Somehow I am saddened to hear of the death of William Thurston, even though I never knew him nor can I even understand his work.
For years I have been pointing people at this article of his, which I find illuminating and inspiring, even in ways that have nothing to do with mathematics: http://arxiv.org/abs/math.HO/9404236
5-paragraph version: http://mathoverflow.net/questions/43690/whats-a-mathematician-to-do/44213#44213
Riffs on his “derivative” exercise: https://profiles.google.com/114134834346472219368/buzz/hTVJiP5LoPb
User profile (http://mathoverflow.net/users/9062/bill-thurston):
“Mathematics is a process of staring hard enough with enough perseverance at at the fog of muddle and confusion to eventually break through to improved clarity. I’m happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarrassment that I might reveal ignorance or confusion. Over the years, this has helped me develop clarity in some things, but I remain muddled in many others. I enjoy questions that seem honest, even when they admit or reveal confusion, in preference to questions that appear designed to project sophistication.”
“At that time, I prided myself in reading quickly. I was really amazed by my first encounters with serious mathematics textbooks. I was very interested and impressed by the quality of the reasoning, but it was quite hard to stay alert and focused. After a few experiences of reading a few pages only to discover that I really had no idea what I’d just read, I learned to drink lots of coffee, slow way down, and accept that I needed to read these books at 1/10th or 1/50th standard reading speed, pay attention to every single word and backtrack to look up all the obscure numbers of equations and theorems in order to follow the arguments.”
“When listening to a lecture, I can’t possibly attend to every word: so many words blank out my thoughts. My attention repeatedly dives inward to my own thoughts and my own mental models, asking ‘what are they really saying?’ or ‘where is this going?’. I try to shortcut through my own understanding, then emerge to see if I’m still with the lecture. It’s the only way for me, and it often works.”
“My first tenure-track job interview was at Cornell. During and after my job talk, most people were pretty quiet, but there was one guy who kept asking very penetrating and insightful questions. And it was very confusing, because I knew all the number theorists at Cornell, and I had no idea who this guy was, or how it was that he obviously understood my talk better than anybody else in the room, possibly including me.”
“When I grew up I was a voracious reader. But when I started studying serious mathematical textbooks I was surprised how slowly I had to read, at a much lower speed than reading non-mathematical books. I couldn’t just
gloss over the text. The phrasing and the symbols had been carefully chosen and every sign was important. Now I read differently. I rarely spend the time and effort to follow carefully every word and every equation in a mathematics article. I have come to appreciate the importance of the explanation that lies beneath the surface. […] I have little expectation for the words to be a good representation of the real ideas. I try to tunnel beneath the surface and to find shortcuts, checking in often enough to have a reasonable hope not to miss a major point. I have decided that daydreaming is not a bug but a feature. If I can drift away far enough to gain the perspective that allows me to see the big picture, noticing the details becomes both easier and less important.
I wish I had developed the skill of reading beneath the surface much earlier. As I read, I stop and ask, What’s the author trying to say? What is the author _really_ thinking (if I suppose it is different from what he put in the mathematical text)? What do I think of this? I talk to myself back and forth while reading somebody else’s writing. But the main thing is to give myself time, to close my eyes, to give myself space, to reflect and allow my thoughts to form on their own in order to shape my ideas.”
“A prominent mathematician once remarked to me that Thurston was the most underappreciated mathematician alive today. When I pointed out that Thurston had a Fields medal and innumerable other accolades, he replied that this was not incompatible with his thesis.”
He believed that this human understanding was what gave mathematics not only its utility but its beauty, and that mathematicians needed to improve their ability to communicate mathematical ideas rather than just the details of formal proofs.
Benson Farb, a mathematician at the University of Chicago and a student of Thurston, said in an email, “in my opinion Thurston is underrated: his influence goes far beyond the (enormous) content of his mathematics. He changed the way geometers/topologists think about mathematics. He changed our idea of what it means to ‘encounter’ and ‘interact with’ a geometric object. The geometry that came before almost looks like pure symbol pushing in comparison.”
Reactions like this are hard to explain: About the Proof&Progress article, http://golem.ph.utexas.edu/category/2006/10/wittgenstein_and_thurston_on_u.html#c005263 says:
“Reading Thurston’s response was one of the most stirring intellectual experiences of my life. It truly struck a cord with my own conception of mathematics. To me, it has the status that the Declaration of Independence has to many Americans, or the U.N. charter has to other global citizens.”
See these for a better summary:
Edit [2012-09-28]: He has also done work in computer science! A fundamental result in data structures (on the number of rotations needed to transform one binary tree into another) was proved in a paper by Sleator–Tarjan–Thurston (with 278 citations). According to DBLP, he has three STOC papers.
See also: http://arxiv.org/abs/math/0503081 (“Mathematical Education”)