Re: Categories ridiculously abstract
Readers of the cat list may be interested in the one meaningful post to Slate's "The Fray" in reply to Holt's MSRI "Diary" article. It was made by David Yetter: category theory David Yetter 28 Nov 2000 20:29 It is sad more than a decade on since the proof of the remarkable categorical coherence theorem of Shum that mathematicians can continue to view category theory as a mere linguistic convention or useless abstraction. Shum's theorem shows that axioms completely natural from the internal dynamic of category theory completely characterize framed tangles, relative versions of the framed knots and links which are central to smooth topology in 3 and 4 dimensions (notice the dimensionality of space and of space-time: hardly divorced from meaning.) Other categories satisfying the same axioms include the categories of representations of quantum groups, physically motivated algebraic structures which have become central objects of study for mathematicians from many old branches of mathematics. Indeed, Shum's theorem, a theorem of category theory, is the only really satisfying explanation for the intimate connection between quantum groups and low-dimensional topology.
I don't think one should blame the guy whose remarks Peter quoted. He is not a mathematician and presumably knows nothing more than some college level mathematics. He has picked up that attitude from the high-powered mathematicians that inhabit places like MSRI (and the CRM, Fields Inst., and PIMS in Canada). Ignoring the fact that category theory was fathered by two of the most eminent mathematicians of the last century and god-fathered by arguably the very greatest, they still go around saying that it is without content and nothing but meaningless abstraction. I was unaware of what David Yetter mentioned, but I am certainly aware of the crucial role categories had in proving the Weil conjectures and the fact that people like John Baez seem to believe that higher dimensional categories will be important in physics. I might also point out that categories were the right framework for Kaplansky's very elegant proof of the Auslander-Buchsbaum theorem. And here is a question: are categories more abstract or less abstract than sets? Michael
Michael Barr wrote:
And here is a question: are categories more abstract or less abstract than sets?
A higher-dimensional category theorist's answer: "Neither - a set is merely a 0-category, and a category a 1-category." There's a more serious thought behind this. Sometimes I've wondered, in a vague way, whether the much-discussed hierarchy 0-categories (sets) form a (1-)category, (1-)categories form a 2-category, ... has a role to play in foundations. After all, set-theorists seek to found mathematics on the theory of 0-categories; category-theorists sometimes talk about founding mathematics on the theory of 1-categories and providing a (Lawverian) axiomatization of the 1-category of 0-categories; you might ask "what next"? Axiomatize the 2-category of (1-)categories? Or the (n+1)-category of n-categories? Could it even be, I ask with tongue in cheek and head in clouds, that general n-categories provide a more natural foundation than either 0-categories or 1-categories? Tom
In "Towards a categorical foundation of mathematics" (Logic Colloquium '95, ed's: J. A. Makowsky and E. V. Ravve, Springer Lecture Notes in Logic no.11, 1998; pp.153-190) and in subsequent work, I am proposing an approach to a foundation whose universe consists of the weak n-categories and whatever things are needed to connect them. This is done on the basis of a general point of view concerning the role of identity of mathematical objects. Readers of said paper who have followed developments on weak higher dimensional categories will note that much has been done since towards fleshing out the program. Michael Makkai On Thu, 30 Nov 2000, Tom Leinster wrote:
Michael Barr wrote:
And here is a question: are categories more abstract or less abstract than sets?
A higher-dimensional category theorist's answer: "Neither - a set is merely a 0-category, and a category a 1-category."
There's a more serious thought behind this. Sometimes I've wondered, in a vague way, whether the much-discussed hierarchy
0-categories (sets) form a (1-)category, (1-)categories form a 2-category, ...
has a role to play in foundations. After all, set-theorists seek to found mathematics on the theory of 0-categories; category-theorists sometimes talk about founding mathematics on the theory of 1-categories and providing a (Lawverian) axiomatization of the 1-category of 0-categories; you might ask "what next"? Axiomatize the 2-category of (1-)categories? Or the (n+1)-category of n-categories? Could it even be, I ask with tongue in cheek and head in clouds, that general n-categories provide a more natural foundation than either 0-categories or 1-categories?
Tom
Tom Leinster wrote:
Michael Barr wrote:
And here is a question: are categories more abstract or less abstract than sets?
A higher-dimensional category theorist's answer: "Neither - a set is merely a 0-category, and a category a 1-category."
There's a more serious thought behind this. Sometimes I've wondered, in a vague way, whether the much-discussed hierarchy
0-categories (sets) form a (1-)category, (1-)categories form a 2-category, ...
has a role to play in foundations. After all, set-theorists seek to found mathematics on the theory of 0-categories; category-theorists sometimes talk about founding mathematics on the theory of 1-categories and providing a (Lawverian) axiomatization of the 1-category of 0-categories; you might ask "what next"? Axiomatize the 2-category of (1-)categories? Or the (n+1)-category of n-categories?
Surely we should start with the set of (-1)-categories? <gd&r> -Robert Dawson
1 ab.stract \ab-'strakt, 'ab-,\ adj (15c) [ML abstractus, fr. L, pp. of abstrahere to draw away, fr. abs-, ab- + trahere to draw -- more at DRAW] 1a: disassociated from any specific instance <abstract entity> 1b: difficult to understand: ABSTRUSE <abstract problems> 1c: IDEAL <abstract justice> 1d: insufficiently factual: FORMAL <possessed only an abstract right> 2: expressing a quality apart from an object <the word poem is concrete, poetry is abstract> 3a: dealing with a subject in its abstract aspects: THEORETICAL <abstract science> 3b: IMPERSONAL, DETACHED <the abstract compassion of a surgeon --Time> 4: having only intrinsic form with little or no attempt at pictorial representation or narrative content <abstract painting> -- ab.stract.ly \ab-'strak-(t)l<e^->, 'ab-,\ adv -- ab.stract.ness \ab-'strak(t)-n<e>s, 'ab-,\ n 1a: Sets and categories as mathematical abstractions are equally disassociated from specific instances. 1b: For almost every interesting known theorem of category theory there is a harder interesting known theorem of set theory, and vice versa. It is plausible that the exceptions from set theory outnumber those from category theory, but it is equally plausible that a majority of mathematical literates judge category theory harder than set theory. No clear winner here. 1c: Sets and categories are both ideal entities. 1d: Set theory and category theory are equally factual, and equally formal. 2: In this sense set theory and category theory are both abstract while sets and categories are objects and so not abstract. 3a: Set theory and category theory deal equally with the abstract aspects of their respective subjects. 3b: The FOM mailing list tends to get worked up much more often and rather more heatedly about the set-vs-category debate than does the categories mailing list. 4. Categories lend themselves better to diagrams than do sets. Conclusions (organized by dictionary meaning of "abstract"): 1 to 3a: No difference. 3b: Category theorists are more abstract than set theorists. 4: Sets are more abstract than categories. -- Vaughan Pratt O res ridicula! immensa stultitia. --Chorus of Old Men, Catulli Carmina
In "Towards a categorical foundation of mathematics" (Logic Colloquium '95, ed's: J. A. Makowsky and E. V. Ravve, Springer Lecture Notes in Logic no.11, 1998; pp.153-190) and in subsequent work, I am proposing an approach to a foundation whose universe consists of the weak n-categories and whatever things are needed to connect them. This is done on the basis of a general point of view concerning the role of identity of mathematical objects. Readers of said paper who have followed developments on weak higher dimensional categories will note that much has been done since towards fleshing out the program.
Michael Makkai
On Thu, 30 Nov 2000, Tom Leinster wrote:
Michael Barr wrote:
And here is a question: are categories more abstract or less abstract
My understanding of the relation between category theory and set theory is that category theory is a formal theory built on abstract concepts (objects and morphisms). The way of defining category theory need a metalanguage which is closed to the logic of set theory language (a particular case of what is called boolean logic). There is a sort of dichothomy between logic behind the two theories. This dichothomy come from our limitation of talking about category theory. We use already two-valued logic (true and false) which we cannot avoid if we need to talk about identity of objects and morphisms. Now a kind of Godel's arguments about natural numbers (If N is consistent, then there is no proof of its consistency by method formalizable within the theory ) is what is going on. This doesn't stop the category theory 'game'. When you give birth to a child you will never know in advance if the child will be an honest person or a criminal. Category theory have generated many structures which can help us to understand why many mathematicians have work differently to describe a same mathematical concept in different ways. As an example we know, from category theory, that Cauchy and Dedekind were defining real numbers from rational numbers but the two definitions are not saying the same thing. ----- Original Message ----- From: "Michael MAKKAI" <makkai@scylla.math.mcgill.ca> To: <categories@mta.ca> Sent: Saturday, December 02, 2000 12:19 AM Subject: categories: Re: Categories ridiculously abstract than
sets?
A higher-dimensional category theorist's answer: "Neither - a set is merely a 0-category, and a category a 1-category."
There's a more serious thought behind this. Sometimes I've wondered, in a vague way, whether the much-discussed hierarchy
0-categories (sets) form a (1-)category, (1-)categories form a 2-category, ...
has a role to play in foundations. After all, set-theorists seek to found mathematics on the theory of 0-categories; category-theorists sometimes talk about founding mathematics on the theory of 1-categories and providing a (Lawverian) axiomatization of the 1-category of 0-categories; you might ask "what next"? Axiomatize the 2-category of (1-)categories? Or the (n+1)-category of n-categories? Could it even be, I ask with tongue in cheek and head in clouds, that general n-categories provide a more natural foundation than either 0-categories or 1-categories?
Tom
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. Nuñez. 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
participants (8)
-
DR Mawanda -
John Duskin -
Michael Barr -
Michael MAKKAI -
Robert J. MacG. Dawson -
Todd Wilson -
Tom Leinster -
Vaughan Pratt