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Tuesday, May 23, 2006

The real entities of mathematics

It's easy to find yourself thinking that there's a natural distinction to be found between entities which are the proper subject matter of mathematics, and those used to give us access to the former. Perhaps the most famous expression of such thinking is Kronecker's "God created the integers, all else is the work of man."

From a discussion after this post over at The String Coffee Table back in February:

DC: In a comment to this post I raised the question of whether we might expect categorification of the special functions to appear with 2-representation theory. Something that might win over more people to higher-dimensional algebra would be the discovery of something as *concrete* as Bessel functions in their role as matrix elements of representations of important Lie groups. In the first few pages of this paper Cherednik exhibits this kind of attitude when he likens the difference between new concepts and new objects to that between the imaginary and the real.

Bruce Bartlett: I think David’s comment is quite relevant. I looked at Cherednik’s paper and found the introduction a most interesting, and shrewd, read. I will certainly keep this in mind from now on! Perhaps some further developments are necessary before the “real projection” of this stuff is nontrivial.

DC: I suppose an obvious question to raise about Cherednik’s real/imaginary dichotomy is whether to take it as timeless. It’s questionable that your average nineteenth century mathematician beamed to the future would take his word for it that characters of Kac-Moody algebras (Fig. 2) are “fundamental objects” because they “are not far from the products of classical one-dimensional theta-functions and can be introduced without representation theory”.

What is clear is that the expression of such a distinction is a part of a mathematician's philosophy. It would be quite consistent to expect such distinctions to alter through the centuries without allowing it to be a wholly subjective matter.

On another point, in the blog discussion, Bruce also made the prediction that:
Somehow, one feels that a lot of roads in this whole business of equivariant string theory, gerbes, elliptic cohomology, higher gauge theory, higher categories, etc. are going to merge at some point.
Something that would interest me in the event of such a merger is the just distribution of credit for any achievements. Can a research programme 'get lucky' by being in the right place at the right time to allow a breakthrough, when with the advantage of hindsight we can see that it would have been fairer had a rival got there first? Would this matter if there weren't the tendency overly to reward the first success?


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