Hi -

I just came across the following pages

http://motls.blogspot.com/2004/11/category-theory-and-physics.html http://motls.blogspot.com/2004/11/this-week-208-analysis.html

written by Lubos Motl, a physicist (string theorist). Some of you may find these articles interesting and probably revealing.

Are we category theorists as a whole going to quietly accept getting discredited by a minority of us presumably applying category theory to string theory?

I can't tell if you're kidding. I'll assume you're not. There's nothing wrong with applying category theory to string theory. The papers by Michael Douglas and Paul Aspinwall cited above by Motl are some nice examples of using derived categories to study D-branes. Further examples: the Moore-Seiberg relations turn out to be little more than the definition of a balanced monoidal category, and the Segal-Moore axioms for open-closed topological strings are nicely captured using category theory here: http://arxiv.org/abs/math.AT/0510664 There were a lot of nice talks on the borderline between category theory and string theory at the Streetfest. Perhaps more to the point, Lubos Motl is famous for his heated rhetoric. He doesn't like me, or anyone else who criticizes string theory. The articles you mention above are mainly reactions to my This Week's Finds. He's actually being very gentle - for him. He even says "the role of category theory can therefore be described as a `progressive direction' within string theory". I'm sure you'll all be pleased to know that. :-)

It is surely not too late to react and point out that this is not what (all of) category theory is about.

I would urge everyone not to react - at least, not until they are well aware of what a discussion with him is like. See his blog and his comments on Peter Woit's blog if you don't understand what I mean. For example: http://pitofbabel.org/blog/?p=51

Please give a thought about what we, as a community, can urgently do to repair this damaging impression.

Since Motl's personality is well known, any damage will be minimal. I think we should relax and take it easy. Best, jb

I also would like to support the remarks of Marta with which I am in full agreement. The category theory community seems happy to accept uncritically, and give centre-stage to, any interest shown by an external field. In this context one should certainly look the gift horse in the mouth. Bob Walters

Hi: I will put quotations from different postings or Molt's writing in between two "**" Well, I can see the classical reaction of some groups when one of its members points out that something is really wrong with the group. Marta will suffer all kind of "polite" (nothing of the sort of the Benabou-Taylor confrontation) attacks, but not for this less devious or sanguine. Typically she will be taken out of context, or get answers to questions she never had asked, or be treated ironically or in disbelief (**I can't tell if you're kidding. I'll assume you're not **) There are two principal points here: 1. The real value of some contributions of category theory to physics. 2. The lot of rubbish written using category theory and which is fashionable because it claims to have applications to physics. Marta was forced to explicit some of the questions we can clearly see in between lines in her original posting: ** I was trying to elicit an open response from those who *do* know about the value (or lack of it) of categorical string theory. In particular, I would like to have an answer to this question. Why is it that anything which even remotely claims to have applications to physics (particularly string theory) is given (what I view as) uncritical support in our circles? Best, Marta ** I will like to see a clear answer to this question. Or a clear refutation proving that it is not the case. Notice that the existence of point 2. above is perfectly consistent with the existence of really valuable contributions of category theory to string theory, which is one of the points treated by Motl. ** There's nothing wrong with applying category theory to string theory. The papers by Michael Douglas and Paul Aspinwall cited above by Motl are some nice examples of using derived categories to study D-branes.** This make us think that they may be some valuable contributions, but this possibility is also left open by Motl himself. Quoting myself: ** I will like to see here a debate about Motls's writing quoted above. Just about this writing, NOT ABOUT Motls himself or other things he may have done or represent !! ** No luck, just discredit Motl, not refute his sayings: ** Perhaps more to the point, Lubos Motl is famous for his heated rhetoric. He doesn't like me, or anyone else who criticizes string theory. The articles you mention above are mainly reactions to my This Week's Finds. ** ** My reaction to the blog posts you cite is that this is a sting theorist holding his breath and refusing to learn category theory. My guess is that Motl wouldn't want to learn the heavily categorical formulations of mirror symmetry that Yan Soibelman uses, even though they are motivated by string theory.** The following is better in answering Motl: ** Categorical ideas are absolutely central to several competitors to string theory: the Barrett-Crane model of quantum gravity (and to a lesser extent 'loop quantum gravity' with which the BC model is often conflated) and Connes' recovery of the Standard Model from non-commutative geometry (a part of mathematics which has obliged reluctant mathematicians to think about categorical ideas deeper than they originally were comfortable with). There is nothing cracked or crackpot about either. ** I am unable to judge, but it seems to me this gives category theory strong support But does not go against what Motl says concerning category theory. Neither against Marta's warning that category theory is being discredited by many (she says a minority) category theory people. Motl writes: ** I've asked the same elementary questions to many people who've been trying to explain me derived categories - some of them with some success, most of them with no success whatsoever: Are these notions and statements of category theory something that you can prove - or at least check in many situations - to be valid for string theory as we know it, or is it just an unproven conjecture that derived categories describe D-branes? ** Can somebody give a an answer ? He also writes: ** I always feel very uneasy if the mathematically oriented people present their conjectures about physics, quantum gravity, or string theory as some sort of "obvious facts". ** I would say that any serious scientist or mathematician would feel the same way !!, and also that this seems to be a common practice in many papers that claim applications of category theory to physics. ** I have this image of differential geometers saying to each other, a century ago, "Don't you think somebody ought to tell that Einstein to stop trying to use differential geometry to explain gravity, before our whole field gets a bad name?" ** Well, Einstein was not "trying to", he was using it, and presented this use as an accomplished fact. Also, you forgot to mention that he flunk a high-school exam or something of the sort proving by this very fact that a lot of people were stupid, just as they are those which have doubts about the real value of some applications of category theory to physics ! ** I do not see how anybody can possibly discredit category theory by applying it to string theory, even inappropriately, any more than "The da Vinci Code" discredits classical geometry and number theory. ** ** Since Motl's personality is well known, any damage will be minimal. I think we should relax and take it easy.** Well, rubbish category theory always discredits the whole of category theory, specially given the fact that it is not yet a prestigious and established subject (think in SGA4 and the introduction of SGA41/2) It will be nice to relax and take it easy. Will all of us do so ? I hope we will read in this cat-list some valuable considerations about Motl's questions and doubts, and about Marta's courageous warnings. e.d.

On Mar 15, 2006, at 5:35 AM, RFC Walters wrote:

The category theory community seems happy to accept uncritically, and give centre-stage to, any interest shown by an external field. In this context one should certainly look the gift horse in the mouth.

i think this is a very nice metaphor. but i am not sure that being critical about science is as easy as looking in horses mouth. already hilbert was largely wrong when he tried to prescribe a shape of a science. and nowadays it is a much harder task. everyone sees just a very small fragment. research advances by evolution, not by intelligent design. the division between pure and applied mathematics is not as simple as it used to be. 20 years ago, if you wanted to work on something that would never ever degrade into applications, then algebraic geometry probably seemed like a good bet. nowadays, at each moment, millions of transactions on the internet are secured using elliptic and hyperelliptic curves; the structure of their picard groups is discussed in standardisation bodies. if a bank protects its customers from phishing by identity-based keys, they are using weil or tate pairing... so the purest math has become the most applied; the most spiritual the most concrete. the other way around, these applications put a babylonian library on everyone's desk. what was picard group again? google for it. biology research is based on large public databases. physics is documented (driven?) by blogs. even category theory is discussed online. so i think it is great that people get nasty, or personal about category theory. the landscape of babylon: "the dog barks while the caravan goes by." just my 2p, -- dusko

Well Robert, 1)

Well, Einstein was not "trying to"; he was using it, and presented this use as an accomplished fact.

I just wanted to put in evidence the following fallacy that you are pushing forward: To attack the use of category theory (by some people) in string theory is at the same level that it would have been to attack the use (by Einstein) of differential geometry in general relativity. General relativity was born with differential geometry; it has no meaning without differential geometry. String theory was already there when a category theory approach began. It is not the same thing. Putting everything in the same bag is a well-known strategy to confuse an issue. Also, to have a poor opinion of many papers on applications of category theory to physics is one thing, to say that category theory has no future in physics is a completely different one. Nobody (including Motl's writing I am discussing (*)) has said the latter!! Quoting now from David Yetter: ** If (I suspect when) the string theory emperor turns out to have no clothes, Category theory will suddenly become de rigeur in physics". ** I start to believe that independently from what it finally happens with string theory; it is possible, even with the emperor well dressed, that category theory will with time become the rigeur in physics. 2)

Also, you forgot to mention that he flunk a high-school exam or something of the sort proving by this very fact that a lot of people were stupid, just as they are those which have doubts about the real value of some applications of category theory to physics !

I can only say that I am sorry about your reaction to this. It was just an irony, and I thought this was evident. e.d. (*) Motl said (if I remember correctly) something of the sort that he thinks that to reach some goals of string theory certain category theory approach will not be helpful.

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

Dear John, Thanks for your candid and informative letter. I feel that few people who so far have responded to me (or to others in the discussion that I started) really understand what my concerns are.

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. :-)

Your comments were very interesting and of course they will not put my worries to rest. I am grateful, though, for your taking this as part of a larger issue, on which I could expand more, but will not, since all I wanted was to raise awareness, not to preach (or even less to police). Best thoughts, Marta --------------------------------------------------------------------------------------------------------

From: "John Baez" <baez@math.ucr.edu> To: categories@mta.ca (categories) Subject: categories: cracks and pots Date: Thu, 16 Mar 2006 12:47:53 -0800 (PST)

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!

... [Further repetition deleted by moderator.]

I join Bob in saying that I fully agree with Marta, and I fully agree with Bob's second sentence. However, I have a problem with "look the gift horse in the mouth", since the horses we get are so often headless... I would also like to make just one comment to Paul's message (although I disagree with most of it; sorry!). Paul says: "Which generation was it that alienated other mathematicians by making outrageous claims about the foundations of mathematics that it never backed up with theorems? Which generation actually got its hands dirty and proved the theorems that relate category theory to other foundational disciplines?" Well, our colleagues active in the 1960s and 70s invented elementary toposes, for example, and proved many theorems about them. Those theorems did not convince set-theorists to forget sets, but are they convinced now? On the other hand those theorems were very beautiful, along with many others from several areas of category theory; I would describe 1960s and 70s as Golden Age of category theory. I am not saying of course that nothing important was discovered after 70s, but I see problems, and growing chaos, often created by ambitiously presented pseudo-relations with "other foundational disciplines". Moreover, talking about "relations": According to the classical work of Sammy and Saunders, the first "relation" was with algebraic topology. As we all know, there are various (co)homology/homotopy functors from topological spaces to groups, or to more complicated algebraic (or coalgebraic, Hopf, etc.) structures. There are also simplicial sets and other combinatorial intermediate players, and the relationship between geometric and combinatorial objects goes back to Euler (if not to Plato...). As we know from 1960s, the universal property of Yoneda embedding yields various adjoint functors, including those between simplicial sets and topological spaces - and this is why combinatorial objects are there! And what do recent algebraic topology text books do instead of explaining this? They are still talking about gluing cells instead. I think if we really care about relations between category theory and "other foundational disciplines", we should begin by explaining that category theory is not just a language allowing one to call homology a functor, but that category theory has beautiful constructions and results (some already from 1940s and 50s!) making enormous simplifications/applications/illuminations in neighbour areas of pure mathematics, such as abstract algebra, geometry, and logic. George Janelidze ----- Original Message ----- From: "RFC Walters" <robert.walters@uninsubria.it> To: <categories@mta.ca> Sent: Wednesday, March 15, 2006 3:35 PM Subject: categories: Re: cracks and pots

I also would like to support the remarks of Marta with which I am in full agreement. The category theory community seems happy to accept uncritically, and give centre-stage to, any interest shown by an external field. In this context one should certainly look the gift horse in the mouth.

Bob Walters

Vincent, you sound like this Beatles song, you know, in the White Album... Fully agree with you on the essentials of point 2 (you know that). However, uncritically referring to vociferations out of a some hate blog only because the blogger is labeled "string theorist" is not unlike point 1, at least in my modest opinion. Among the problems with the way research is sponsored there is this particularly modern one: the commitment to the short-term. It is quite similar to what happens in other sectors of the globalised society (of "high civilisation index" as L.Motl would presumably say -:( ) and leads to a growing disbalance in the allocation of resources. Cats are a very fine tool to organise concepts and proofs. Surprisingly enough, most mathematicians are quite reluctant or openly hostile. On the high-end, cat theory is crucial when it comes down to unify seemingly disparate areas of maths (which is unlikely a goal for itself) and this kind of work is quite clearly long-term. My 2 p: cat theory needs to be demystified in first place rather than to be sold. In particular, I think that the (still somehow ongoing) debate if it is a better foundation for maths or not is absolutely pointless.

Category theory is just not very trendy at the minute and to get the money one needs to do theoretical physics (there had been also Computer Science at some point - that was poor is not it?).

Now we have the best of both worlds: quantum computing :-)) Cheers Krzysztof -- my government will categorically deny the incident ever occurred

jim stasheff wrote:

Robert J. MacG. Dawson wrote:

And, as you know, there are still scales, almost a century later, on which its predictions are unsatisfactory.

For us ignorant of these, please explicate.

(1) At the very small scale, nobody has really managed to unify quantum mechanics (which is as thoroughly tested on its home turf as relativity is on its own) with general relativity. QM works astonishingly well on the atomic scale, GR works astonishingly well on the astronomical scale, but there is a big gap, "in which we live", in which neither is particularly evident and classical Newtonian mechanics works pretty well for most purposes. (2) At very large scales there is some question as to whether additional forces, not predicted by general relativity, are needed to explain some cosmological observations. This is more speculative, but a lot of physicists seem to think *something* needs to be done. -Robert

I think knot theory is particularly helpful here but I'll let Yetter and Freyd reply further. Jim Stasheff jds@math.upenn.edu Home page: www.math.unc.edu/Faculty/jds On Thu, 16 Mar 2006, Vaughan Pratt wrote:

Category theory, and for that matter modern (as opposed to elementary) algebra, is to mathematics as mathematics is to physics, and for that matter to computer science. Whereas mathematics organizes reasoning about the phenomena studied by physicists and computer scientists, algebra and category theory perform a similar function for mathematics.

In any setting organization is desirable, and arguably necessary on occasion. But the use of algebra and category theory to organize physics and computer science is a double whammy here. One should therefore be doubly sympathetic of those physicists and computer scientists who want to know what substantive contribution is being made to their subject and can't evaluate the answers because they are one if not two levels removed from the necessary abstractions.

Vaughan Pratt

On 17 Mar 2006, at 09:36, George Janelidze wrote:

... I think if we really care about relations between category theory and "other foundational disciplines", we should begin by explaining that category theory is not just a language allowing one to call homology a functor, but that category theory has beautiful constructions and results (some already from 1940s and 50s!) making enormous simplifications/applications/illuminations in neighbour areas of pure mathematics, such as abstract algebra, geometry, and logic.

Dear George, I think the straight answer is that it is genuinely difficult. Even for elementary applications it is not easy. Try asking non- categorical topologists how they explain the product topology to students. Many will say, "This definition may look odd, but it turns out to work best." Others will produce various ad hoc justifications, such as "It's the definition that makes Tychonoff's theorem true." (Though that may be at least historically correct.) You point out that the product topology is the unique one such that projections are continuous and tupling preserves continuity, but they still don't see that as anything special. But with regard to certain advanced applications, there are pictures in the minds of the category theorists that do not translate at all easily to paper. Even the master expositors find it hard. I'm thinking for example of the idea of topos as generalized space. I have been working seriously with toposes (usually as generalized spaces) for about 15 years now and in some respects my understanding of them is quite deep. Yet there is still a huge gap in my understanding when it comes to their applications in algebraic geometry, Galois theory and algebraic topology, the kind of fields that gave rise to toposes in the first place. Somehow when I read the accounts I see a mass of machinery but no clear intuitions for what it is doing. This surprises me. A characteristic strength of category theory is that it is particularly good at explaining the underlying meaning of constructions, with its notion of universal properties, and with some beautiful tricks of categorical logic. So is it possible to explain, or illuminate, those particular categorical applications to someone like me? (Perhaps the challenge has already been met, and I've just missed the right book; and of course I eagerly await vol. 3 of the Elephant.) Here's a sample question where my categorical understanding falls short of the applications. If A is an Abelian group, then the space ^A of A-torsors is also a group (modulo canonical isomorphisms - the equational laws of group theory do not necessarily hold up to equality). The identity element is the regular representation of A on itself, group multiplication is "tensor product" of torsors, and inverses are got by inverting the A- action. It follows that if X is any space, then the collection of continuous maps from X to ^A is also a group, and this construction is self- evidently contravariant in X. Obviously it takes topos theory to formalize this, but already we can paint a picture. For example, suppose A is the cyclic group C_2 of order 2 and X is the circle. Then there are (classically) two isomorphism classes of maps T: X -> ^A, essentially because in going once right round the circle the variable torsor T(x) can come back either just how it started or with an automorphism swapping its two elements. The corresponding group is C_2. This looks like some kind of cohomology, so is it already part of the standard theory? I've never managed to follow all the machinery through. All the best, Steve Vickers

Dear Steve, My answer to your straight answer is "yes and no"; that is: First of all I suggest to avoid discussing the reasons why some mathematicians do not see the categorical definition of cartesian product as anything special: this discussion continues, stops, and comes back again for a half of century, and I don't think you and I can make any further progress in it. Next, the idea of topos as generalized space is very nice and important, but it is too far. Long before it one learns easier things, such as, say, adjoint functors - which do not have a non-categorical definition! And then, immediately, there are amazing applications, such as seeing the geometric realization of a simplicial set as an outcome of the universal property of Yoneda embedding (which I mentioned in my previous message). Category theory is not a religion, and if someone discovers tomorrow something better than category theory, I shall be happy to study it. But this has not happened yet, and at the moment category theory provides the unique way to unify mathematical theories - and if one has new ideas or constructions in algebra, geometry, or logic - they should be understood and presented categorically - not with "ad hoc justifications" (using your expression). Unfortunately the number of mathematicians that were/are either ignorant or careless about this, is much larger then the number of those who tried to clean things up. The result is a dangerously growing chaos in abstract mathematics - and we certainly do not want applied mathematics to contribute to this chaos by telling us science fiction stories about things like string theory mixed up with operads and higher-dimensional categories! A simple categorical theory of torsors (via monads and resulting cohomology), and Galois theory in general categories (I mean what I call Galois theory) do exist, and your example of C_2 can be described via any of them. I am one of many people who can explain this to you if you are really interested. With best regards, George ----- Original Message ----- From: "Steve Vickers" <s.j.vickers@cs.bham.ac.uk> To: <categories@mta.ca> Sent: Sunday, March 19, 2006 8:25 PM Subject: categories: Re: cracks and pots

On 17 Mar 2006, at 09:36, George Janelidze wrote:

...

... I think if we really care about relations between category theory and "other foundational disciplines", we should begin by explaining that category theory is not just a language allowing one to call homology a functor, but that category theory has beautiful constructions and results (some already from 1940s and 50s!) making enormous simplifications/applications/illuminations in neighbour areas of pure mathematics, such as abstract algebra, geometry, and logic.

Dear George,

I think the straight answer is that it is genuinely difficult.

Even for elementary applications it is not easy. Try asking non- categorical topologists how they explain the product topology to students. Many will say, "This definition may look odd, but it turns out to work best." Others will produce various ad hoc justifications, such as "It's the definition that makes Tychonoff's theorem true." (Though that may be at least historically correct.) You point out that the product topology is the unique one such that projections are continuous and tupling preserves continuity, but they still don't see that as anything special.

... [further quoted material removed by moderator.]

Hi, I thought that my intention in raising the issues that I did in my original posting of March 12 were clear enough. Now it seems that they were not, to some. 1. I find ridiculous the suggestion put forward by Robert Dawson (March 23) that my presumed "call for collective action against an entire field of research seems uncomfortably close to an organized boycott, an extreme breach of tradition that only an emergency -if that - could justify it". The invention of an alleged "boycott" plot seems aimed at dismissing the questions that I (and other concerned mathematicians who joined the discussion) have raised. Anybody who, like Robert Dawson, resorts to such inventions appears to be panicking in that he is trying to divert attention from, rather than help, a healthy discussion. 2. Eduardo Dubuc writes: "I do not agree necessarily with Marta's implicit views". There is nothing implicit in my views. Just take a second look at my various postings of March 14, 15, and 17 in reply to some people. If Eduardo refers to my bringing in the Templeton Foundation into the discussion, then I would like to add some comments, partly expanding (and correcting) my reply to Vincent Schmitt (March 17). I can back up my contentions in reference to the the Goedel Centenary Symposium in Vienna http://www.logic.at/goedel2006/ and the workshop organized by A. Connes at the Sir Isaac Newton Institue in Cambridge (Non Commutative Algebra) http://www.newton.cam.ac.uk/programmes/NCG/ncgw02 I should, however, make more precise my reference to the Perimeter Institute for Mathematical Physicts. What sems clear is that one of its most prominent long-term researchers is at the same time one of the prominent particpants in Templeton funding and activties, for instance the Foundational Questions Institute. I quote from the last issue of Nature http://www.fqxi.org/about.html "Phycists to confront those big questions. Time travel, multiple universes and extraterrestrial intelligence seem more the purview of Star Trek scriptwriters than of serious researchers. (...) The FQI was set up last October with a grant from the Templeton Foundation, which promotes research at the boundary of religion and science. With US$8 million in seed money, the FQI will fund dozens of researcher's part-time work on these questions. (...) "I am very happy to see that a project has started to address these needs" says Lee Smolin of the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, who is also on the FQI's scientific advisory board. -- Geoff Brumfield. Nature. 2 March 2006." I stated incorrectly that the Pi is devoted to String Theory, when it seems, judging from the work of Lee Smolin, that Pi rather promotes Loop Quantum Gravity, a competitor to String Theory. By the way, an article by Lee Smolin entitled "Atoms of Space and Time" on LQG has been issued already three times (with minor variations) in Scientific American (200, 2004, 2006), so many of you must have seen it. 3. I have never suggested that "an entire field of research" should be suspect of constituting bad mathematics. If by this entire field of research it is meant n-categories, theta-categories, operads, topological quantum theories, and so on, there is, as in any other field, good and bad mathematics. Perhaps I should bring to your attention my comments to the organizers of the StreetFest, requested by them of all participants, and posted in their website as http://streetfest.maths.mq.edu.au/feedback?lastname=Bunge&firstname=Marta I stand by this, and only hope that my remarks in the "cracks and pots" postings have not been misinterpreted by the people mentioned in my comment above, and by others, like Ieke Moerdijk, not mentioned in it since they were not there. 4. I also think that a problem persists in the emphasis given to the "you do not want to know" general message in Baez postings, not because of them intrinsically, or of himself, but of the use others (for what purposes, I do not know) are making of this general trend. One instance of this trend (although in a different casting) is the following http://www.math.uchicago.edu/~eugenia/morality/ of a lecture that Eugenia Cheng gave in Cambridge last year. With best wishes for (and absolute faith in) category theory, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/bunge/ ************************************************

From: Eduardo Dubuc <edubuc@dm.uba.ar> To: categories@mta.ca Subject: categories: re: cracks and pots Date: Thu, 23 Mar 2006 13:50:45 -0300 (ART)

Hi

To follow are the contents of two postings that Bob (always vigilant, ja!) thought best to concatenate in only one.

On spite of Robert's erudition and his knowledgeable discourse, I still think Einstein using differential geometry to develop general relativity is not at the same level that John Baez using category theory to develop and/or understand string theory. His arguments are valid in a court of law, but do not convince me. I imagine John himself is probably the first to laugh at such a comparison.

But this is not the issue of my present posting. He touches also some pertinent points that go more to the core of the "cracks_and_pots" debate.

(In between ** are Robert words)

What Motl says certainly does not make people using category theory in string theory laugh. Applications of category theory to string (or to other physical theories competing with string theory ?, see Yetter's posting, it is all very confusing !!) may be valuable or may not. I (and a lot of us) can not tel.

** In which case demands that they ($) be read out of the meeting are premature. ($) papers that claim applications to physics **

This is a difficult question.

Marta was saying (and Bob Walters and others agree) that when a paper was claiming applications to physics it was easily accepted without knowledgeable and close examination, and that there were a lot of them.

Probably a lot of them should be read out, but not by policy against (as it was erroneously interpreted in these postings). Serious refereeing is a healthy practice that should not be equaled with censorship.

**Remember - in mathematics it's a matter of "In God We Trust, everybody else must provide a proof."**

This is not so much so. Speculations in math are very difficult. If not well founded they are vacuous. Only great mathematicians can do them (example close to us, Grothendieck), the rest of us must provide a proof.

**If the math itself meets mathematical standards of rigor, its application to physics need surely only meet the standards appropriate to that subject.**

The math itself must also meet standards of quality, not only of rigor. Besides that, "standards appropriate to that subject" does not mean "free for anything". Motl writes:

"I always feel very uneasy if the mathematically oriented people present their conjectures about physics, quantum gravity, or string theory as some sort of "obvious facts."

Clearly he is saying that these standards are not being fulfilled (in his opinion of course) by claimed applications of math to physics.

Motl may be wrong or he may be right, what we have not seen yet in these postings is a convincing or clear answer to the questions he arises. I would say, not even an answer at all.

These questions triggered Marta's original posting, which in turn was arising other (not exactly the same) questions. I do not agree necessarily with Marta's implicit views, what I support is her courage to point out that they are serious problems in the category theory community (for example, quality of the publications, abuse of fashionable topics to get grants, invited speakers in CT meetings).

Best wishes e.d.

Dear Category Theorists, I have begun trying to compile a list with information (mainly links to reviews and other literature) on applications of categories in mathematical physics and string theory. (It is not finished yet, though.) See http://golem.ph.utexas.edu/string/archives/000775.html . Best regards, Urs Schreiber

Vincent Schmitt wrote in part:

Now that theoretical physics, computer science, phylo., a mix of those, or whatever? , is used to justify poor "categorical" work is, in my view, an existing problem. More or less everyone is conscious of it (come on!...) but so far that has not been publically debated. I am happy that it happens now.

Actually, I've had great difficulty with this thread [*] because I am ~not~ conscious of this (justification of poor work). It seems all too obvious to many of the posters here; you are probably more familiar than I with the bulk of the literature. But unless I've missed it, nobody has given an example of this. (The closest is that John Baez's work has too much prominence, but nobody wants to claim that his work is poor, quite the opposite. And there was a work by a philosopher that was cited, but that did not pretend to be mathematics.) I would understand your concerns much better if I knew a few examples, hopefully from various fields, of poor work that has been unjustifiably accepted. I know that it may be hard to give specific examples without running the risk of insulting colleagues, and I'm sorry about that; but without them, I really don't have any idea what you're all complaining about. (Not just Vincent, but Marta and all of the others supporting her are included in this request, please!) [*] Incidentally, "thread" is an old Internet term for a discussion resulting from a single "original post" ("OP"); the thread consists of the OP, every post written in reply to the OP, everything written in reply to those posts, and so on (recursively). So Marta's first email on this topic is the OP, and the 100 or so public emails since constitute the thread. -- Toby

I just returned from a vacation and caught up with this thread, so please bear with me as I back up to the central question posed by Marta Bunge. She suggested that

anything which even remotedly claims to have applications to physics (particularly string theory) is given (what I view as) uncritical support in our circles.

Is there any evidence to support this claim? I.e., actual examples where such research was disproportionally supported that was uncritical and perhaps unwarranted? There have been several posts seemingly agreeing that this is the case, but none have given concrete evidence. I feel that it is necessary to establish that such practices indeed exist, before discussing what, if anything, needs to be done about it. Can one rule out another possibility, namely that such research is supported because it is original, timely, and interesting? -- Peter Marta Bunge wrote:

Robert Dawson wrote:

...

It is not clear to me that the majority of theoretical physicists agree with the negative view of categorical string theory held by the cited blog writers; and in the absence of a consensus among the physicists, I for one (with an undergradate degree and some graduate courses in physics) do not feel qualified to take sides; if anything, errors should be on the side of trying out too many ideas, not too few.

I was trying to elicit an open response from those who *do* know about the value (or lack of it) of categorical string theory. In particular, I would like to have an answer to this question. Why is it that anything which even remotedly claims to have applications to physics (particularly string theory) is given (what I view as) uncritical support in our circles?

Best, Marta

Quoting dusko <dusko@kestrel.edu>:

but at the end of the day, i think, we'll all agree that the source of the unreasonable effectiveness of categorical algebra is its foundational content (although there is probably a lot of it that we dont understand yet); and the other way around. eg, if you look at grothendieck's work, he started working in algebra, and ended up developing foundational structures, because he needed them. and a lot on the "algebra" side now is built upon them. ok, then for a while it was thought that he exaggerated with foundations, and that a more direct approach "could have been in better taste" (to cite eilenberg). but maby the fermat theorem would have a more useful proof if it was developed in grothendieck style. and nowadays, there is a lot of foundational content in tannaka duality etc, in TQFT in general ...

TQFT!? It seems dusko has finally discovered the shift key on his keyboard. Alex -- Alex Simpson, LFCS, School of Informatics, Univ. of Edinburgh, UK Email: Alex.Simpson@ed.ac.uk Tel: +44 (0)131 650 5113 Web: http://homepages.inf.ed.ac.uk/als Fax: +44 (0)131 667 7209

Dear Peter, You were lucky to have been away on vacation, but perhaps quickly reading (how else) the mass of postings in the "cracks and pots" postings has caused you intellectual indigestion. Your reaction is therefore quite understandable. For your sake (and that of others in similar situations), I will sum up what caused my postings, and be more explicit concerning

Is there any evidence to support this claim? I.e., actual examples where such research was disproportionally supported that was uncritical and perhaps unwarranted? There have been several posts seemingly agreeing that this is the case, but none have given concrete evidence.

...

From the many responses that I got (some public, and many more privately), I

1. On March 13, I shared with you all a disturbing posting in Motl's blog, criticizing category theory in its applications to physics, and more particularly, John Baez. My concern was based on the possibility that any of this criticism might be justified because I could not failed to notice how John Baez had become more or less a prominent figure (as speaker/member of the scientific committee) in recent(ly announced) meetings in CT. Explicitly, I was thinking of Firenze, Ramifications of CT, 13-Nov, 2003, Sydney, StreetFest, July 11-16, 2005, Union College, UC Mathematical Conference, December 3-4, 2005, Chicago, MacLane Memorial Conference (Unni Nambondiri Lectures), April 7,10,11, 2006, Halifax (near), CT'06, June 25-July 1, 2006. 2. On March 14, and in response to some, I asked more explicitly what caused organizers of meetings to bring to center stage one aspect of CT over others, particulaly one which seemed to me not to be in good standing after Motl's postings. Was it because it is indeed the case that CT is in disrepute, and if so its reputation needs to be restored, this being the best way to do it? Was it because it is funding for CT (notoriously lacking in the USA) that may be more easily secured that way? I wanted to know myself, but also possibly alert organizers of meetings to reflect on this issues, since their power and responsibility is indeed enormous in promoting a certain kind of research over another. 3. picked on one (March 17) to add some information that I had just come across by reading Nature (on our coffee table, along with a dozen or so scientific journals), in an article which connected Lee Smolin of the Perimeter Institute with the Templeton Foundation, the latter a promoter of anything they can in the borderline of science and religion. In the Scientific American articles by Lee Smolin on Loop Quantum Gravity and the discreteness of the universe, a paper John Baez is quoted among the few references given at the end of the article. This, in turn, led me to research the Templeton Foundation itself, and with some help from a fellow categorist who seemed to know a lot about it, I easily located references to Templeton funding to the Goedel Centenary Symposium in Vienna, and to the A.Connes workshop on NCA at the Sir Isaac Newton Institute in Cambridge. I was, however, relieved not to find any direct connection between Templeton and Category Theory. Still, I meant to warn those unaware of this easy source of funding (with strings attached). In a subsequengt posting (March 27) I gave explicit references to these claims in response to some queries. 4. In short, I do not think that I can be blamed for not being explicit enough in matters that I could be explicit about. I still do not have all the answers to my questions. As I mentioned on March 27, I was mistaken in thinking of John Baez as a promoter of string theory when, in fact, he promotes a competitor thery, LQG. But the general question of categorical applications to physics remained. Why are they promoted now? As you, Peter, kindly offer as a possible explanation,

Can one rule out another possibility, namely that such research is supported because it is original, timely, and interesting?

No, of course not -- one cannot rule it out. Here, I am ignorant of physics so I cannot answer this question (David Yetter has supported the view that they are original, timely and interesting, and has contrasted "algebraic" to "foundational" aspects of CT). But even if the answer were "yes", I would welcome responses to the question which still remains unaswered (except that most of us surely have a formed opinion) -- is CT in such a poor state that it needs revamping? Sould we not wait a few years until several original and interesting (maybe not timely) contributions to CT in connection with other fields of mathematics are appreciated and incorporated into the mainstream? What do we gain by pushing those under the rag? To imply, perhaps, that we lourselves do not value them? These, I believe, are crucial and timely questions, and I do not regret unwilingly having brought them up 5. I take this opportunity to thank Bill Lawvere for his first posting "Why are we concerned? I", in which the lucid article by Saunders MacLane (Synthese, 1997) is recalled in connection with the discussions that arose in the "cracks and pots" so-called-thread (why "thread"?). I am sure that most of you have read it, but just in case you have not, I attach it here it in pdf form. This is very timely in view of the upcoming MacLane Memorial Conference in Chicago. Peter, I hope that I have answered your questions. I can't speak for the others who have contributed to this "thread". Unlike what has been suggested, what I originated on March 13 was far from a "complot". It was a genuine concern of mine and I see now, by many of the responses, that it is also a concern of others. On the other hand, getting personally attacked (for the wrong reasons, to boot) is a necessary price that I have to pay and it does not concern me as much. Yours, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/bunge/ ************************************************

From: selinger@mathstat.dal.ca (Peter Selinger) To: categories@mta.ca Subject: categories: Re: cracks and pots Date: Mon, 27 Mar 2006 10:28:57 -0400 (AST)

I just returned from a vacation and caught up with this thread, so please bear with me as I back up to the central question posed by Marta Bunge. She suggested that

...

anything which even remotedly claims to have applications to physics (particularly string theory) is given (what I view as) uncritical support in our circles.

Is there any evidence to support this claim? I.e., actual examples where such research was disproportionally supported that was uncritical and perhaps unwarranted? There have been several posts seemingly agreeing that this is the case, but none have given concrete evidence. I feel that it is necessary to establish that such practices indeed exist, before discussing what, if anything, needs to be done about it. Can one rule out another possibility, namely that such research is supported because it is original, timely, and interesting?

-- Peter

Marta Bunge wrote:

...

Robert Dawson wrote:

...

It is not clear to me that the majority of theoretical physicists

...

...

with the negative view of categorical string theory held by the cited blog writers; and in the absence of a consensus among the physicists, I for one (with an undergradate degree and some graduate courses in physics) do not feel qualified to take sides; if anything, errors should be on the side of trying out too many ideas, not too few.

I was trying to elicit an open response from those who *do* know about

agree the

...

value (or lack of it) of categorical string theory. In particular, I would like to have an answer to this question. Why is it that anything which even remotedly claims to have applications to physics (particularly string theory) is given (what I view as) uncritical support in our circles?

Best, Marta