It talks about what it means in mathematics to “understand” a subject, and the fact that proofs have an importance that goes beyond certifying that a result is true. On the matter of proofs as certificates, and the possibility of truly formal proofs, he considers the plausibility of interactive computer systems to construct formal proofs, and has an interesting, dystopian, view of where the quest for fully formal proofs might lead us
A very similar [to what is done in the engineering of a large piece of computer software] kind of effort would have to go into mathematics to make it formally correct and complete. It is not that formal correctness is prohibitively difficult on a small scale—it’s that there are many possible choices of formalization on small scales that translate to huge numbers of interdependent choices in the large. It is quite hard to make these choices compatible; to do so would certainly entail going back and rewriting from scratch all old mathematical papers whose results we depend on. It is also quite hard to come up with good technical choices for formal definitions that will be valid in the variety of ways that mathematicians want to use them and that will anticipate future extensions of mathematics. If we were to continue to cooperate, much of our time would be spent with international standards commissions to establish uniform definitions and resolve huge controversies.
It also discusses the way in which progress is a collective activity.
I agree with all the many good points that he makes, and I highly recommend the essay to those who have not read it.
Interestingly, the essay was written as a response to an essay by Jaffe and Quinn on rigor versus speculation in mathematics, in turn a response to the use of non-rigorous arguments from physics in geometry. (Here are more responses.) Probably it wouldn’t have occurred to Thurston to collect his thoughts on these issues and write them down if he had not been bothered by the Jaffe-Quinn essay, and so I am glad that Jaffe and Quin wrote their essay, even if I did not find it very interesting.
Indeed one of the good things about controversies (at least when they happen among thoughtful and intellectually honest people) is to make people question why they think the thoughts that they do, and to find a heart-felt way to articulate their reasons. I remember that, when I studied philosophy in high school, the various doctrines seemed equally boring and indistinguishable in the textbook treatment; the few times we read primary sources, however, and saw how a given philosopher would defend his views and trash other doctrines, the subject did come alive.
Perhaps this is why blogs that have controversial posts have the most interesting comments.
And, by the way, Vista is better than OS X.