Dear Peter Woit,
You wrote, "In this essay I'll argue that unified theories of fundamental physics are closely linked with some of the great unifying structures that mathematicians have found to underlie much of modern mathematics." If nature is infinite, then G2, F4, E6, E7, and/or E8 might be the basis for a unified theory. If nature is finite, then the monster group and the 6 pariah groups might be the basis for a unified theory. The space roar and the photon underproduction crisis suggest that nature might be finite (or at least that our universe has a finite wavelength and undergoes cycles of expansion and instantaneous quantum collapse).
In "Is String Theory Even Wrong?", "American Scientist", March-April 2002, you wrote: "... string theory predicts that the world has 10 space-time dimensions, in serious disagreement with all the evidence of one's senses. Matching string theory with reality requires that one postulate six unobserved spatial dimensions of very small size wrapped up in one way or another. All the predictions of the theory depend on how you do this, but there are an infinite number of possible choices, and no one has any idea how to determine which is correct." Your objection does not apply to my physical interpretation of string theory. Suppose that string vibrations are confined to 3 copies of the Leech lattice. Imagine 36 different quarks moving in 36 different particle paths. These 36 dimensions might be approximately isomorphic to 26 dimensional bosonic string theory with 10 dimensions for a general relativistic model. Such an approximation scheme might yield a physical interpretation of string theory that takes the place of curling up of extra spatial dimensions. The idea is to replace supersymmetry with some version of Wolframian pseudo-supersymmetry.