Question I'm a mathematician looking at some of the work of Leonhard Euler on the "pentagonal number theorem". My question is about how we can know some statement is true. Euler had found this theorem in the early 1740s, and said things like "I believed I have concluded it by a legitimate induction, but at the same time I haven't been able to find a demonstration" (my translation), and that it is "true even without being demonstrated" (vraies sans etre demontrees). This got me thinking that "knowing" something is not really a mathematical question. A proof lets us know a statement is true because we can work through the proof. But a mathematical statement is true whether we know it or not, and if you tell me you know that a statement is true, and then in fact someone later proves it, I can't show mathematically that you didn't know it all along. This isn't something I have thought about much before, and my question is are there any papers or books that give some ideas about this that would be approachable by someone who has not studied philosophy at the university level.

Accepted:
August 3, 2008

Comments

Perhaps there are two different questions here. There's a very general question about truth and proof; and there's a much more specific question about the sort of case exemplified by Euler, where a mathematician claims to know (or at least have good grounds for) a proposition even in the absence of a demonstrative proof.

Let's take the specific question first, using a different and perhaps more familiar example. We don't know how to prove Goldbach's conjecture that every even number greater than two is the sum of two primes. Yet most mathematicians are pretty confident in its truth. Why?

Well, it has been computer-verified for numbers up to the order of 1016. But so what? After all, there are other well-known cases where a property holds of numbers up to some much greater bound but then fails. [For example, the logarithmic integral function li(n) over-estimates the number of primes below n but eventually under-estimates, then over-estimates again, flipping back and forth, with the first tipping point now thought to be in the order of 10316. See here.] We know, then, that extrapolation from (merely!) the first 1016 cases is dangerous -- for that "sample" is biased towards relatively tiny numbers. Yet, as I said, mathematicians do all the same tend to be confident in Goldbach's conjecture. Which suggests that they have other non-demonstrative grounds for their belief, grounds better than mere extrapolation from the initial cases. For a discussion of what these might be, and some evaluative comments, see e.g. Alan Baker's paper "Is there a problem of induction for mathematics?" in M. Leng, A. Paseau & M. Potter (eds.) Mathematical Knowledge. That's a pretty accessible read, and one of the few recent papers I know about the interesting question of the role of non-demonstrative reasoning in mathematical thought.

Now, suppose NM (for "naive mathematician"!) does believe Goldbach's conjecture just on the basis of extrapolation from small cases. Then, if a proof were discovered, could NM claim to have known the conjecture to be true already? Surely not so. NM's mathematical reason for his belief was (demonstrably!) a bad reason; and a true belief held for a bad reason isn't a case of genuine knowledge (it's more like a lucky guess). So actually, I think it is wrong in general to say "if you tell me you know that a statement istrue, and then in fact someone later proves it, I can't showmathematically that you didn't know it all along." NM didn't know Goldbach's conjecture all along, and that's because his grounds don't pass muster as good mathematical reasons.

Now a nod towards the much more general question here. Certainly, it is very plausible to say that when we prove a mathematical proposition we aren't (so to speak) creating a new truth, but discovering something that was true all along -- "a mathematical statement is true whether we know it or not". Even when mathematicians invent a new branch of mathematics -- as Eilenberg and Mac Lane did in founding category theory -- it remains tempting to say that they are discovering pre-existent patterns and structures. But is this right? For what is the nature of this supposed realm of pre-existing mathematical structures, and how do we get epistemic access to it? Well, they are two of the Big Questions in the philosophy of mathematics. And here, I can probably do no better than just point to Stewart Shapiro's fine introductory book, Thinking About Mathematics.