### The Two Cultures of Mathematics Revisited

#### Posted by David Corfield

Where did we get to in our discussion of the two cultures of mathematics? To explore the possibility that interaction may be possible between what Gowers called ‘combinatorics’ and our Café subculture we were set the challenge of categorifying instances of the Cauchy–Schwarz inequality, which, unless I missed something, didn’t result in any noticeable success.

Now, an extreme wing of our subculture would take Urs’ remark

One knows one is getting to the heart of the matter

whenthe definitions in terms of which one conceives the objects under consideration categorify effortlessly.

and replace the **when** by **when and only when**.

I was probing in this direction in my paper, Categorification as a Heuristic Device, when I put:

A happenstantial equation is one which cannot be categorified productively.

The term ‘happenstantial’ is supposed to contrast with ‘law-like’. I had written something on the idea of mathematical laws inspired by reading Poincaré say

Les faits mathématiques dignes d’être étudiés, ce sont ceux qui, par analogie avec d’autres faits, sont susceptibles de nous conduire à la connaissance d’une loi mathématique de la même façon que les faits expérimentaux nous conduisent à la connaissance d’une loi physique. Ce sont ceux qui nous révèlent des parentés insoupçonnées entre d’autres faits, connus depuis longtemps, mais qu’on croyait à tord étrangers les uns aux autres.

Now, if we can’t categorify the Cauchy–Schwarz inequality, this would seriously weaken the extreme view. But perhaps we can think of more subtle ways of bridging the gap between the cultures. Terence Tao reports on Shing-Tung Yau’s lectures in UCLA’s Distinguished Lecture Series in a field “adjacent” to his own areas of expertise. Plenty of Yau’s themes over the three lectures are adjacent to Café interests too.

## Re: The Two Cultures of Mathematics Revisited

Yau’s extraordinarily verbal lecture notes are here: 1, 2, 3.