From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5279 Path: news.gmane.org!not-for-mail From: Andre.Rodin@ens.fr Newsgroups: gmane.science.mathematics.categories Subject: Re: categorical foundations Date: Sun, 15 Nov 2009 20:02:16 +0100 Message-ID: References: Reply-To: Andre.Rodin@ens.fr NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1258328392 1681 80.91.229.12 (15 Nov 2009 23:39:52 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 15 Nov 2009 23:39:52 +0000 (UTC) To: Colin McLarty , categories@mta.ca Original-X-From: categories@mta.ca Mon Nov 16 00:39:45 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 1N9oh6-0000HU-NR for gsmc-categories@m.gmane.org; Mon, 16 Nov 2009 00:39:45 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N9oJW-0003iG-0H for categories-list@mta.ca; Sun, 15 Nov 2009 19:15:22 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5279 Archived-At: Hi Colin, here are my answers to questions you asked me in your last two postings (= living now our terminological misunderstanding aside). CM: Do you mean that every formalized axiom system uses arithmetical notions such as "finite string of symbols." =A0This is why that formal axioms cannot be the real basis of our knowledge of math, but it has no more bearing on categorical axioms than any others. AR: No I did not mean this. Agree that this argument has no more bearing,= etc. CM: Or do you think that pre-formal notions of "set" or "collection" are all based on iterated membership and Zermelo's form of the axiom of extensionality, so that CCAF is less basic than ZFC? =A0That is a common belief among logicians who have not read Zermelo's critique of Cantor (where Zermelo points out that Cantor did not hold these beliefs) and who know a great deal more of ZFC than of other mathematics. AR: No. I certainly do NOT think that pre-formal notions of "set" or "collection" are all based on iterated membership and Zermelo's form of t= he axiom of extensionality. I explain in the next entry what I do think abou= t this matter. CM: In fact, long before mathematicians could analyze the continuum into = a discrete set of points plus a topology, they were well aware of collections like the collection of rigid motions of the plane -- and that "collection" is a category. =A0It is not just a ZFC set of motions but comes with composition of motions and with an object that the motions act on. AR: True, the most general notion of =93collection=94 one can imagine may= cover =93category=94 and whatnot. But, I claim, the preformal notion of colecti= on *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. The idea of *this* axiomatic method (= not to be confused with other versions of axiomatic method like Euclid=92s) is, = very roughly, this. One thinks of collection of =93bare=94 unrelated individua= ls and then introduces certain relations between these individuals through axiom= s. Objects of a theories obtained in this way are sets provided with relatio= ns between their elements, i.e. =93structured sets=94 (or better to say =93s= tructured collections=94. The principal feature of the preformal notion of collection involved here= is that elements of such a collection are unrelated. Because of this feature= the collection in question is not a general category. (It might be perhaps th= ought of as a discrete category but this fact has no bearing on my argument.) The idea of building theories *of sets* using the version of axiomatic me= thod just described is in fact controversial: it amounts to thinking of sets a= s bare preformal sets provided with the relation of membership. I mention this l= atter problem (which is not relevant to my argument) only for stressing that th= e notion of set or collection I have in mind talking about categorical foun= dation is NOT one that has any specific relevance to ZFC or any other axiomatic. 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). The notion of collection involved in this construction is MORE = BASIC than the resulting notion of category simply because this very axiomatic = method is designed to work similarly in different situations - for doing axiomat= ic theories of sets and of whatnot. Even if there are pragmatic reasons to b= uild 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. CM: What is a "formal basis" of a theory T? =A0 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 BC. BC is = ETC plus some other axioms. Conceptually the order of introduction of these axioms matters. My point (or rather guess) is that BC involves a prototype of a = new axiomatic method (different from one I described above), which, however, doesn=92t work in the given form independently. I=92m not quite prepared = to defend any general notion of formal basis - I didn=92t mean to introduce such a = general notion and didn=92t think about a general rule. CM: The Eilenberg-MacLane axioms are a subtheory of CCAF and also have a natural, conceptually central interpretation in CCAF. =A0I consider this an insight, Bill's insight, and I do not see how it becomes any kind of objection to CCAF. AR: The subtheory you are talking about is what I call ETC in these posti= ngs, right? I hope I understand it coorectly what you mean by "natural, concep= tual central interpretation in CCAF" - the fact that any object in CCAF is a m= odel of ETC, right? Now, the objection is this: ETC involves the preformal notion of collection that can NOT be thought o= f as a category (for the reason I tried to explain above). In addition to the above argument my conclusion about CCAF is also based = on the following historical observation. Every major historical shift in founda= tions of mathematics so far involved a major change of the notion of axiomatic method. (I can substantiate the claim if you'll ask.) But ETC (and, forma= lly speaking, the whole of CCAF) relies on the old Hilbert-Tarski-style axiom= atic method. best, Andrei [For admin and other information see: http://www.mta.ca/~cat-dist/ ]