Tuesday, August 30, 2011

Bill Thurston - some thoughts about mathematics

These are exercepts from Bill Thurston's notes for the seminar Topics in Topology which started on August 29, 2011 (used by permission from the author).
Mathematics often progresses in waves, where flocks of mathematicians cluster around topics that are currently undergoing rapid development, or have recently seen big developments. Group cohesion is maintained by conferences they attend, as well as visits, talks, and collaborations. They often know many of the same mathematical ideas, and it is common for mathematicians to mainly tune out for topics outside their shared knowledge of their flock. Some mathematicians participate in two or more flocks. Going against the flocking trend, there are also many mathematicians who work in a more solitary style, often persevering doggedly on topics they've become fascinated with. They often participate in one or more flocks, but they maintain an interest not widely shared by their peers.

Here is one downside of the flocking behavior: a great part of the propogation of mathematical understanding is by face-to-face discussions and oral tradition, because mathematical papers are often much more technical, tedious, and denatured compared to how people really think about the ideas when they are actively working on them and developing them, Participants in a flock read each other's papers, but often this is made much easier and more natural because fo their shared background. It is much harder for someone not already versed in the basics of a topic to understand from a technical paper what it is all about. Different flocks use different language to discuss similar mathematical phenomena; people have a big decoding problem when they jump to papers written in a different tradition.

We take in things geometrically bu do not really have adequate ways of communicating these things in the same way.

Mathematical tradition has a vast breadth. My experiences have led me to believe that in principle mathematics is quite unified, with almost any topic potentially connected to almost any other topic, but that the connections are often disguised and undeveloped.
I didn't start out with this mental image of mathematics as highly connected. As an undergraduate and graduate student, I had many interests that would distract and divert me, and that I wanted to pursue. I became frustrated by all the loose ends, and I had a mental image of mathematics (and science) as something like a tree, or perhaps like a hyperbolic space, spreading out exponentially in a quickly unworkable number of directions. .... I often found myself getting sidetracked by some question or remark a professor or fellow student would throw out, and I'd spend days trying to work out an understanding of a small question or topic thjat I became fascinated by. I'd chide myself for wasting my time, instead of following along and giving more mind to the important things I was supposed to be learning.
At the time, I didn't think the side topics led anywhere special, but over the course of my career, again and again, the mathematical side-branches that I pursued enough to gain some insight have connected, often in significant ways, to other more mainstream branches of mathematics that I was interested in. I've seen enough occasions where I or others developed surprising interconnections that I think potential connections are quite common.
A major difficulty in the development of connections is not only that there is limited intermingling between different flocks of mathematicians, but also flocks, like individuals, have a limited life-span: topics and points of view often disappear from sight as flocks of mathematicians dissipate and move on to fresher feeding rounds, and solitary mathematicians also move on or go out of circulation.
Mathematics is about teaching human brain how to think. When your brain is educated, your can see much more interesting things and connections.

These thoughts echoed with the Forward Bill Thurston wrote to my book Crocheting Adventures with the Hyperbolic Planes - it is possible to read it by clicking Look Inside!
W. Thurston On Proof and Progress in Mathematics
Mathematical Education

About photos: first one is in Stewart Park, Ithaca, NY, the others are from my recent trip to Canada.