Eduardo J. Dubuc wrote:
I wish you (V.P.) were more clear. I can not see what is your point. Witty msages for the Illuminati don't serve any pourpose, except amusement to some.
Yes, sorry about that. A couple of others wrote privately with the same request. I have no problem with the notion of mathematical truth per se, which I imagine to be what all mathematicians seek, along with mathematical tools and a consensus thereon by their colleagues. What I had in mind by the "haunting" remark is that the implications of Goedel's incompleteness results don't immediately leap out at one, and there is a certain optimistic tendency to minimize those implications and continue to argue the issues as though Goedel's theorems weren't relevant. We can't *define* mathematical truth (Tarski may have been the first to enunciate that implication most clearly), yet we can often recognize it when we see it. Learning to do mathematics amounts to learning how to find and communicate those mathematical truths that are easily recognized as such by other mathematicians according to community standards. We imagine that mathematics on Arcturus must be like ours, but mathematics is an intrinsically cultural subject and I don't see why Arcturan mathematics should be like ours. Do Arcturans have logic? Do they have algebra? Do they draw a distinction between the two? Do they believe in either? Add category theory as a third framework and ask the same questions of it. Do they know about initial algebras and final coalgebras, and if so which came first for them? Do they know about monads and adjunctions, and if so which came first? That's surely too brief to be clear. I'd be happy to engage further in this sort of speculation on the practice of mathematics as a cultural issue. I have less to contribute on the intrinsic nature of mathematics itself for lack of insight into its scope. Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]