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Mathematics and Skepticism by Alex Kasman

This is a copy of an article of 23 Aug 2004 by Alex Kasman, originally at the website lowcountryhumanists.net.


I would be very curious to see how the percentage of mathematicians who are skeptics, atheists and/or agnostics compares to the percentage in the general population. Though no official study of this nature has ever been conducted, to the best of my knowledge, my own experience leads me to suspect it would be a good deal higher. That is, although there are certainly many religious mathematicians, as well as mathematicians who unskeptically believe in anything from alternative medicine to flying saucers, I believe that there would be a much higher proportion of skeptics among mathematicians than in just about any other group. [Note added later: I have been shown statistics indicating that although the percentage of atheist mathematicians is higher than the percentage among the average population, it is somewhat smaller than the percentage among scientists in other fields.]


Advanced mathematics appeals to skeptics in a fundamental way: more than any other human endeavor, mathematics seems to have a real handle on proof and truth. The job of a mathematician involves discovering facts (about mathematics) that were previously unknown, and proving that they are true. If you cannot completely prove your claims in mathematics, the new results will not be accepted by the mathematical community, they will not be published in a journal, and - to be blunt - you won’t be a mathematician for long. A valid proof of a mathematical theorem is most certainly a more rigorous and certain thing than what passes for proof in the other sciences. One of the most elementary examples of a fact which has been proved in this way is the claim that you cannot find two integers x and y so that the fraction xy is the square-root of two. (This fact was proven quite long ago.) A famous more recent example is Andrew Wiles’ proof that there are no integer solutions to the equation xn + yn = zn for any integer n > 2. Because of these proofs, mathematicians know that the claims are true - we know that it would be a waste of time to search for numbers satisfying these equations since they have been proven to not exist. (If you want to see examples of what mathematicians have proved THIS month, check out the e-print archive of current mathematical papers.) In contrast, something which seems true (such as the apparent fact that any even integer can be written as a sum of two prime numbers) is not given the status of a fact at all unless it has been proven (which the “fact” in the previous parenthetical remark has not… although it has been checked and has been found to be true for any even number that anyone has ever tried).


So, you can see that people who are skeptical, who find themselves doubting things unless they are familiar with good evidence, would feel at home in the world of professional mathematics. Perhaps then it is no coincidence that more than a handful of the members of the SHL are mathematicians, nor that the famous mathematician Descartes is remembered for his skeptical observation that the only thing one can really be certain of is one’s own existence (“I think therefore I am.”)


I had special cause to think about all of this last week when a colleague, a fellow mathematician but not anyone I would have ever described as a skeptic due to his strong religious convictions and unfailing faith in a just and loving god, came to my office to discuss a question of faith and mathematics. It seems that he, like myself, felt originally drawn to mathematics because of its certainty, its firm grasp on truth. However, my friend’s confidence in mathematics was shattered by his discovery of some old results by Kurt Gödel.


You see, at the turn of the 20th century, the mathematical community was attempting to show that mathematics really was our best glimpse of ultimate truth. They were trying to show, using the same mathematical methods of proof which so effectively show that the square root of two is not a rational number, that mathematics is a tool which really allows one to figure out if a given statement is true or if it is false. Unfortunately, these dreams were shattered by Kurt Gödel’s famous proof (using a new formal algebra that he invented for this very purpose) that there was no hope of doing this. I will refer the interested reader to other sources for the details of this interesting period in mathematical history (see, for instance, “Godel’s Proof” by J.R. Newman). However, let me briefly paraphrase it here by saying that the best we can hope for is that mathematics is consistent (which is to say that there are no facts which are both true and false at the same time), but that there is no way it will allow us to necessarily figure out if a given statment is true or false. Moreover, we cannot even be certain that it is consistent -- at least not without resorting to things outside of mathematics.


So, having just heard of this recently, my colleague was a bit shaken. He had always believed that mathematics, the subject to which we both have devoted our professional lives, was beyond skepticism. He had always believed that the one thing we know for certain is that mathematics is “really true”. Now, Gödel (posthumously) has taken this away from him. I think he came to me hoping that I would either be able to explain how we still have some guarantee of objective truth, despite Gödel… but I disappointed him.


In fact, as I told him, I would think that even if we did have a mathematical proof of the consistency of mathematics - contrary to Gödel’s theorem, I would still have to be skeptical of it! There may be good reasons to believe in mathematics (the proofs sure do look like valid proofs to us, and it has produced very useful results such as aerodynamic airplane wings, the theory of relativity, the ability to predict eclipses and the collision of a comet with a planet) but there is no way we can ever be completely certain that it is true. Suppose, for instance, that we all have a flaw in our brain that makes us think that 2+2 is 4 when it is really 3.99. Just because we all agree it is true and it seems to make sense doesn’t really mean that it must be correct… you need to have faith that our brains are capable of really differentiating between truth and falsehood in this mathematical setting. In other words, though I consider myself to be a skeptic and an atheist, I fully admit that when I’m working on my mathematics, I am acting out of faith.


This is, I believe, a good model of what we should mean by “skeptic”. As Descartes argued so long ago, if you really are willing to consider the question: “What can I be absolutely sure of?” the answer will be “All I know for sure is that I, myself, exist. Everything else could be my own delusion… but there must be a ‘me’ there to have that delusion!” (Actually, Descartes also concluded that God must exist, which I obviously don’t agree with, but you get the point…) However, I don’t want to reserve the word skeptic for people who don’t bother going to work because they can’t be completely certain that they’ll starve to death if they have no food. Those of us who live normal lives have to take some things for granted (such as the existence of the other people around us) and that involves some degree of faith. So, the moral I take away from this story is this: skepticism does not mean freedom from faith, but rather that you should consciously question your faiths, and make an effort to support (or even disprove) them with the best available evidence, even if not by an unquestionable mathematical proof.

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