"The real satisfaction from mathematics is in learning from others and sharing with others." William Thurston
, one of the greatest mathematicians of the 20th century, has died
. He revolutionized topology and geometry, insisting always that geometric intuition and understanding played just as important a role in mathematical discovery as did the austere formalism championed by the school of Grothendieck. Thurston's views on the relation between mathematical understanding and formal proof are summed up in his essay "On Proof and Progress in Mathematics."
In the 1970s, Thurston proposed the astonishing "geometrization conjecture," asserting that every possible 3-dimensional geometry arose by composition of elements from a short list of fundamental types (which formed the basis for a memorable Paris runway show
which Thurston helped design) Geometrization formed the template for much of the progress of topology over the succeeding decades, culminating in Perelman's proof of the geometrization conjecture
(and its tiny dangling corollary, the Poincare conjecture) Thurston gave an hour lecture
on Perelman's proof in Paris in 2010.
Thurston has won the Fields Medal, been a director of the Mathematical Sciences Research Institute, and held faculty positions at Princeton, the University of California, and Cornell. He trained many of today's leading geometers
In recent years, Thurston became a frequent contributor to the math Q-and-A site MathOverflow
, answering many questions on subjects technical and philosophical. In answer to the question "how can I contribute to mathematics?" he wrote
"It's not mathematics
that you need to contribute to. It's deeper than that: how might you contribute to humanity, and even deeper, to the well-being of the world, by pursuing mathematics? Such a question is not possible to answer in a purely intellectual way, because the effects of our actions go far beyond our understanding. We are deeply social and deeply instinctual animals, so much that our well-being depends on many things we do that are hard to explain in an intellectual way. That is why you do well to follow your heart and your passion. Bare reason is likely to lead you astray. None of us are smart and wise enough to figure it out intellectually.
The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding.
The world does not suffer from an oversupply of clarity and understanding (to put it mildly). How and whether specific mathematics might lead to improving the world (whatever that means) is usually impossible to tease out, but mathematics collectively is extremely important.
I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Dehumanized mathematics would be more like computer code, which is very different. Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind...
In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining --- they depend very heavily on the community of mathematicians."