Tuesday, July 22, 2014

Jaffe and Quinn: "Theoretical Mathematics" (1993)

Shaking the moralistic finger at sloppy mathematics, this paper suggests that the field should be subjected to certain "prescriptions" in order to avoid errors, priority disputes, and confusions about the reliability of proofs.

As a means to this end, Jaffe and Quinn suggest that we understand the activity of writing mathematical proofs according to the analogy
theoretical physics : experimental physics = speculative math : "rigorous" math
This leads them to adopt the odd and misleading term "theoretical mathematics" for what is essentially mathematics without detailed proofs. They suggest that some time in the future, "theoretical mathematics" might branch off from rigorous mathematics and become a discipline of its own, like theoretical physics.

This seems to misconstrue the motivations people have for writing heuristic or intuitive proofs in mathematics.

Most professional mathematicians prefer to give detailed and formal proofs whenever they can, and they only deviate from that norm when they absolutely have to. For a theoretical physicist, on the other hand, there is no shame in having no experimental expertise. The analogy thus seems quite misleading, and trying to drive a wedge in between the practice of conjecturing and the practice of proving mathematical claims consequently misconstrues the ethos of mathematics.

Another thing which is never dwelled upon in the paper (but mentioned in Jeremy Gray's response to the paper) is that even "rigorous" proofs can contain mistakes.

This can be the case even for computer-verified proofs, since there is a very realistic chance of bugs, typos, or even misinterpretations of the code (e.g., subtle differences between an axiom system and the geometric domain it is intended to formalize).

When Jaffe and Quinn assume that we can recognize a "rigorous" proof when we see it, and that rigorous proofs are always correct, they are thus thinking in terms of about a perfect-world mathematics, not actual-world mathematics. As a consequence, they imagine that the whole enterprise of mathematics could be "corrected" into a steadily advancing literature of absolutely certain theorems piled on top of each other.

Of course, everybody wants to give the best arguments they can, and honesty about the quality of the argument would be laudable. But in a world where quality is gradable, it would seem that any step taken toward a realization of Jaffe and Quinn's dream would kill off the whole range of mutant forms in between complete guesswork and completely "rigorous" math (however we measure rigor). Since some of these mutants would have gone on to become the standard of future mathematics, this would almost certainly be a scientific loss, in addition to being very boring.

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