I'd like to thank the contributors to what has turned out to be another Category Theory vs Set Theory thread -- Steve Vickers, Michael Barr, Colin McLarty, Charles Wells, Zinovy Diskin, and John Duskin -- especially Steve and Zinovy for their lengthy replies. My original post was, in a way, meant to be a step in the direction of ending such CT vs ST debates, but there still seems to be some interest in establishing "oppositions" between the two theories. Here is my attempt to frame the debate and to respond to some comments of the contributors in the process. I apologize for the length of this post. Zinovy Diskin observes that the debate involves "methodologically quite different problems whose merging under the same title CTvsST may be misleading", echoing the earlier statement by Bill Lawvere that "[s]o much confusion has been accumulated that an opposition of the form `set-theoretical versus non-set-theoretical' has at least seven wholly distinct meanings." I quite agree, and this was exactly the point of the end of my original post: And, finally, shouldn't "better" really be "better for what"? In other words, aren't the two communities really just arguing past one another, like people arguing over types of automobile? What really is the issue here? Colin McLarty has asked for clarification of these questions. My point was simply that it is pointless to claim that one theory or subject or approach is better than another without specifying what it is supposed to be better *for*. (Is a pick-up truck better than a 2-seater sports-car? Is a Mercedes Benz better than a BMW?) Here is a list of eight possible uses to which CT and ST may be put: 1. Offering an axiomatic foundation upon which all of mathematics may be developed, with a view towards a. establishing or making manifest its consistency b. providing a standard of rigor c. providing a common framework for the cooperation between different mathematical domains 2. Acting as a language in which certain mathematical ideas can be expressed, so that a. better use can be made of them (in applying them to specific instances, seeing connections between them, highlighting their more important features, etc.) b. they can be communicated more effectively to other mathematicians c. they better please our aesthetic sense 3. Providing models for specific phenomena (physical or computational), with a view towards a. informally illuminating their properties and connections b. predicting the outcomes of experiments involving them All of these uses are more or less concrete enough that comparisons between CT and ST could be undertaken empirically using these criteria (although some of them, for example 2a and 2c, are clearly more subjective than the others). My expectation is that CT and ST would each be "winners" on some non-trivial subset of the criteria. In addition to these eight uses for CT and ST, there is another important role that these theories play, namely, as formal theories of pre-mathematical concepts -- "collection" and "membership" in the case of ST, and "transformation" and "comparison" in the case of CT. From this point of view, I am left somewhat mystified by Lawvere's reference to the "totally arbitrary `singleton' operation of Peano with the resulting chains of mathematically spurious rigidified membership". Whatever its other "faults", ST seems to me to be a pretty accurate theory of "recursive collections" (or "classes" or "containers"), that is, collections of collections of .... Indeed, we can easily play with real containers of various sizes, and put them inside each other in various ways, and ST can be seen to be an accurate description of these configurations. The part of our conceptual apparatus that deals with container relations such as these is real and deserves to be modeled by a formal theory, and I find it hard to criticize ST on its appropriateness for this role. (Of course, ST is also about infinite collections, and its role there is more open to debate.) Continuing from this point of view, one feature of ST that I find especially interesting is that, although it is ostensibly a theory of "recursive collections", it nevertheless is quite successful, without any substantial additions, in many other roles -- see the list of eight uses above. It is an interesting philosophical and practical question to ask why this might be so, but one can hardly dispute this success. Thus, I also find it difficult to appreciate claims (such as Lawvere's) that set theory is "ill-suited for mathematics"; doesn't a simple comparison of the mathematical achievements before and after its "arithmetization" by set theory suggest otherwise? Turning to more specific topics, several respondents to my original post realized that my discussion of Cartesian products was a bit muddled, and that my suggestion of an "unordered product" was incoherent. Colin McLarty answered my question about the technical work involved in dealing with the a/the distinction by suggesting that "nothing is involved if we introduce a product operator". However, if we have a category with many non-trivial isomorphisms, then the task of introducing a product operator does involve something: making some arbitrary choices. Without getting into the details, I just wanted to ask whether such choices themselves added a kind of "spurious element" to the construction at least qualitatively similar to those in ST referred to by Lawvere. Steven Vickers nicely spelled out in more detail some of the ideas behind Lawvere's "cohesive and variable sets". However, I didn't mean to suggest in my post that I didn't appreciate the difference in approach (for example, I did my PhD thesis on frame theory, an algebraic underpinning for point-free topology). Rather, I fail to understand the point of criticizing ST for presenting a *particular* view of cohesion. Isn't it interesting that a surprisingly general and mathematically fruitful definition of cohesion/nearness among a set of points can be given in terms of just a single element of the double powerset of those points? The many achievements of "point-set" or "general" topology would suggest so. Now, I certainly agree that this view of cohesion may not be the best one for some, or even many, applications (which is what initially led to my interest in frame theory), but why does that have to turn into a criticism of the set-theoretic framework? We don't criticize the rectangular representation of complex numbers because we also have a polar representation. Vickers also makes the point that set-theoretic topology "does not generalize well to situations where you want to vary the set theory," and also mentions situations where it is fruitful to vary the underlying logic as well. These topics are among my favorites in all of mathematics, and I have been a proselytizer for these ideas in other forums, but the reaction of set theorists is understandable: varying the ST and logic makes it harder to relate what you are doing in the different "universes". A mathematician wanting to take advantage of a shift in ST and logic is faced with the problem of (re-)interpreting the results in the original framework. The extra overhead involved in moving between universes (not to mention the large "start-up costs" involved in learning the framework in the first place) is seldom ever justified by the advantages that ones gains, however real they can be. This is a very practical matter, involving mathematicians' choice of where to invest their time, and, again, I don't see what is to be gained by criticizing them for their choices. CONCLUSION: PLURALISM AND A CHALLENGE To sum up, I think that any debate on CT vs ST should take place in the context of a concrete and particular use of the two theories, where it is possible to investigate, more or less empirically, the advantages and disadvantages of each. In any other context, the debate reduces to a battle over personal preference, artistic sense, working habits, and other such subjective issues, and is unlikely to get anywhere. Second, I think we ought to foster a more pluralistic viewpoint. Each theory has its strengths and weaknesses, and we should choose the most appropriate tool for whatever job it at hand. If someone contends that there is a significant difference in appropriateness between two approaches, then, for this contention to be taken seriously, the difference has to be made clear and concrete for the "worker in the field". I would make the same point to computer scientists who are involved in the endless "Language Wars" over which programming language is the "best". And finally, I would like to offer a challenge (or challenges). For those mathematicians and computer scientists enamored with the vistas opened up by category theory, and topos theory in particular (and I count myself as among these), - Can we build computer-implemented formal systems that make it easier to navigate through several universes, work simultaneously with several logics, and help with re-interpretation when necessary? - Can we write books that help reduce the start-up costs involved in "outsiders" learning and using the framework? - Can we discuss in public places and in detail the importance of the topos-theoretic or category-theoretic outlook in obtaining our mathematical results? - And can we all the while hold off on our criticism of other approaches and instead let the results speak for themselves? -- Todd Wilson A smile is not an individual Computer Science Department product; it is a co-product. California State University, Fresno -- Thich Nhat Hanh