What Godel Means by Time
At issue is the leitmotif of Godel’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 grace 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 “Godel program” of assessing the extent to which intuitive ideas can be captured by formal concepts. This is what Godel 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, Godel 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 four-dimensional 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 Godel 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 Godel, 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 Godel 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 Godel 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.
Godel’s Dialectical Dance with Time
As he had previously done in his incompleteness theorem, Godel 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. Godel’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 Godel’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 Godel, is inescapable: if relativity theory is valid, intuitive time disappears.
Step two takes place when Godel 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 Godel’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 Godel 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 Godel discovers a relativistically possible universe {actually, a set of them)-now known as the Godel 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 Godel 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, Godel 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 Godel universe signals the unreality of time. Once again, time disappears.
But the dance is not over. For the Godel 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 Godel 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 Godel’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, Godel, 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 Godel universe, in which time disappears, to the nonexistence of time in the actual world, Godel was employing a mode of reasoning in which he had more confidence than most of his philosophical colleagues. In the case of the Godel 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, Godel 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 Godel universe. It cannot, therefore, exist in our own. The final step is taken; the curtain comes down: time really does disappear.
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