I must have read Thurston’s excellent essay “On proof and progress in mathematics,” when it came out, but I don’t have any memory of it. I re-encountered it the other day while playing with Springer’s eBook service, and flipping through the chapters of the recent collection 18 Unconventional Essays on the Nature of Mathematics.
Thurston makes a passionate case against theorem-proving as the measure of a mathematician’s contribution:
In mathematics,it often happens that a group of mathematicians advances with a certain collection of ideas. There are theorems in the path of these advances that will almost inevitably be proven by one person or another. Sometimes the group of mathematicians can even anticipate what these theorems are likely to be. It is much harder to predict who will actually prove the theorem,although there are usually a few “point people”who are more likely to score. However, they are in a position to prove those theorems because of the collective efforts of the team.The team has a further function,in absorbing and making use of the theorems once they are proven. Even if one person could prove all the theorems in the path single-handedly,they are wasted if nobody else learns them.
There is an interesting phenomenon concerning the “point”people. It regularly happens that someone who was in the middle of a pack proves a theorem that receives wide recognition as being significant. Their status in the community—their pecking order—rises immediately and dramatically.When this happens,they usually become much more productive as a center of ideas and a source of theorems.Why? First,there is a large increase in self-esteem, and an accompanying increase in productivity. Second, when their status increases,people are more in the center of the network of ideas—others take them more seriously. Finally and perhaps most importantly, a mathematical breakthrough usually represents a new way of thinking,and effective ways of thinking can usually be applied in more than one situation.
This phenomenon convinces me that the entire mathematical community would become much more productive if we open our eyes to the real valuesin what we are doing. Jaffe and Quinn propose a system of recognized roles divided into “speculation”and “proving”. Such a division only perpetuates the myth that our progress is measured in units of standard theorems deduced. This is a bit like the fallacy of the person who makes a printout of the first 10,000 primes. What we are producing is human understanding. We have many different ways to understand and many different processes that contribute to our understanding. We will be more satisfied, more productive and happier if we recognize and focus on this.
Thurston concludes with some very interesting and frank reminiscences, including some regrets, about the way certain parts of topology bent around his gravitational field in the 70s and 80s.
By the way, some libraries have stopped buying new physical books from Springer in favor of access to the e-books. If you’re at an institution that’s gone this route, tell me about it in comments!