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March 7, 2010

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 0T_0); traces are usually faithful; etc.)

Posted at March 7, 2010 10:11 AM UTC

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17 Comments & 0 Trackbacks

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?

Posted by: Burhan on March 7, 2010 3:32 PM | Permalink | Reply to this

How many dimensions? Re: The Two Cultures Again

This is another wonderful thread on a very important metaphysical meta-analysis. In the original “two cultures of Mathematics” statement, an axis was implicitly defined, and mathematicians and their word apread long it. Other axes have since been introduced. How many dimensions do we now have, in the metaphysical stances for Mathematics, the armamentaria of methodologies, the social networks of coauthorship? And what is the limit of this process of introducing new axes? The ∞-cultures of Mathematics?

Posted by: Jonathan Vos Post on March 7, 2010 5:38 PM | Permalink | Reply to this

Two Cultures: Mathematics and Physics

Sorry if this is off topic: While I do not have much to say about different cultures in mathematics, I certainly feel the dichotomy of mathematics and theoretical physics: While I can try to understand some topics in mathematics and in theoretical physics, I’m unable to do both at the same time:

- If I turn on the math part of my brain I’m unable to read the introductory chapter of any QFT text about canonical quantization of a scalar field (they introduce functions but never specify domain or codomain, the “Hilbert space” is rather a rigged Hilbert space etc.).

- If I turn on the physics part of my brain I find it hard to accept nonconstructive proofs or statements like the Banach–Tarski paradox.

This is similar to talking in foreign languages: Maybe you are fluent in two foreign languages, but try to do a conversation where you constantly have to translate from one to the other: This is much harder to do than sticking to one.

(One of my favourite anecdotes is this (from a QFT class):
mathematician:”So the Feynman integral is not a well defined mathematical object, why didn’t you say so?”
physicist: “That’s exactly the reason why I did not mention it, there is nothing to say!”)

Posted by: Tim van Beek on March 8, 2010 9:52 AM | Permalink | Reply to this

Re: The Two Cultures Again

I’m surprised Tao missed the simplest example of different conventions about degenerate objects: is 0 a natural number? (In my personal experience, most algebra texts say yes, most analysis texts say no.)

Posted by: Mark Meckes on March 8, 2010 5:00 PM | Permalink | Reply to this

Re: The Two Cultures Again

A closely related example is the ‘terminological calamity’ Tom discussed in this post: Is 0 a positive number? It depends whom you ask.

Posted by: Mark Meckes on March 8, 2010 5:04 PM | Permalink | Reply to this

Re: The Two Cultures Again

Strangely, when I posted the above, I failed to notice that the particular instance that came up in Tom’s post is a counterexample to the trends Tao pointed out. In this case, many analysts call 0 positive, while most mathematicians do not.

Posted by: Mark Meckes on March 8, 2010 11:40 PM | Permalink | Reply to this

Re: The Two Cultures Again

And we had that confusion with (strictly) positive (semi)-definite. If I remember right (see this) the analysts’ unmarked term includes the boundary case. So that resembles what you say about integers.

Posted by: David Corfield on March 9, 2010 10:36 PM | Permalink | Reply to this

Quantifiers and Pullbacks?

How does this viewpoint interact with Lawvere’s characterization of the existential and universal quantifiers as adjoints of pullbacks?

Posted by: Scott McKuen on March 11, 2010 2:57 AM | Permalink | Reply to this

Re: Quantifiers and Pullbacks?

I suppose as left and right adjoints, they will preserve colimits and limits respectively. E.g.,

x(F(x)G(x))xF(x)xG(x), \exists x(F(x) \vee G(x)) \Leftrightarrow \exists x F(x) \vee \exists x G(x),

and

x(F(x)&G(x))xF(x)&xG(x). \forall x(F(x) \& G(x)) \Leftrightarrow \forall x F(x) \& \forall x G(x).

That seems to tally with what Tao says.

Posted by: David Corfield on March 12, 2010 1:38 PM | Permalink | Reply to this

Re: The Two Cultures Again

One of the things I find interesting about the two cultures is that it is hardly ever mentioned to students of mathematics. We are all aware of the two branches of math, i.e. pure and applied, yet nobody tells us about the two cultures. Somebody quoted Noam Elkies, where he mentioned that Harvard is more theory oriented. Would it help students to be made aware of these sorts of things rather than it being allowed to be discovered by chance?

Posted by: kim on March 14, 2010 7:07 AM | Permalink | Reply to this

Re: The Two Cultures Again

Somebody quoted Noam Elkies…

That would be me here.

Posted by: David Corfield on March 14, 2010 8:33 PM | Permalink | Reply to this

Re: The Two Cultures Again

Would it help students to be made aware of these sorts of things rather than it being allowed to be discovered by chance?

If it’s a robust a distinction as it appears, then I think it would be a good thing to point out at some stage. As an undergraduate I remember being in a state of almost complete ignorance as to the large-scale structure of mathematics.

Posted by: David Corfield on March 16, 2010 10:29 AM | Permalink | Reply to this

Re: The Two Cultures Again

Seven years after finishing my PhD I still feel pretty ignorant of the large-scale structure of mathematics. But I’m filling in my understanding of it bit by bit.

I think the two-cultures distinction is quite robust, albeit to some extent necessarily an oversimplification just like the pure/applied dichotomy. Going back to kim’s question, it probably would be interesting to students to have it pointed out after they’ve seen enough instances of it to corroborate from their own experience, which probably isn’t until graduate school for many students. I’m not sure whether pointing out the distinction explicitly would “help” students in a concrete way. However, in my own teaching of more “problem-solving”-oriented subjects, I do make a point of emphasizing the general principles that come up again and again, and tell the students that understanding those principles is at least as important as knowing the big theorems. So I’m at least trying to help the students understand that culture in a more explicit way.

Posted by: Mark Meckes on March 16, 2010 2:07 PM | Permalink | Reply to this

Re: The Two Cultures Again

David wrote:

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

You’ve raised this kind of question before. I find it interesting that I haven’t seen you raise the obvious complementary question. Isn’t it equally interesting to wonder whether some present-day “theory-building”-oriented field will begin to develop in some direction in which many concrete problems will be solved by the use of powerful general principles that resist being summarized in powerful general theorems?

Posted by: Mark Meckes on March 16, 2010 2:20 PM | Permalink | Reply to this

Re: The Two Cultures Again

You have detected in me a bias towards theory-building.

Posted by: David Corfield on March 16, 2010 4:04 PM | Permalink | Reply to this

Re: The Two Cultures Again

Aha! Got you!

Actually, I was kind of hoping for an answer like, “Actually, the first question seems more natural because…” that might lead to some interesting discussion.

Posted by: Mark Meckes on March 18, 2010 2:25 PM | Permalink | Reply to this

Re: The Two Cultures Again

Can I ask whether there exists a similar situation in mathematical physics where there are two cultures? It’s interesting to compare the situation with mathematics.

Posted by: kim on March 16, 2010 8:21 PM | Permalink | Reply to this

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