The Two Cultures of Mathematics: A Rebuttal

Gowers writes thoughtfully about combinatorics here, in an essay which references Snow’s famous lectures (or famous amongst mathematicians – I’ve never met anyone else who has ever heard of them). The trouble, however, starts (as it often does) with the invocation of the word “obvious”:

It is equally obvious that different branches of mathematics require different aptitudes.

I do not think that this claim stands up to scrutiny. By “aptitude,” Gowers specifically distinguishes the following two abilities: problem-solving and theory-building. Here algebraic number theory is singled out as area which is firmly tilted towards theory-builders. Yet the vision of algebraic number theory as a rising sea with progress signaled by the application of (to quote Gowers) deep theorems of great generality is not, in my opinion, an accurate reflection on reality.

A good lemma is worth a thousand theorems. Gowers describes various principles of combinatorics which (he suggests) play the role of (a direct quotation again) precisely stated [general] theorems. Yet examples similar to his are readily available in algebraic number theory. Consider, for example, the following Lemma of Ribet (modified from its original formulation):

If a reducible representation $U \oplus V$ of a group $G$ deforms continuously into an irreducible representation of $G$, then either there exists a non-trivial extension of $U$ by $V$, or an extension of $V$ by $U$.

As a mathematical result, this is not particularly deep. For example, if $G$ is finite, it relates two well known facts: there are no extensions between irreducible representations (Maschke’s theorem), and representations of finite groups are defined over number fields (and so do not deform). Yet this lemma is a crucial ingredient behind many key results (Ribet’s construction of unramified extensions, the proof of the main conjecture of Iwasawa Theory by Mazur-Wiles, the non-triviality of the Selmer group of an ordinary Elliptic curve with $L(E,1) = 0$ by Skinner-Urban, and many more). It seems to me (as in the examples Gowers discusses) that the value of this lemma is not in its difficulty, but in the principle it encapsulates: in order to construct extensions of $U$ by $V$, try to deform $U \oplus V$.

It is a common graduate student error to imagine that mathematics consists merely of judicious applications of highly technical machinery. But I am not accusing Gowers of making this mistake; I would expect him to argue that algebraic number theory is not exclusively the domain of theory-builders, but rather only strongly slanted in that direction, and to that end, he might point to Grothendieck. A fascinating essay on Grothendieck may be found here. Grothendieck contrasts his own way of thinking with that of Serre, whom he describes as using the hammer and chisel approach, which might loosely be considered synonymous to “problem-solver” (and I would count Serre as someone who has worked in algebraic number theory). Note that, despite the merits of Grothendieck’s work, he famously failed to prove the Weil conjectures (by “failing” to prove the standard conjectures) and it required Deligne’s use of the tensor power trick (a problem solving technique par excellence) to finish the argument. Thus, while Grothendieck’s role in modern number theory is significant, it would be an error to imagine that it constitutes the whole subject.

Perhaps Gowers would instead argue that what set combinatorics apart from (say) algebraic number theory is not that it requires problem solvers while the latter field does not, but that (in contrast) it is the exclusive domain of problem solvers. There’s a hint of this opinion in the following quote:

One will not get anywhere in graph theory by sitting in an armchair and trying to understand graphs better.

Why is this claim any more convincing than the same statement with the word “graphs” replaced by the word “rings”? I don’t see any a priori reasons why there cannot be a Grothendieck of graphs. If the history of mathematics teaches us anything, it is that the nature of a subject can change quite radically over a relatively short period of time (say 30 years). I am not claiming that there is no difference between combinatorics an algebraic number theory. There may well be a difference in the overall structure of the field, the level of background, the need to understand ideas in a broader conjectural framework, etc. And I might also consider agreeing to the claim that these fields, as they are currently constituted, may well be better suited to different personalities. But it is my opinion that the divide between the type of mathematics required for either subject is not as great as Gowers claims it is.

Gowers main point is that a significant part of the mathematical establishment looks down on combinatorics as not being “deep”, and that this attitude is both harmful and ignorant. On this point, I think that Gowers criticisms are fair, accurate, and valuable. It’s undeniably true that there are many graduate students who fall in love with formalism to the detriment of content, and milder forms of this predujice are pervasive throughout mathematics. To this end, I think Gowers’ essay is timely and relevant. However, I can’t help but sense a little that, perhaps after having spent a career defending combinatorics against ignorant snobs, Gowers suffers from the opposite prejudice, where “theory-builders” are a short distance away from empty formalists, sitting comfortably in their armchairs thinking deep thoughts, studying questions so self referential that they no longer have any application to the original questions which motivated them (this sense also comes from reading some of the remarks on the Langlands programme here).