Gödel and the unreality of time

In 1949, in a festschrift devoted to Einstein, Kurt Gödel published a very short but profound paper titled “A Remark About the Relationship Between Relativity Theory and Idealistic Philosophy.” It has since become well-known as a defense of the possibility in principle of time travel in a relativistic universe. But in fact that is not exactly what Gödel was trying to show. He was trying to show instead that time is illusory. He was using Einstein to revive the timeless conception of reality defended historically by thinkers like Parmenides and McTaggart.

Gödel had discovered solutions for the field equations of the general theory of relativity (GTR) that allow for the possibility of closed causal chains in a rotating universe, where the “backward” part of such a chain can be interpreted as an object’s revisiting its earlier self. As Einstein acknowledged in his response to Gödel in the festschrift, in such a causal chain – in which an apparently earlier event E leads to an apparently later event L, but where L in turn leads back to E– you may with equal justice regard L as the earlier event and E the later. The relations “earlier than” and “later than” cease to be objective features of the situation. Now, as even the B-theory of time acknowledges, the objective reality of the relations “earlier than” and “later than” is essential to the reality of time. Hence Gödel concluded that in a universe of the sort he describes, time is illusory.

Now, whether our universe is of the rotating kind that would allow for such causal chains depends on the distribution of matter within it, which is an empirical consideration that cannot be settled from the armchair. But Gödel thought this irrelevant. As Palle Yourgrau has emphasized, Gödel intended his scenario as a limit caseof GTR’s spatialization of time, which shows what follows if that spatialization is pushed through consistently. He also thought that the existence of something as purportedly metaphysically fundamental as time could not plausibly depend on a contingent matter such as the precise distribution of matter in the universe. Hence his judgement that the possible scenario allowed by GTR that he uncovered casts doubt on the reality of time in our world.

Yourgrau has long rightly complained that the common tendency to present Gödel as a defender of the possibility of time travel distorts his actual intentions. He writes:

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.

End quote. Now, among the assumptions you have to make in order to accept Gödel’s argument is that GTR provides an exhaustivedescription of the nature of time and space. (This is an assumption that you would not make if you gave GTR either an instrumentalist or an epistemic structural realist interpretation.) That is to say, you’d have to assume that if GTR doesn’t capture some purported aspect of time and space, then that aspect just isn’t really there. (You have to assume more than this too, since Gödel’s argument can also be challenged at other places. But I’m not getting into that here.)

Now, Yourgrau notes that Gödel’s argument is in one respect interestingly parallel to, but in another respect interestingly departs from, his famous Incompleteness results in mathematics. The parallel is this. The Incompleteness Theorem shows that arithmetical truth cannot be captured within a formal system (because there will be propositions that are true but not provable within the system). The argument about relativity, meanwhile, shows (so Gödel thought) that time, in the strict sense, is not definable within GTR. The departure is this. In the case of arithmetic, Gödel’s conclusion was not that there is no such thing as arithmetical truth, but rather that since there is such a thing, formal systems of the sort in question are incomplete. But in the case of relativity, Gödel’s conclusion was not that GTR is incomplete if it fails to capture time, but rather that time must be unreal.

As Yourgrau asks, why this asymmetry in Gödel’s conclusions? Why wouldn’t he conclude instead that GTR is simply incomplete if it fails to capture time?

Yourgrau’s answer is to suggest that there are philosophical problems with our commonsense understanding of time, and with the A-theory of time that is its philosophical expression, that might be taken independently to cast doubt on its reality, whereas there are no similarly formidable objections to the notion of arithmetical truth. Perhaps that was part of Gödel’s motivation, though in my own view the purported difficulties with the A-theory are vastly overstated.

But I would conjecture that the deeper explanation lies in Gödel’s Platonism. For the Platonist, the highest degree of reality is to be found in the realm of abstract objects conceived of as denizens of an eternal “third realm” over and above the spatiotemporal world of concrete particulars on the one hand and the mind on the other. And mathematical objects are the gold standard instances of such abstract objects. The empirical world of time and space has, on this view, only a second-rate kind of reality, and the temptation is strong to dismiss it as altogether illusory.

If you buy this general picture, then the asymmetry in Gödel’s thinking noted by Yourgrau is quite natural. If a formal system doesn’t capture arithmetical truth, then since such mathematical truth is the gold standard of Platonic reality, the problem must be with the formal system. But if a mathematicized picture of nature such as GTR doesn’t capture time (as Gödel thought it did not), then since mathematics is the gold standard of reality and time is a second-rate kind of reality at best, then the problem must be with time.

Much more on time, the A-theory versus the B-theory, time travel, and related matters in my forthcoming philosophy of nature book. Stay tuned.

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