


Thurston in His Own WordsBelow are some representative quotes from Bill’s own writing and speaking, assembled by his son Dylan. Many people have an impression that mathematics is an austere and formal subject concerned with complicated and ultimately confusing rules for the manipulation of numbers, symbols, and equations, rather like the preparation of a complicated income tax return. Good mathematics is quite opposite to this. Mathematics is an art of human understanding. … Our brains are complicated devices, with many specialized modules working behind the scenes to give us an integrated understanding of the world. Mathematical concepts are abstract, so it ends up that there are many different ways they can sit in our brains. A given mathematical concept might be primarily a symbolic equation, a picture, a rhythmic pattern, a short movie — or best of all, an integrated combination of several different representations. (Foreword to Crocheting adventures with hyperbolic planes by Daina Taimina) The aesthetic goals and the utilitarian goals for mathematics turn out, in the end, to be quite close. Our aesthetic instincts draw us to mathematics of a certain depth and connectivity. The very depth and beauty of the patterns makes them likely to be manifested, in unexpected ways, in other parts of mathematics, science, and the world. To share in the delight and the intellectual experience of mathematics — to fly where before we walked — that is the goal of a mathematical education. (Mathematical Education, Notices of the AMS 37:7 (September 1990), pp 844–850) When the idea is clear, the formal setup is usually unnecessary and redundant — I often feel that I could write it out myself more easily than figuring out what the authors actually wrote. It’s like a new toaster that comes with a 16page manual. If you already understand toasters and if the toaster looks like previous toasters you’ve encountered, you might just plug it in and see if it works, rather than first reading all the details in the manual. … We mathematicians need to put far greater effort into communicating mathematical ideas. To accomplish this, we need to pay much more attention to communicating not just our definitions, theorems, and proofs, but also our ways of thinking. We need to appreciate the value of different ways of thinking about the same mathematical structure. We need to focus far more energy on understanding and explaining the basic mental infrastructure of mathematics — with consequently less energy on the most recent results. This entails developing mathematical language that is effective for the radical purpose of conveying ideas to people who don’t already know them. … What we [mathematicians] are producing is human understanding. We have many different ways to understand and many different processes that contribute to our understanding. (On proof and progress in mathematics, Bull. Amer. Math. Soc. (N.S.) 30 (1994), 161–177) It is easy to forget that mathematics is primarily a tool for human thought. … The most important thing about mathematics is how it resides in the human brain. … When mathematics loses its connection to our minds, it dissolves into a haze. … In mathematics, what is intriguing, puzzling, interesting, surprising, boring, tedious, exciting is crucial; they are not incidental, they shape how we think. (Foreword to Teichmuller theory and applications to geometry, topology, and dynamics; Volume I: Teichmuller theory, by John Hubbard) 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. … Mathematics is about teaching the human brain how to think. When your brain is educated, you can see much more interesting things and connections. (Seminar notes taken by Daina Taimina, Fall 2011) The product of mathematics is clarity and understanding. … The real satisfaction from mathematics is in learning from others and sharing with others. … The question of who is the first person to ever set foot on some square meter of land is really secondary. (Math Overflow answer to the question “What can one contribute to mathematics?”) Mathematics is a process of staring hard enough with enough perseverance 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. (MathOverflow “About me”) Further contributions to MathOverflow I used to feel that there was certain knowledge and certain ways of thinking that were unique to me. It is very satisfying to have arrived at a stage where this is no longer true — lots of people have picked up on my ways of thought, and many people have proven theorems that I once tried and failed to prove. (Steele Prize response, Notices of the AMS, April 2012, page 565) Last modified:September 4, 2012 