### The Two Cultures Again

#### Posted by David Corfield

People may have noticed that the two cultures of mathematics idea has a certain grip on me. In one culture there’s all the mathematics we love here at the Café, which by 2050 will be condensed into some beautiful statements about $\infty$-adjunctions between $\infinity$-toposes of space and quantity. Algebraic geometry and homotopy theory will find themselves simple consequences of this grand theory.

But will it make inroads into the other culture? Will it penetrate into combinatorics to cover, say, the Erdös discrepancy problem? You’d think not. So what makes for the difference?

Terry Tao has made some interesting comments on the subject, which I’ve gathered here. The first explains why it is difficult to establish a general theory of nonlinear partial differential equations. The second discusses the relationship between ‘structure’ as understood by the combinatorialist and the theory-builder, and offers the prospect of a degree of convergence. The third concerns the different kinds of condition on the entities dealt with in algebra and analysis. This last comment appeared on Buzz, which, as he tells us at *What’s New*, Tao uses as “an outlet for various things I wanted to say or share, but which were too insubstantial to merit a mention on this blog”.

I’ll reproduce the third comment here in case people would like to discuss it:

When defining the concept of a mathematical space or structure (e.g. a group, a vector space, a Hilbert space, etc.), one needs to list a certain number of axioms or conditions that one wants the space to satisfy. Broadly speaking, one can divide these conditions into three classes:

1.

closed conditions.These are conditions that generally involve an $=$ or a $\ge$ sign or the universal quantifier, and thus codify such things as algebraic structure, non-negativity, non-strict monotonicity, semi-definiteness, etc. As the name suggests, such conditions tend to be closed with respect to limits and morphisms.2.

open conditions.These are conditions that generally involve a $\neq$ or a $\gt$ sign or the existential quantifier, and thus codify such things as non-degeneracy, finiteness, injectivity, surjectivity, invertibility, positivity, strict monotonicity, definiteness, genericity, etc. These conditions tend to be stable with respect to perturbations.3.

hybrid conditions.These are conditions that involve too many quantifiers and relations of both types to be either open or closed. Conditions that codify topological, smooth, or metric structure (e.g. continuity, compactness, completeness, connectedness, regularity) tend to be of this type (this is the notorious “epsilon-delta” business), as are conditions that involve subobjects (e.g. the property of a group being simple, or a representation being irreducible). These conditions tend to have fewer closure and stability properties than the first two (e.g. they may only be closed or stable in sufficiently strong topologies). (But there are sometimes some deep and powerful rigidity theorems that give more closure and stability here than one might naively expect.)Ideally, one wants to have one’s concept of a mathematical structure be both closed under limits, and also stable with respect to perturbations, but it is rare that one can do both at once. Instead, one often has to have two classes for a single concept: a larger class of “weak” spaces that only have the closed conditions (and so are closed under limits) but could possibly be degenerate or singular in a number of ways, and a smaller class of “strong” spaces inside that have the open and hybrid conditions also. A typical example: the class of Hilbert spaces is contained inside the larger class of pre-Hilbert spaces. Another example: the class of smooth functions is contained inside the larger class of distributions.

As a general rule, algebra tends to favour closed and hybrid conditions, whereas analysis tends to favour open and hybrid conditions. Thus, in the more algebraic part of mathematics, one usually includes degenerate elements in a class (e.g. the empty set is a set; a line is a curve; a square or line segment is a rectangle; the zero morphism is a morphism; etc.), while in the more analytic parts of mathematics, one often excludes them (Hilbert spaces are strictly positive-definite; topologies are usually Hausdorff (or at least $T_0$); traces are usually faithful; etc.)

## Re: The Two Cultures Again

Hi, I’ve never commented here before. But I felt compelled to add two things:

(1) Prof. Gower’s Two Cultures always reminded me of the ontological distinction that Gilles Deleuze gave between axiomatics and problematics. According to Daniel W. Smith in “Mathematics and the Theory of Multiplicities”, Deleuze’s philosophy was more on the problematics side while Alain Badiou’s is more on the axiomatics side.

(2) I liked Prof. Tao’s various ways of conceptualizing the distinction between the algebraic and the analytic approach – and also the random, ergodic or probabilistic approaches. Gower’s Two Cultures essay (and Rota’s work in Indiscrete Thoughts) says a bit about what kind of methodology and point of view is peculiar to graph theory and combinatorics. As it is well-known, Atiyah has also written some stuff a few decades ago on the distinction betweeen the algebraic and the geometric point of view.

What about the other fields of mathematics? Like geometry, topology, mechanics, statistics, number theory, dynamics, etc?