## Thurston on proof and progress in mathematics

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!

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## 8 thoughts on “Thurston on proof and progress in mathematics”

1. I agree that Thurston’s essay is definitely worth reading. I think he’s right that status in the mathematical community is too tied to getting “theorem credits”, while other very valuable things which advance understanding, like expository works (or exposition generally), are ignored. This is probably tied into the way mathematicians seem to use the $L^\infty$ norm in judging each other, which in and of itself may have something to do with the human cognitive flaw favoring concise justifications when making decisions.

On to books and their virtualizations: Here at Illinois we’ve stopped getting the paper versions of most journals, but while we also have the Springer e-book thing, we’re still getting the physical books as well. While I think going purely electronic for journals is fine — I can’t imagine why I’d ever want/need to look at the paper version — not getting the physical books makes me oddly squeamish. The exception is books, like computer manuals, that have very short life-spans; we only get the O’Reilly series computer books electronically now, I think. On the other hand, I’ve gotten used to being able to search the PDFs and its rare that I checkout a book to read anything like the whole thing; usually I’m just trying to find a reference for some “standard” fact. So maybe it’s just a matter of time before I come around on this. Indeed, as I write this, I should be preparing my lecture for tomorrow out of the electronic version of Neukirch’s book which I got as an “add on” after purchasing the physical book on Amazon; it’s not as nice as having the book in front of me, but it’s good enough that I’m not going to lug the book back and forth from school.

It’s actually amazing how infrequently books are checked out from a math library. When I was at Caltech, I looked at the statistics, and the median number of times a math book was checked out over it’s lifetime was either 1 or 2. Of course a few books, like Thurston’s “3-dimensional geometry and topology” were checked out dozens of times.

If most libraries shift to all-electronic books, I wonder if more people will just post their books on their webpages/the arXiv/lulu.com instead of messing with a traditional publisher. Sure journals still exist, but that’s entirely because their filtering and status-granting functions. Since books barely count toward ones status in mathematics, the slight extra imprimatur of having one published by Springer or the AMS doesn’t seem worth the hassle or forgoing the widest possible dissemination by making it free.

2. John Cowan says:

See Hilary Putnam’s Martian mathematicians in his paper “What is mathematical truth?” (Collected Papers (1979), p. 60.) You can read the early part, at least, on Google Books.

3. Richard says:

I’ve read Thurston’s essay twice within the last four years, and now for a third time, and it never ceases to be thought provoking.

In regard to Springer eBooks, and electronic subscriptions in general, I would like to know to what extent unaffiliated people will be locked out of access to information. AMS’s MathSciNet is a prime example. If you are not officially connected to an institution and not fabulously wealthy, it is totally out of reach.

4. […] 9, 2009 in math, philosophy I have just read (via quomodocumque) a beautifully written and reasoned old essay by William Thurston on “proof and progress in […]

5. Graham says:

I was but a babe in arms when Thurston’s article came out, so thanks for pointing to it.

6. […] assignment for yours truly Posted by toomuchcoffeeman under academe, research   Via Quomodocumque, an old essay by Thurston which I remember noticing but whose content I’d forgotten, assuming […]

7. Scott Carnahan says:

His writing near the end really rubbed me the wrong way. In particular, his discussion of geometrization seemed a lot like, “I’m so awesome, I don’t need to write proofs. Details should be left to the peasants. I learned my lesson from the foliations fiasco.”

There were a couple other strange details, like his mention of Wiles’s proof (at the time, it had a hole) and his discussion of Godel’s incompleteness without mentioning completeness.

8. Scott Carnahan says:

I should mention that my feelings toward this article are certainly not a big ball of hate. I agree with Thurston that there is much value in spreading ideas and sketching out mathematical architectures in broad strokes. My troubles lie in trying to reconcile his “soccer team” model of progress with his description of his own experiences, and the way he justifies his lack of published precision as a positive contribution. It smells like a broken-window fallacy.