Tim May wrote: > As I hope I had made clear in some of my earlier posts on this, mostly > this past summer, I'm not making any grandiose claims for category > theory and topos theory as being the sine qua non for understanding the > nature of reality. Rather, they are things I heard about a decade or so > ago and didn't look into at the time; now that I have, I am finding > them fascinating. Some engineering/programming efforts already make > good use of the notions [see next paragraph] and some quantum > cosmologists believe topos theory is the best framework for "partial > truths." > > The lambda calculus is identical in form to cartesian closed > categories, program refinement forms a Heyting lattice and algebra, > much work on the fundamentals of computation by Dana Scott, Solovay, > Martin Hyland, and others is centered around this area, etc.
FWIW, I studied category theory carefully years ago, and studied topos theory a little... and my view is that they are both very unlikely to do more than serve as a general conceptual guide for any useful undertaking. (Where by "useful undertaking" I include any practical software project, or any physics theory hoping to make empirical predictions). My complaint is that these branches of math are very, very shallow, in spite of their extreme abstractness. There are no deep theorems there. There are no surprises. There are abstract structures that may help to guide thought, but the theory doesn't tell you much besides the fact that these structures exist and have some agreeable properties. The universe is a lot deeper than that.... Division algebras like quaternions and octonions are not shallow in this sense; nor are the complex numbers, or linear operators on Hilbert space.... Anyway, I'm just giving one mathematician's intuitive reaction to these branches of math and their possible applicability in the TOE domain. They *may* be applicable but if so, only for setting the stage... and what the main actors will be, we don't have any idea... -- Ben Goertzel