Saturday, November 29, 2014

What is a Proof, Really?

[ original thread ]

On Sat, Nov 29, 2014 at 11:12 AM, Joe Niederberger wrote:
<< SNIP >>

> Finally, I'm happy to accept your chess problem as mathematical. Frankly, I
> don't know what a survey on that question, given to working mathematicians,
> would turn up. And, any mathematically acceptable way of arguing it would
> have to be logical in my opinion. (If you have an illogical approach that
> is also mathematical I'd be fascinated to hear about it.)
> Cheers,
> Joe N

I think we mostly agree. Criteria apply.

A proof is not a recipe nor even algorithm.

An algorithm tends to have proofs in the background, to back it up as it were, e.g. we make use of V + F = E + 2 in some step in a computer program, e.g. we get E from V + F - 2, but then why is it safe to get E in this way?

In the background: Euler's Theorem for Polyhedrons and the many proofs thereof, my favorite probably the one by G. K. C. Von Staudt:

[ however this is not my favorite forumulation of it; that would be in Peter Cromwell's Polyhedra, cite ]

The other thing I'd say is: lets not go overboard in assuming some finite roster of individuals tagged as "mathematician" truly owns or controls or governs the discipline and shared heritage we loosely call mathematics ("loosely" because anything tighter would be clearly too tight and therefore outright wrong).

Innovations come in from left field all the time e.g. most naturally from closely neighboring disciplines, and those self-identifying as official spokespersons for mathematics, i.e. mathematicians, must scramble to keep their background cosmetically acceptable i.e. the pros keep it looking professional, add the right panache (sometimes a little lipstick on the pig is all one needs).