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." This 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? That 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 the axiom of extensionality. I explain in the next entry what I do think about 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. It 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 collection one can imagine may cover category and whatnot. But, I claim, the preformal notion of colection *relevant to the axiomatic method in its modern form* is more specific, and does NOT cover the preformal notion of category. Im talking about systems of things in the sense of Hilbert 1899 rather than sets in the sense of ZFC 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 Euclids) is, very roughly, this. One thinks of collection of bare unrelated individuals and then introduces certain relations between these individuals through axioms. Objects of a theories obtained in this way are sets provided with relations between their elements, i.e. structured sets (or better to say structured collections. 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 thought 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 method just described is in fact controversial: it amounts to thinking of sets as bare preformal sets provided with the relation of membership. I mention this latter problem (which is not relevant to my argument) only for stressing that the notion of set or collection I have in mind talking about categorical foundation 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 Bills 1966 paper) categories are conceived as collections of things called morphisms provided with relations called domain, codomain and composition (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 axiomatic theories of sets and of whatnot. Even if there are pragmatic reasons to build theories of sets like ETCS and other mathematical theories on the basis of ETC rather than use axiomatic theories of sets like ZFC for doing category theory and the rest of mathematics, this doesnt change the above argument. CM: What is a "formal basis" of a theory T? AR: I called ETC formal basis of BT (Basic Theory of Categories in the sense of Bills 1966s 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, doesnt work in the given form independently. Im not quite prepared to defend any general notion of formal basis - I didnt mean to introduce such a general notion and didnt 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. I 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 postings, right? I hope I understand it coorectly what you mean by "natural, conceptual central interpretation in CCAF" - the fact that any object in CCAF is a model of ETC, right? Now, the objection is this: ETC involves the preformal notion of collection that can NOT be thought of 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 foundations 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, formally speaking, the whole of CCAF) relies on the old Hilbert-Tarski-style axiomatic method. best, Andrei [For admin and other information see: http://www.mta.ca/~cat-dist/ ]