The Monster Group is well-known to mathematicians as the largest “sporadic group” (that is, a simple group that doesn’t fall into one of the few easily-defined infinite classes of simple groups), and was first generated as a group of rotations in 196,883-dimensional space.
The j function is something that I don’t really know anything about, but it apparently has important connections to the study of elliptic curves, and was studied by Gauss, Hermite, Dedekind, and Klein among others. It turns out that the j function can be represented as:
1/q + 744 + 196,884q + 21,493,760q^2 + …
In 1979, Conway and Norton noticed that 196,884 = 196,883 + 1, suggesting some connection between representations of the Monster Group and coefficients of the j function. This was confirmed when they noticed that the next smallest representation of the Monster Group is in 21,296,876-dimensional space, and that 21,493,760 = 21,296,876 + 196,883 + 1.
Investigation of this connection, and the search for the explanation of this regularity led to the development of the (apparently extremely productive) field of Monstrous Moonshine. (Don’t ever say mathematicians only have boring names for their theories.)
Now let’s consider another mathematical regularity (one I’ve mentioned before) that cries for an explanation. Hartry Field claims (in the introduction to Realism, Mathematics, and Modality) that if we are platonists and believe that most mathematical beliefs held by prominent mathematicians are true, then we should be able to explain this regularity. However, there is no plausible epistemology for platonic mathematics (unlike for the rest of the natural sciences), so we don’t seem to be able to explain this regularity. John Burgess and Gideon Rosen (in A Subject With No Object) give what Øystein Linnebo (in his excellent paper “Epistemological Challenges to Mathematical Platonism”) calls a “boring explanation” of this regularity. That is, they give a historical explanation of how mathematicians came to believe each particular mathematical fact they do, and then are done because the truth of the claims isn’t contingent.
However, note what would happen if we tried this in the Monstrous Moonshine example – instead of explaining why there’s a connection between the coefficients of the j function and the dimensions of representations of the Monster Group, we’d just explain why each particular coefficient is what it is, why each representation has the dimension it does, and then the numerical relations between them are just identities, and so don’t need further explanation, being necessary. If this sort of explanation were sufficient, we’d never have developed Monstrous Moonshine with all its benefits.
All this just goes to support Field against Burgess and Rosen in that we are correct to ask for a deeper explanation of mathematical facts than just the boring one. Linnebo points out why it’s so troublesome to say just what such an explanation would be like, and suggests that it might require a thicker notion of truth than many people want to deal with, but seems neutral about whether such an explanation is actually possible.
In other words, mathematical explanation is still a problem, and has important philosophical consequences for mathematics. But there are clear examples within mathematics itself, and we can look at how these work to give some guidance to a theory of explanation.