From: james dolan Subject: Re: Category theory vs. Set theory questions Date: Thu, 01 Feb 2001 02:33:40 GMT Newsgroups: sci.math Summary: Comparing category theory and set theory as foundational approaches bill taylor wrote: |james dolan writes a very useful essay on "the |other side": | |||> the category philosophy here is that if you want to think of the |||> elements of a set s as complex structures of some kind (and it's |||> often crucial to do so), you accomplish this _not_ by making any |||> attempt to describe any sort of "intrinsic internal structure" of |||> the elements of s, but rather by putting explicit external |||> structure on s, in the form of maps to and from s and (possibly) |||> other sets. | |Yes, that sums up in a nutshell what it is all about. An excellent |summary, and very helpful. Though again we note it implicitly seems |to assume something has to "already be there" in order to *have* any |structure. Cat is the study of functional structure per se, and as |such it is good. But I repeat, that doesn't really seem all that |foundational; that depends on what goals you expect a "foundation" to accomplish. "pulling itself into existence by its own bootstraps without circularity" and "eliminating doubt about its consistency" and so forth are the sort of "sacred" and probably unrealistic goals one might naively have hoped a foundation could accomplish before they found out the way things actually are. the only real goals for a foundation these days are the "profane" ones: 1. provide a language in which the things working mathematicians want to say can easily be said and: 2. make the circularity in its formulation blatant and explicit enough that interesting theorems can be stated and proved about this circularity. (#2 here is actually a special case of #1.) the traditional epsilon-universe-of-sets foundation succeeds quite well in making the circularity in its formulation blatant and explicit. the basic foundational cycle here goes something like this: we believe (or formalisticly pretend to believe) in the epsilon-universe of sets. we talk about this epsilon-universe in the first-order logical language of epsilon-universes. the efficacy of first-order logical languages is based on their having interpretations in the epsilon-universe of sets. the interpretations of the first-order logical language of epsilon-universes in the epsilon-universe of sets are more or less hoped to have as an ideal limiting case the "intended interpretation", which is not itself a "genuine interpretation". the foundational cycle in the category-of-sets foundation is almost identical, just very slightly tighter: we believe (or formalisticly pretend to believe) in the category of sets. we talk about this category in the first-order logical language of categories. the efficacy of first-order logical languages is based on their having interpretations in the category of sets. the interpretations of the first-order logical language of categories in the category of sets are more or less hoped to have as an ideal limiting case the "intended interpretation", which is not itself a "genuine interpretation". to put it more sloppily, the epsilon-universe of sets is itself more or less a glorified set, and this conceptual and foundational circularity in the epsilon-universe-of-sets foundation is as "bad"/"good" as the almost identical circularity in the category-of-sets foundation, that the category of sets is itself more or less a glorified set. the cycle in the category-of-sets foundation is slightly tighter than the cycle in the epsilon-universe of sets foundation because the concept of interpreting a first-order logical language in the epsilon-universe of sets really only depends on the category abstracted from the epsilon-universe. not surprising, since just about everything interesting to the working mathematician about an epsilon-universe of sets really only depends on the category abstracted from the epsilon-universe. it might be interesting to hear intelligent challenges to this idea, but the challenges to it that you usually hear, even from people who ought to know better, tend to be based on silly misconceptions. there may be all sorts of other interesting and not-so-interesting issues that can get confused with the issue that i've been discussing here, of whether the category-of-sets foundation for mathematics is an adequate and good replacement for the epsilon-universe-of-sets foundation. i've tried to restrict my discussion here to this one relatively minor issue because i think it is to a great extent a separate issue from all those other issues, and i think it's good for people to see that. it may well be that the category-of-sets foundation is, in contrast to the epsilon-universe-of-sets foundation, the foundation for people who aren't very interested in the particular foundational preoccupations of certain bygone eras. i tend to agree with whoever it was (maybe mark kac?) that said that as long as mathematics has upward motion like a rocket ship it doesn't need a foundation the way a fortress needs one. "foundational" concerns of a sort could conceivably become important in the development of stuff like artificial mathematical intelligence, but i'd be surprised if something like an epsilon-universe-of-sets turned out to be more useful than something like a category-of-sets in this context. meanwhile, category theory itself continues to develop as a branch of mathematics which is mainly "panoramic" rather than "foundational", the branch of mathematics for people who don't like to confine themselves to one branch (but who also don't like to settle for vague generalities as a substitute for understanding). it's far more than just a "language for talking about mathematics", in the same way that mathematics is far more than just a "language for talking about physics and other branches of science", but i probably don't have the time to spare to discuss much about that at the moment. Sent via Deja.com http://www.deja.com/