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Bill Thurston on collective progress in mathematics

by Michael Nielsen on February 9, 2009

Apropos the polymath project, a nice quote from Bill Thurston on how progress is made collectively in mathematics (via Cosma and Quomodocumque):

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 values in 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.

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One Comment
  1. That was a terrific Thurston quotation!

    By coincidence, last week our UW QSE Group sent that *same* Thurston article to all our students.

    And so that our physics-oriented students wouldn’t feel left-out, that same email included a (Google Books) link to the Introduction to Bob Geroch’s excellent Mathematical Physics. Fantastic!

    And so that our engineering-oriented students wouldn’t feel left-out, we included a link to the April 29, 1957 Time Magazine article The New Age, featuring analysis by Simon Ramo and Dean Wooldridge. Wonderful!

    The common theme of all three articles is simply this: confidence in the future of mathematics, science, and engineering. Because as Warren Buffett has taken to saying:

    “Confidence in institutions is a lot like oxygen. When you have it, you don’t even think about it. When it’s gone for five minutes, it’s the only thing you think about.”

    Not many other human institutions can convey the same confidence as mathematics, science, and engineering … and herein is a great opportunity for the present generation of mathematics, science, and engineering students.

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