“Scientific people,” proceeded the Time Traveler, “. . . know very well that time is only a kind of space.”
H.G. WELLS, THE TIME MACHINE
Albert Einstein: Philosopher-Scientist, dedicated by P.A. Schilpp to the physicist on the occasion of his seventieth birthday, was a great success. It remains the most influential volume in the Library of Living Philosophers, not least because of its wonderful debate between Einstein and Niels Bohr, friends and adversaries, on the future of the Copenhagen interpretation of quantum mechanics. Einstein’s walking companion, in turn, had worked intensely on his own contribution, writing to his mother that the work left him little leisure for correspondence. That summer, he canceled as well his accustomed vacation trip to the seashore. When the auspicious volume finally appeared, he cannot have failed to be disappointed by the near silence with which his essay was greeted.
To be sure, there was a minor stir among astrophysicists and cosmologists concerning the validity of Gödel’s construction of new world models for general relativity. The initial response, however, was that he had simply made a mistake in his physics. No less a physicist than S. Chandrasekhar, who had attended a talk Gödel had given on his new models at Princeton, published an article with J.P. Wright in the Proceedings of the National Academy of Sciences claiming that Gödel had made an error when he described the possibility of time travel as along a geodesic—the path of inertial motion, or free fall—in the Gödel universe. This no doubt contributed to the lack of interest in Gödel’s essay among philosophers. If the physical premise was faulty, why bother to examine the philosophy?
But was Gödel really in error? Amazingly, the editors of the Proceedings had not seen fit to consult the author himself before publishing a report of his alleged error concerning an elementary concept of relativity theory. Might it not have been Chandrasekhar and Wright, not Gödel, who had made a mistake? This possibility seems not to have occurred to the editors, yet it turned out to have been the case, a fact demonstrated not by a physicist but by a philosopher, Howard Stein, who showed clearly that time travel in the Gödel universe could take place only under great acceleration, which could be provided by a spaceship, not along the free-fall path of a geodesic. More astonishing yet, however, Stein could not get the correction of Chandrasekhar and Wright accepted for publication. Only when Gödel himself intervened did the fact finally make it into print that his argument for the possibility of time travel was relativistically valid.
What had gone wrong? Clearly, regardless of Gödel’s reputation as a great logician, the astrophysics community saw him as an outsider, and moreover as attempting to swim against the intellectual tide. But the scandal of disregard extended to philosophy as well. Gödel’s contribution to the Schilpp volume had almost no impact on the community of philosophers. Except for a few highly technical discussions of the physics, with some brief though poignant glances at Gödel’s philosophical goals, his argument that relativity theory, correctly understood, provides strong support for the great philosophers throughout history who were skeptical of the objective reality of time, went unheeded. Naturally, there was some interest in the question of time travel. There always is. It is a topic of perennial fascination among thoughtful and imaginative people, and the fact that Gödel had derived such an exotic conclusion from the respectable equations of relativity inevitably raised a few eyebrows. But on the question of whether he had succeeded in showing that time is ideal there was a profound silence. If Gödel had not been taken seriously as a philosopher before his contribution to the Schilpp volume, nothing changed after its appearance.
Quite simply, he had never been a member of the club: he was out of touch and out of step with the philosophical establishment, in Princeton as elsewhere, and the reason was not hard to fathom. Just as Wittgenstein’s language-centered early work, the Tractatus, had helped set the philosophical agenda following World War I, not least in Gödel’s Vienna, in the aftermath of the Second World War, Wittgenstein’s later, still linguistically oriented philosophy came to dominate again, this time in Gödel’s newly adopted country. To many philosophers, it must have seemed as if Gödel had slept through not one but two Wittgenstein revolutions. It added insult to injury that W.V.O. Quine, the dominant figure for years in American philosophy and the most analytic of analytical philosophers, was also absent from Gödel’s thinking. Gödel himself was acutely aware of this alienation. When the time came for his essay on Cantor’s continuum problem to be reprinted in a now classic collection coedited by his Princeton colleague Paul Benacerraf and Harvard’s Hilary Putnam, he would not agree to the republication until he had been convinced that the editors would not deride his essay. Shameful it may have been that coming out of nowhere in every sense his highly compressed, paradoxical-sounding philosophical defense of temporal idealism, based on an arcane new cosmological model for an abstruse physical theory, failed to arouse more than a murmur. But it was not surprising. Yet why, more than half a century after it was proposed and ignored, is Gödel’s argument still worthy of attention? What had Gödel really accomplished?
The problem Gödel inherited from Einstein had been understood for centuries to concern the most fundamental aspect of human experience. For Kant, space and time are the two essential “forms of human sensibility,” with time, as the form of both “inner” and “outer sense,” being the more basic. Yet time is far more elusive than space. Capturing time through mathematics—a form of thought from which philosophers since Plato have taken pains to remove anything remotely temporal—is like trying to trap water with a net. With the advent of Einstein’s theory of relativity, however, the mystery of this form of being was widely taken to have been resolved. Philosophers could finally relax. Einstein had taken care of business.
Appearances, however, can be deceptive. The universe, for example, as everyone knows, is very old. Its exact age is a matter for debate, but there is no disagreement that it runs to billions of years. We marvel that as frail and isolated a species as we are can have achieved such impressive wisdom about the origins of everything that is. In truth, however, it is more than marvelous to have discovered the age of the universe. It is impossible. For if the universe is n years old, its present state comes n years after the moment when it all began. In 1905, however, Einstein had demonstrated in the special theory of relativity that there is no such thing as “the present state of the universe,” that is, what would be revealed by a snapshot of the universe as it exists at this very moment. The relativity of simultaneity implies that what is taken to be “now” relative to one inertial frame will differ from what is “now” in another frame if the second frame is in motion relative to the first. It follows immediately that if the theory of relativity is correct, there simply is no such thing as “the present state of the entire universe” of four-dimensional space-time. Einstein himself said this quite clearly: “The four-dimensional continuum is now no longer resolvable objectively into sections, all of which contain simultaneous events; ‘now’ loses for the spatially extended world its objective meaning.”
None of this was lost on Gödel. To him, there was an inconsistency between Einstein’s theory and the everyday belief that time, unlike space, “passes” or “flows.” On this question, two assumptions dominated, then as now, in the popular as well as the scientific consciousness. Both of them are faulty. The first is that special relativity is compatible with the passing of time, as long as it is acknowledged that this flow has only local, as opposed to global, significance. The other is that the world according to special relativity is a fixed four-dimensional space-time “block,” but that this does not conflict with the deliverances of ordinary experience. The former fails because whatever the flow of time is, it is not a merely geometrical fact and thus cannot enjoy only local existence. A river’s course, for example, may curve locally—may be serpentine near us but straight elsewhere—but what would it mean to say that the river flowed only in our neighborhood?
Concerning the second assumption, one need only recognize the befuddlement that would ensue if one were to try to act on the assumption that today’s breakfast is no more actual than yesterday’s or tomorrow’s, that the future, like the present, has already arrived. (“The future is now,” reads the logo of Hudsucker Enterprises in the film The Hudsucker Proxy. This makes for an entertaining story but an unconvincing metaphysics of everyday life.) Should I still be wondering what to order for breakfast yesterday, as I am for tomorrow, or should I cancel both orders because the meals have already arrived? And since the present is no more real than the past and I am still lying on the beach as I was last summer, why am I identifying only with the “I” that is presently shivering in the cold? Am I simply making a mistake? Or are there as many “I’s” as there are moments in time, and if so, are they all me, or only parts of me? (I have spatial parts, of course: head, hands, feet; do I also have “temporal parts”?) The confidence of the popular (and not so popular) mind is misplaced when it clings to the belief that all is well, temporally speaking, between the universe and Dr. Einstein. All is not well at all.
But as indicated by its name, special relativity is not the full theory of relativity. Its validity is restricted to so-called inertial reference frames, those that are unaccelerated and move in straight lines. The final, comprehensive theory, general relativity, has no such restrictions. It includes an account of gravity, the first theory of gravity to replace Newton’s. Since it is gravity that governs the universe as a whole, general relativity is the foundation of the modern science of cosmology. If special relativity, moreover, introduced the discovery that matter is equivalent to energy, the general theory announced the identity of gravity with space-time curvature. Matter in motion determines the shape of space-time. The possibility arises that some reference frames might be privileged, namely those that follow, as Gödel put it, the mean motion of matter in the universe. Time relative to those frames of reference bears the designation “cosmic time,” and this opens up the possibility that time in something like the pretheoretical sense might after all be consistent with relativity, in particular with general relativity. It is time in this sense that is (or should be) invoked when cosmologists speak of the age of the universe.
The question remains, however, how closely this new concept of time resembles what time was thought to be before Einstein. The astrophysicist James Jeans, whom Gödel would cite by name when he came to discuss these issues, thought the resemblance was very close indeed. With the advent of general relativity and cosmic time, “time regained a real objective existence, although only on the astronomical scale.” Since, moreover, every known relativistically possible universe “makes [in this way] a real distinction between space and time,” Jeans believed, “[we have] every justification for reverting to our old intuitional belief that past, present, and future have real objective meaning.” In short, “we are free to believe that time is real.” Just this Gödel would put to the test.
At issue is the leitmotif of Gödel’s lifework, the dialectic of the formal and the intuitive, here, of formal versus intuitive time, between what remains of time in the theory of relativity and the time of everyday life. The difference between these two conceptions is crucial. It can be illuminated by considering what the early-twentieth-century philosopher J.M.E. McTaggart called the A-series and B-series. The B-series is founded on the characterization of dates and times in terms of the fixed relationship of “before” and “after.” It is a structurally or “geometrically” defined series, analogous to a space. It is the temporal series captured by calendars and by history books. The year 1865, for example, comes—now and forever—before 1965 and after 1765, and these structural, “geometric” facts are fixed and unchangeable. The A-series, in contrast, is essentially fluid or dynamic. It contains the “moving now,” i.e., the present moment, which is always in flux. That your dentist appointment is at 3 p.m. on May 19 is a B-series fact that has been marked on your calendar for months. It will remain a fact after the appointment is long forgotten. That now, however, is the very date and time of the appointment is a scary A-series fact that has not obtained until this very moment, and will happily no longer obtain tomorrow. (It is no accident that a famous philosophical essay on the A-series is entitled “Thank Goodness That’s Over.”)
Though the A-series represents, intuitively, the most fundamental aspect of time—indeed, what distinguishes time from space—it is marked by several concomitants, each one difficult to capture in the formal language of mathematics. First is the fact that one time—now—is privileged over all others. This privilege passes from time to time. What is now will soon be then. Second, according to this conception, time passes, or flows, or lapses, and in a certain “direction”: what is future becomes present, then past. Third, unlike both space and the B-series, “position” in the A-series is not ontologically neutral. Whereas to exist in New Jersey is to exist no less than in New York (protests by New Yorkers notwithstanding), to “exist in the past” is no longer to exist at all. Socrates had his time on stage, but it passed, he died, and his name has been removed from the rolls. (It follows that there is nothing subjective or mind-bound about the A-series, i.e., about what is happening now. If there is such a thing as “inner time”—the subject, it would appear, of Husserl’s investigations—then this must be distinguished from the A-series.) Fourth, while the past has passed and is now forever fixed and determinate, the future remains, as of now, open. Simultaneity, finally, since it determines what really exists at the same time as other things exist, is absolute and nonrelative. We cannot, merely by choosing a frame of reference, determine what really exists at this moment. Either my friend in Paris is speaking on the phone at very same time at which I am writing this, or she isn’t, regardless of how I try to determine, via synchronized clocks, whether her speaking is occurring at the same moment as my writing.
Intuitively, time is characterized by both the A- and the B-series. If time as we experience it in everyday life, however, is to be identified with formal time—time as it is studied in physics—a problem arises. What we call “t,” the temporal component of relativistic space-time, can be consistently interpreted as representing the B-series. The problem lies with the A-series. Since, as Einstein put it, in special relativity “‘now’ loses for the spatially extended world its objective meaning”—that is, there is no objective, worldwide “now”—it appears that “t” cannot represent the A-series, in which there is a single worldwide “now” whose “flux” constitutes the change in what exists that characterizes temporal, but not spatial, reality. This should come as no surprise. One of the most striking characteristics of relativistic space-time is that space and time are no longer to be considered independent beings but rather two inextricably intertwined components of a single new kind of being, not space or time but rather space-time.
The A-series cannot be made to resemble space. What keeps this seemingly obvious fact hidden from many formal thinkers, whether physicists or logicians, is that in special relativity, “t” is formally distinguished from the three spatial dimensions. In the definition, for example, of the space-time “interval”—the unique relationship between any two space-time events that is frame-invariant, hence agreed upon by all observers, no matter their state of motion—the temporal variable, “t,” is distinguished from the three spatial variables by being preceded by a negative sign. All this demonstrates, however, is that time in special relativity has a different “geometry” from the spatial dimensions, not that it is a qualitatively different kind of being, namely something that “flows.” To be blind to this fact is to confuse the formal with the intuitive.
It is not for nothing that with the theory of relativity Einstein is said to have accomplished the geometrization of physics (an achievement for which, as we have seen, he owed a great debt to the mathematician Minkowski, his long-suffering teacher at the Technical Institute in Zurich, who took the bold step of re-creating special relativity in a four-dimensional geometric framework). It is not just that Einstein reconceived the geometry of the universe. Rather, in special relativity, he made the defining characteristic of time not its qualitative distinction from space, as Kant and Newton had done, but rather its contribution to the geometry of four-dimensional space-time. Similarly, in general relativity, he not only provided a new geometry for the laws of gravity, he defined gravity itself geometrically, as space-time curvature. One of Einstein’s claims to fame, after all, is his uncanny ability not only to provide new descriptions of old phenomena but new definitions as well. In this, as in many other aspects of his discoveries, he is as much philosopher as physicist. The coup de grâce came when he replaced Newton’s intuitively evident Euclidean mathematics with unintuitive non-Euclidean geometry.
Time as it appears in relativity theory, then, was ripe for consideration in the “Gödel program” of assessing the extent to which intuitive ideas can be captured by formal concepts. This is what Gödel had in mind when he titled his contribution to the Schilpp volume, “A Remark About the Relationship Between Relativity Theory and Idealistic Philosophy.” The “idealistic philosophers” he was referring to were thinkers like Parmenides, Plato and Kant, who questioned whether our subjective experience of the flow of time has an objective correlative. To such thinkers, time was always an ontological suspect. As before, when he examined the relationship of intuitive arithmetic truth, or big “T,” to its representation as formal mathematical proof in Russell’s Principia Mathematica, Gödel would begin by clarifying the distinction between intuitive time and little “t,” its formal representation in Einstein’s theory of relativity as the temporal component of fourdimensional Einstein–Minkowski space-time. Drawing from his contribution to the Schilpp volume as well as the longer versions of this essay that have now been published, we can say that Gödel characterized intuitive time—“what everyone understood by time before relativity theory”—as “Kantian,” or “prerelativistic.” Time in this intuitive sense, he said, is “a one-dimensional manifold that provides a complete linear ordering of all events in nature.” This “objective lapse of time” is “directly experienced” and “involves a change in the existing [i.e., in what actually exists].” Time in the intuitive sense, for Gödel, is something “whose essence is that only the present really exists.” In particular, it “means (or is equivalent to the fact) that reality consists of an infinity of layers of ‘now’ which come into existence successively.” These features Gödel took to be essential properties of time in the intuitive sense, since “something without these properties can hardly be called time.” Clearly, time so characterized is reflected in the A-series, and indeed Gödel refers to McTaggart by name in his essay. The question that remains is whether this intuitive concept can be captured by the formal methods of relativity.
As he had previously done in his incompleteness theorem, Gödel demonstrated that those who fail to grasp the distinction between the intuitive and the formal concept are not in a position to make a proper assessment of their relationship. Having made that distinction with remarkable clarity, he was able to establish, by an ingenious and entirely unsuspected formal argument—which in itself, as Einstein pointed out, was a major contribution to relativity theory—the inability of the formal representation to capture the intuitive concept. Gödel’s dialectical dance with intuitive and formal time in the theory of relativity contained an intricate series of steps. We begin with a large-scale view of the structure of Gödel’s argument, then move on to a closer examination. First the forest, then the trees.
The opening move concerns the more limited special theory of relativity. Given that the A-series contains the flux of “now,” the absence of an objective, worldwide “now” in special relativity rules out its existence. But absent the A-series there is no intuitive time. What remains, formal time as represented by the little “t” of Einstein–Minkowski space-time, cannot be identified with the intuitive time of everyday experience. The conclusion, for Gödel, is inescapable: if relativity theory is valid, intuitive time disappears.
Step two takes place when Gödel reminds us that special relativity is “special” in that it recognizes only inertial frames in constant velocity relative to each other. It does not include an account of gravity. Einstein’s general theory of relativity, in contrast, of which the special is a special case, does. In general relativity, as we have seen, gravity itself is defined as space-time curvature, determined, in turn, by the distribution of matter in motion. It follows that whereas in special relativity no frames of reference or systems in motion are privileged, in the general theory some are distinguished, namely those that, in Gödel’s words, “follow the mean motion of matter” in the universe. In the actual world, it turns out, these privileged frames of reference can be coordinated so that they determine an objective remnant of time: the “cosmic time” we encountered earlier. In general relativity, then, time (of a sort) reappears.
But no sooner has time reentered the scene than Gödel proceeds to step three, where he exploits the fact that Einstein has fully geometrized space-time. The equations of general relativity permit alternative solutions, each of which determines a possible universe, a relativistically possible world. Solutions to these complex equations are rare, but in no time at all Gödel discovers a relativistically possible universe (actually, a set of them)—now known as the Gödel universe—in which the geometry of the world is so extreme that it contains space-time paths unthinkable in more familiar universes like our own. In one such Gödel universe, it is provable that there exist closed timelike curves such that if you travel fast enough, you can, though always heading toward your local future, arrive in the past. These closed loops or circular paths have a more familiar name: time travel. But if it is possible in such worlds, Gödel argues, to return to one’s past, then what was past never passed at all. But a time that never truly passes cannot pass for real, intuitive time. The reality of time travel in the Gödel universe signals the unreality of time. Once again, time disappears.
But the dance is not over. For the Gödel universe, after all, is not the actual world, only a possible one. Can we really infer the nonexistence of time in this world from its absence from a merely possible universe? In a word, yes. Or so Gödel argues. Here he makes his final, his most subtle and elusive step, the one from the possible to the actual. This is a mode of reasoning close to Gödel’s heart. His mathematical Platonism, which committed him to the existence of a realm of objects that are not accidental like you and me—who exist, but might not have—but necessary, implied immediately that if a mathematical object is so much as possible, it is necessary, hence actual. This is so because what necessarily exists cannot exist at all unless it exists in all possible worlds.
This same mode of reasoning, from the possible to the actual, occurs in the “ontological argument” for the existence of God employed by Saint Anselm, Descartes and Leibniz. According to this argument, one cannot consider God to be an accidental being—one that merely happens to exist—but rather a necessary one that, if it exists at all, exists in every possible world. It follows that if God is so much as possible, He is actual. This means that one cannot be an atheist unless one is a “superatheist,” i.e., someone who denies not just that God exists but that He is possible. Experience teaches us that ordinary, garden-variety atheists are not always willing to go further and embrace superatheism. Following in the footsteps of Leibniz, Gödel, too, constructed an ontological argument for God. Then, concerned that he would be taken for a theist in an atheistic age, he never allowed it to be published.
In arguing from the mere possibility of the Gödel universe, in which time disappears, to the nonexistence of time in the actual world, Gödel was employing a mode of reasoning in which he had more confidence than most of his philosophical colleagues. In the case of the Gödel universe, he reasoned that since this possible world is governed by the same physical laws that obtain in the actual world—differing from our world only in the large-scale distribution of matter and motion—it cannot be that whereas time fails to exist in that possible world, it is present in our own. To deny this, Gödel reasoned, would be to assert that “whether or not an objective lapse of time exists (i.e., whether or not a time in the ordinary sense exists) depends on the particular way in which matter and its motion are arranged in this world.” Even though this would not lead to an outright contradiction, he argued, “nevertheless, a philosophical view leading to such consequences can hardly be considered as satisfactory.” But it is provable that time fails to exist in the Gödel universe. It cannot, therefore, exist in our own. The final step is taken; the curtain comes down: time really does disappear.
Such, in broad outlines, is the structure of Gödel’s argument. Even from this brief sketch, it should be apparent how complex and subtle was the case Gödel made for the ideality of time, a far cry from the amateurish philosophical fumblings with which he is frequently credited. To appreciate the full force of his reasoning, however, it is necessary to look more closely at the details of his argument, to get close to the trees in the forest. His very first step, from little “t” and special relativity to temporal idealism, went unappreciated, in part because, as he remarked about his incompleteness theorem and big “T,” mathematical truth, there was a widespread failure to appreciate the distinction between the formal and the intuitive. There still is. Even today, one can find distinguished proponents of the view that special relativity implies only that the flow of time must be tied to a frame of reference, and that the relativity of simultaneity—combined with the fact that the progress of “now” represents the flux of reality—simply means that reality itself must be relativized to a frame of reference. The question not asked is this: does this conclusion make any sense? Fifty years ago Gödel had the answer: “the concept of existence . . . cannot be relativized without destroying its meaning completely.”
How does Gödel know this? Perhaps relativity has revised what we mean by existence? This Gödel considered nonsense. Science, he maintained in his discussions with Hao Wang, does not analyze concepts, as does philosophy. It applies them. “The notion of existence,” in particular, “is one of the primitive concepts with which we must begin as given. It is the clearest concept we have.” To appreciate the force of Gödel’s reductio ad absurdum, then, it is first necessary to recognize the absurdum. Not everything can be relativized. You can relativize velocity to a frame of reference. You can recognize that what’s on my left is on your right and that what is here for me is there for you, that is, that when I say it is raining here, you agree that it is raining there. But reality as such is absolute. One cannot speak coherently of “my reality” or “your reality,” “reality here” versus “reality there.” When people say things like “my reality is a world in which people care for each other,” they mean— or should mean—that this is their subjective view of the world, how it is or should be. But there is still only one objective reality, which includes the fact that this is your view of the world. If a doctrine implies the opposite, it is that doctrine that has to go. We can have a world in which there is time or a world in which there is existence, but not both. Gödel made the only rational choice: a world without time.
Since there is no single objective worldwide “now” in special relativity, and since there cannot be multiple rivers of time each of which determines the advance of reality, it follows that there simply is no such thing as the universal, worldwide flux of “now” or lapse of time consistent with relativity. As Gödel put it, “each observer has his own set of ‘nows,’ and none of these various systems of layers can claim the prerogative of representing the objective lapse of time.” Special relativity, then, is not simply “incomplete” with respect to intuitive time. Einstein’s theory is inconsistent with the existence of the A-series, with the reality of time in the intuitive sense. There is simply no way around it: if time as it is experienced in ordinary life is to be not ideal but fully real, Einstein must be wrong. And so he is. Or rather, the special theory of relativity is to be replaced by the general theory, which contains a universal theory of motion, including acceleration due to gravity. Hope remains. But this hope too Gödel will quash, beginning with the second step of his argument.
In general relativity, as we have seen, one can define, if not time itself, at least a kind of simulacrum of the real thing, namely, “cosmic time,” determined by those frames of reference whose motion follows the mean motion of matter in the universe. This was a possibility opened up by Einstein’s geometrization of space and time. The only constraints placed on this geometrization are those determined by the laws of general relativity. Any possible universe that obeys these rules must be, by the letter of relativity, physically possible. What Gödel discovered—by the judicious use of ingenious new geometrical methods that themselves constituted an important advance in relativistic mathematics—was that there are solutions to the equations of general relativity that provide world models in which all matter is rotating. Yet absent Newton’s absolute space, with respect to what is the universe supposed to be rotating? “As a substitute for absolute space,” said Gödel, “we have a certain inertial field which determines the motion of bodies upon which no forces act. . . . This inertial field determines the behavior of the axis of a completely free gyroscope.” This is what Gödel used to define universal rotation: “It is with respect to the spatial directions defined in this way (by a free gyroscope . . . ) that matter will have to rotate.” (Gyroscopes, it will be recalled, entered Einstein’s thought elsewhere, when he helped improve their design for use on U-boats during WWI.) In these rotating or “Gödel universes,” Gödel proved, no single objective cosmic time can be defined. The last remnant of something even approximating intuitive time cannot be introduced into these Gödel universes, on pain of contradiction.
If one stake in the heart is good, two are better. Gödel discovered that in a subclass of the rotating universes, those that are not expanding, the large-scale geometry of the world is so warped that there exist space-time curves that bend back on themselves so far that they close; that is, they return to their starting point. A highly accelerated spaceship journey along such a closed path, or world line, could only be described as time travel. And it would be some spaceship. Gödel worked out the length and time for the journey, as well as the exact speed and fuel requirements. The top speed would be a significant fraction of the velocity of light, and the fuel requirements, too, would be enormous. (One theorist has calculated that even with a perfectly efficient rocket engine, the spaceship would require 1012 grams of fuel for every 2 grams of payload.) Paradoxically, however, the very fact that this inconceivably fast spaceship would return its passengers to the past demonstrated, by Gödel’s lights, that time itself—hence speed and motion—is but an illusion. For if we can revisit the past, it still exists. How else could it be revisited? You can’t revisit New Jersey if New Jersey is no longer there, and you can’t return to time t if t has departed from the realm of existence. Thus temporal distance—past and future—turns out to be as ontologically neutral as the measure of space. This is something that even the “friends of Gödel,” who in recent years have stepped forward to defend his account of time travel as logically and physically coherent, have failed to note. For Gödel, if there is time travel, there isn’t time. The goal of the great logician was not to make room in physics for one’s favorite episode of Star Trek, but rather to demonstrate that if one follows the logic of relativity further even than its father was willing to venture, the results will not just illuminate but eliminate the reality of time.
Such, in essence, was the argument put forward by Gödel in “A Remark About the Relationship Between Relativity Theory and Idealistic Philosophy,” a gift for his friend Albert Einstein in the Schilpp volume dedicated to the great physicist on the occasion of his seventieth birthday. Six pages was all Gödel needed to defeat time. Over fifty years later, however, what Gödel really accomplished in this brief compass remains hidden. He had once again constructed a surprising “limit case,” a formal structure whose “geometry” or “syntax” limited the possible interpretations it could be invested with. In the case of big “T,” arithmetic truth, he was able to prove in his incompleteness theorem that the logical system he had constructed could successfully capture the concept of formal proof but could not, on pain of contradiction, represent truth. Before the incompleteness theorem, it was possible to mistake proof for truth. Afterward, with Gödel’s introduction of the “syntactically extreme” conditions of his formal system—the conceptual analogue of an atom smasher—no reasonable person could fail to see the distinction. In his contribution to relativity theory, Gödel, once again, constructed a limit case, this time for the relativistic geometrization of time. That is, he had demonstrated that in the mathematical construction of the Gödel universe, little “t,” the variable that represents the temporal component of four-dimensional space-time, cannot bear the standard interpretation of time in the intuitive sense. Indeed, he proved that it cannot even be interpreted as “cosmic time,” itself at most a simulacrum of the real thing.
Once again, he had been able to make a discovery because he had used his philosophical eye to isolate the essential properties that distinguish the intuitive from the formal concept, in this case, the properties that make intuitive time time, and was thus in a position—as those who took little “t” to be an analysis of intuitive time were not—to prove that these features were excluded by the very geometrical structure of the Gödel universe. Whereas in our world, it was possible—if you didn’t look too closely—to confuse formal, relativistic time with time as ordinarily conceived, this identification became patently unacceptable in the extreme geometrical environment represented by the Gödel universe. What once was hidden was now revealed.
The similarities continue. Just as David Hilbert tried at first to avoid the consequences of the incompleteness theorem by inventing a new rule of logical inference out of whole cloth, so too the relativistic establishment, in the person of Stephen Hawking, tried to get around the embarrassing consequences introduced by the Gödel universe. If the annoying Gödel universe was consistent with the laws of general relativity, why not change the laws? Hawking thus introduced what he called the “chronology protection conjecture” (though a better name would have been the “anti-Gödel amendment”), which proposed a modification of general relativity whose primary goal was to rule out the possibility of world models like Gödel’s, with their awkward chronologies permitting closed temporal loops and causal chains with no beginning. Despite having, as Russell noted in a different context, all the advantages of theft over honest toil, Hawking’s chronology protection conjecture has won few adherents, its ad hoc character betraying itself.
If it is shocking that such a profound insight into the philosophical implications of the theory of relativity has had little impact on physicists, it is dismaying that Gödel’s ideas have failed to catch the attention of philosophers. In this atmosphere of neglect, it is hardly surprising that the striking dissimilarity between Gödel’s two great contributions to the dialectic of the formal and the intuitive has also gone unnoticed. Gödel was at once a mathematical realist and a temporal idealist. He concluded from the incompleteness of Hilbert’s proof-theoretic system for arithmetic that the Platonic realm of numbers cannot be fully captured by the formal structures of logic. For Gödel, the devices of formal proof are too weak to capture all that is true in the world of numbers, not to say in mathematics as a whole. When it came to relativistic cosmology, however, he took the opposite tack. The consequence of his discoveries for Einstein’s realm was not that relativity was too weak to encompass all that is true about time, but rather that relativity is just fine, whereas time in the intuitive sense is an illusion. Relativity, by Gödel’s lights, does not capture the essence of intuitive time, because when it comes to time, our intuitions betray us. “As we present time to ourselves,” he said, “it simply does not agree with fact. To call time subjective is just a euphemism.” This, for Gödel, was the point of intersection between Kant’s idealism and the temporal idealism implicit in Einstein’s physics.
Having failed to notice the asymmetry between the two incompletenesses Gödel discovered, his colleagues in the relativistic and philosophical establishments were of course in no position to comprehend it. It remains one of the most important unanswered questions in our understanding of Gödel’s philosophy. A promising line might proceed as follows. In the case of his incompleteness theorem, Gödel could compare the well-determined set of theorems of formal arithmetic with the equally well founded deliverances of intuitive, i.e., unformalized, arithmetic, accumulated over millennia by the world’s great mathematicians, from which no contradictions have been derived. Even the concept of set, as employed “naïvely” by mathematicians, has not led to paradoxes. “This concept of set,” Gödel pointed out, “according to which a set is anything obtainable from the integers (or some other well-defined objects) by iterated application of the operation of ‘set of,’ and not something obtained by dividing the totality of all existing things into two categories, has never led to any antinomy whatsoever.” Russell’s Paradox, in contrast, arose precisely from attempts like Frege’s to formalize Cantor’s intuitive theory of sets by “dividing the totality of all existing things into two categories,” those that fall under a given concept and those that don’t. “These contradictions,” Gödel reminded us, “did not appear within mathematics but near its outermost boundary toward philosophy.” It is formalisms like Hilbert’s and Russell’s that are problematic; everyday mathematical practice is not founded on a mistake.
Things stand otherwise with time. Whereas special and general relativity are coherent, well-formulated, well-understood physical theories that have enjoyed extensive empirical confirmation, our ordinary, pretheoretical, conceptions of time, i.e., of the A-series, cannot be trusted. The proof of this comes from the fact that our own experience of time in the actual world as something that lapses might well be indistinguishable from how one would perceive “time” in the Gödel universe, in which intuitive time, which lapses, is provably absent. If a form of experience is compatible with both a thesis and its antithesis, it cannot be taken as reliable testimony for either. The fact, then, that the theory of relativity fails to account for the deliverances of our everyday experience of time suggested to Gödel not that Einstein’s theory is incomplete, but rather that our sense of intuitive time is founded on a misunderstanding or misapprehension. In the clash between Einstein and everyday experience, it is experience that has to yield.
Such an answer to the fundamental question of Gödel’s asymmetrical responses to his two incompletenesses has not heretofore been proposed, for the very simple reason that the question itself has never been raised. The failure of his contemporaries—and ours—to appreciate what Gödel has accomplished with his Einsteinian inheritance is a sad tale indeed. Rarely have so many understood so little about so much. Gödel’s “detour” into relativity has been dismissed as a bit of intellectual dabbling by someone outside his field and out of his depth. No one has seen this work for what it was: a continued development of Gödel’s program of probing the limits of formal methods in capturing intuitive concepts, a move from the big “T” of mathematical truth to the little “t” of relativistic time. As a consequence of this failure, no one asked why Gödel’s responses to his incompleteness results in the two cases were diametrically opposed.
The details of Gödel’s conclusions about little “t” were also neglected. Cosmologists questioned whether the possibility of time travel in the nonexpanding Gödel universe was consistent with relativity, but made little note of the primary purpose for which he had constructed these world models, which was to show that since time travel was possible, time was not. And when it became clear that his new world models were indeed relativistically consistent, attention was diverted once more from the essential to the inessential: now cosmologists asked whether the actual world is an expanding Gödel universe. The foundation of Gödel’s case for temporal idealism, his modal argument from the possibility of the Gödel universe to the nonexistence of time in the actual world, disappeared from sight.
Though misunderstood and underappreciated, Gödel’s birthday present for Einstein did attract some immediate attention, if not from philosophers then at least from the guardians of relativity. Of the two great theories of modern physics, general relativity is clearly the philosophical cousin, leading naturally to speculation on the origin, shape and fate of the universe—a highly theoretical, not to say metaphysical, preoccupation of philosophers from time immemorial—whereas quantum mechanics has immediate implications for technology and practice, from lasers and microchips to the whole panoply of information-theoretic hardware. Within the confines of general relativity itself, moreover, the question of time represents an especially elusive philosophical corner. What to do with time in special relativity is easy (if you know what to look for); what to do with it in general relativity is something else entirely. Since Gödel’s discoveries concerned an even more isolated niche of this already remote corner—namely, speculations about geometrically extreme cosmologies with bizarre chronological consequences, not to mention Gödel’s even more arcane philosophical reflections based on these monstrous models—it was to be expected that the small ripple raised by his rarefied achievements would soon fade away.
Into this quiet pond, one spring day decades later, stepped the physicist John Wheeler, a colleague of Gödel’s at Princeton. He was with his friends and fellow physicists Kip Thorne and Charles Misner, with whom he was completing what would become one of the great texts in general relativity, called simply Gravitation. The sunshine beckoned and the three betook themselves across campus to the grassy knolls of the institute, there to meet Wheeler’s friend Kurt Gödel. The warmth of the day notwithstanding, the old logician was found wrapped in his overcoat, the electric heater in his office turned on. Wheeler and friends had a question. Could Gödel shed light on the relationship between his incompleteness theorem and Heisenberg’s uncertainty principle? No. For Gödel, it was bad taste even to pose such a question. Heisenberg’s principle was the finest flower of the Copenhagen interpretation of quantum mechanics, itself the blue-eyed boy of positivism. It represented the high-water mark of indeterminism in physics—in effect, a rejection of Leibniz’s principle of sufficient reason, so beloved of Gödel—and the acme of irrealism in physical science. As such, it was the very thorn on the rose for both Einstein and Gödel. As Gödel put it, “in physics . . . the possibility of knowledge of objectivizable states of affairs is denied, and it is asserted that we must be content to predict the results of observations. This is really the end of all theoretical science in the usual sense.” The incompleteness theorem, in contrast, was a definitive refutation of positivism. Its methods and formal conclusions, though positivistically acceptable, were of a piece with classical mathematics. Moreover, the proof itself, by Gödel’s lights, constituted strong evidence in favor of realism in mathematics. To have suggested a connection or correlation between Heisenberg and Gödel was a major faux pas.
Gödel notwithstanding, however, Wheeler and his friends were not far off the mark. It cannot be denied that there are striking parallels between Gödel’s incompleteness and Heisenberg’s uncertainty (though tact should have counseled against pointing this out to Gödel). For one thing, both thinkers were at pains to use methods that would be epistemologically acceptable to the most hard-headed positivist: formal systems in the case of Gödel’s theorem, direct empirical observations in the case of Heisenberg’s principle. Furthermore, each theorist drew ontological conclusions from epistemological premises, conclusions that established the intrinsic limitations of the epistemologically acceptable methods they had employed. This form of argument is the very hallmark of positivism. It is also characterizes Einstein’s special theory of relativity, a fact with which Heisenberg tried (unsuccessfully) to impress Einstein. That the conclusions Gödel drew pointed to mathematical realism, while Heisenberg made the case for physical irrealism, does not alter the fact that both thinkers blazed an ontological trail through the thickets of epistemology, and that each inaugurated thereby an intellectual revolution whose full implications are yet to be realized. Not for nothing did Gödel’s colleague at the institute, Freeman Dyson, remark that “the two great conceptual revolutions of twentieth century science [are] the overturning of classical physics by Heisenberg and the overturning of the foundations of mathematics by Gödel.”
Now Gödel himself had a question. Would there be a discussion in their new text of the rotating universes he had discovered in relativity? No. Gödel was disappointed. He was still seeking to discover whether the actual world is a (expanding) rotating Gödel universe. The evidence for universal rotation, should it exist, would be found in the axes of rotation of the surrounding galaxies. Wheeler was taken aback by the practical astronomical preoccupations of the great logician. Gödel, he noted, “had taken down the great Hubble photographic atlas of the galaxies, lined up a ruler on each galactic image to estimate the galaxy’s axis of rotation, and compiled statistics of the orientation.” The results, however, were negative.
That Gödel had made discoveries about rotating universes in general relativity had been known to Wheeler for many years. He was present in 1949 when Gödel lectured on the subject at Einstein’s seventieth birthday celebration. Yet he too, despite his impressive credentials, seems to have misunderstood what Gödel was saying. “In a universe with an overall rotation,” he wrote, attempting to summarize Gödel’s lecture, “. . . there could exist world lines (space-time histories) that closed up in loops. In such a universe, one could, in principle, live one’s life over and over again.” Wheeler, unfortunately, has conflated a temporal circle with a cycle, precisely missing the force of Gödel’s conclusion that the possibility of closed, future-directed, timelike curves, i.e., time travel, proves that space-time is a space, not a time in the intuitive sense. Whereas a circle is a figure in space, a cycle is a journey undertaken along a circular path, one that can be repeated, in Wheeler’s words, “over and over again.” Exactly how many times, one wants to ask Wheeler, is the journey supposed to be repeated? The question clearly cannot be answered, since the time traveler’s journey is not over time, along the closed timelike curve: it is the curve itself. Just as one cannot ask of a circle how many times the points that constitute that figure have gone around, one cannot sensibly ask how often the time traveler in the Gödel universe has made his or her trip.
Wheeler should have known better. As he himself pointed out, an “unsettling consequence of Einstein’s 1905 special theory of relativity is that time is relative.” And not just relative, but “static,” for “the other thing that special relativity did for time is join it with space into the four-dimensional entity space-time . . . [and] a consequence of this new space-time view is that motion through time, or motion of time . . . is replaced by static time.” But, as Gödel showed, a time that is relative or static is no time at all. Wheeler seems reluctant to call a spade a spade. Yet he does entitle his chapter “The End of Time,” so perhaps he does, after all, recognize this. Not at all. What Wheeler means by “the end of time” is not that it disappears in Einstein’s theory as a consequence of being relative and static, but rather that, as he sees it, when the “Big Crunch” comes, after the “Big Bang,” time will come to an end. “There was no ‘before’ the Big Bang,” he writes, “and there will be no ‘after’ after the Big Crunch.” Moreover, “every black hole brings an end to time and space . . . as surely as the Big Crunch will bring an end to the universe as a whole.” What Gödel has seen, it seems, Wheeler has not.
A year after he introduced Misner and Thorne to Gödel, Wheeler found himself in the office of a colleague, the cosmologist James Peebles. In walked Peebles’ student Dan Hawley, announcing that he had just completed his dissertation on the question of a preferred rotation among the galaxies. Gödel, Wheeler commented, would be pleased. “Who’s Gödel?” asked Hawley. “The greatest logician since Aristotle,” Wheeler replied. And much more. A phone call to Gödel allowed Wheeler to apprise the greatest logician since Aristotle of the new work being done in Princeton on the rotation of the galaxies. Gödel’s queries, however, were soon too demanding for the physicist, so Wheeler handed him over to the student of cosmology. The questions quickly exhausted him, too, so the phone was passed yet again, this time to Peebles. When the conversation finally concluded, there was just one thing Peebles had to say: “My, I wish we had talked to him before we started this work.”
Though the world at large had not yet taken note of what Gödel had accomplished in Einstein’s backyard, there were rumblings among the cosmologists that something new was brewing. Just what this was, however, would remain hidden for years to come. That a noted cosmologist was moved as recently as the 1990s to protect chronology from the Gödel universe suggests that the world is still not ready for Gödel. Yet the mere fact that as distinguished a theorist as Stephen Hawking believed protection was needed, combined with the fact that his chronology protection conjecture has so far failed to attract a significant number of adherents, suggests that readiness may be near. The zeitgeist, as Gödel noted, has its own time and agenda.