There's a curious sense in which almost no one really feels comfortably mainstream, regardless of how they stand with respect to the cohomological divide, or even of their community status. The Grothendieck phenomenon is rather an obvious example, but there are many others. If we venture outside of mathematics proper, Noam Chomsky, often referred to as the most cited intellectual alive, frequently speaks of himself as an outsider. (Specifically in relation to his linguistics, not his politics.)

Of course, it’s tempting to speculate about the honesty of such self-perception, but I tend to think of it as largely reflective of the human condition. It may also be that this kind of view goes well with a sort of rebellious energy conducive to creative intensity. For people who like literature, the sensibility is wonderfully captured in the novella `Tonio Kroeger’ by Thomas Mann. The irony is that almost anyone who reads the story is able to relate to the loner, as is also the case with the typical rebel in simpler dramas.

Why go far? Here we have Tim Gowers, an enormously respected mathematician by any standard, apparently presenting himself as a spokesman for the tributaries. In his case, I take it as the prototypical gentlemanly self-effacement one finds often in Britain.

At the very least, the whole picture is complicated.

The point is it’s probably not worth spending too much energy on this question. Administrative constraints, classifications, and selections are a real enough part of life within which we have to find some equilibrium, but serious mathematics has too much unity to be divided by the watery metaphor.

David Corfield once (good-humouredly) misquoted me with regard to the perceived distinction:

'Which do you like better, the theorem on primes in arithmetic progressions or the one on arithmetic progressions in primes?’

The original context of that dichotomy, however, was a far-fetched suggestion that there should be a common framework for the two theorems.

*Added*: The more I think of it, the more it seems that the original thrust of cohomology was very combinatorial, as might be seen in old textbooks like Seifert and Threlfall. The way I teach it to undergraduates is along the lines of:

space $X$ --> triangulation $T$ --> Euler characteristic $\chi_T(X)=V_T+F_T-E_T$ --> $T$-independence of $\chi(X)$ --> dependence of $V_T$ etc. on $T$ --> 'refined incarnation' of $V_T, E_T, F_T$ as $h_0$, $h_1$, and $h_2$, which are independent of $T$--> refined $h_i$ as $H_i$.

The emphasis throughout is on capturing the combinatorial essence of the space.

with what purpose? – Mariano Suárez-Alvarez♦ Mar 5 '10 at 21:52