Tim May wrote >(I was struck by the point that the sequence "1, 2, 4, 8" is the only >sequence satisfying certain properties--the only "scalars, vectors, >quaternions, octonions" there can be--and that the sequence "3, 4, 6, >10," just 2 higher than the first sequence, is closely related to >allowable solutions in some superstring theories, and that these facts >are related.)

That's indeed what amazes me the more. I always thought that the dimension justification in string theories was unconvincing, but with the octonion apparition there, I must revised my opinion. Needless to say I hope octonions will appear in the Z1* semantics! (so we could extract string theory from comp directly). Do you know that Majid found a monoidal category in which the octonions would naturally live, even (quasi)-associatively, apparently. I think the sedenions (16 dim) could play a role too, even if they do not make a division algebra. cf the (not really easy) 1998 paper by Helena Albuquerque and Shahn Majid "quasialgebra structure of the octonions". For the paper and some other see http://arXiv.org/find/math/1/ti:+octonions/0/1/0/1998/0/1 All that gives hope for finding the generalized statistics we need on the (relative) consistent histories or observer-moments (i.e, with AUDA, a Z1* semantics). Well... let us dream a bit... ;-) Bruno