Dan, I already knew what you said about real division algebras and topology. However, I do not understand those topological proofs. I in fact asked Milnor for help at one point. I have a conjecture that we ought to be able to use "generalized smoothness" (a new topological notion I introduced in this paper) to prove that the only real division algebras, (wide sense) were 1,2,34,8,16 dimensional. Milnor and another Topologist, Denis Sullivan;, agreed this was interesting problem. However, Milnor said he no longer was able to do this stuff and no longer remembered how the topological proofs he was so famous for, work. Sullivan claimed he was going to work on it, but if so, he didn't accomplish anything, at least not that he told me about. I suggest reading my paper, especially the parts about division, e.g. sections 19-23. Here are some quotes: nnacd = not necessarily associative, commutative, or distributive. 2^n-ons: for n=1,2,3,4 these are the reals, complexes, quaternions, octonions, and my 16-ons. 19. What is a “division algebra” and are the 2^n-ons one? One (boring) definition of “division algebra” is “an algebra in which xy = 0 implies x = 0 or y = 0.” By that definition, all of our 2^n-ons are nnacd “division algebras” simply because of the multiplicativity of Euclidean norm. A stronger definition of “division algebra” is “an algebra in which, given b and given a not= 0, 1. there exists z so that az = b, and 2. there exists q so that qa = b.” Finally, a still stronger definition would require, not only that the solutions z and q of these two division problems must exist, but that, furthermore, they must be unique. The first of these division problems, for our 2^n-ons, trivially does have a unique solution z... The second division problem, however, is not so trivial because there is no right-cancellation law in the 16-ons... Theorem 70 (16-on division). There exists a unique solution to any generic 16-on division problem. -- Note also, my 16-ons do NOT satisfy the distributive law ("nonlinear algebra") although they do satisfy right-distributivity and weakened versions of left-distributivity. The 16-ons are very interesting and I would like to see some use found for them. In any event, they do not deserve to be lightly dismissed as though I were ignorant. Had you bothered to look at my paper, you would have found I'd already said everything you just told me, plus considerably more, as part of my review of that area. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)