Proof, Progress and Programs.

William Thurston is a prominent mathematician. I know nothing about his research except that his work is held in high regard by professional mathematicians. He has received several awards for both his research and expository work.  He has also written a few essays about the philosophy of mathematics, the most prominent of these I am aware of is titled On proof and progress in mathematics. Much of what he says is applicable to any form of intellectual discourse, particularly of a mathematical nature.

Mathematics in some sense has a common language: a language of symbols, technical definitions, computations, and logic. This language efficiently conveys some, but not all, modes of mathematical thinking. Mathematicians learn to translate certain things almost unconsciously from one mental mode to the other, so that some statements quickly become clear. […]

We mathematicians need to put far greater effort into communicating mathematical ideas. To accomplish this, we need to pay much more attention to communicating not just our definitions, theorems, and proofs, but also our ways of thinking. We need to appreciate the value of different ways of thinking about the same mathematical structure.
We need to focus far more energy on understanding and explaining the basic mental infrastructure of mathematics—with consequently less energy on the most recent results. This entails developing mathematical language that is effective for the radical purpose of conveying ideas to people who don’t already know them.

These thoughts are preceded by an example of different ways of thinking about the derivative of a function. When I first read this essay, I greatly appreciated the emphasis on different ways of thinking about the same concept. I feel there is at times a tendency for an entire field to adopt a certain mode of thought to the detriment of other views.  Such an intellectual monoculture is to our own detriment, as a community, because different modes of thought bring different perspectives and allow a field to progress in different ways. In future posts, I will examine different conceptions of ideas in logic and computer science. To conclude this post, I will recall Thurston’s comments about rigour and dealing with a computer. It was heartening to see a prominent mathematician appreciate the challenge involved in writing correct software and the even greater challenge of producing a proof of correctness.

I have spent a fair amount of effort during periods of my career exploring mathematical questions by computer. In view of that experience, I was astonished to see the statement of Jaffe and Quinn that mathematics is extremely slow and arduous, and that it is arguably the most disciplined of all human activities. The standard of correctness and completeness necessary to get a computer program to work at all is a couple of orders of magnitude higher than the mathematical community’s standard of valid proofs. Nonetheless, large computer programs, even when they have been very carefully written and very carefully tested, always seem to have bugs.

When one considers how hard it is to write a computer program even approaching the intellectual scope of a good mathematical paper, and how much greater time and effort have to be put into it to make it “almost” formally correct, it is preposterous to claim that mathematics as we practice it is anywhere near formally correct.

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