Sunday, December 24, 2006

Deep mathematics (vs. the other kind)

Let me belatedly join in the fray on Scott's blog, revolving around what counts as "deep" math. The discussion at times degenerates into a micturition contest, as such things are liable to, but interesting points are certainly raised. First, I have to take Scott to task for the comment, "The theorems in real analysis all seemed like painstaking formalizations of the obvious". I don't know what material you covered in Real, but: quick, without consulting textbooks, notes or the web (1) does there exist an everywhere continuous and nowhere differentiable function f:R->R? (2) what if we add the additional restriction that f be monotone?

If the answers to these were obvious to you before taking Real, then it's a real shame you didn't pursue analysis as a specialty. The year I'd graduated with a math degree from Princeton, I still made the shameful blunder of convincing myself that I'd constructed a continuous, monotonic, nowhere differentiable function (fortunately, I recalled the Lebesgue-Radon-Nikodym theorem before I had a chance to really embarrass myself). The latter, btw, is an example of a Truly Deep (tm) theorem from undergrad analysis.

Of course, it's silly to debate the relative "depth" of different mathematical disciplines. I'll grant you that P?=NP occupies a central place because of its philosophical implications (and it's a real pleasure to read your expositions on these). But I tend to avoid mine-is-deeper-than-yours contests as they produce nothing useful and occasional animosity. A "deep" result is any claim that's surprising, nontrivial, and has rich implications. It need not be paticularly difficult (the Gelfand-Mazur theorem is plenty deeep, but has a 1-line proof).

The "two cultures" paper (linked by Elad Verbin) is certainly relevant here (on top of being an educational and enjoyable read). I've always found discrete math more difficult to reason about than continuous. What makes combinatorics so difficult is that "continuity of intuition" fails. Things often go wrong not because you mis-formulated some technical condition but because the "structure" of the problem is so difficult to grab hold of. The hardest things I've proved haven't been terribly technical -- indeed, the techniques I used were elementary. The key part has always been was understanding the elusive structure of the objects I was working with. Alexandre Borovik refers to these two types of thinking by "switches" and "flows" in his book. I'd like to read the cognitive explanation more closely sometime (and of course would be thrilled if someone would summarize it).

PS. Don't mind Conway's off-handed dismissal of The Central Question. When you have a group named after you, you can pretty much say anything at all.


Scott said...

Thanks, Leo -- the question about an everywhere-continuous nowhere-differentiable function is really nice! I thought about it in the shower, and convinced myself that such functions must exist -- since (for example) you could take an infinite sum of periodic functions, with both the period and the amplitude decreasing exponentially, in order to get a fractal-like deal. Then I googled it, and found that was basically the example given by Weierstrass. I wish we'd done more problems like that one in my analysis class!

Regarding Conway's dismissal of the Central Question, I would slightly rephrase your comment: when you have a group named after you, you think you can pretty much say anything at all. :-)

Alexandre Borovik said...

This is the problem with blogs -- the best bits are frequently buried in archives. Thanks to your post I red the old dispute on Scott' blog -- a fascination discussion which leaves the matter unresolved. So, what is 'deep'? Thanks for re-gniting the dispute.

Aryeh said...

I guess deep math is like pornography -- hard to define, but I know it when I see it!

Aryeh said...

Scott -- in my humble view, analysis, at its best and "deepest", is about developing the right intuition about the behavior of un-visualizable quantities. Note that in your (er, Weierstrass's) example, the function you produce is very jumpy (everywhere so). Is that necessary? Could a monotonoe function be nowhere differentiable? (It's easy to see that it can have at most countably many discontinuities.)

Having spent a few years with Rudin's books at my bedside (and even making occasional use of the stuff in my work), it takes an effort to recall that this intuition is anything but natural or obvious. The name of this blog is partly a reflection of this view of analysis.

Alexandre, was I basically correct in my understanding of your "switches and flows" terminology? So the construction of the Lebesgue integral by approximating measurable functions by simple ones is a "flows" argument (and a very intuitive one at that!). Proving, say, Vandermonde's convolution formula (by the combinatorial method) is an example of a "switches" argument -- right?

gudipudi said...
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