Dear Marta - You write:
I am relieved to learn (from the postings by David Yetter and John Baez) that Motl's blog on the issue of categories and string theory is based on 1) (Yetter) Motl's reluctance, as is the case with many string theorists, to refuse to learn category theory, and 2) (Baez) Motl's personal dislike of John Baez and of many other people, so that since Motl's personality is well-known, any damage will be minimal.
Good!
I thank David and John for taking the trouble to respond in detail to what may have seem as a "provocation" on my part (well, perhaps it was...).
By the way, I should explain why I thought you might be kidding in your original post. I had never heard anyone before suggest that category theory could be discredited by applications to string theory. It completely surprised me. I'm used to the opposite complaint: that category theory is discredited by its *lack* of applications. Of course, this always comes from people who 1) haven't taken the time to learn of its applications, 2) don't know enough category theory to appreciate its *intrinsic* interest. But it's good to hear your real concern:
But these informative responses do not address my main concern, which is one that others (publicly, as Eduardo Dubuc, but several others privately) have expressed to me following my posting. I was aiming at the fact that there is a certain trend within category theory (when did it start?) to consistently give center stage to anything that claims to have connections with physics (in particular string theory). Is this because (it is believed that) the state of category theory is now so poor (as "evidenced" by the lack of grants) that they (the organizers of meetings) want to repair this image at any cost?
Since I began as a mathematical physicist and got interested in n-categories for their applications to topological quantum field theory, only later falling in love with category theory per se, I'm the wrong one to answer this question. I don't even know if it's true that applications to physics are given center stage, much less when this started, or why. I know a bit more about how people in differential geometry and differential topology got excited about work with links to physics. This trend probably started around the time of the Atiyah-Singer index theorem, which uses characteristic classes to compute the Euler characteristics of certain chain complexes built using differential operators. At the time this result was proved (1962-1965), it seemed an audacious blend of analysis and topology. That's one reason it caught people's interest. Another reason people liked the index theorem so much was that it turned out to be related to "anomalies" in quantum field theory, a phenomenon discovered by Adler, Bell and Jackiw around 1969. These nasty "anomalies" are actually a very practical issue in particle physics: they're related to the lifetime of the pion, and you can rule out field theories that have certain kinds of anomalies. I guess the relation between the index theorem and anomalies only became clear in the late 70's. I guess people were shocked and excited when it turned out that such sophisticated topology had practical applications to physics. Most topologists didn't know any quantum field theory, and most quantum field theorists didn't know that much topology. So, a kind of mutual fascination developed: both sides began learning about each other. People gave lots of proofs of the index theorem that illustrated very different ways of looking at it. The first proof had used a lot of K-theory and cobordism theory; later proofs used more facts about the heat equation, but by the time I was in grad school (1982-86) Quillen was giving lectures in which he tried to find a proof that only used multivariable calculus and "super" reasoning - i.e., lots of Z/2-graded linear algebra. This was when supersymmetry was just hitting the shores of mathematics, and Witten was starting to work his wonders. Anyway, index theory is just one of the first of many developments where ideas from physics met ideas from branches of math that seemed to have nothing to do with physics. In the heyday of Bourbaki, I guess pure mathematics seemed very removed from physics. It's fun to read what Dieudonne says about mathematical physics in his "Panorama of Pure Mathematics". By now, the situation has completely reversed in many fields, starting with differential geometry and topology, but then moving on to certain areas of algebra, and algebraic geometry, and now category theory, especially higher category theory.... This process has caused friction at every stage. Physicists don't always enjoy the intrusion of more mathematics into their various fields! Mathematicians don't always enjoy the intrusion of more physics - or the fast-paced, exploratory, sometimes sloppy cognitive style of physicists. You may recall Jaffe and Quinn's worries about the impact of physics on mathematics: http://www.arxiv.org/abs/math.HO/9307227 and how Atiyah in reply called for mathematicians to adopt the more "buccaneering" style of physics: http://www.ams.org/bull/pre-1996-data/199430-2/199430-2TOC.html which led Mac Lane to respond with the ballad of Captain Kidd: http://www.math.nsc.ru/LBRT/g2/english/ssk/proof_is_necessary.pdf The interesting big question is: how has this increased interaction both helped and hurt mathematics and physics? Clearly there are benefits. But does math become too "trendy" by chasing after links with the latest ideas of string theory? Does physics lose sight of its real purpose by focusing too much on mathematical elegance? There are lots of issues here. I've gone on too long already to want to tackle them now. But I think it's fair to say that that mathematics has benefited more than physics. One reason is that theories of physics do not need to be correct - i.e., apply to this particular universe of ours - to be mathematically interesting. Indeed, the funny thing about string theory is that while leading to an abundant harvest of rigorous mathematical results, it has not yet correctly predicted a single result from a single experiment, even after more than 20 years of work on the part of many smart people. This is part of a more general malaise in the theoretical side of fundamental physics, which various people have been commenting on recently: http://www.math.columbia.edu/~woit/wordpress/?p=307 http://www.nyas.org/publications/UpdateUnbound.asp?UpdateID=41 http://math.ucr.edu/home/baez/where_we_stand/ So, it's possible that string theory will eventually fall out of fashion. This could change the current dynamic between math and physics. A lot will depend on the results from the LHC particle accelerator, due to start operation in 2007. It may get evidence for string theory; it may not. Anyway, I'm sure these comments won't put your worries to rest! They're not really meant to. I just think it's good to see the issue of "category theory and string theory" as part of a much bigger and more complicated mess. :-) Best, jb