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Wednesday, December 14, 2005

The aim of mathematics

There's often much to think about in even brief pieces of mathematical writing. Take the following extract from The algebraic meaning of genus-zero, where Terry Gannon claims:
At the risk of sending shivers down Bourbaki’s collective spine, the point of mathematics is surely not acquiring proofs (just as the point of theoretical physics is not careful calculations, and that of painting is not the creation of realistic scenes on canvas). The point of mathematics, like that of any intellectual discipline, is to find qualitative truths, to abstract out patterns from the inundation of seemingly disconnected facts. An example is the algebraic notion of group. Another, dear to many of us, is the A-D-E metapattern: many different classifications (e.g. finite subgroups of SU2, subfactors of small index, the simplest conformal field theories) fall unexpectedly into the same pattern. The conceptual explanation for the ubiquity of this metapattern—that is, the combinatorial fact underlying its various manifestations — presumably involves the graphs with largest eigenvalue |λ| ≤ 2.

Likewise, the real challenge of Monstrous Moonshine wasn’t to prove Theorem 1, but rather to understand what the Monster has to do with modularity and genus-0. The first proof was due to Atkin, Fong and Smith [25], who by studying the first 100 coefficients of the Tg verified (without constructing it) that there existed a (possibly virtual) representation V of M obeying Theorem 1. Their proof is forgotten because it didn’t explain anything.

By contrast, the proof of Theorem 1 by Borcherds et al is clearly superior: it explicitly constructs V = V natural, and emphasises the remarkable mathematical richness saturating the problem. On the other hand, it also fails to explain modularity and the Hauptmodul property. The problem is step (iii): precisely at the point where we want to identify the algebraically defined Tg’s with the topologically defined Jg’s, a conceptually empty computer check of a few hundred coefficients is done. This is called the conceptual gap of Monstrous Moonshine, and it has an analogue in Borcherds’ proof of Modular Moonshine [3] and in H¨ohn’s proof of ‘generalised Moonshine’ for the Baby Monster [15]. Clearly preferable would be to replace the numerical check of [1] with a more general theorem.
Even with the advances Gannon describes in the paper:
...the resulting argument still does a poor job explaining Monstrous Moonshine. Moonshine remains mysterious to this day. There is a lot left to do — for example establishing Norton’s generalised Moonshine [24], or finding the Moonshine manifold [14]. But the greatest task for Moonshiners is to find a second independent proof of Theorem 1. It would (hopefully) clarify some things that the original proof leaves murky. In particular, we still don’t know what really is so important about the Monster, that it has such a rich genus-0 moonshine. To what extent does Monstrous Moonshine determine the Monster?
Clearly you have to see the original paper to understand what he's talking about, but already there are some clear opinions about the aims of mathematics and the means to achieve these ends. Gannon is a very good expositor, see, e.g., Monstrous moonshine and the classification of CFT.

Bourbaki is presented as opposed to his view of the aim of mathematics, but perhaps this understanding of Bourbaki is gleaned only from their textbooks. Leo Corry, the historian of mathematics whose views I have been discussing in previous entries, has written extensively about aspects of Bourbakian philosophy in his Modern Algebra and the Rise of Mathematical Structures. Remember it was Andre Weil's views about how mathematicians were best equipped to write history of mathematics that form the departure point for Corry's paper. An example of a Bourbakian history is Jean Dieudonne's A History of Algebraic and Differential Topology, 1900-1960. Presumably, for some people a book like this answers the problem of how to write a history of a research programme which runs over decades, and involves a host of different contributors. Given its scale it could hardly deal with individual research philosophies, or the practices of institutions, even in the unlikely event that the author would have wanted to do so.

By the way, returning to reality and 'reality', the other day I was helping my 11-year-old daughter with some mathematics homework concerning adding volumes in litres and millilitres. The advice to the helper began: In 'real life' we often have to deal with measured quantities. What on earth are those scare quotes doing there? Is there a hint here of a pristine, unspoilt life in which we didn't have to deal with measured quantities? Perhaps life in Eden, before we were forced out and had to build things like Arks.


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