Following the award of the Abel Prize to Endre Szemerédi, there have been some fairly hot-blooded comments on a couple of blogs. One person even put a comment on Tim Gowers’ blog challenging him to respond to one of these comments.
Humans are ineluctably tribal. At my primary school, the children divided ourselves into those who rode horses or bikes to school (this was before there was a school bus). The most homogeneous group of people will find something to create a division. A feature of this tribalism is that we always see the divisions as much more absolute than they really are. In north Oxford, a wall was built to separate the council tenants from the residents of private houses, and remained for many years.
There are old stand-bys for dividing people: race, religion, and so on. And if all else fails, there is also the division by gender (which I always annoy people by mentioning).
Traditionally, mathematicians have been divided into pure and applied. According to the above principle, it is not surprising to find that G. H. Hardy’s pride in the complete uselessness of number theory appeals to quite a few people (even though it is no longer true; for forty years, number theory has been the basis of public-key cryptography, and it also has strong links with dynamics). Others divide us into discrete and continuous (prickly and gooey, as Aldous Huxley described them). I will yield to the temptation to give two terrible old jokes:
- There are three kinds of people: those who can count, and those who can’t.
- There are 10 kinds of people: those who understand binary notation, and those who don’t.
About ten years ago, Tim Gowers was brave enough to put forward another dividing line, in his essay on “The two cultures of mathematics”: there are the theory-builders and the problem-solvers. To the first group, the most important thing is a big theorem; to the second, a useful technique. I suspect that the division is not as clear-cut as Gowers made it, since when I try to explain it, I always get tangled up in apparent exceptions.
Anyway, Gowers’ two cultures, or tribes, seem to have evolved out of his control. Now, according to some commentators, “problem-solvers” means people who do “Hungarian combinatorics”, and Gowers thinks that they, and only they, are worthwhile mathematicians. (I exaggerate, only slightly, some positions I have seen expressed.) The deep-seated prejudice of many mathematicians against combinatorics is not dead, it seems.
Let me state as clearly as I can my view: there are not two cultures at all; there is no part of mathematics which is not intimately linked with the whole.
I remember encountering this prejudice from a mathematician who shall be nameless, who could not believe that Jean-Pierre Serre and Paul Erdős had anything in common, and challenged me to find a path between them in the collaboration graph. On the spot, I found a path of length 5 passing through me. If I recall correctly, it was Serre, Borel, Springer, Cameron, Erdős. (The true distance is smaller; go to MathSciNet and see for yourself.) Indeed, a glance at the data on collaboration on the Erdős number website should dispel any illusion that there is a cutoff, or even a bottleneck, between “real mathematics” and combinatorics, or indeed between any two artificial subdivisions of our subject. Of the roughly 400000 published mathematicians in 2004, in very round figures 20% have no joint papers, 10% belong to small components (mostly of size 2), and 70% belong to a single giant component which seems very highly connected and hard to subdivide.
Of course this graph is dynamic, and is constantly changing. When a young mathematician writes her first joint paper, it will probably be with someone in the giant component (maybe her PhD supervisor), and there is a pendant vertex for a time. But, if there are really two cultures in mathematics who find it difficult to collaborate, we would expect to find a large subset (a substantial fraction of the whole) with small boundary. I don’t know if there has been a systematic search for such a subset, but I would be prepared to bet that no such subset exists.
Since the collaboration graph captures at least some of the pairs of mathematicians who are sufficiently in tune to be able to work together, I contend this is as near as we will get to an objective test.
As for myself, my job title is “Professor of Mathematics”, which seems to me to describe what I do very well. I have published in the Journal of Mathematical Psychology, and more recently in Communications in Mathematical Physics, as well as the “usual places”. I do not think any more restrictive term would really describe what I do.
Peter Cameron's Blog 
See
http://www.epsrc.ac.uk/ourportfolio/themes/mathematics/Pages/default.aspx
for the current EPSRC view of how mathematics connects up.
There is only one kind of person. Those who divide people into two groups.
Somewhere near the heart of our mathematics is a dialog between the discrete and the continuous. The dichotomy — or whatever one wishes to call it — appears everywhere, beginning with the deceptively obvious counting numbers, the divisive rationals, Pythagoras’ casually stated theorem, the insidious square root of two ones. Consider the stubbornly integral three dimensions within which the stubbornly finite equivalence classes of mutually-similar Platonic solids are forever frozen.
Convergent series. Derivatives (first, second, third…) of continuous functions. The nameable, the computable and the forever inaccessible other 100% of the real numbers.
Ergodicity yes and no. Fourier’s Theorem.
One might argue that all such troubles begin with the insatiable human urge to declare two or more somethings equal — or rather — congruent. With that rash assertion we crumple our conceptual universe by attaching two of its points together. Only thus can anything become discrete. By our intervention.
The first word in human language must have been “no,” for “no” is the primal gesture which begins every characteristically human endeavor. “No” casts aside. By saying No to this, Yes to that, we instantly divide the universe. Good and bad, heretofore lovers, at once draw back from each other as enemies.
Blake, of all the artists, all the writers, all the dreamers, understood the bewildering necromancy of “No”:
I asked a thief to steal me a peach;
He turned up his eyes.
I ask’d a lithe lady to lie her down;
Holy & meek she cries.
As soon as I went
An angel came;
He wink’d at the thief
And smil’d at the dame,
And without one word said
Had a peach from the tree,
And still as a maid
Enjoy’d the lady.
That angel was never a particle! He was pure wave.
The discrete and the continuous are the nearest thing to a dichotomy that we have, but I completely agree with you that it is actually a dialogue rather than a division. Your examples are spot on too. Wave-particle duality is like Fourier series duality, which is the fact that a continuous group can have a discrete dual, or said otherwise, an operator on continuous functions can have a discrete spectrum. There are other points of contact too. John Maynard-Smith pointed to Hopf bifurcation as a process for producing discreteness from continuity, or quality from quantity as he put it. Indeed, you might say that this was the rather shocking message of catastrophe theory.
Children learn “no” long before they learn “yes”, not because “no” is more important, but because they accept everything until they learn the word “no”. Indeed, if you make a decision, the inputs are usually continuous (such as the degree of aggravation that various management diktats cause), but the output is certainly discrete.
I do like that Blake poem. Why didn’t they give us that to read at school, instead of (or as well as) the tiger and the lamb?
Negative operations (NOs), if not more important than positive operations (POs), are at least more powerful or generative, because the right NOs can generate all POs, but the reverse is not so.
Which brings us to Peirce’s amphecks, NAND and NNOR, either of which is a sole sufficient operator for all boolean operations.
In one of his developments of a graphical syntax for logic, that described in passing an application of the Neither-Nor operator, Peirce referred to the stage of reasoning before the encounter with falsehood as “paradisaical logic, because it represents the state of Man’s cognition before the Fall.”
Here’s a bit of what he wrote there —
C.S. Peirce • Relatives of Second Intention
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I re-read this today, as it was back in the “top ten”. One small correction: “prickles and goo” is due to Alan Watts, not Aldous Huxley. Huxley quotes Kabir by saying “Behold the One in all things; it is the second that leads you astray.”