### The Two Cultures of Mathematics

#### Posted by David Corfield

Part of what intrigues me about reading Terence Tao’s blog is that he displays there a different aesthetic to the one largely admired here. The best effort to capture this difference is, I believe, Timothy Gowers’ essay The Two Cultures of Mathematics, in which the distinction is made between ‘theory-builders’ and ‘problem-solvers’. I think we have to be very careful with these labels, as Gowers himself is.

…when I say that mathematicians can be classified into theory-builders and problem-solvers, I am talking about their priorities, rather than making the ridiculous claim that they are exclusively devoted to only one sort of mathematical activity. (p. 2)

To avoid misunderstanding, then, perhaps it is best to give straight away paradigmatic examples of work from each culture.

Theory-builders: Grothendieck’s algebraic geometry, Langlands Program, mirror symmetry, elliptic cohomology.

Problem-solvers: Combinatorial graph theory, e.g, Ramsey’s theorem, Szemerédi’s theorem, arithmetic progressions among the primes.

Gowers mentions Sir Michael Atiyah as a prime example of a theory builder, and recommends his informal essays, the ‘General papers’ of Volume 1 of his Collected Works. Indeed, they convey an aesthetic which I came to admire enormously as a PhD student in philosophy. On the other hand, Paul Erdös was a consummate problem-solver. What then of the corresponding aesthetic?

One of the attractions of problem-solving subjects, which Gowers collects under the loose mantle ‘combinatorics’, is the easy accessibility of the problems.

One of the great satisfactions of mathematics is that, by standing on giants’ shoulders, as the saying goes, we can reach heights undreamt of by earlier generations. However, most papers in combinatorics are self-contained, or demand at most a small amount of background knowledge on the part of the reader. Contrast that with a theorem in algebraic number theory, which might take years to understand if one begins with the knowledge of a typical undergraduate syllabus. (p. 12)

For someone who had recently won a Fields’ Medal, it would seem strange to feel the need to defend one’s interests, but after describing a problem involving the Ramsey numbers, Gowers writes:

I consider this to be one of the major problems in combinatorics and have devoted many months of my life unsuccessfully trying to solve it. And yet I feel almost embarrassed to write this, conscious as I am that many mathematicians would regard the question as more of a puzzle than a serious mathematical problem. (p. 11)

Two types of appeal which are commonly made to warrant the importance of one’s field are its connections to other fields and its applicability. Now,

As for connections with other subjects, there are applications of combinatorics to probability, set theory, cryptography, communication theory, the geometry of Banach spaces, harmonic analysis, number theory … the list goes on and on. However, I am aware as I write this that many of these applications would fail to impress a differential geometer, for example, who might regard all of them as belonging somehow to that rather foreign part of mathematics that can be safely disregarded. Even the applications to number theory are to the “wrong sort” of number theory. (p. 13)

The Green-Tao theorem might be a good candidate to illustrate this “wrong sort” of number theory.

Now, it’s not that, on the theory-building side, all number theoretic results emerging from the “right sort” of number theory are deemed important. Indeed, in TWF 217, John writes:

Now, personally, I think Fermat’s Last Theorem is a ridiculous thing. The last thing I’d ever want to know is whether this equation:

$x^n + y^n = z^n$

has nontrivial integer solutions for $n \gt 2$.

Rather, it was the activity behind the scenes leading to the proof of the Taniyama-Shimura conjecture that is generally regarded as the major achievement. So, even were results about the existence of arithmetic progressions amongst the primes to be judged similarly as ‘ridiculous’, there might again be some general result lurking behind the scenes. However, according to Gowers, in combinatorics one deals not so much with general theorems, but rather broad principles, such as:

if one is trying to maximize the size of some structure under certain constraints, and if the constraints seem to force the extremal examples to be spread about in a uniform sort of way, then choosing an example randomly is likely to give a good answer. (p. 6)

Several similar principles are given by Tao in this talk.

If one is according importance to mathematical activity in terms of its impact on mathematics as a whole, then rather than the transfer of theoretical results and apparatus between fields, it may be necessary to look to more subtle relationships, such as when:

Area A is sufficiently close in spirit to area B, that anybody who is good at area A is likely to be good at area B. Moreover, many mathematicians make contributions to both areas. (p. 14)

Finally, there remains the question of whether there are missed opportunities arising from the presence of a barrier between the two cultures. Gowers ends his essay by encouraging dialogue. Perhaps blogs are the right arenas in which such dialogue might take place.

## Re: The Two Cultures of Mathematics

Erdos was considered the Prince of Problem Solvers and the King of Problem Posers.

One of the things I like in Terrence Tao’s informal essays on his formal solutions is the narrative. He provides emotional texture to the subproblems. Numbers “want to be primes” but are prevented by “conspiracies.” This, to me, suggests that this particular genius has learned a method to use both left-brain and right-brain methodologies in optimum combination.

This tracks back, in this blog, to the threads on Story in Mathematics.