From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1724 Path: news.gmane.org!not-for-mail From: Todd Wilson Newsgroups: gmane.science.mathematics.categories Subject: Re: Categories ridiculously abstract Date: Thu, 30 Nov 2000 12:52:58 -0800 (PST) Message-ID: <14886.48682.377864.620074@goedel.engr.csufresno.edu> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1241018047 32617 80.91.229.2 (29 Apr 2009 15:14:07 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:14:07 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Dec 1 15:15:17 2000 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB1IXvj09440 for categories-list; Fri, 1 Dec 2000 14:33:57 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f In-Reply-To: X-Mailer: VM 6.75 under 21.1 (patch 8) "Bryce Canyon" XEmacs Lucid X-MIME-Autoconverted: from quoted-printable to 8bit by mailserv.mta.ca id eAUKrFt03291 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 55 Original-Lines: 93 Xref: news.gmane.org gmane.science.mathematics.categories:1724 Archived-At: On Wed, 29 Nov 2000, Michael Barr wrote: > And here is a question: are categories more abstract or less > abstract than sets? There is a trap lurking in this question, and it has to do with understanding the term "abstract": different notions of "abstract" can lead to different answers to the question. In the case of sets and categories, since these are of different similarity types, something other than inclusion of classes of models is meant. For example "abstract", applied to sets and categories, might mean: 1. Having wider applicability. In this case, we can observe that the theorems of category theory (e.g., products are unique up to unique isomorphism) are generally more widely applicable than theorems of set theory (e.g., the powerset of a set has greater cardinality than the set itself), and so we would be inclined to say that categories are more abstract than sets on this criterion. 2. Having more general conditions for being an instance. In order to specify a set, we need only give (list, characterize) its members. To specify a category we need to do the same thing for both the collection of objects and the collection of arrows, and then we need to specify the composition law. (Even in an arrows-only formulation of category theory, we still need to specify both the collection of arrows and the composition law.) So, on this criterion, sets come out as more abstract. Some time ago, on the Foundations of Mathematics mailing list (FOM), there was a long and sometimes heated debate on alternative foundations of mathematics (where alternative meant non-set-theoretic) -- in particular on the viability of some kind of category-theoretic foundation for mathematics (e.g., elementary topos theory + some additional axioms) -- and the majority view seemed to be that - Set theory is more all-encompassing. The standard arguments about the bi-interpretability of category theory and set theory were met with the challenge (unanswered, as far as I know) to produce, in a category-theoretic foundation, a natural linearly-ordered sequence of axioms of higher infinity that can be used to "calibrate" the existential commitments of extensions to the basic axioms comparable to the large cardinal axioms of set theory, where the naturality requirement supposedly precludes the slavish translation of these large cardinal axioms into the language of category theory. (Recall that all known large cardinal axioms for set theory fall into a very nice linear hierarchy that can be used to gauge the consistency strength of a theory.) - Set theory is conceptually simpler. Set theory axiomatizes a single, very basic concept (membership), expressed using a single binary relation, and posits a natural set of axioms for this relation that are (more or less) neatly justified in terms of a fairly (some would say perfectly) clear semantic conception, the cumulative hierarchy. Category theory, the view goes, could only approach the scope of set theory, if at all, by adding many axioms that are unnatural and quite complicated to state and work with without the aid of multiple layers of definitions and definitional theorems (for products, exponentials, power-objects/subobject classifier, higher replacement-like closure conditions on the category, etc.). The arguments put forward in support of these views were very similar to those that are implicit in the labeling of category theory as "ridiculously abstract", and there are no doubt many readers of this list who would disagree with part or all of these views (me, for one). However, my intention in reporting them here is *not* to start another set-theory vs category theory thread, but rather to point out that, although category theorists have yet to make a convincing case -- at least I haven't seen one -- that category theory is more fundamental or foundational in any important sense (sorry, Paul), recent research in cognitive science on the embodied and metaphorical nature of our thinking indicates that category theory may well be able to make such a claim after all. See the books G. Lakoff and M. Johnson. Philosophy in the Flesh: The Embodied Mind and Its Challenge to Western Thought. Basic Books, 1999. G. Lakoff and R. Nuņez. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books, 2000. for a popular account of this research. I should mention, of course, that, closer to home, the book F.W. Lawvere and S.H. Schanuel, Conceptual Mathematics: A First Introduction to Category Theory. Cambridge University Press, 1997. is certainly a step in this direction. -- Todd Wilson A smile is not an individual Computer Science Department product; it is a co-product. California State University, Fresno -- Thich Nhat Hanh