Like most individuals, and probably, most mathematicians outside his field, I do not deeply understand much of Thurston's work. But he did have a rather large impact on my life insofar as a paragraph of his has stayed with me several years (not many do, I'm afraid), and has been my reminder to slow down whenever I find myself saying "psh, that was easy, all I had to do was browse through the documentation":
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.
If a Fields medalist needed to slow down to read some maths, I can slow down to really understand whatever it is that I'm doing. And when I tell myself "I've learned that already!" I stop and ask whether I learned it at a "1/50th pace."
I don't print ANYTHING. Except papers. If I find a good one, I print it out, read it with my red pen, and mark all over it. When I'm done, I print it out again and start over. I do this until there is no longer a need to use my red pen. Only then do I consider that paper to have been "read".
Thank you for this. Every once in a while one stumbles upon a bit of writing that instantly changes the context in which they frame aspects of their lives. This very much had that effect on me – As well as bringing up fond memories of academia. =)
Fields medalist Terence Tao writes about earlier Fields medalist William Thurston, "In addition to his direct mathematical research contributions, Thurston was also an amazing mathematical expositor, having the rare knack of being able to describe the process of mathematical thinking in addition to the results of that process and the intuition underlying it. His wonderful essay “On proof and progress in mathematics“, which I highly recommend, is the quintessential instance of this; more recent examples include his many insightful questions and answers on MathOverflow."
From his "about" page on his MathOverflow profile:
"Mathematics is a process of staring hard enough with enough perseverence 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 embarassment 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."
Such beautiful modesty from a brilliant mathematician who inspired many.
I like the embarrassing arguments from MathOverflow moderators re. not closing Mr. Thurston's (vague, subjective, "not a question") question on MathOverflow.
Nicely put by Georges Elencwajg in one of the comments:
...posed by somebody else, this question would have been closed within a picosecond. The rationale, ironically, being that professional mathematicians of quality would flee this site in droves because the question is "discussiony", "vague", "has no right answer", etc. This is the Matthew effect in all its glory: "For to all those who have, more will be given, and they will have an abundance; but from those who have nothing, even what they have will be taken away" (Matthew 25:29) Needless to say, I'm euphoric at the thought of Bill's participation in MathOverflow.
The parts about how communication works inside mathematical specialties and about the difference between math and computer programming are worth their weight in gold.
Really, everyone, just do yourself a favor and read it – all the way to its astonishing final section and beautiful ending. It doesn't require you to remember any math.
It's been a while since I've done serious mathematics so I'm almost certainly wrong about this, but the "logical" definition for a derivative looks incorrect under the section "How do People Understand Mathematics?" (pg 3).
Shouldn't the distance between the quotient and d be less than ɛ?
If you've wondered what topology was about and wanted to get a better feel for it than just a dictionary definition, then see the 20-minute video in the article.
A blurb about the video:
The computer animation "Outside In" explains the amazing discovery, made by Steve Smale in 1957, that a sphere can be turned inside out by means of smooth motions and self-intersections. Through a combination of dialogue and exposition accessible to anyone who has some interest in mathematics, "Outside In" builds up to the grand finale: Bill Thurston's ``corrugations'' method of turning the sphere inside out. Along the way, the narrators discuss the related case of closed curves and why they generally cannot be turned inside out. Everyday analogies such as train tracks, belts, smiles and frowns are used throughout, all richly animated and complete with sound effects. (quoted from http://www.geom.uiuc.edu/docs/outreach/oi/ )
I just watched this on Youtube and was not disappointed. There was just a great clarity to the question-and-answer format of the dialogue; the animations and sound effects fit the exposition to a tee. I hope they're still making brilliant educational videos like this, somewhere.
I suppose you could say that he changed over to the world of "not existing anymore." Or you could just state the tragedy outright, and say that he's dead. Please excuse my bluntness; a person's obliteration is something I'd prefer not to swathe in soft euphemisms.
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.
If a Fields medalist needed to slow down to read some maths, I can slow down to really understand whatever it is that I'm doing. And when I tell myself "I've learned that already!" I stop and ask whether I learned it at a "1/50th pace."
http://matrixeditions.com/Thurstonforeword.html