[FOM] What is a proof?

Charles Silver 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  
> of
> 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).

Charlie Silver

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