why mathematicians\' proofs compel agreement

Abstract for POM-SIGMA-MAA at San Antonio, January 13, 2014

by Reuben Hersh

Title: Mathematicians' proof: "The kingdom of math is within you"

Abstract: A mathematician's informal proof works by enabling others to perceive internally what he/she is trying to show them. I give a simple example, that Sp(n), the sum of the p'th powers of the first n integers. is a polynomial in S1(n), if p is an odd number. (Experiencing mathematics, starting on page 89.)

[[[[[[[ English philosopher Brendan Larvor asks, "What qualifies mathematicians' informal proofs as proofs?" A mathematician seeking a proof is working with internal mental e have direct access to your own internal mental models. You observe some properties of theirs, you manipulate them, you relate them to each other and to other mathematical entities. Your separate individual internal mental models match mine well enough that we communicate about them successfully. In mental struggle with your internal mental models, you notice something interesting. Then you want me to "see" what you "see". You hunt for a sequence of steps which will lead me to share your insight. That sequence of steps, which enables me to "see" what you "see", is what mathematicians call "a proof". ]]]]]]]]]]
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Formal proof, explained in formal logic, and realized in existing computer proofs, is conclusive, it it compelling. But it is universally recognized that mathematicians' proofs, in ordinary teaching and exposition, and in nearly all research publication, are not formal proofs. They are something else.

Mathematicians' proofs, while grossly incomplete and deficient as formal proofs, still are compelling and conclusive. In fact, a proof is accepted, the "result" becomes "established" and available for use in future proofs, if and only if qualified readers find it compelling. Since it isn't acceptable as a formal proof, what makes it compelling? As an English philosopher, Brendan Larvor, put it, "What qualifies mathematicians' informal proofs as proofs?"

G.H. Hardy had already answered this question some 80 years earlier! But his answer was given metaphorically, and its merit is not yet recognized. So I restate Hardy's answer, in direct psychological terms, without metaphor. My explanation is backed up by quotes from living mathematicians, including Andrew Wiles, David Ruelle, and Alain Connes.

When you are seeking a proof, you are working with your internal mental models of the relevant mathematical entities.

You have direct access to your own internal mental models. You observe some of their properties, you manipulate them, you relate them to each other and to other mathematical entities.

When you and I are thinking about the same mathematical entities, our separate individual internal mental models match well enough that we communicate about them successfully.

In mental struggle with your internal mental models, you notice something interesting. You want me to "see" what you "see". You hunt for a sequence of steps which will lead me to share your insight. That sequence of steps, which enables me to "see" what you "see", is what mathematicians call "a proof".

When you lead me through a sequence of mental steps or operations with my own internal models, and thereby lead me to "see" internally, in my own mind, the result that you are claiming, then I am compelled to accept your result.

This process of leading me to "see" your result is what mathematicians' mean by a "proof."

To recognize the veracity of this story, we are obliged to think and talk about our own thoughts and ideas as we do mathematical work. Our thoughts and ideas as we work mathematically are not fictions nor illegitimate nonentities. They are what mathematical proof is about. To explain the nature of mathematical proof, it is necessary to think and talk about such thinking and ideas. Such a kind of thinking and talking is not very popular, or even very legitimate, in current Anglophone "analytic philosophy."


REFERENCES

Anonymous
Alain Connes
G. H. Hardy
Reuben Hersh
Michael Harris
David Ruelle
Andrew Wiles