In the last couple of weeks, I kept catching myself in --- uhm --- philosophical discussions re. category theory. They were a bit hectic, and didn't get very far... You know, for most mathematicians, philosophy is a no-go area; one takes care not to get too involved. But category theorists seem to be driven to it, somehow. (Sometimes, at least.) Anyway, hoping for comments from more experienced people, and apologizing to those who don't want to be bothered, I would like to put forward a couple of themes that keep coming back to me. The starting point for one of those discussions was my paper Maps II. Why is that simple characterisation so complicated in details? Doesn't that mean that the setting is wrong? In a more extreme and less serious form, I've heard a similar objection to somebody else's work: "It is too complicated to be of interest". Yes, category theory is the discipline of conceptual mathematics: it helps us extract the relevant structure and discard the contingent. "Mental hygiene" is a nice metaphor. Categories provide means for analyzing complex situations into conceptual parts. In a sense, they embody the first commandment of the Cartesian Method: DIVIDE. They are tools of understanding --- so they should at least be transparent and easy to understand themselves, shouldn't they? But they are not. The complexity issue arises on a rather general plan. For instance, in a recent issue of the Advances in Mathematics, in the review of our 1991 Montreal proceedings, Gian-Carlo Rota says that it's good that categories are still alive and well, but suggests that all those papers should be written in two versions --- the second one for people who might need some encouragement before "swallowing this morass of definitions". I don't suppose you need to hear the more extreme and less serious objections this time. Everybody knows that most mathematicians think of category theory just as their yoga instructors, their grandmothers and so on think of mathematics: it's complicated and useless. Of course, yoga instructors are less likely to be right about maths than mathematicians about categories; and those objections to my work are probably right, at least to some extent. So I am not trying to transform a criticism of me into a criticism of the State. They are quite different --- but somehow analogous, and I am trying to understand what lies behind. Is the complexity phenomenon just subjective (and more so in the narrower part of the telescope) or not? I mean, does the morass arise just from our wrong settings and overly complicated proofs --- feeding people's prejudice --- or is there more to it? Well, I think there is often less! I want to argue for the possibility that morass is at some places a natural state of ground, AND not always bad or dangerous. But let me try to put this in less extreme and more serious terms. (OK, I won't use this figure any more.) Categories are about understanding. Understanding is a process in time; but when it is achieved, it can often be summarized in a moment, captured by a picture. For instance, the periodic system of elements. In category theory, instead of wandering among equations, we draw a diagram to see what happens. Even for a hardest theorem, one often finds a crucial point or two, which can be retained in our mind's eye. Then we say we understand. Sometimes, however, understanding is essentially dynamical, and cannot be fixed. You may understand topology, but you can hardly survey it in a glimpse, compress it into a single idea. Or take natural language: we understand sentences as they go, without memorizing them or trying to capture the structure. In the morass of language, we move like centipedes, not worrying which leg comes after the 32nd. In mathematics and exact sciences, one tends to be sceptical towards this dialectical kind of understanding. The imperative of logical certainty seems to preclude it. "Clare et distincte", commanded Cartesius, and his voice still resounds very clearly. Mathematics takes good care not to become a natural language! Things should be cristal clear. Even a philosopher like me gets this. But when I look at our concrete mathematics --- there are all those strange phenomena, more and more of them, leading people to question the feasibility of the current criteria of logical certainty. Every month, an important new result is announced, with a proof that either just computers can check, or less than 7 people around the globe can hope to understand. Many of these proofs come with gaps --- but then there is this new branch of computational proof theory which measures how economic it is to allow such gaps. To a pure pure mathematician, this is ultimate horror, although it is perhaps better than the current practice of judging the validity of proofs by the concensus of experts. (However, some rather serious people argue that this practice cannot and should not be abolished: Deligne, Thomas Kuhn.) Experts usually agree whether a gap is easy to be filled or not, but for that long sought result about packing spheres like cannonballs, achieved three years ago by Hsiang (I think), the experts have not been able decide if there is a gap or not. Two equally large groups are still arguing that there is and that there ain't. So what can category theory hope for in this heavy-weight world of gaps and cannonballs? Not only in the granting sense, but also --- forgive me --- philosophically? Of course, I don't know the answer. What I wanted to say is that category theory might be some kind of a natural language, whether we like it or not. Hence the morass. But, like sentences in natural language, complicated categorical derivations are usually "simpler than they look", say I. Children make and understand long sentences, without being able to capture their structure. Laws of objective dialectics take care that they don't get lost in the morass. (Otherwise, they could never be careful enough anyway.) A very distinct experience that I had in categorical proof theory: derivations get lengthy, sometimes scary; you get surprised every once in a while --- but in the end, things always turn out to be tame and natural. I wish more people would visit the area. Simplicity, hygiene and logical certainty are, of course, good things when you can afford them, but "some books would be shorter if they were not so short", wrote Kant in the foreword of his first Critique. (Young Wittgenstein did not listen, and wrote that very hygienic book of his, which ended up in silence.) Ata ny rate, we should pay more attention to the dialectical contents of our science, as Lawvere and MacLane have been saying for a long time, more or less explicitly. This becomes even more important now, when it has become clear that the dream of powerful, problem-solving category theory won't come true. Complexities are not all due to a lack of category theory, and cannot be conceptualized away. We should not try to simplify the world, but... (But perhpaps I shouldn't philosophize it too much either.) Cordially, Dusko Pavlovic
A few brief thoughts on Dusko's discussion: First, almost all definitions in mathematics when done from the ground up are hopelessly complex. Ask your skeptical friends to define a continuous real-valued function on a topological space from first principles (for them, I suppose, from axiomatic set-theory alone). The only reason real numbers (pairs of subsets of the set of equivalence classes of formal ratios of formal differences of things satisfying the Peano axioms, modulo... and satisfying.... ) don't give the same impression is that we are used to them and have consequently forgotten the formal definition and the proceed to use only the properties. One way of making progress in mathematical understanding, indeed the way most congenial to categorists, is to get the definitions right so that one can forget the morass of quantifiers in the definition, and use only the essential properties of the objects under consideration. (Deep results, though, usually come from doing this, then briefly returning to the less abstract level to pull out an unexpected relation between the more abstract and the more concrete: e.g. the Atiyah-Singer Index Theorem, or Shum's freeness result for framed tangles.) It has been a persistent problem for category theory that we are interested in foundational questions (which most mathematicians abhor), and have not made it generally understood that we are really doing a kind of algebra (albeit a kind of algebra so potent that it can be applied to foundations). You might therefore turn aside much criticism by pointing out that category theory is no longer abstract in the sense of having no really interesting concrete examples: the work of Joyal/Street, Freyd/Yetter, Shum, Turaev, Reshtikhin, Majid, Lyubashenko, (the physicists) Moore/Seiberg, Kapranov/Voevodsky, and Fisher (to name but a few) are full of remarkable categories (with additional structure requiring yet more complex definitions!) with intimate connections to low-dimensional topology and Hopf algebra theory. Similarly, coming from the more foundational uses of category theory, the realizability topos is surely one of the most beautiful objects anywhere. (Indeed, topoi in general are quite beautiful.) --David Yetter
Dusko Pavlovic raises some interesting points - I just want to comment quickly on a couple of them. Firstly, it may be that (like those annoying "semiconvergent" series in numerical analysis), attempts to clarify something pass through a point of maximum simplicity after which they get worse again. Given all the indications that we can't have the kind of foundations that were once sought after (Goedel and All That), perhaps we are just looking for that point? The other comment addresses the idea that children use complex grammatical structures without completely understanding them. Sure they do - but they don't always use them *correctly*. Neither, of course, do adults. The trouble with mathematics is that unlike most of what we use natural language for, its groundrules don't (despite certain recent articles) permit statements on the 'well, it's mostly right and everybody knows what I mean' level. Most natural language usage jumps out of the system all the time... which gets into the AI debate. In mathematics, the medium is much closer to being the message. We try not to get into situations similar to the old chestnut "Time flies like an arrow, while fruit flies like a banana." , supposed by some to show that computers can't parse English. (I'm not sure that it does, any more than if I say 'Professor Smith is visiting next week' and you don't know whether to ask 'Is he?' or 'Is she?' , that shows that *you* don't know how to parse English...) But natural languages do, because in "natural" situations there is something outside the linguistic universe that can be referred to usefully. -Robert Dawson
Dusko Pavlovic offers, in passing, several descriptions of category theory: Yes, category theory is the discipline of conceptual mathematics: it helps us extract the relevant structure and discard the contingent. Categories provide means for analyzing complex situations into conceptual parts. Categories are about understanding. We know how to respond to students who tell us that they understand the material but can't do the problems. So how does one respond to this? I would suggest that any branch of mathematics is justified finally only because it allows us to prove things previously unprovable. If we were restricted to doing things we understand I don't think we would have gotten very far. Indeed, the most conspicuous function of mathematics I know is that it allows us to do things we don't yet understand. In my experience the feeling of understanding doesn't come until after the problems are solved. And sometimes long after. I think there are strong cases to be made for the presence of categories in mathemamatics (beginning at least with the Adams operations). And I think there are good cases to made even for the _theory_ of categories in mathematics. But -- something none of us expected -- the best cases to be made for the theory of categories are not in mathematics but in applied areas. It is there that we are finding the refutation of Dusko's sad assertion "now when it has become clear that the dream of powerful, problem-solving category theory won't come true."
participants (4)
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David Yetter -
Dusko Pavlovic -
Peter Freyd -
RDAWSON@HUSKY1.STMARYS.CA