The n-Category Café

Well, I’ll repeat the question at the link. Does anyone have a reference on “Transformation Dialectics”, in the math teaching context?

Does John have something of the mystic in him? From his 30 September entry of his diary:

The material [categorifying everything associated to simple Lie groups and quantum groups] is getting really fascinating at this point, after several years of work. Enormously deep patterns. It’s like peering down into the foundations of the universe and seeing the mysterious machinery that makes it tick.

Urs wrote:

In fact, trying to identify underlying structures in math is indeed a lot like trying to identify a secret conspiracy.

Once you see the same pattern hidden in a couple of apparently unrelated contexts, you begin to wonder if you have stumbled upon footprints left by an unknown institution, which secretly governs all these fields.

Heh - that’s a fun way to put it!

Louis Crane compared it to digging up the fossil of an enormous dinosaur.

I sometimes say that math is the field where it most pays to be paranoid, because in math it’s really true that nothing is a coincidence.

However, this leads to a discussion of whether it’s really really true that nothing is a coincidence in math. People like Chaitin make a good case that there’s a lot of algorithmic randomness built into math, so that lots of things must be true “by coincidence”.

For example, if you treat the primes as randomly chosen with a density roughly like 1/log($n$), a simple calculation shows that Goldbach’s conjecture has a very high chance of being true after you’ve checked it for small even numbers - and you can make this chance as high as you like, by checking enough small even numbers. It’s possible that this is all there is to it.

But, you could try a similar probabilistic approach to Fermat’s last theorem, and you would miss out on the Taniyama-Shimura conjecture. In other words: there really is a cosmic conspiracy ruling out nontrivial integer solutions of $x^n + y^n = z^n$, but you have to dig extremely deep to see it.

The same kind of thing could be lurking beneath Goldbach’s conjecture, only so deep we can’t even sense it yet. There’s never any way to rule this out. After all, you can prove that if Goldbach’s theorem is true, you can never prove you can’t prove it!

The situation with the 26 sporadic finite simple groups is touch-and-go: we have some people here arguing that some of these exist “by chance”, while others are holding out for a deep “reason” we haven’t seen yet.

But I’m digressing a bit - the patterns Urs is talking about are quite different: they are deep “structural” patterns relating different branches of mathematics. Nobody has ever made a convincing case that these are “coincidental”. The only two choices I’ve heard are that 1) they reflect something deep about the mathematical universe, or 2) they reflect something deep about how we think.

Unfortunately, it’s hard to tell the difference between 1) and 2). It may always be hard - at least until we meet, build or breed nonhuman mathematicians.

Alas, right now the closest thing to a nonhuman mathematician is a African grey parrot named Alex who can count, read numbers and may have a primitive notion of zero. It’ll be a while before Alex learns category theory.

On doing mathematics for spiritual reasons: I had one undergrad pure math prof who began every lecture with five minutes of chanting in order “to get us in a receptive mood” for the math, and several others who spoke of us finding proofs for already-proved theorems “for the good of our souls”. Although the language they used was (most likely) metaphorical, it is interesting that the metaphor was spiritual rather than instrumental.