The n-Category Café

Contemporary differential geometry is dramatically broadening its horizons. For a taste see Non Abelian Differential Gerbes, “We develop a differential geometry theory of non-abelian differential gerbes over stacks using Lie groupoids”, and Higher Gauge Theory “We describe a theory of 2-connections on principal 2-bundles and explain how this is related to Breen and Messing’s theory of connections on nonabelian gerbes”.

What is very noticeable is the number of routes which seem to be leading in the same direction. One shouldn’t underestimate, however, the work of reconciling different viewpoints. Great rewards are due for work which, although it proves nothing new, performs this reconciliation well. In his Racah-Wigner quantum 6j Symbols, Ocneanu Cells for AN diagrams and quantum groupoids, Robert Coquereaux claims:

Our purpose in this paper is very modest. Indeed, all the objects that we shall manipulate have been already introduced and studied in the past, sometimes long ago: 6J symbols, quantum or classical, are considered to be standard material, cells and “double triangle algebras” have been invented in [28], [31] and analyzed for instance in [5], [37], [14] or [39], finally, quantum groupoids are studied in several other places like [7], [25] or [26]. However, it is so that many ideas and results presented in these quoted references are not easy to compare, not only at the level of conventions, but more importantly, at the level of concepts, despite of the existence of the same underlying mathematical “reality”.

Now, why the scare quotes? There are two types of philosophical position that require them. One is a form of idealism which would want scare quotes to be used at the mention of any form of reality. Even the reality of chairs and tables needs putting into question. This is presumably not what Coquereaux believes. What I take it that he is implying is that just as there is a physical world which places severe constraints on what we can and can’t do - we can swim in a river, we can’t walk through trees, we can’t jump up 10 metres, etc. - there is something not so very different which forces mathematicians to work along similar lines, even if this is not always obvious, and this something is not merely logic. In this quotation of Connes, again we see ‘mathematical reality’ in scare quotes. Again, mathematicians often meet each other in the same places:

whatever the origin of one’s itinerary, one day or another if one walks long enough, one is bound to reach a well known town i.e. for instance to meet elliptic functions, modular forms, zeta functions

There is a danger in confusing this mathematicians’ realism (remember not all mathematicians are convinced that this convergence is so important - Zeilberger’s Opinion 49, Ruelle’s Is Our Mathematics Natural?, Bull. AMS 19, 259-268, 1988), with what is at stake when analytic philosophers of mathematics take up realism. Here there is no interest in specific concepts like 6j symbols or elliptic functions. Where the mathematicians will be able to point to concepts that although consistent are not a part of their reality, philosophers generally argue for or against realism across the set theoretic board.

And John replied:

There is a danger in confusing this mathematicians’ realism […]

This is a very important point. Presumably this is because most philosophers don’t actually do mathematics (or even carefully watch people who do).

When you do mathematics, you keep “bumping up against” certain concepts. When you bump up against something - make contact with it unexpectedly, not by design - you start treating it as real. It’s like Boswell’s old anecdote, dating back to 1763:

After we came out of the church, we stood talking for some time together of Bishop Berkeley’s ingenious sophistry to prove the non-existence of matter, and that every thing in the universe is merely ideal. I observed, that though we are satisfied his doctrine is not true, it is impossible to refute it. I never shall forget the alacrity with which Johnson answered, striking his foot with mighty force against a large stone, till he rebounded from it, ‘I refute it thus.’

Are any philosophers of mathematics willing to grant more “reality” to “natural” concepts than “artificial” ones? A “natural” concept is one you bump up against in the course of doing mathematics, while an “artificial” concept is something you could make up just for fun, but wouldn’t ever feel the need to study.

It’s not clear that “reality” is the right word for what the former concepts have that the latter don’t - but whatever you call it, it’s very important to most mathematicians, and any philosophy that neglects it is not a philosophy of real mathematics.