Gödel, Pascal, and the Rationality of Unprovable Beliefs

Some people seem to go through life under the confident delusion that all of their most important beliefs can be absolutely proved. Others have not only given up on the idea that their beliefs can be proved, but are therefore troubled that perhaps no belief is really anything but arbitrary and irrational in the end anyway, so that we may as well believe anything. Is either of these two assumptions reasonable?

We begin our analysis with the consideration of an intriguing logical proposition, namely “this statement cannot be proved.” At first this might sound like a paradox, similar to “this statement is false.” The reason “this statement is false” is a logical paradox is that it can neither be true nor false, so it is not really a logical proposition at all. But “this statement cannot be proved” is not a paradox. It is a real proposition which must be either true or false, according to the rules of logic. So which is it? It cannot be false, for to say that it is false is to say that the proposition actually can be proved, which is to say that it is not false after all. Therefore, we must instead believe it to be true even though in believing it to be true we also believe it cannot be proved. It almost sounds like we have proved the unprovable!

Some readers may recognize this proposition as essentially the Gödel sentence in the mathematician Kurt Gödel’s first incompleteness theorem. This theorem states that no formal system sufficiently complex to express elementary arithmetic can be both consistent and complete.[1] In other words, if we assume the system is consistent, meaning free of internal contradictions, then it must be incomplete, meaning there must be some proposition that is true within the system but cannot be proved within the system. The Gödel sentence is just such a proposition. The less formally defined proposition that we have been discussing is essentially the same thing: We can see that it must be true within the framework of our basic logical beliefs, yet we cannot prove that it is true within that framework of our basic logical beliefs, for that would involve a contradiction.

How then are we able to “see” that the proposition is true if we cannot prove it? The answer lies in our assumption that our basic logical beliefs are consistent. If they were not consistent, then we would have no way of “seeing“ that the proposition could not be proved, as it states. In other words, if our basic logical beliefs could potentially lead to internal contradictions, then they could also potentially lead to the proposition actually being proved, even though that would be just such an internal contradiction. But when we assume they cannot lead to such an internal contradiction, we have to admit that they also cannot prove our proposition even though our assumption of consistency says the proposition is true. So what Gödel’s incompleteness theorem shows, in effect, is that it is pointless to look for proof that our basic logical beliefs are consistent.[2] Assuming these basic beliefs are sufficiently complex to express elementary arithmetic (which I think we all acknowledge since I have not yet encountered anyone who truly doubted that 2+2=4), it can’t be done. We must accept our basic logical beliefs as true, and therefore consistent, on faith; we have no proof that they are even consistent, let alone true.

Does this mean our basic logical beliefs are arbitrary and irrational? Not at all. In fact, I think it is safe to say that every rational person finds the basic beliefs of logic and mathematics to be very reasonable indeed. This is exactly what enables us also to see that the proposition we’ve been discussing is presumably true even though it cannot be proved. We have good reasons to believe both that this proposition is true and that our basic logical beliefs are true, even though we can’t prove them to be true. For one thing, these basic beliefs just “feel right” to us. For another, they seem to be eminently practical, serving us quite well in the kinds of trivial and not-so-trivial puzzles we routinely solve every day as we navigate life in the real world. But for those who do not find these to be terribly convincing arguments, I suggest there is also a deeper, more purely rational reason than just these, and this reason is what I will call the truth wager.

The truth wager is modeled after a famous argument for Christianity known as Pascal’s wager. Blaise Pascal was a 17th century mathematician who pioneered probability theory. He pictured the question of whether or not to believe the basic claims of Christianity as an all-important gamble. If a person is still undecided on this question after considering all the relevant evidence, then there is one more thing he should consider: If Christianity is true, then the “payoff” for believing it (presumably eternal bliss) is infinite and the “price” for not believing it (presumably eternal punishment) is also infinite. But if Christianity is false, then the “price” for believing it is minimal and the “payoff” for not believing it is also minimal, at least by comparison. Therefore, there are two possible outcomes if a person chooses to believe in Christianity: Either he gets an infinite payoff or he pays a minimal price. And there are two possible outcomes if he chooses not to believe in Christianity: Either he gets a minimal payoff or he pays an infinite price. The rational gambler will therefore choose to believe in Christianity even if he can’t estimate the odds for it being true, or for that matter even if he considers Christianity more likely to be false than true. The potential payoff compared to price still makes the wager favorable whatever the odds.

Pascal’s wager is rather obviously a poor argument for believing in Christianity, at least when made to stand on its own. It can only be persuasive if one first assumes that the only two possibilities are Christianity (conceived of as entailing an infinite reward for those who believe and an infinite punishment for those who don’t) and some form of materialism (conceived of as entailing no life after death and thus no potential reward or punishment beyond this finite life). Pascal apparently intended his wager argument for those who were already convinced that these were the only two possibilities worth considering. But for those who are open to other possibilities as well, his wager is not convincing at all, since it gives no reason for choosing to believe Christianity instead of some other religion, for example. It is also questionable, to say the least, whether we can simply “choose” to believe something on the basis of such an argument, especially if we are really convinced that the odds are against it even though the payoff is for it. Our beliefs seem to be what we actually think is most likely true, not what we would like to bet on.

The truth wager is much more general in nature than Pascal’s wager, however, and hence not subject to the same criticisms. The idea is this: We cannot prove that our basic logical beliefs are really true and consistent. But there seems to be only two possibilities. Either these beliefs are in fact true, and therefore consistent, in which case logical propositions have genuine meaning, or else they are not true, in which case logical propositions really don’t have any property we could consistently consider meaningful at all. The same applies to our basic mathematical beliefs. For example, if it was not true that one was different from zero, how could it be meaningful to speak of even one proposition? Our one proposition might as well be no proposition at all. So the only rational option is to bet that the basic axioms of logic and mathematics are indeed true. If they are not true, then it is not even meaningful to disbelieve them anyway, so we may as well assume they are true. Indeed, it is extremely doubtful that we are even capable of betting on the alternative, which would be meaninglessly believing they are false, which is meaningless. The wager for their truth appears to be so obviously reasonable that our minds simply can’t refuse. So we do take the truth of logic and mathematics on faith rather than absolute proof, but it is a very reasonable faith nonetheless.

This gives us a valuable insight into how our most basic beliefs work, and how they can be rational even though not provable. It is clear enough that we cannot have absolute logical proof for what we believe, since that would lead to an infinite regression of beliefs. Sooner or later our beliefs must rest on unprovable assumptions, such as the basic axioms of logic and mathematics. But the basic assumptions our beliefs rest on are not arbitrary, or at least they do not have to be arbitrary. They can instead be very reasonable wagers, wagers that we can’t sensibly bet against. We therefore don’t have to be bothered by the fact that our beliefs are ultimately unprovable. Even though they are unprovable we can still strive to ensure that they are rational, in the sense of being ultimately based on what cannot be meaningfully wagered against. Most of our beliefs will admittedly not follow with deductive precision from such basic assumptions, but if they can at least be probabilistically inferred from such assumptions, we are still on quite rational grounds. Unprovable beliefs do not have to be arbitrary or irrational.


[1] The bulk of the first incompleteness theorem is occupied with the task of showing that any formal system complex enough to express elementary arithmetic is also complex enough to express a self-referential Gödel sentence.
[2] It may appear that the assumption of consistency could just be included as one of the axioms in a formal system. This doesn’t work, as Gödel’s second incompleteness theorem shows that such an addition actually makes the system inconsistent. The net result is that the only way a system can possibly be consistent is if it can’t prove its own consistency.

This page copyright © 2012-2014 Edward A. Morris.  Created July 10, 2012.  Last updated July 21, 2014.

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