms does no good. The enriched system is also incomplete, the infection moving upward by degrees.

A number of mathematicians knew of Gödel’s achievement in 1941, but word of his genius had not left the cloister, where it was still conveyed in whispers. Einstein, on the other hand, was sixty-two, and one of the century’s mythic figures, his plump sad face known throughout the world.

The difference in their public stature were reflected in the nature of their friendship. In letters to his mother, Gödel took pleasure in affirming that through his friendship with Einstein, he was basking in reflected light. “I have so far been to his house two or three times,” he wrote (in 1946), “all for scientific discussions. I believe it rarely happens that he invites anybody to his house.”

Yet in the grandeur of their achievements, Einstein and Gödel stood alone, and so must have turned to one another at least in part because they could turn to no one else.

They were in their personalities quite different. Einstein was a man of unshakeable self-confidence, intellectually massive, unafraid. Gödel was, by way of contrast, both delicate and diffident. He loathed criticism and shrank from controversy. His life was hardly an exercise in vigour. He was under the best of circumstances a valetudinarian, and under the worst, a hypochondriac. Often both.

And yet his philosophy reveals currents that move against the grain of what seemed to be his personality. His experiences in Europe notwithstanding, he was an optimist by conviction and a theist by inclination; he took seriously speculations about the after-life; he was sceptical about the Darwinian theory of evolution. And he was a voluptuous Platonist, arguing with great boldness and ingenuity that the human intellect is capable of directly grasping pure mathematical abstractions. If during his life he chose to keep his philosophical views largely to himself, perhaps this is because he was persuaded that criticism would serve only to impede his tranquillity while doing nothing to advance his intellectual agenda.

The general theory of relativity is Einstein’s supreme creation, and it is to general relativity that Gödel made an unexpected contribution in 1948. The idea governing general relativity is not difficult to grasp. Space and time are fused within the theory, but, in truth, 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 — at roughly 1.30 EST on the afternoon of November 22nd, 1963. Three numbers suffice to mark Dallas, Texas on any map that indicates height as well as longitude and latitude; one number is need to mark the time. Four numbers identify the event precisely.

If events are brief bursting episodes, processes in the most ordinary sense are sequences of such events, the events trailing one another like elephants marching trunk to tail. It is processes that comprise the fundamental objects treated in general relativity, where they are called world-lines, and the theory’s considerable mathematical apparatus is put in place to shed an analytic light on their behaviour.

We may leave the mathematical details to the mathematicians who cherish them. For all its complexity, the theory subordinates itself nicely to a number of homely metaphors. Imagine a marble placed on a mattress. Given a tap, the marble will move in a straight line. The mattress is, after all, flat. A heavy bowling ball is now placed on the mattress, its weight deforming the mattress by means of an obvious dimple. Given precisely the same tap, that marble flows downward toward the dimple, its path changing from a straight to a curved line. The weight of the bowling ball, the shape of the mattress and the path of the marble are obviously co-ordinated. The bowling ball deforms the medium of the mattress, and the deformed medium in turn controls the way in which the marble moves.

Those metaphoric crutches may now be withdrawn, marbles and mattresses replaced by the universe itself, with its stars, planets, wheeling galaxies and clouds of cosmic dust. The happy co-ordination just scouted reappears in this more general setting, no worse for wear. Massive objects deform the medium of space and time; and the deformed medium in turn influences the processes that take place within it, the field equation of general relativity co-ordinating that deformation with those processes.

Co-ordinate? Not quite. In its control over the cosmos, the field equation of general relativity sets the stage, but it does not determine the drama. The Russian mathematician Alexandr Friedmann provided the first realistic solution of Einstein’s field equation in 1922; and by the 1930s, his work had been assimilated into a general analytic structure — Friedmann-LeMaitre cosmology. It is this work that suggested the now familiar picture of an expanding universe, one moving explosively outward from a dense initial singularity.
But a universe proceeding from nothing to nowhere by means of an enthusiastic expansion — our universe, apparently — is but one possibility; and there are others. Some interpretations of the field equation are realised in a static but unstable universe, one that simply hangs around for all eternity if it manages to hang around at all. Early on, Einstein had committed his allegiance to a universe of this sort; he came to regard the universe of contemporary cosmology as inevitable, given the facts, but somewhat vulgar.

Until Gödel’s work, the universes available to cosmologists, although different in some respects, were all of them well-behaved and roly-poly, with the great imponderables of time, space and cause arranged with consideration for the common intellectual decencies.

Gödel succeeded in coaxing a new and certainly a flamboyant universe from the alembic of Einstein’s symbols. His analysis reflects the distinctive characteristic of all his work. It is highly 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 very and distinctly odd.

The concept of time now occupies centre stage. A number of philosophers are standing by. And what they are saying, those philosophers, is that change is an illusion. Things do not become, they have not been, and they will not be: they simply are. Human beings reach events in the future by displacing themselves in time just as they reach places on the earth by displacing themselves in space. They do not bring those places into being, nor those events. It is thus that time dwindles, and thus that time disappears, replaced by an entirely more arid notion, that of position along a temporal stream.

Einstein’s special theory of relativity, Gödel observed, was widely thought to support this view. Imagine a group of observers scattered carelessly throughout the cosmos. Each is able to organise the events of his life into a linear order; and as a result each is persuaded that his life consists of a series of nows, moving moments passing from the past to the present to the future. I might as well dismiss those observers before they do any real harm. This is how we see things. Now is after all now, it is not? Right now.

Apparently not. Simultaneity, special relativity revealed, depends on the speed at which we are moving with respect to one another. Moving at different speeds, the two of us, it is entirely possible that my now might be your past or your future.

It follows that what is becoming for me may have become or may become for you. But then Gödel asks, very reasonably, how something can become for me when it has already been for you? The idea is if not absurd then deeply unattractive. What is left when becoming is subtracted from the cosmic account is time — that remains. But change has disappeared. A philosophical conjecture has been ratified by a great physical theory.

This is a view that Gödel found congenial; in fact, it is view that Einstein also embraced, writing to the widow of his old friend Michael Besso, and writing with great poignancy, that “for us believing physicists, the distinction between the past, the present and the future is only an illusion.”
And now the odd point. However much the illusory nature of time is suggested by special relativity, it is, in fact, slyly contravened by the dominant interpretation of general relativity. There is at least one universal system of time that provides a compelling standard of simultaneity throughout the cosmos, and that is the system provided by cosmology itself. The expansion of the universe is universal, space and time stretching the very fabric of creation. This means that there is a universal reference frame as well — it is the frame provided by the behaviour of matter. Physicists talk, after all, about the first three minutes. If this makes sense, it makes sense as well to talk of times after the first three minutes. It makes sense to talk of the time after the first three minutes everywhere in the universe, and if time has an origin, and a uniform measure, then we are again within the sounds of Newton’s universal clock. It is everywhere approximately fifteen billion years after the Big Bang, and it is that time now.

It is against this interpretation that Gödel turned his face, his work arising from his desire to reaffirm the very deepest insights of Einstein’s special theory of relativity.

In an expanding universe, space and time rush outward in a hot gush from a primordial explosion. So, too, world-lines. They have no choice. There is a profound connection between the structure of an expanding universe and the nature of those processes taking place within it. Processes are, of course, nothing more than events following one another in a series, but it is useful to give them a visual incarnation, if only for purposes of illustration. Imagine those processes as strands in a great rope, a cosmic hawser. In a simply expanding universe, such as our own, the cosmic hawser is twist-free. The strands do not snake around one another, as they do in an ordinary rope. To say that they are twist-free is just to say that a cosmic knife could slice through them all at a right angle. It is this that makes for a global sense of simultaneity. It is now throughout the universe wherever the cosmic knife cuts the cosmic hawser, and it is now throughout the universe just because a twist-free cosmic hawser can be cut and cut completely by a cosmic knife.

The assumption of a simply expanding universe is now allowed to lapse. If not expanding, just what is the universe doing otherwise? It might, of course, be doing nothing whatsoever, as Einstein had originally expected; but then again it might be rotating in the void, turning serenely like a gigantic pinwheel, and it this idea that Gödel found compelling. In a universe of this sort, each observer sees things as if the whole universe were rotating about him. This strange assumption, Gödel demonstrated, satisfies the field equation of general relativity exactly.

Rotating universes may now be added to the catalogue of possible things.

A pause to collect ourselves. A rotating universe might suggest, at first glimpse, nothing more than the universe scouted by ancient astrologers, with observers clustered on the earth, and the celestial sphere turning around them. There is surely something to this analogy, a kind of historical wormhole snaking from one speculative enterprise to another; but the analogy is flawed inasmuch as it suggests that it is the galaxies alone that are rotating. Not so. Everything else goes along for the ride. If space and time can expand, as they do in Big Bang cosmology, they can also assume other geometrical properties. Which properties they assume depends on the behaviour of material objects, and if those objects are turning in circles, space and time must follow. This is precisely what happens in Gödel’s universe. As the galaxies rotate, they drag space and time with them, the medium in which all processes take place crumpling before and after those fugitive galaxies. 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 it works to profoundly different effect.

Rotating universes, most notably, permit travel in time. By moving in a large enough circle around an axis of rotation, an observer might in fact catch his own temporal tail, returning to his starting point some time earlier than his departure. Force is required, but not speeds in excess of the speed of light. The requisite paths are known as closed time-like curves. Their existence is guaranteed in Gödel’s rotating universe.

Needless to say, neither Gödel nor anyone else succeeded in making sense of the idea of time travel, whatever the unexpected possibilities his solutions suggest. The most obvious problem is well known. Were time travel possible, it would also be possible mischievously to influence the causal stream, say by assassinating one’s own grandfather or by otherwise causing upheavals in the flux and fleen of things. Star Trek notwithstanding, these practical problems are both bizarre and uninteresting. Gödel’s crucial point lies elsewhere.

In a universe containing closed time-like curves — Gödel’s universe — the cosmic hawser is twisted, the strands looping over one another like snakes, and no cosmic cut of any cosmic knife could possibly cut them all. With the cosmic hawser irreparably twisted, time completely loses its significance as a form of change.

Inferences and assumptions now arrange themselves in a delicate array. Imagine a grand cosmological division, with rotating universe on the right, and non-rotating universes on the left. Gödel was able to demonstrate that on the left, where the non-rotating universes are collected, and the cosmic hawser is twist free, it is always possible to define an ever moving now and so a natural temporal order. On the right, where there are rotating universes, time undergoes its fateful dissolution and change disappears.

So much the worse, one might think, for rotating universes. Our universe is blessedly twist free, time a vehicle for change as it has always been.

But this, Gödel observed, is an accident of creation. The equations of general relativity are compatible with other possibilities. If in other possible universes no cosmic or global time is accessible, this might suggest that on the very deepest level, the features of time that we take for granted are also accidents of creation. It is this idea that Gödel found objectionable. If time exists, wherever it does exist, it must exist simply in virtue of “the particular way in which matter and its motions are arranged in the world.” A philosophical view leading to this conclusion, he added dryly, “can hardly be considered satisfactory.” Time and change demand a deeper explanation. And this physical theory does not provide.

They were close. This much we know, Gödel regarding the avuncular Einstein with appreciation, a sense of indebtedness for his robust psychological health.
And both men admired one another. This we know, too.

And yet curiously enough, the circumstances of their lives revealed countervailing currents. Einstein struggled to purge himself of the ties of family and friends, seeking solace not only in solitude but in a deliberate, carefully contrived release from the ordinary human bonds of family and affection. He had married as a young man, and divorced his first wife; he was hardly an inspired father. He lived within himself.

Like Einstein, Gödel found ordinary social intercourse an immense chore. He was notoriously reclusive, working at the Institute for Advanced Study in a darkened room, never attending other men’s 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. He died of “inanition,” in the lapidary words of the Princeton health examiner: he had refused to eat.

And yet Gödel spent much of his adult life in state of matrimonial contentment. He had chosen his wife, Adele Porkert, by dropping from his own social class, scandalising his parents and his brother. She had been a dancer in a Viennese cabaret; and if she gave any thought to incompleteness, it must have been expressed by a typical Viennese expression of amused indulgence. But she was obviously a woman with a powerful will and great self-possession, and she made Gödel’s life possible by making it bearable.

Gödel spent the second half of his life absorbed by philosophy. In the end, he concluded that his efforts had been unavailing. “I did not,” he remarked to Hao Wang, “find in philosophy what I was looking for.” Much the same is true for Einstein. The great unified theory for which he had searched for more than thirty years eluded him.

Did Einstein and Gödel discuss issues such as this? I do not know. I suspect that the depth of their friendship made what diplomats often call frank discussions unnecessary. Gödel was sceptical of Einstein’s quest for a unified theory; and Einstein, we may be sure, must have regarded Gödel’s philosophical investigations with detachment, the deep and ineradicable melancholy in his personality making it impossible for him to regard optimism or theism with anything more than a sense of tolerant scepticism.

The issues between the men remain unsettled. But however they may be settled, neither Einstein nor Gödel found the arch that would have completed their lives.

But then again, none of us do.