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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeOct 6th 2010
    • (edited Oct 6th 2010)

    Recently we had discussion of "subculture"; problems of communication between category theorists and other mathematicians, and between strains of category theorists like those on the interface with homotopy theory and those in pure category theory, or those on the interface with algebraic geometry and so on; and even between different schools and practitioner circles like Canadian/American (MacLane, Lawvere, Joyal...), Australian (Kelly, Street), French (Charles Ehresmann/Benabou), French (Grothendieck), French (Simpson, Toen et al.), Russian (Beilinson and followers, Kontsevich and followers), John Baez "circle", Georgian school (Janelidze, Jibladze...), nlab...

    I have moved a query box discussion from TAC entry to here:

    Urs Schreiber: Lately I have been wondering what will be happening to this unpopularity of category theory among AMS in light of recent developments. Before long and opposing category theory in math will be a bit like opposing the use of complex numbers (there was a time when that was vehemently opposed by some, too) and currently the impetus of this development comes notably from US researches, and there notably from AMS grantees.

    Zoran Škoda: I recall from 1990s that the whole Grothendieck school was at that time very unpopular in US, and recall mean jokes about Grothendieck (like "the French mathematician whose only example of a big theory was a circle") and even recall being scorned by an algebraist because of using schemes in a discussion; the popularity of stacks (and similar notions) in recent mathematical physics changed the balance in the geometric part of the story since then. But I am not optimistic that so soon we could see more general change beyond central parts of pure mathematics and formal mathematical physics. I mean, we do see the huge coming influence of category theory in central parts of pure mathematics (algebra, topology, modern geometry), but not much in most of analysis, including so popular PDEs as well as in stochastics and probability; then influence in one direction of mathematical physics, but I would not say in mainstream theoretical physics (to name few major areas say highTc superconductivity, perturbative analysis of standard model and its extensions, turbulence theory, plasma physics and so on) either. Combinatorics is in between, it is huge area where most often deep knowledge of categories will not help radically, but there are so many examples of wonderful cooperation. We should keep in mind that modern geometry, algebra and topology while so central to us they are not nearly a half of an average department in math, the rest being mathematical biology, financial math, PDEs, ODEs, probability, algorithms, complexity theory, operator spaces, metric geometry, complex systems, game theory and so on...

    • CommentRowNumber2.
    • CommentAuthorHarry Gindi
    • CommentTimeOct 6th 2010

    I feel like category theory is in some sense one of the main areas of intersection of pure algebra and pure combinatorics. Topology is in some sense dual to algebra, and analysis is emergent from the study of topology and algebra together. Expecting category theory to lead to a very deep understanding of fundamental algebraic and combinatorial structures seems a bit naive.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 6th 2010

    It surely depends on what kind of algebra, and what kind of combinatorics, and what one means by “understanding”, and what one’s expectations are.

    There is so much to say here that I for one am almost dumbstruck. Just one comment is that the introduction of species in combinatorics is a general advance wherein the “art” of combinatorics begins to be transformed into a “science”. I think this transformative quality of species has been widely acknowledged in combinatorial circles (even among those who seem generally distrustful of abstract methods, for example Zeilberger).

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 6th 2010
    • (edited Oct 6th 2010)

    With regard to #1: I am no longer in academe, so I am not well placed to observe the general Zeitgeist with regard to category theory, but my observations of math on the internet lead to me believe that general hostility towards making a career of research in category theory is still very much alive and well in the US.

    For instance, there is Walt Pohl, the author of Ars Mathematica, who opined a few years ago, “The attempt to rewrite the foundations of mathematics in terms of category theory is evil and wrong. It is the fourth great evil we have been called to face down: after Nazism, Communism, and Islamo-fascism, it is our destiny to confront Catego-fascism. I’m not surprised to see dsquared on the side of the catego-fascists.” (He is at most half-joking, I’m pretty sure. Most of his posts having to do with the interface between category theory and foundations look like they have some kind of “agenda” to them.)

    A recent sample comment comes from Greg Kuperberg, who wrote on MO, “In my opinion, category theory is to mathematics what garlic is to cooking. It is a widespread ingredient that adds a very important flavor. But, usually, it should be minced and mixed in, and used with restraint.” The implication would seem to be that pure research in category theory is in bad taste, and one must be “restrained”, or face pariahdom. This (IMO lame-ass) comment was met with great celebration and jubilant up-voting. (So nice to see the gatekeepers and guardians doing their duty!)

    These are just two of many, many comments one can find if one looks around.

    Perhaps others can speak more directly to the funds and grants situation for category theory (in the US). My impression is that it’s godawful.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 6th 2010
    • (edited Oct 6th 2010)

    This summer in Oberwolfach one day over lunch I happened to chat with Michael Douglas – famously one of the central figures in string theory. He asked me, quoting from memory:

    It seems that we need to start teaching our graduate students higher category theory. Which texts could you suggest?

    Last month in Vienna at the ESI institute, Calin Lazaroiu, string theorist, former student of Edward Witten, taught an introducory course on category theory to an audience of theoretical physicist, announcing it to them as the inevitable toolset necessary to even handle the sophisticated structures that they claim to want to be dealing with nowadays.

    Sooner or later this new wind will become a storm.

    At the same time I saw various people pull the interesting trick of telling me how they despise category theory, only to manifestly get lost in the next talk on superfields (via Yoneda lemma) on sigma model target spaces (via Lie n-groupoids) and so on.

    I watch this in an entirely relaxed spirit. I don’t need the random guy from the street (or from some institute, for that matter) to confirm to me the value of category theory. It’s their problem, not mine.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeOct 7th 2010
    • (edited Oct 7th 2010)
    Urs. We all agree with you in values of category theory. We also agree that in the immediately affected areas of homotopy theory, formal geometrical physics and alike this is going to be recognized soon and that the process is visible. But how to tell this to most mathematicians who never use words bundle, connection, homotopy, cohomology, TQFT or adjointness, and talk instead about american option (kind of model in financial mathematics), asymptotics of prime numbers, martingales, inequalities, subtle properties of Banach spaces, integer sequences, weak solutions of hyperbolic PDEs, approximation schemes, Navier-Stokes equation, percolation, entropy, biostatistics, DNA computing etc. ? These people form most of the departments and will hate if they advise a graduate student who is fullfilling their graduate algebra course requirement with more usage of category theory than they remember from their own, often odius in their memory, algebra course from their own graduate school ?

    I met Lazaroiu 6 years ago and he was equally in category camp then. He is a very bright guy which seems to lack deserved support of the surrouding physicists. String theory is also now loosing its popularity among mainstream theoretical physicists, unlike the situation about a decade ago.

    As far the situation in US, I was told by successful grant applicants who do mathematical physics that one should not mention words "mathematical physics" in an NSF physics proposal (maybe among math proposals it is OK). Say more concretely "integrable models" or "percolation models" or "Conformal Field Theory" and it is more or less OK. Though all the mentioned are areas of mathematical physics. This says how much of the thing is an ideological balast.
    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 7th 2010

    String theory is also now loosing its popularity among mainstream theoretical physicists

    Or has already. Which I also find very relaxing. Let them all go and do something else, if they don’t understand it.

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeOct 7th 2010
    • (edited Oct 7th 2010)

    Todd, the example of Zeilberger is an extreme, as the latter is an extremist and likes to publicly expose the tirades of his extremism and hi opinions. Zeilberger is one of the sharpest and most successful combinatorialists, e.g. he has very important series of papers advancing and abstracting the Gian-Carlo Rota's idea of umbral calculus.

    On the other hand, he assumes his ways of doing things and the ways of some people who are closest to his interests must be the standard, best, everything else is less effective, it is boring or unimportant. Here he easily gives lectures to anybody. If Gauss were alive, with contemporary topics of research, I am sure he would get his own portion of instruction from Zeilberger. For example, few days ago, on his homepage-blog, Zeilberger gives their portion to the newest Fields medalists. He likes Smirnov's work as it has aspects close to what he does, but to Ngo and Villani he admits that they do something what is deep and he does not understand, but asks them to quit that and stop doing boring stuff. Halo ?? Deep and boring ?? How can somebody who ever felt a beauty of mathematics say such a nonsense. That something is deep and boring ? Then he says all fine with the prizes, but I do not see any of important problems solved and he lists several problems, mostly from number theory. These are THE problems. Anything not preconcieved in his, it seems simplistic, view of math does not belong into important and interesting camp.

    Well, I can understand that some people like applications or problem solving and do not feel importance of abstract mathematics. But then, what is the usage of engineer by having a proof of Goldbach conjecture or even more the Fermat's theorem which was empirically known for all reasonably small numbers before ? On the other hand, he complains why Villani got Fields for work related to Boltzmann H-theorem as basically physicists work with versions of it anyway. So proof of one empirical fact in number theory would be important but in mathematical physics, something what is exactly known in too simple situations or not rigorously enough is worthless ? This is clearly uneven thinking, and choices are made by taste and not by true comparison of internal value or of true and cumulative external impact.

    Similarly, Zeilberger gave a lecture to Grothendieck. He said, well the guy left because he was not like his department colleague Gel'fand. Like, Gel'fand collaborated with people, was concerned with applications, was diversifying, and so on, so says Zeilberger, he did not get burnt out. Grothendieck was doing his own solemn big abstract programs so of course he got burnt out and left young while Gel'fand was working up big math till late age. What kind of shamelessness from Zeilberger in doing such an assertion, giving a recipe for Grothendieck...First of all every person has his natural mode in which he gives most of it. Most humans know this, and one does not need to be a very successful mathematician like Zeilberger to learn this, and even with having that advantage of life and mathematical experience, Zeilberger fails to see with the eyes of understanding some life and mathematical life. If you do things which you are not inclined to it is hard to be truly creative. Second, everybody has their own psychological needs. It is personal if one discovers the beauty outside of math and decides to live there. Sometimes an excellent sportsman leaves her sport to go to school and become a surgeon, what she dreamed in a childhood. Somebody does not have that inclination, strength, or support to do it.

    Who knows all the inner feelings AG might have had in deciding where to direct his energy at points of rethinking its human wishes and philosophical role. Finally, it is not true that Grothendieck did not work with others, according to the records he was sharing his thoughts generously with surrouding mathematicians and with big stamina for long communication sessions. Some of them published volumes on what is essentially Grothendieck's work (e.g. Hartshorne's book on duality). This is not changed by AG's decision to find his peace elsewhere at certain point in life.

    Life rules, and a genius can not be reduced to little of external information. But Zeilberger clearly knows what is wrong with everybody...and made 100% self-assured assertions on everybody's standing (including that Terry Tao is clearly the greatest mathematician of the present time, what may be true, but the essence is that the bidder does not know or does not want to acknowledge or understand other possibilities).

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeOct 7th 2010
    • (edited Oct 7th 2010)
    > Or has already. Which I also find very relaxing. Let them all go and do something else, if they don't understand it.

    I agree. Nice spirit behind the thought!

    Though people like Lazaroiu who have the heart and wit at the right side may suffer as well. Now graphen is a big thing in physics, this was a dominant statement even before the most recent Nobel Prize.
    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeOct 7th 2010

    how to tell this to most mathematicians who never use words bundle, connection, homotopy, cohomology, TQFT or adjointness

    Sheaf theory had similarly cristalized around 1950 with big impact in central fields of mathematics in 1950-s and 1960-s. Then it penetrated say complex analysis. From 1970s the influence of sheaf theory in mainstream mathematics started decreasing and retreated from most of complex analysis books since. In most areas of mathematics, now 50 years since its golden era people despise the term and most mathematicians do not know what it is. Higher category is as ubiquitous as sheaves, and I think its standing among mainstream mathematicians will be the same in 50 years as it is now of sheaves. Really both still have unlimited possibilities to grow, while in some areas of math of course less than in others.

    • CommentRowNumber11.
    • CommentAuthorEric
    • CommentTimeOct 7th 2010

    Here are three important factors:

    1. Funding
    2. Funding
    3. Funding
    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeOct 7th 2010
    • (edited Oct 7th 2010)
    No, Eric, far from that. Many of the opinions fights are among thinkings of regular tenured professors with stable jobs. There is much of ideology, overspecialization and subdivision of knowledge, and pressures from external system of commercial journals and so on as well as systematic brainwashing from political side. You think that *lack* of money is the reason that many states fund teaching creationism and other pseudoscientific theories ? Or it is easier to govern a mass who is brainwashed ?
    • CommentRowNumber13.
    • CommentAuthorEric
    • CommentTimeOct 7th 2010

    From wikipedia:

    While tenure protects the occupant of an academic position, it does not protect against the elimination of that position. For example, a university that is under financial stress may take the drastic step of eliminating or downsizing some departments.

    As to your other comments, I have no response because I have no idea where they are coming from or if they are even directed at me.

    • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeOct 7th 2010
    • (edited Oct 7th 2010)

    Well, we are talking about undeserved unpopularity of category theory among mathematicians and physicists. Once unpopular, it may correlate to other biases and pressures, however the genesis and main sources are in lack of understanding and in social affirmation of naive imagery about science and about abstraction which eventually comes from brainwashed society in which the antiscientific revolution as Arnold would put it has already rich history, and which inside the math circles comes from overspecialization (which has related causes) and from lack of true deep and wide understanding among the scientists in general.

    It is certainly not in any major way influenced by the fear that a tenure position would be eliminated because the research flavour of a mathematician favors or appreciates category theory to some extent.

    Thus the arena in which even some good mathematicians like Zeilberger scorn colleagues who delve into non-obvious depths of abstraction is not the same arena in which mathematicians prove that mathematics is useful or important. At least not directly. It is like bringing the non-relevant argument that mathematicians are not spending enough time for refereeing because their spouses want them home rather than staying at work too late. While such an effect may as well be a part of a general scene, it is not the central issue and not directly related to problem with the fall of refereeing quality. Similarly with mutual understanding of mathematicians in different fields and with understanding of deep tendencies and needs for architecture of mathematics which brought big programs like motives, higher categories and so on.

    So my comment is about that the heart of the issue is very far from your exclamation on funding.

    • CommentRowNumber15.
    • CommentAuthorEric
    • CommentTimeOct 7th 2010

    True true. I agree with you. I didn’t suggest funding is the sole or even the most important issue. But it is an issue that I think is important and does determine behavior to a large extent that many on the fronts lines of science rarely see.

    Schools are mostly publicly funded. The funds are limited, so choices need to be made as to who gets support. Many times the people doling out resources do not have a clear understanding of the underlying scientific merit and must rely on observable factors, e.g. who is the loudest, who is the most cited, who are the experts, etc. Sadly, the first item “loudest” is weighed more than it should be.

    The way funds are allocated impacts the behavior of participants who depend on those funds. There are pure scientific differences for sure, but I’m just posing funding as an “important” and largely under-discussed factor.

    • CommentRowNumber16.
    • CommentAuthorzskoda
    • CommentTimeOct 7th 2010
    • (edited Oct 7th 2010)

    So why do you think that a category theorist would be in worse position as far as funding is concerned than a number theorist. Both sciences are equally not considered "applied" from the point of view of non-mathematicians. The reason for more doubt about category theory content among mainstream mathematicians than about number theorists must be far predominantly in the ideology within mathematics. It is of course different to some extent if we compare say fluid mechanics and category theory.

    • CommentRowNumber17.
    • CommentAuthorEric
    • CommentTimeOct 7th 2010
    • (edited Oct 7th 2010)

    I never mentioned anything about applied being better than pure as far as funding is concerned, but in this particular example I think number theory is applied to important security applications. A friend of mine who graduated from Harvard with a PhD in number theory now works for Microsoft.If you push me, I’ll admit it was algebraic geometry, but… :)

    As far as category theory itself within the broader picture, I think many category theorists (not all of course) display antisocial behavior, which is another important factor.

    Edit: I chose bad wording there. Maybe “lack of political savvy” is better than “antisocial behavior”. Also my evidence is anecdotal based on historical references.

    • CommentRowNumber18.
    • CommentAuthorTim_Porter
    • CommentTimeOct 7th 2010
    • (edited Oct 7th 2010)

    Various independent comments on the above:

    (i) I agree with Eric that it is funding (and related things) that count. I had tenure, so don’t count on that! Over a period of 15 years I wrote and submitted a large number of categorically based proposals and got 1 very small one (for a visit by Turaev). The proposals used category theory in topology, and related areas, but the hostility was evident, even virulent vitriolic rhetoric sometimes. It was not simply, however, hostility to category theory but (as usual in the UK) a mix of that and prestige of the institution. One referee’s report more or less said that if the project was funded he would blacklist anyone who took up a postdoc position on it! Perhaps unbelievable, but true. Within the UK system, tenure does not mean much.

    (ii) Zoran was talking about AG earlier in this thread. The idiot who attacked him failed to take into account the historical context of AG’s childhood and career (and shows all the signs of being himself a mathematico-fascist). Mathematics and mathematicians do live in a historical context although sometimes that is forgotten by them.

    (iii) I heard of a ’well known’ mathematician who was asked at a conference : ’Professor X, how do you decide on what is good mathematics?’ His reply: ’easy, it is the mathematics done by good mathematicians.’ Later in the week the same person asked him’ Professor X, how do you decide who is a good mathematician?’ His reply… you can probably guess. ’Easy, it is one who does good mathematics.’

    On that line, If David Corfield is reading this, can I ask him if he knows any attempt by, say, mathematical philosophers to draw up a list of criteria to help evaluate mathematical areas. I do not mean attaching a number, or even attaching a ’good’/’bad’ label. For instance, I once asked Dieudonné how the seminar Bourbaki organisers evaluated topics for inclusion in the series. I did not like the list he provided, but he suggested some criteria such as ’interaction with adjacent areas of mathematics’ . I replied to a related recent MO question with part of a list that Ronnie and I put forward some years ago, (In our paper on Math Methodology, I think.)

    I have always liked category theory exactly because it interacted with other areas in a very constructive way (and category theorists tend to be fairly open in their evaluation of other areas). This means meeting lots of interesting ideas and seeing lots of new concepts from a reasonably unified point of view.

    (v) The comment made on MO on garlic was I think flavoured partially by the context which was about where in a course of algebra should category theory be taught. The comment suggested that its introduction should not be made in an indigestible block but rather as a natural way of doing things as the level of abstraction needed was greater. (NB. I am not defending the overall thrust of the discussion there.) There were comments made about Mac Lane CWM, and I think they may be valid. It is a book that can be daunting the first time you see it. As category theorists (and related), it is perhaps important to think how we present categorical material to the outside mathematical world. If there is hostility to category theory, some of it may be due to attitudes that we have when presenting the subject to others. (There is also anti-category theory indoctrination of course which is also evident sometimes.)

    • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 7th 2010

    Tim, yes I’m reading. You ask whether philosophers have attempted lists of criteria for evaluation. The short answer is ’no’, apart perhaps from my own attempt in my chapter on groupoids in my book, which was closely related to that one of Ronnie’s you mentioned on MO. From my current perspective, I think the attempt to give anything very prescriptive is misguided. Here’s something from a piece I’m working on:

    I have mentioned Atiyah’s reliance on the noun ’story’ in his survey, but it is clear from his article that it is inseparable from a form of judgement. He is telling us there what he considers to have been the most important developments through the past hundred years, and what he considers to be promising for the future. Recently Terence Tao has written a piece ’What is good mathematics?’, which after making a list of what ’good mathematics’ can mean, continues by telling us the story of what he assesses to be some good mathematics. The list, consisting of twenty items, received this harsh assessment from Alain Connes:

    It is hard to comment on Tao’s paper, the second part on the specific case of Szemeredi’s theorem is nice and entertaining, but the first part has this painful flavor of an artist trying to define beauty by giving a list of criteria. This type of judgement is so subjective that I really had the impression of learning nothing except the pretty obvious fact about arrogance and hubris…

    The story, the ’second part’, escapes censure.

    I think Connes is right to point to what is troublesome about lists of criteria. Criteria seem clumsy. Also in mathematics at least it seems rather easy to do well. In chapter 8 of my book, the Methodology of Mathematical Research Programs, it was too easy to give an account of a program scoring well. What are we left with in Tao’s piece then is a good story. Would this be such a bad thing to conclude: Good mathematics is that which can be described by good mathematical stories? But what is a good mathematical story? Can we capture this except by listing qualitities we would expect to find in it? Fortunately assessment is made easier by a phenomenon Tao notes that when good things happen in a piece of research other good things follow in its wake.

    It may seem from the above discussion that the problem of evaluating mathematical quality, while important, is a hopelessly complicated one, especially since many good mathematical achievements may score highly on some of the qualities listed above but not on others. However, there is the remarkable phenomenon that good mathematics in one of the above senses tends to beget more good mathematics in many of the other senses as well, leading to the tentative conjecture that perhaps there is, after all, a universal notion of good quality mathematics, and all the specific metrics listed above represent different routes to uncover new mathematics, or difference stages or aspects of the evolution of a mathematical story.

    …the very best examples of good mathematics do not merely fulfil one or more of the criteria of mathematical quality listed at the beginning of the article, but are more importantly part of a greater mathematical story, which then unfurls to generate many further pieces of good mathematics of many different types. Indeed, one can view the history of entire fields of mathematics as being primarily generated by a handful of these great stories, their evolution through time, and their interaction with each other. I would thus conclude that good mathematics is not merely measured by one or more of the “local” qualities listed previously (though these are certainly important, and worth pursuing and debating), but also depends on the more “global” question of how it fits with other pieces of good mathematics, either by building upon earlier achievements or encouraging the development of future breakthroughs.

    The primary problem with the list of criteria is that we should expect what it is to be a good story to change. We need a meta-level story of how we have moved on from old stories. These days we expect surprise, such as when Vaughan Jones working on von Neumann algebras very unexpectedly realises he has his hands on a new knot invariant. It would be a very worthwhile task to examine the way overviews have changed in style over the years. For example, how does Klein’s Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert compare with modern surveys?

    END

    That’s about as far as I’ve got with it. Any thoughts?

    • CommentRowNumber20.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 7th 2010

    Am I correct in thinking that I’m hearing a difference between views on the reasons for the patchiness of the penetration of category theory into mainstream mathematics, between one which attributes it to the nature of the material (PDEs aren’t fertile territory) and one which attributes it to human failings (lack of effort, lack of receptivity to new ideas)?

    I don’t know if you’ve looked at the Two Cultures page I’m keeping at my wiki, but there are some interesting thoughts from Terry Tao, beginning here. On the one hand, there’s the line that areas which involve open or hybrid conditions are different from areas which involve closed and hybrid conditions. One might imagine category theory doing better on the latter. On the other hand, in his comments on nonstandard analysis, perhaps we see a clue (and I wished I’d pushed him harder on this) that the status as open, closed and hybrid may be modified by a different framework. A sign of that might be category theory doing better with nonstandard analysis than ordinary analysis, though perhaps SDG is a sign already.

    • CommentRowNumber21.
    • CommentAuthorTim_Porter
    • CommentTimeOct 7th 2010

    I think that I was not wishing for a definitive list of criteria but rather some dialectic relating to the methodology of mathematics. I personally have suffered at the hands of people who never question the criteria on which they judge the mathematics of others (’not mainstream’ being one very doubtful term used to shoot down some of my research proposals). One comes to the conclusion that many of the top ranking practitioners in many subjects are merely interested in power, but they suffer from self delusion pretending that it is ’for the good of the subject’.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeOct 7th 2010
    • (edited Oct 7th 2010)

    Category theory is like a high-level computer language. You need it to design a big software, not necessarily to code the routine for a given algorithm (though it may help there, too).

    That explains most of the difference in people feeling the need to use category theory. For instance the math I see discussed on Tim Gowers’s blog seems to be very remote from the software-design end and all into the subtleties of coding explicit routines. It should be no wonder that there is no category theory appearing in that context.

    According to the nice report by Tim Gowers here, at the last ICM Jacob Lurie said the following nice sentence about higher category theory

    I don’t want you to think all this is theory for the sake of it, or rather for the sake of itself. It’s theory for the sake of other theory.

    That’s the point. You need it to build other theory. If your other theory is all set in stone and you are only turning the crank on the machine, then that may not necessarily involve any further category theory.

    So the people at that end of the spectrum will begin to think that the cat theory is all useless. Which is only understandable. Not everbody needs to have a view of the big picture.

    • CommentRowNumber23.
    • CommentAuthorEric
    • CommentTimeOct 7th 2010

    @Urs #22: If only some of the early pioneers had taken that attitude, maybe things would be different. It is hard to change impressions once established.

    I can imagine someone whose spent years studying foundations in mathematics without ever feeling the need to learn category theory and then seeing foundations people in category theory come along. It is kind of like, “I don’t need your stinking foundations. I already have one thank you.”

    I could be wrong and most of what I know I’ve learned second and third hand from discussions here and the nCafe, but I think some of the early people were a lot more “in your face”. “Category theory is the greatest thing since sliced bread”. Then the natural response would be, ’What have you done for me lately?”

    I like your description. Maybe a slogan is, “Category theory is a meta theory.”

    • CommentRowNumber24.
    • CommentAuthorAndrew Stacey
    • CommentTimeOct 7th 2010
    • (edited Oct 7th 2010)

    I just wrote my thoughts in the MO question that Todd linked: http://mathoverflow.net/questions/41057/categories-first-or-categories-last-in-basic-algebra. When Todd explained what he didn’t like about Greg’s comment, I decided that I agreed with Todd. Then I went back and read the other answers and found that I didn’t much like them either.

    (Added in edit: with the exception of Tim’s answer, of course! And that’s not to say that I disagree vehemently with everything in the other answers there, but I didn’t like the underlying assumption: that category theory was like one of these advanced topics that only people who were that way inclined should study. So I felt that there was more to say and I said it.)

    • CommentRowNumber25.
    • CommentAuthorTim_Porter
    • CommentTimeOct 7th 2010

    Andrew. I just read your comment. I liked it (and voted it up!) Perhaps something along those lines could be put somewhere accessible and more visible.

    • CommentRowNumber26.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 7th 2010
    • (edited Oct 7th 2010)

    Tim #21. Unfortunately there’s not much one can do with people who won’t enter into dialogue about the ends of the activity they’re engaged in. You can comfort yourself with the thought that they’re acting irrationally, but that’s little consolation. Leaving a trail for the next generation to follow is a greater consolation.

    • CommentRowNumber27.
    • CommentAuthorTim_Porter
    • CommentTimeOct 7th 2010

    When Ronnie and I were pioneering Maths in Context, we discussed the course with several people. One, a very well known Spanish mathematician, said that before you tried to teach students about the Context of mathematics, you had to teach the professors and lecturers. Of course, you can’t do that can you! but in discussing the issues with the next generation of teachers, and lecturers perhaps the questioning can be handed on.

    Another person, this time an Economics professor, said that too few subjects actually talked about the context and methodology of their subject area with their students. (But I should add that just as talking about category theoretic arguments in the process of discussing algebra or topology etc., talking about methodology per se is not going to do the trick, it is rather a willingness to discuss the whys and wherefores of proof, analogy, discovery etc as part of ordinary content heavy courses.)

    • CommentRowNumber28.
    • CommentAuthorAndrew Stacey
    • CommentTimeOct 7th 2010

    I don’t know about accessible and visible, but I’ve copied it to http://www.math.ntnu.no/~stacey/CountingOnMyFingers/TeachingCategories.html as a starting place. I can envision polishing it a little so putting it a little nearer to hand makes some sort of sense.

    • CommentRowNumber29.
    • CommentAuthortrent
    • CommentTimeOct 13th 2014
    • (edited Oct 13th 2014)

    I’m just starting Andy Clark’s Supersizing the Mind* and came across the following passage on pages 15-16: “consider the now-classic example of running to catch a fly ball in baseball. Giving perception its standard role, we might assume that the job of the visual system is to transduce information about the current position of the ball so as to allow a reasoning system to project its future trajectory. Here, too, however, nature looks to have found a more elegant and efficient solution: You simply run so that the optical image of the ball appears to present a straight-line constant speed trajectory against the visual background (McBeath, Shaffer, and Kaiser 1995). This solution (the so- called LOT, for Linear Optical Trajectory, model) exploits a powerful invariant in the optic flow, discussed in Lee and Reddish (1981). There is, however, now some debate concerning the precise nature of the simple invariant we lock onto in solving this kind of problem. Thus, McLeod, Reed, and Dienes (2001, 2002) reported data that conflict with the predictions of the simple LOT model and that seem better predicted by an Optical Acceleration Cancellation (OAC) model first suggested by Chapman (1968). Shaffer et al. (2003) offer a mixed model combining uses of both strategies. For present purposes, however, the point is simply that the canny use of data available in the optic flow enables the catcher to sidestep the need to create a rich inner model to calculate the forward trajectory of the ball.”

    I see theory as helping one adopt elegant solutions like that, and, more generally I think that the more physicists understand the role things seemingly outside of the mind such as notation play in cognition, the more they will see the importance of work in mathematical physics which places physics in the most elegant possible notation. It’s not just theory addicts trying to justify their work when they say that it aids in problem solving, it’s how cognition works.

    *the main point of the book: “When historian Charles Weiner found pages of Nobel Prize-winning physicist Richard Feynman’s notes, he saw it as a “record” of Feynman’s work. Feynman himself, however, insisted that the notes were not a record but the work itself. In Supersizing the Mind, Andy Clark argues that our thinking doesn’t happen only in our heads but that “certain forms of human cognizing include inextricable tangles of feedback, feed-forward and feed-around loops: loops that promiscuously criss-cross the boundaries of brain, body and world.” The pen and paper of Feynman’s thought are just such feedback loops, physical machinery that shape the flow of thought and enlarge the boundaries of mind.”

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeOct 13th 2014

    Regarding this:

    In Supersizing the Mind, Andy Clark argues that our thinking doesn’t happen only in our heads but that “certain forms of human cognizing include inextricable tangles of feedback, feed-forward and feed-around loops: loops that promiscuously criss-cross the boundaries of brain, body and world.”

    This is an old observation, Heinrich von Kleist famously argued for this around 1805 in “Über die allmähliche Verfertigung der Gedanken beim Reden”.

    Then some time between 1879 and 1970 E.M. Forster famously wrote “How can I tell what I think till I see what I say?”

    I’d be surprised if some ancient Greek didn’t already make a comment to this extent – but this last thought only occured to me while typing this! :-)

    But, so, yes, I suppose I agree with what I gather is your point. Some years back sombody used to bug me by insisting that “Higher category theory is just a language!” I always found this a curious statement from a mathematician.

    • CommentRowNumber31.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 13th 2014

    From what I understand of phenomenology (in the sense developed by Husserl et al.), it seems to be a prevalent observation there too.

    • CommentRowNumber32.
    • CommentAuthortrent
    • CommentTimeOct 20th 2014

    The observation that “external” media feedback with our thought is old, but its much more important now that we’re entangled with things like iphones, film, visual media in general, more advanced mathematical notation etc. As far as I’m aware, phenomenology focuses on embodiment, which is an important part of but just a subset of the extended mind idea.

    On a super trivial note, regarding the unpopularity of category theory: a way to (aesthetically) improve category theory’s image would be to frame physics and more concrete mathematics done in light of hyper-abstract theory like Lurie’s as architecture designed in light of clouds: http://www.archdaily.com/549665/cloud-citizen-awarded-joint-top-honors-in-shenzhen-bay-super-city-competition/.

    • CommentRowNumber33.
    • CommentAuthortrent
    • CommentTimeFeb 2nd 2015

    a general comment on the technological infrastructure of society and category theory:

    the notion of “platform” (something which instead of creating value acts as infrastructure governing the conditions allowing things external to the infrastructure to come in and create or consume value https://www.youtube.com/watch?v=J1BVrsAtkWE, http://platformed.info/platform-stack/ http://platformed.info/holiday-longreads-best-platform-thinking-2014/) is extremely popular in the tech-business world. as platforms become more and more prominent to the extent that people learn to think about how platforms function in general, what makes good platforms …. I think it will be more and more natural for people to think about “theory that exists for the sake of other theory” rather than simply particular theories, solutions to particular problems, particular results in theories. (though, contra the type of platforms that exist in the tech-business world, theory that exists for the sake of other theory is valuable in itself, there’s (much more of) a blurring between the platform and what is hosted on the platform, and the emphasis isn’t on making it possible for just any random person to design/discover a theory but rather making it easier for mathematicians/physicists to discover high quality theory)

    • CommentRowNumber34.
    • CommentAuthortrent
    • CommentTimeNov 15th 2016

    with regards to getting people at large more interested in category theory, i think a good opportunity just came up with the release of the movie adaptation of ted chiang’s Story of Your Life [pdf] - Arrival [trailer].

    Without spoiling the story, I’ll just say that with the way the aliens think, I think Grothendieck’s method of immersing problems in general contextual seas then arriving at a natural and general solution would come very naturally to these aliens. (Definitely read the story before seeing the movie, I think it does a better job of portraying how the aliens think in general, and it undeniably does a better job of portraying how they think mathematico-physically, as that component was almost entirely left out of the movie. Still definitely worth watching the movie though, especially b/c a strange form of graphical notation that is not captured by what we call text (or speech) plays such a huge role. Given the role that notation plays, I actually think that film is a better medium than text for portraying this story … but unfortunately people making movies face much more constraints (appealing to a popular audience to generate revenue) than do people writing stories. Good luck getting nontrivial math/physics into a movie when our culture is such that you even have to fight for a female lead and elements (flashbacks) without which the story makes literally no sense!)

    • CommentRowNumber35.
    • CommentAuthorMike Shulman
    • CommentTimeNov 15th 2016

    rot13 to avoid spoilers: Ohg qb orvatf jub guvax va Urcgncbq O rira pner nobhg “fbyivat ceboyrzf” va gur frafr gung jr qb?

    • CommentRowNumber36.
    • CommentAuthortrent
    • CommentTimeNov 15th 2016

    Jryy, V thrff gurl pbhyq whfg ybbx vagb gur shgher naq frr jung gur nafjre gb n tvira ceboyrz vf. Ohg gurl nyfb unir gb neevir ng gung nafjre fbzrubj (rira vs gurl qba’g unir n yvarne rkcrevrapr bs jbexvat ba gur ceboyrz), naq gur tebguraqvrpx fglyr frrzf gb or gur bar gurl jbhyq zbfg bsgra rzcybl. Nyfb, gurer’f gur dhrfgvba bs inelvat cebsvpvrapl va urcgncbq o: cerfhznoyl gur zber cebsvpvrag bar vf va urcgncbq o gur zber bs gvzr bar creprvirf va n zber havsvrq jnl. Erfrnepuvat qrrc fgehpgheny vainevnagf va zngurzngvpf pbhyq or n terng jnl gb vapernfr cebsvpvrapl va urcgncbq o.

    • CommentRowNumber37.
    • CommentAuthorMike Shulman
    • CommentTimeNov 16th 2016

    V unira’g frra gur zbivr lrg, ohg bar bs gur znva cbvagf bs gur fubeg fgbel vf gung rira jura gur cebgntbavfg yrneaf gb guvax va urcgncbq o, vg qbrfa’g punatr ure npgvbaf: fur fgvyy qbrf nyy gur fnzr guvatf gung fur jbhyq unir qbar rira vs fur qvqa’g xabj gur shgher, orpnhfr rnpu bs gurz vf jung unf gb or qbar. Fb vg’f abg pyrne gb zr gung fcrnxref bs urcgncbq o jbhyq arprffnevyl nccrne gb neevir ng nafjref nal qvssreragyl guna jr jbhyq. (Ba gur bgure unaq, gur npghny urcgncbqf qvq qrirybc n culfvpf jvgu n qvssrerag rzcunfvf guna bhef, juvpu vzcyvrf gung xabjvat gur shgher qvq fbzrubj nssrpg gurve npgvbaf…)

    I suppose we should probably cut this out before we annoy everyone else around here too much…

    • CommentRowNumber38.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 16th 2016

    rot13 is fine by me. Film kinda-spoiler ahead

    Va gur svyz, gurer ner guvatf gur cebgntbavfg nccneragyl pbhyqa’g unir qbar rkprcg ol rkcrevrapvat guvatf va n abayvarne jnl - V’ir bayl frra gur svyz naq nz nobhg gb ernq gur fgbel, fb V pna’g fnl ubj guvf punatrf guvatf.

    (Man, why does that feel soo Lovecraftian…)