Recently Isabel Lugo asked about problems that are hard for intermediate values of some parameter, and in discussing the question I got to thinking about exceptional structures in mathematics such as the sporadic groups. In 2006 David Corfield asked about how “natural” the sporadic simple groups are at the n-Category cafe. In that discussion and more generally there seem to be approximately two extremes in perspective:
- Exceptional structures represent a lack of room for asymptotic behavior to occur; thus they are distractions from the “generic” case. This seems to be the case for certain exceptional isomorphisms; there are only so many groups of a particular small order. It also seems to be a good way to think about objects that behave fine in characteristic zero or high characteristic but behave badly in low characteristic, characteristic
usually being the worst offender.
- Exceptional structures represent the deepest part of a theory, and the exceptional structures in different fields are often related; thus understanding exceptional structures is crucial. This seems to be the case for the octonions, which can be thought of as an underlying cause of Bott periodicity. It also seems to be a good way to think about objects related to the number
; John Baez tells a great story about connections between the Leech lattice, the Dedekind eta function, string theory, and elliptic curves all centered around this mysterious number.
So what do you think? Are exceptional structures accidents or miracles? (Or, as a third option: am I failing to distinguish carefully enough between interesting and uninteresting exceptional structures?)
Annoying Precision 
[…] also have a few draft articles for the blog, in various stages of completion. One grew out of my comment about exceptional structures and has to do with mathematical beauty. It’s pretty […]
Take, for example, the outer automorphism of
. Do you think its existence is a “coincidence” or do you think there is some “deeper meaning” behind it?
From a philosophical standpoint, what do you mean by an “accident” or a “miracle” in mathematics (not a discovery, but about something that already exists)? I’m having difficulty understanding your question.
I’m going to play the “false dichotomy” card here (:P) but not before first pointing out Doron Zeilberger’s opinion on necessary vs. contingent beauty.
So let’s take as an example the cluster of mindblowing exceptional objects (the Golay code, the Leech lattice, the Conway and Mathieu sporadic groups among others) that you and Baez mention surrounding the number 24. I don’t claim to fully understand all the mathematics behind the connections between these things (I doubt anyone seriously could), but let’s take Baez’ word that it has to do partially with the fact that
is a perfect square. (Here’s hoping Latex works in comments…) Now of course using some pretty basic number theory, you can show that this is a perfect square (replacing 24 by 
iff one of 
is 6 times a perfect square, and both the other one and 
are perfect squares.
So let’s treat the perfect squares as random, modulo what we needed for the number theory — then given some
, there’s a 
probability that 
is a perfect square. Integrating that from 1 to infinity, you come out with an expected value of 2/9.
Now, of course, 2/9 isn’t great — I wouldn’t bet my life savings that there’s some number with property P given that the expected number of integers with property P is 2/9 or anything — but it’s not terrible, either, and a priori there’s no reason to think that there aren’t numbers with the perfect-squares property. But, of course, here’s the thing — there are lots and lots of such nice numerical properties that would lead to other exceptional objects, and we’re bound to see some of them actually occur!
This is sort of the metamathematical corollary of Littlewood’s law; mathematical “miracles” should happen by chance alone. They may have particularly beautiful explanations (as in the case of the Leech lattice, the sporadic groups, and the Golay codes) or they may not (as in the case of, say, the four-color theorem) but it might be best to call them all accidental miracles, or maybe miraculous accidents, instead of trying to arbitrarily distinguish them.