The Mind of a Mathematician

Tao's Solving Mathematical Problems: A Personal Perspective (https://www.amazon.com/Solving-Mathematical-Problems-Persona...) mentioned here is an excellent book that provides keys to mathematical approach to problems and is now a classic ranking there with Polya's How to Solve It.

One thing the article doesn't touch on is Tao's prominence in massively collaborative math research through his blog (latest such work from there was discussed on HN a day ago https://news.ycombinator.com/item?id=21542054). This new approach, first proposed by Tom Gowers (https://gowers.wordpress.com/2009/01/27/is-massively-collabo...) in 2009 has successfully solved a number of open problems in the past ten years (https://polymathprojects.org/2019/02/03/ten-years-of-polymat...).

I would argue that Terrence Tao is overrated. He has solved tons of problems in mathematics, but none groundbreaking.

Either this comment, or its parent, is a joke.

Perhaps like Erdos, he could be called more of a problem solver than a theory builder. In contrast to a figure like Cantor, Galois, or Grothendieck whose work has more fundamental ramifications. I think this raises interesting questions about the continued viability of the "great man" theory of methematical/scientific progress going forward. As Tao and Gowers exemplify, the trend is towards increasing collaboration on problems as they became ever more complex and individual mathematicians become ever more specialized. Could the age of the great theory builders one day come to an end? Is it already over?

Not that I'm trying to downplay Tao's achievements. How overrated can a Fields medalist really be? They don't hand it out for nothing.

I'll quote my own Amazon review of Tao's book:

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First, the bad news.

The front cover of this book gives its author as "Terence Tao, Fields Medal winner 2006". Well ... yes, and no. The thing is, this book was written when Tao was 15 years old. It reflects the precocious skill and insight of an outstandingly gifted 15-year-old, who had won a gold medal at the International Mathematical Olympiad at age 13 (most participants are 17-ish), but not really those of the outstandingly gifted 31-year-old who won the Fields.

It's only about 100 pages long; the problems it discusses are mostly relatively easy (meaning, say, national high-school mathematics competition level or thereabouts, rather than IMO, so not that easy). It doesn't give away any very deep secrets (if there are any) about how to solve such problems. Write down what you know, look for symmetries, simplify step by step, etc.; the real rocket science, as it were, is hidden away in the bits of Tao's brain that instinctively know what symmetries to look for, what steps are likely to lead in the right direction, and so on.

The good news: You wouldn't know it was written by a 15-year-old if the preface didn't tell you. It is a book about mathematics written by someone with a Fields-medal-quality brain, and a book about Olympiad-style problems written by one of the greatest-ever exponents of that art. It contains some nice problems, with solutions by (I repeat myself) one of the finest minds in the business. It's also quite cheap.

If you're interested in this, you should also look at Paul Zeitz's "The art and craft of problem-solving"; it has more pages and more substance to it, but it's twice the price and wasn't written by a Fields medalist.

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Pedantic note: Tim Gowers, not Tom.