From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5283 Path: news.gmane.org!not-for-mail From: Colin McLarty Newsgroups: gmane.science.mathematics.categories Subject: Re: Re: categorical foundations Date: Mon, 16 Nov 2009 09:54:20 -0500 Message-ID: Reply-To: Colin McLarty NNTP-Posting-Host: lo.gmane.org Content-Type: text/plain; charset=windows-1252 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1258417027 8099 80.91.229.12 (17 Nov 2009 00:17:07 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 17 Nov 2009 00:17:07 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Tue Nov 17 01:17:00 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1NABkh-0002U9-RC for gsmc-categories@m.gmane.org; Tue, 17 Nov 2009 01:17:00 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NABLB-0006QR-G0 for categories-list@mta.ca; Mon, 16 Nov 2009 19:50:37 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5283 Archived-At: 2009/11/15 : Suggests a better take on CCAF than the one he has been taking. That would be a take based more on Bill's published work on CCAF, and less on the philosophical objection that Geoff Hellman used to make about CCAF. Geoff himself has given up this objection. > AR: True, the most general notion of =93collection=94 one can imagine may= cover > =93category=94 and what not. But, I claim, the preformal notion of collec= tion > *relevant to the axiomatic method in its modern form* is more specific, a= nd > does NOT cover the preformal notion of category. I=92m talking about =93s= ystems of > things=94 in the sense of Hilbert 1899 rather than sets in the sense of Z= FC or of > any other axiomatic theory of sets. This is the Hilbert conception where axioms are not asserted as true but offered as implicit definition; and so they are not about any specific subject matter but may be applied to whatever satisfies them. Lawvere from 1963 on has always been clear that his first order axioms ETCS and CCAF can be taken this way for metamathematical study -- but that he does assert them as true specifically of actual sets and categories. (Now Bill is not talking about any idealist truth or objects. He takes a dialectical view. But that is another topic.) > In ETC (the Elementary Theory of Categories in the sense of Bill=92s 1966= paper) > categories are conceived as collections of things called =93morphisms=94 = provided > with relations called =93domain=94, =93codomain=94 and =93composition=94 = (I hope I nothing > forgot). This is one use of ETC, and indeed a use made daily in mathematics. But it is not the use in CCAF. The fragment of CCAF you are calling ETC is asserted of specific things. Bill says it deals with: "the category whose maps are =91all=92 possible functors, and whose objects are =91all=92 possible (identity functors of) categories. Of course such universality needs to be tempered somewhat." The requisite tempering is very like that familiar in set theory, and Bill describes it. (The quote is his dissertation p. 26 of the TAC reprint.) > Even if there are pragmatic reasons to build > theories of sets like ETCS and other mathematical theories on the basis o= f ETC > rather than use axiomatic theories of sets like ZFC for doing category th= eory > and the rest of mathematics, this doesn=92t change the above argument. What does change it though, is the interpretation of ETC in CCAF. That interpretation does not use The "Hilbert conception." Actually, it is best regarded as a single interpretation with a parameter: interpret "object" in the ETC axioms as "functor from 1 to X" where X is a fixed free variable of identity functor type in CCAF, interpret "morphism" as "functor from 2 to X" and so on always with the same free variable X. Interpreting the ETC axioms in CCAF this way is not at all treating them in the Hilbert way. But even take the interpretation corresponding to any one object A of CCAF. That amounts to specifying X as A in the parametrized interpretation. This interpretation does not deal with "the collection of objects of A" and "the collection of morphism of A". It never refers to any such collections. It deals with categories A,1,2,3, and functors among them. If you want to push this line: > Every major historical shift in foundations > of mathematics so far involved a major change of the notion of axiomatic > method. (I can substantiate the claim if you'll ask.) Then you would do better to notice the novelty of these parametrized and single-category interpretations of ETC in CCAF and take this as the kind of major change that you expect to see. > AR: I called ETC =93formal basis=94 of BT (=93Basic Theory of Categories= =94 in the > sense of Bill=92s 1966=92s paper) meaning the two-level structure of BT. = BT is ETC > plus some other axioms. Conceptually the order of introduction of these a= xioms > matters. My point (or rather guess) is that BT involves a prototype of a > new axiomatic method (different from one I described above), which, howev= er, > doesn=92t work in the given form independently. This different axiomatic method is explicit in CCAF, and does work independently there. Specifically what is supposed to "not work" about it? Is it supposed to be formally inadequate to interpreting mathematics? (That is a non-starter, and even Feferman only made vague hints that it was so and never tried to fill them in.) Is it not really comprehensible? (Bill comprehended it already around 1960, and so do many of us now. Feferman argues well that he does not comprehend it, but falsely concludes that no one can.) best, Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]