[FOM] What is a proof?
silver_1 at mindspring.com
Mon Jan 26 16:07:21 EST 2009
Jesse Alama wrote:
> A related, quite different question is whether the positive judgment
> the community of mathematicians *constitutes* the correctness of a
> proof. That is a question not about feelings, but about the
> epistemology or metaphysics of proof.
Agreed. A proof is correct, independently of whether anyone
believes the proof to be correct or not. It's thus metaphysical (not
epistemological). A proof is deemed correct by the mathematical
community if they judge it to be correct. This is epistemic. And,
they can all be wrong.
[Issues related to this cropped up a couple of years ago concerning
Reuben Hersh's book _What Is Mathematics, Really?_. He argued (I
thought incorrectly) for the social character of mathematical
*truth*. Perhaps I'm exaggerating, but I think he argued that a
proof was what was called a proof by the mathematical community, which
I seem to recall (??) was analogous to a baseball being out of bounds
because the referees agreed that it was out of bounds.
Harvey got sick of the extensive wave of argumentation on this
restricted topic and asked for those interested to please form their
own group discussion. A group was formed, and George Lakoff and
Rafael Núñez joined the fray (They'd co-authored a book _Where Does
Mathematics Come From?_) From my point of view, Lakoff & Núñez
similarly disregarded the metaphysical fact of the *correctness* of a
proof or the *truth* of a mathematical result (though on somewhat
different grounds than Hersh).]
In short, I think your question of what "*constitutes* the
correctness of a proof" is an important one (however, I feel that
metaphysical questions are at bottom unanswerable).
I'd be very interested in others' opinions (though I wouldn't want
this topic to dominate the discussion as it did a few years before).
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