Friday, November 16, 2012

On Human Understanding of Science and Mathematics


William Thurston was a Fields medalist who pioneered a lot of work in topology. His geometrization conjecture was what Grigori Perelman proved in 2003 to also prove the Poincare Conjecture.

I found this beautiful writeup by Thurston about the human and the social dimension of mathematics:

The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true.
But in any field, there is a strong social standard of validity and truth. Andrew Wiles’s proof of Fermat’s Last Theorem is a good illustration of this, in a field which is very algebraic. The experts quickly came to believe that his proof was basically correct on the basis of high-level ideas, long before details could be checked. This proof will receive a great deal of scrutiny and checking compared to most mathematical proofs; but no matter how the process of verification plays out, it helps illustrate how mathematics evolves by rather organic psychological and social processes.
Thurston also has a beautiful explanation of how mathematics (and I would generalize this to all of fields of science) needs a community of people, that it's not just about individuals (despite the popular press' efforts in hyping up individuals):

Even if one takes the narrow view that what we are producing is theorems, the team is important. Soccer can serve as a metaphor. There might only be one or two goals during a soccer game, made by one or two persons. That does not mean that the efforts of all the others are wasted. We do not judge players on a soccer team only by whether they personally make a goal; we judge the team by its function as a team.

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