Continuing my flogging of the issue: is it the essence of a mathematical proof to be persuasive, such that someone who fails to persuade has failed to engage in “mathing?”
I ran across this fun little Numberphile video which raises in passing an interesting and important point. Fermat came up with an idea (not his super-famous one) about some primes being the sum of two squares (like 17=16+1). What the video goes on to mention is that many mathematicians after Fermat–the super-heavyweights like Euler and Gauss and Dedekind and Co.–all came up with proofs of this idea.
Each of those proofs is different. Very different. If the goal of mathematical proof were simply to persuade, the proofs would be valued for getting different “mathematical demographics” to agree to the truth of the conclusion. Or perhaps, even more simply, one could insist that everyone should agree to the conclusion of the first, rational proof and then get on with life.
But this is not the role of mathematical proofs, any more than it is the role of the scientific method or logical argumentation. The various proofs are valued because each of them illuminates different aspects of the problem as well as different areas of the wide world of mathematics.
Sometimes you will hear–ok, you will hear if you follow math news like a giant nerd like me–mathematicians talk about proofs being exciting because of all the new questions they raise or the new applications in other, often unexpected branches of the realm. The idea is similar: mathematical proof has as its nature the illuminating of the causal connections of mathematical ideas.
Ok, enough of my flogging. Here’s the video, which ends with a continuation to an even simpler, recent proof of the same: