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Tuesday, February 07, 2006

In praise of exposition

If one takes seriously my philosophical position, that understanding is the aim of mathematics, one is led to think that expository writing is crucial to the mathematical enterprise, and that in many ways it is more important than the journal articles which rigorously prove the results covered by that writing. To forestall the usual criticisms, let me be clear that this is not to deny the importance of proof, or to be in any way dismissive of the achievement of a robust rigour over the centuries. Indeed, the level of rigour achieved by contemporary mathematicians provides a stable framework in which understanding can flourish. It is to say, however, that without the understanding transmitted face-to-face from teacher to student, and all too rarely captured in print, mathematics collapses.

I come then to praise exposition, and for an example will consider a piece recently brought to my attention, written by Jacob Lurie and titled A Survey of Elliptic Cohomology . If we take this survey as at least as worthy of philosophical attention as any stretch of definitions, lemmata, and theorems from a journal paper, we are brought to pose ourselves a rather different set of questions. First, what kind of writing does it most resemble? A story might be the best one word answer. Second, who is it addressed to? Clearly not just anyone. You must have some training in various branches of mathematics, although the audience ranges from those like me whose mathematical understanding is being stretched to breaking point during various passages (even with the help of Week 197, and references therein), to those for whom it is just an exercise in shaping what they largely already know. Either way, it is clear that to participate as a mathematician you need to train in it as you would a craft. Third, what are we to make of the language employed by Lurie? To take a couple of examples from page 10:
Many of the cohomology theories which appear "in nature" extend in a natural way to equivariant cohomology theories.

There are some respects in which Borel-equivariant cohomology is not a satisfying answer to our question.
What needs to be undertaken here are textual analyses. I made a start on the use of in nature and natural in chapter 9 of my book. For a longer passage, one might look at Lurie's final paragraph:

Unfortunately, our algebraic perspective does not offer any insights on the problem of where to find such a cohomology theory in geometry. Nevertheless, it seems inevitable that a geometric understanding of elliptic cohomology will eventually emerge. The resulting interaction between algebraic topology, number theory, mathematical physics, and classical geometry will surely prove to be an excellent source of interesting mathematics in years to come.

'Insights', 'inevitable', 'geometric understanding', 'interesting' - there's plenty of work to be done.


urs said...

Interesting point of view. Probably Jacob Lurie's piece has by now received a completely unexpected ration of blogwise attention compared to its online lifetime. ;-)

May I remark that those interested in the context that this is taken from might want to have a look at this?

February 07, 2006 3:01 PM  
A.J. said...

Hi David,

You might also find it worthwhile to take a look at the Encyclopedia of Mathematical Sciences. These are little yellow hardbacks, translated from Russian. They don't contain many proofs, but they're chock full of the sort of mathematical exposition you're talking about here. Gelfand & Manin's contribution (Algebra V) occasionally rises to the level of poetry: "Algebraic wisdom tells us that.."

February 08, 2006 2:22 AM  
Anonymous said...

In 2001, I heard a speaker from the US National Science Foundation claim that some 250,000 mathematical theorems were published each year. If so, then more exposition and less new research is probably warranted.

February 11, 2006 1:19 PM  
david said...

The curious thing is that so many senior mathematicians know that exposition should be rewarded. To give a single example, William Thurston writes in his contribution to the Jaffe-Quinn debate:

"I think that our strong communal emphasis on theorem-credits has a negative effect on mathematical progress. If what we are accomplishing is advancing human understanding of mathematics, then we would be much better off recognizing and valuing a far broader range of activity."

Pierre Cartier told me last Summer that he wouldn't have written his two expository pieces 'A Mad Day's Work' and 'Mathemagics (A Tribute to L. Euler and R. Feynman)' had they not be conmissioned, for fear that journals would not accept them. This is irrationality.

February 11, 2006 3:34 PM  

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