The Mystery of Time: A Gödelian Perspective
If we view Time as the consistent order of events, a surprising link might connect Time to Gödel’s Incompleteness Theorems. The result? The future is intrinsically open.
John Wheeler (1990a, p. 10) famously quipped: “Time is nature’s way to keep everything from happening all at once.” But what would happen if everything happened at once? Many contradictions will arise. Imagine a father and son both dying at the same time as they are born. Time, therefore, is what prevents these contradictions from taking place.
“Time is nature’s way to keep everything from happening all at once.” — John Wheeler
This reminds us of “time paradoxes,” including the famous grandfather paradox when the son travels back in time and kills his grandfather. If he does, he will not be born, and so he cannot kill his grandfather … a contradiction!
Do the laws of physics allow for such paradoxes? Stephen Hawking (1992) proposes what he calls the “chronology protection conjecture.” It basically says that the laws of physics conspire to prevent anything from traveling backward in time, thereby keeping the universe “safe for historians.” Nobel laureate Kip Thorne (2009) tends to agree. He notes that “just as the laws of physics must be logically consistent with each other, so also the evolution of the Universe, as governed by the laws of physics, must be fully consistent with itself” (Thorne 1994, p. 510). Hence, time seems to be the key to preserving the consistency of the universe.
Time as a Consistent Order
This brings up the issue of the nature of time: Is it a substance? Or is it a relationship? Gottfried W. Leibniz is of the view that time and space are orders of things, not things, and Albert Einstein agrees (Wheeler 1990b, p. 5).
How about space? Hans Reichenbach (1958, p. 168) argues that time is logically prior to space. If time precedes space, and space and time are the orders of all events of the universe that keep it consistent, then time seems to be the ultimate source of consistent order in the universe.
If so, this would allow us to link Gödel’s Incompleteness Theorems to time. Gödel’s Second Theorem, in particular, dictates that, roughly, no consistent formal system rich in arithmetics can prove its own consistency. To prove the consistency of a (consistent) system S, we need a higher system, Sʹ, that includes S together with the necessary axioms for proving the consistency of S. But how do we verify that the new system, Sʹ, is consistent? We need to build an even higher system, Sʹʹ, the consistency of which can be proved through an even higher system … ad infinitum. Thus, rich axiomatic systems are not completable; mathematical truth is inexhaustible.
It is important to emphasize that the Second Theorem does not deny the possibility of proving the consistency of a formal system; it only states that if the system is consistent, then it is not possible to prove its consistency within the system (Smullyan 1992, p. 109). While local consistency of parts of the system can be proved within the system, to prove the universal consistency of the system as a whole, we need to go “outside” the system. The distinction between the views from within and without is a crucial insight of the Incompleteness Theorems (Dawson 1997, p. 263; Smith, 2013, pp. 236–237).
Global time is invisible within the universe
Accordingly, if we model the physical universe as a (gigantic) formal axiomatic system, then we can conclude that global time, just like global consistency, cannot be formally characterized within the universe. Global time is invisible within the universe. We can formalize global time only if we imagine ourselves outside the universe.
Endogeneity of Time
The distinction between local and global time can be further elaborated if we carefully consider how clocks work. A clock requires two essential properties:
1. It must be based on regular motion. The word “regular” might involve circularity if it implies activities carried out in equal periods. For the sake of argument, let us assume that this condition can be somehow satisfied.
2. The clock must be independent of, or exogenous to, the observer using the clock. For example, one may use his heartbeats as a clock, but his heartbeats are not exogenous to the person: If he starts running, his heart will beat faster, and so time for him will flow faster. The opposite is also true. To avoid this circularity, the clock must be exogenous to the observer.
So far, so good. But can we have a clock that is exogenous to all observers within the universe? To assume that we can have a clock exogenous to all observers is to commit the fallacy of composition. We can have a clock that is exogenous to some observers, but it is impossible to have one that is exogenous to all observers. The reason is simple: A clock operates within the universe. Accordingly, there is no clock within the universe to measure global or absolute time. Interestingly, this is precisely the conclusion of the Special Theory of Relativity (Taylor and Wheeler, 1992).
The distinction between local and global time is analogous to the distinction between local and global consistency. Just as global consistency cannot be proven within a system, global time cannot be quantified by any clock within the universe.
Time, Uncertainty, and the Second Theorem
Time is delicately implied by the Second Theorem. To see this, note that the Second Theorem says that, within a sufficiently rich axiomatic system, we cannot prove that we will never prove a contradictory statement (see Boolos, 1994). This is implicitly a statement about the future.
So the Second Theorem, in essence, says that if the system is consistent, we cannot prove from within that we will never reach a contradiction. This implies uncertainty about the future development of the system, which is analogous to our uncertainty about the global future of the universe. While we can make statements about the local future, uncertainty about the global future is inevitable.
Irreversibility
There is another important aspect that time and logical consistency share. We know that time marches in one direction. Critical natural processes are mostly irreversible. How about consistency?
Irreversibility in formal systems is shown by Gödel’s First Incompleteness Theorem. The Theorem argues that, in a sufficiently rich formal system S, a sentence can be constructed that roughly says: “I am not provable within S.” This sentence, known as Gödel’s sentence, cannot be proved within the system. This introduces irreversibility in the system. How?
In formal systems, we may derive theorems from axioms and vice versa. The process is reversible. But the emergence of Gödel’s sentence in rich systems breaks this reversibility: The sentence cannot be proved within the system. While the sentence “I am not provable” might appear ambiguous, it is actually general to include many system-level features. One clear application of Gödel’s sentence is the sentence: “System S is consistent.” This sentence is not provable within S since, as indicated above, consistency is not provable within the system. From this perspective, the consistency of a rich system is not only invisible within the system but is irreversible.
Conclusion: The Mystery of Time and The Mystery of Consistency
Relativity Theory argues that there is no global clock: time is relative. Yet, we have this deep feeling of the “pulse of the universe.” Time has the strange feature that “we can sense it but cannot prove it.” St. Augustine remarked: “What, then, is time? If no one asks me, I know. If I wish to explain it to one that asks, I know not.” This is another way to say that time is not something that can be concretely formulated, despite the fact that we deeply feel it. But this is the same position regarding consistency: We do believe that mathematics is consistent, but we cannot prove it mathematically.
The invisibility of global time and of global consistency conspire to render the future effectively open.
The mystery of time seems to be of the same nature as that of consistency:
- Both are relational
- Both are global properties that cannot be pinned to any portion or region within the system
- Both are uncertain from within, and
- Both involve irreversibility.
But the mysteries of time and consistency might have a direct implication on how we view the future: From our perspective, the long-term future is logically undecidable. The invisibility of global time and global consistency conspire to render the future effectively open.
References
- Boolos, George. 1994. “Gödel’s Second Incompleteness Theorem Explained in Words of One Syllable.” Mind, New Series, 103, no. 409: 1–3.
- Dawson, John. 1997. Logical Dilemmas: The Life and Work of Kurt Gödel. A K Peters.
- Hawking, Stephen. 1992. “Chronology Protection Conjecture.” Physical Review D 46: 603.
- Reichenbach, Hans. 1958. The Philosophy of Space & Time. Dover Publications.
- Smith, Peter. 2013. An Introduction to Gödel’s Theorems. 2nd edition. Cambridge University Press.
- Smullyan, Raymond. 1992. Gödel’s Incompleteness Theorems. Oxford University Press.
- Taylor, Edwin, and John Wheeler (1992). Spacetime Physics. 2nd Edition. W. H. Freeman.
- Thorne, Kip. 1994. Black Holes and Time Wraps. W.W. Norton.
- Thorne, Kip. 2009. “Is time travel allowed?” Plus, plus.maths.org. Accessed April 30, 2023.
- Wheeler, John A. 1990a. “Information, Physics, Quantum: The Search for Links.” In Complexity, Entropy, and the Physics of Information, edited by W. Zurek. CRC Press.
- Wheeler, John A. 1990b. A Journey into Gravity and Spacetime. Scientific American Library.
