From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1817 Path: news.gmane.org!not-for-mail From: "zdiskin" Newsgroups: gmane.science.mathematics.categories Subject: Re: Michael Healy's question on math and AI Date: Tue, 30 Jan 2001 16:21:41 -0800 Message-ID: <004901c08b1b$e880d8f0$822f1b3f@cpu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 X-Trace: ger.gmane.org 1241018123 616 80.91.229.2 (29 Apr 2009 15:15:23 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:15:23 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Wed Jan 31 12:15:57 2001 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f0VFk0w08188 for categories-list; Wed, 31 Jan 2001 11:46:01 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Priority: 3 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 48 Original-Lines: 109 Xref: news.gmane.org gmane.science.mathematics.categories:1817 Archived-At: It seems that a discussion "CT vs. ST " (despite some silent resistance at the beginning :) has nevertheless started in this list too, and is going in a few different directions ranging from rather technical thru methodological to philosophical aspects. Probably it's unavoidable (and useful) when we talk about math in general and its applications but, as Bill Lawvere pointed, too much quite different stuff is packed into the same title of CTvsST and the type mismatch in the very discussion and its comprehension by the (rather diverse as I could guess) audience is very possible. (Under type mismatch I mean something like this. Suppose we discuss the result of multiplication 3 x 3. If the alternatives are 8,9 and, say, 12, we have good chances for a reasonable discussion, and if even the alternatives are 12, 37 and 111, type correctness is still respected and we have chances to achieve useful results, but if the alternatives to be discussed are 8, 9, that triangle and that , the situation is hopeless). So, some methodological arrangement and figuring out explicitly the relevant meanings of the CTvsST-problem would be useful. Here are a few contexts already touched in the discussion, where we deal with methodologically quite different problems whose merging under the same title CTvsST may be misleading. a). Ways of setting/defining math structures. The actual meaning of CTvsST here is the opposition between the following two ways (described by Steve Vickers yesterday, I'll just rephrase slightly his presentation). --(1) Math structure Carrying set (of abstract elements) + Structure over it defined in terms of the elements (so, structure resides in elements of the carrier). --(2) Math structure Category (collection of abstract objects and their morphisms) + Structure over it defined in terms of the morphisms (and then structure on a carrier object resides in morphisms adjoint to it). The title CTvsST may be misleading here since in the both ways we use sets, their elements, mappings between them. To name the two ways somehow, I'd propose to call the 1st one Boubakian (since this way of setting math structures got its classical completion in Bourbaki's volumes), and the 2nd one categorical. So, the actual opposition here is CatStr vs. BrbStr (but, of course, a category with structure is itself a Bourbakian math structure) This opposition is of extreme great relevance for software engineering, business modeling, knowledge representation ... but even brief outline of the reasons needs a more detailed description of the issue. Let's postpone that for a next posting. b). Modeling (formalization) of set theory underlying the setting for math structures and reasoning about them (usually referred to as foundations). The actual meaning of CTvsST here is "categorical set theory" vs. "formal set theory(ies)" (ZF, NBG,...), or CatST vs. FrmST. This context and meaning are totally irrelevant to applications i question and asking about what is better, CT or ST, for applications in AI,SE, ... is very much like asking what is better for applications in mechanical engineering: the differential/integral calculi or ST; or what is better for general theory of relativity, the tensor calculus or ST etc. c). Modeling (formalization) of reasoning about math structures (often called metamathematics) Of course, what we have here depends on which way of setting math structures we consider: Bourbakian or categorical. Nevertheless, it's possible and make sense to treat reasoning about Bourbakian structures categorically and, say, to treat reasoning about categories in a elementwise fashion. We again have two different approaches. Historically first (originated by Tarsky and Mal'cev) was formalization of first order logic (FOL, including syntax and semantics) in a quite immediate way now well known to a wide audience of quite different backgrounds including computer scientists and philosophers. This way is often referred to as "Tarskian formalization of the notion of truth", and let's call this entire style Tarskian MetaMathematics, TarMMath. The other approach was developed in CT and is usually called categorical logic, CatLog. So, the opposition we actually have here is CatLog vs. TarMMath, or, if you prefer, CatMMath vs. TarMMath. This oppostion, though of course connected with that in (a), CatStr vs. BrbStr, has its own peculiarities which are of extreme high relevance for knowledge representation, business modeling and similar domains. So, details here would be very useful but I'd again postpone them for a next posting. So, as it was said in Lawvere's note, there are a few aspects of CTvsST (understood in the wide sense), some of them may be not relevant for applications (for example, b) while others are of great importance and deserve more detailed exposition (a,c). On the other hand, all the three context we have considered are special cases of a quite general intellectual activity usually referred to as modeling. Mathematics is itself a special (very refined) discipline of modeling, and foundations and meta-mathematics are nothing but modeling math by math means (well, there are other means, say, philosophical :). It seems that to make these contexts more understandable for a wider audience (and for myself :), not only more details should be provided but some sketch of what modeling in general and math modeling in particular are, would be also useful. So, I'd ventured to sketch some general tutorial on applying math to engineering domains and to itself but certainly it's for a next posting. --Zinovy Diskin