Proofs and Guarantees | Math - Scientific American

Photo: James Robert Brown

We can prove things in math, but does that mean they’re true? argues James Robert Brown, The Mathematical Intelligencer.

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Let us assume what most mathematical readers would take for granted anyway: There are mathematical objects such as numbers and functions and there are objective facts about these objects, such as 3 < 5 and the set of primes is infinite. Truth on this view is banal. ‘‘3 < 5’’ is true because the objects 3 and 5 are in the less than relation to one another, just as ‘‘Bob is shorter than Alice’’ is true, because Bob and Alice stand in the shorter than relation.

Why bother assuming this? There are plausible alternatives. We say: ‘‘Bishops move diagonally’’ is true and we say: ‘‘Sherlock Holmes lives at 221B Baker Street’’ is true. What makes them true, however, is a conventionally adopted rule in one case and a literary fiction in the other. Truth in mathematics, on the account I’m taking for granted, is no different than truth as normally understood in, say, physics. A proposition is true when it correctly tells us how things objectively are. I hope most readers are still with me, in spite of the mundane mathematical metaphysics so far. The interesting point comes next.

Why do we believe that 3 < 5 and that there are infinitely many primes? Most would say that’s an easy question with an obvious answer: proof. Here is a tougher question: Is proof the only sort of legitimate evidence in mathematics? Many will say—indeed, they will shout—yes, proof and proof alone is the source of mathematical evidence. Proofs are both necessary and sufficient. We know a theorem is true, they might add, when we have a proof, or we don’t know it’s true when we lack a proof—it’s all or nothing...

Timothy Gowers has written interestingly and extensively on the philosophy of mathematics in various places. His views on evidence have been summed up in a slogan suitable for a bumper sticker: Proof = explanation + guarantee.

Gowers himself and those who have discussed his work have focussed on ‘‘explanation,’’ which is a hugely interesting and important notion in mathematics and philosophy. A proof provides evidence that a theorem is true, but some proofs also produce insight into what is going on. Gowers is trying to understand this phenomenon when he discusses explanation. I, however, will take a different tack: I will focus on ‘‘guarantee,’’ which Gowers and others take to be the evidence that shows the theorem is certainly true. A proper proof that there are infinitely many primes is a guarantee that this is true. As proof is normally envisaged, we couldn’t ask for better than this sort of guarantee. This is the gold standard. The natural sciences don’t have a hope of matching it.
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Reprinted with permission from the Mathematical Intelligencer.
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Source: Scientific American