Wednesday, March 13, 2019

Horgan admits math proofs are not dying

SciAm writer John Horgan finally concedes
Okay, Maybe Proofs Aren't Dying After All

Two experts argue that proofs are doing fine, contrary to a controversial 1993 prediction of their impending demise
It appears that a famous mathematician led him astray:
But influential figures were behind the changes. One was William Thurston, who in 1982 won a Fields Medal — the mathematical equivalent of a Nobel Prize — for delineating links between topology and geometry.

Thurston, who served as a major source for my article, advocated a more free-form, “intuitive” style of mathematical research, communication and education, with less emphasis on conventional proofs. He sought to convey mathematical concepts with computer-generated models, including a video that he called “Not Knot.”

“That mathematics reduces in principle to formal proofs is a shaky idea” peculiar to the 20th century, Thurston told me. Ironically, he pointed out, Bertrand Russell and Kurt Godel demonstrated early in the century that mathematics is riddled with logical contradictions. “Set theory is based on polite lies, things we agree on even though we know they're not true,” Thurston said. “In some ways, the foundation of mathematics has an air of unreality.”
The Fields Medal is not really the mathematical equivalent of a Nobel Prize. The Abel Prize is much closer.

No one showed that mathematics is riddled with logical contradictions. Thurston was not knowledgeable about the foundations of math. He was a brilliant mathematician, and he was good at explaining his work to others, but he was lousy at writing up his proofs. Some of his best work was written up by others.

Thurston's ideas were not accepted until proofs were written and published. Probably his biggest idea was that all three-dimensional manifolds could be decomposed into one carrying one of about eight geometric structures. This was always called a conjecture, until Perelman published what appeared to be a proof, and others filled in the gaps so that everyone was convinced that it really was a proof.

Russell showed that certain set theory operations led to contradictions, and then showed how an axiomatic approach could resolve them. Goedel gave much better axiomatizations of set theory, can examples of undecidable statements. An undecidable is the opposite of a contradiction.

Horgan's concession is based on quoting two bloggers. It would have been better if he had asked someone who was trained in mathematical foundations, instead of computer science and particle physics.

BTW, Scott Aaronson comments:
More importantly, I’ve been completely open here about my unfortunate psychological tic of being obsessed with the people who hate me, and why they hate me, and what I could do to make them hate me less. And I’ve been working to overcome that obsession.
I seem to be one of his enemies, but I do not hate him. I don't disagree with his comments about proof, but he is not a mathematician and he does not speak for mathematicians.

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