Mathematics

In writing mathematical proofs, I've been struck that direct proofs often seem to offer a kind of explanation for the theorem in question; an answer the question, "Why is this true?", as it were. By contrast, proofs by contradiction or indirect proofs often seem to lack this explanatory element, even if they they work just as well to prove the theorem. The thing is, I'm not sure it really makes sense to talk of mathematical "explanations." In science, explanations usually seem to involve finding some kind of mechanism behind a particular phenomenon or observation. But it isn't clear that anything similar happens in math. To take the opposing view, it seems plausible to suppose that all we can really talk about in math is logical entailment. And so, if both a direct and an indirect proof entail the theorem in question, it's a mistake to think that the former is giving us something that the latter is not. Do the panelists have any insight into this?

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You've asked a terrific question! I wish I were more qualified to venture an answer to it. As you suggest, a sound direct proof of a theorem shows that the theorem must be true, in the broadest possible sense of "must." But a sound indirect proof shows the same thing. The difference, if any, seems purely psychological: some people find one proof psychologically more satisfying than the other. My sense is that some philosophers of math take this psychological difference very seriously and propose far-reaching revisions to classical math on the basis of it. You might take a look at the SEP entry on intutionism in the philosophy of math, particularly the discussion of constructive and nonconstructive proofs. The entry includes other helpful links and references too.

Anyone with any mathematical training will be familiar with the fact that proofs in mathematics do much more than just show that the statement proved is true. One way this manifests itself is that we often value different proofs of the same theorem. Thus, as Jamie Tappenden once pointed out, Herstein's Topics in Algebra, which was the standard algebra text when I was a student, contains three different proofs of the Stone Representation Theorem. Boolos, Burgess, and Jeffrey's Computability and Logic, one standard text for an intermediate logic course, similarly contains multiple proofs of several of the key results, including Church's Theorem on the undecidability of first-order logic and Goedel's First Incompleteness Theorem. And, oddly enough, I myself have just re-proven an existing result in a way that, I think, is clearly better. But not because the original proof wasn't convincing!

It's an interesting question, though, why we value different proofs. Somehow, they seem to throw different light on the result proved, but how? Frege mentions this sort of fact in Foundations of Arithmetic, saying:

The aim of proof is...not merely to place the truth of a proposition beyond all doubt, but also to afford us insight into the dependence of truths upon one another. After we have convinced ourselves that a boulder is immovable, by trying unsuccessfully to move it, there remains the further question, what is it that supports it so securely? (§2)

This idea that a proof reveals relationships between truths might be helpful. One thing people often say about good explanations in science is that they are "unifying", and maybe a good proof is "unifying" in a similar way.

If you want to know more, then check out the SEP article on Explanation in Mathematics. It was written by someone who has spent a lot of time studying this issue, both from an historical and from a contemporary point of view.

And, by the way: There are similarly interesting questions to be asked about such things as "elegance", "beauty", and "depth".

I probably should have noted before that, in the case of the different proofs of the first incompleteness theorem in Boolos, Burgess, and Jeffrey, the first proof they give is indirect or, as it is sometimes put, non-constructive: The proof shows us that, in any given consistent theory of sufficient strength, there is an "undecidable" sentence, one that is neither provable nor refutable by that theory; but the proof does not actually provide us with an example of an undecidable sentence.

The second proof, which is closer to Gödel's own, is direct and constructive: It does give us such a sentence, the so-called Gödel sentence for the theory. By doing so, it gives us more information than the first proof. It shows us, in particular, the there will always be an "undecidable sentence" of a very particular form (a so-called Π1 sentence).

This is a good example of why constructive proofs are often better than non-constructive proofs: They often give us more information. But it does not directly address the issue about explanation.

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