I think the premise of the work is silly. I knew Gödel personally and many people who knew him. My long-time friend Robin Gandy was a close friend and associate of Turing. I think it is true that Godel was very frustrated by not being able to settle the status of the Continuum Hypothesis, but to say that his mathematics drove him mad is terrible pop psychology. In the case of Turing, the matter is even less clear. For him, persecution may have been a big factor in his suicide -- which also could have been an accident. Gödel did suffer from paranoia, and Cantor was oppressed by religious questions, which he did relate to his theories of the infinite. But to conclude cause and effect seems pretty hard to prove. And what is the point? [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Hi - I think I watched this documentary a while ago. I watched only a few minutes of it today and then realized that I should probably be doing something else. However, I was interested and want to comment briefly.

In this one-off documentary, David Malone looks at four brilliant mathematicians - Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing - whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide.

...

From what I can gather in this summary and my recollection of the film, the

I am not sure what it means for one's genius to drive him or her insane. director seems to exploit the illnesses of some of these men to provide some theatrical drama to the story. If this is so, then it seems rather irresponsible. If I am mistaken and have not watched enough of the video, then my apologies to the director.

The film begins with Georg Cantor, the great mathematician whose work proved to be the foundation for much of the 20th-century mathematics. He believed he was God's messenger and was eventually driven insane trying to prove his theories of infinity.

This statement does not directly blame the illness on his struggle with mathematics, but it seems dangerously suggestive of this conclusion. Here is a quote from Wikipedia: Cantor's recurring bouts of depression<http://en.wikipedia.org/wiki/Clinical_depression>from 1884 to the end of his life were once blamed on the hostile attitude of many of his contemporaries,[9]<http://en.wikipedia.org/wiki/Georg_Cantor#cite_note-daub280-8>but these episodes can now be seen as probable manifestations of a bipolar disorder <http://en.wikipedia.org/wiki/Bipolar_disorder>.[10]<http://en.wikipedia.org/wiki/Georg_Cantor#cite_note-bipolar-9> This seems like the more reasonable, albeit possibly less glamorous description of the causes of Cantor's illnesses. Since I did not watch further I cannot comment on the others, except that I did not know of any mental disorders associated to Turing. Best, Alex Hoffnung [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

The graphic novel Logicomix by Apostolos Doxiadis and Christos H. Papadimitriou has the relation between being logicians and madness as one of its themes. The novel is actually quite good, and I recommend it, but they make too much of the insanity stuff. One saving grace is that the novel has interludes featuring the authors and the artists arguing about that theme and other aspects of the novel. Charles Wells 2009/11/30 Mike Stay <metaweta@gmail.com>:

Here's the summary from BBC's site:

In this one-off documentary, David Malone looks at four brilliant mathematicians - Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing - whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide.

The film begins with Georg Cantor, the great mathematician whose work proved to be the foundation for much of the 20th-century mathematics. He believed he was God's messenger and was eventually driven insane trying to prove his theories of infinity.

Ludwig Boltzmann's struggle to prove the existence of atoms and probability eventually drove him to suicide. Kurt Gödel, the introverted confidant of Einstein, proved that there would always be problems which were outside human logic. His life ended in a sanatorium where he starved himself to death.

Finally, Alan Turing, the great Bletchley Park code breaker, father of computer science and homosexual, died trying to prove that some things are fundamentally unprovable.

The film also talks to the latest in the line of thinkers who have continued to pursue the question of whether there are things that mathematics and the human mind cannot know. They include Greg Chaitin, mathematician at the IBM TJ Watson Research Center, New York, and Roger Penrose.

Dangerous Knowledge tackles some of the profound questions about the true nature of reality that mathematical thinkers are still trying to answer today. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com

-- professional website: http://www.cwru.edu/artsci/math/wells/home.html blog: http://sixwingedseraph.wordpress.com/ abstract math website: http://www.abstractmath.org/MM//MMIntro.htm astounding math stories: http://www.abstractmath.org/MM//MMAstoundingMath.htm personal website: http://www.abstractmath.org/Personal/index.html sixwingedseraph.facebook.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Hi, A while ago, I started to watch a BBC programme on a mathematical Olympiad camp, but gave up when it was clear that they were concentrating on a participant with peculiar behaviour and hardly noticed the several pretty girls around! That also sounds like peculiar behaviour. Perhaps we need a TV programme about producers of TV programmes? Ronnie Brown Alex Hoffnung wrote:

Hi -

I think I watched this documentary a while ago. I watched only a few minutes of it today and then realized that I should probably be doing something else. However, I was interested and want to comment briefly.

...

In this one-off documentary, David Malone looks at four brilliant mathematicians - Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing - whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide.

...

From what I can gather in this summary and my recollection of the film, the

I am not sure what it means for one's genius to drive him or her insane. director seems to exploit the illnesses of some of these men to provide some theatrical drama to the story. If this is so, then it seems rather irresponsible. If I am mistaken and have not watched enough of the video, then my apologies to the director.

...

The film begins with Georg Cantor, the great mathematician whose work proved to be the foundation for much of the 20th-century mathematics. He believed he was God's messenger and was eventually driven insane trying to prove his theories of infinity.

This statement does not directly blame the illness on his struggle with mathematics, but it seems dangerously suggestive of this conclusion. Here is a quote from Wikipedia:

Cantor's recurring bouts of depression<http://en.wikipedia.org/wiki/Clinical_depression>from 1884 to the end of his life were once blamed on the hostile attitude of many of his contemporaries,[9]<http://en.wikipedia.org/wiki/Georg_Cantor#cite_note-daub280-8>but these episodes can now be seen as probable manifestations of a bipolar disorder <http://en.wikipedia.org/wiki/Bipolar_disorder>.[10]<http://en.wikipedia.org/wiki/Georg_Cantor#cite_note-bipolar-9>

This seems like the more reasonable, albeit possibly less glamorous description of the causes of Cantor's illnesses. Since I did not watch further I cannot comment on the others, except that I did not know of any mental disorders associated to Turing.

Best, Alex Hoffnung

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Actually, I think Dana understates the problem with this program. It suffers from what I call the "PBS documentary syndrome" (equally afflicting the BBC, however, so the name is not universal enough!): it repeatedly tells you what's cool about its topic, without ever actually telling you what the topic really is. Afraid to scare viewers away with the actual details of the topic, it just talks about it in terms so general (and often over-inflated or sensationalized, which was Dana's point) they are really quite meaningless. Though not perfect by any means, I think a recent 4-part series "The Story of Maths" narrated by Marcus du Sautoy does better - he even tries to sketch some proofs. (The episode closest to "Dangerous Knowledge" would be the fourth.) Even better is an old series (but still to be found on Youtube!) called Mathematical Mystery Tour. But generally, science documentaries are disappointing, and maths ones even more so. It's a pity, because you actually can get an audience of non-specialists to understand (at least a little) what mathematical results etc are about. I teach an honours Liberal Arts maths & logic class, and a surprisingly large percentage can actually appreciate the beauty of (eg) natural deduction proofs in predicate logic, basic theory of natural numbers (infinitude of primes, irrationality of primes, the fundamental theorem of arithmetic, ... ), simple axiomatics (we do Boolean algebras as an example), and even Godel's theorems, and the "Lambek calculus" for linguistics (even a bit of category theory there!). This isn't a mickey mouse course (sample class tests available on request!), and it's a challenge to many of the students. The point is: they are willing to make the effort if they know you're not being condescending, and that you are giving them "the real thing", not some pablum that only looks good in the box. I wish more TV documentary producers took that attitude - they might get a slightly smaller audience, but their audience will appreciate their efforts more. -= rags =- On Mon, 30 Nov 2009, Dana Scott wrote:

I think the premise of the work is silly. I knew Gödel personally and many people who knew him. My long-time friend Robin Gandy was a close friend and associate of Turing. I think it is true that Godel was very frustrated by not being able to settle the status of the Continuum Hypothesis, but to say that his mathematics drove him mad is terrible pop psychology. In the case of Turing, the matter is even less clear. For him, persecution may have been a big factor in his suicide -- which also could have been an accident. Gödel did suffer from paranoia, and Cantor was oppressed by religious questions, which he did relate to his theories of the infinite. But to conclude cause and effect seems pretty hard to prove. And what is the point?

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear Robert, The "The Great Global Climate Swindle" is an example of anti-scientific propaganda in the name of science: http://en.wikipedia.org/wiki/The_Great_Global_Warming_Swindle You can watch it on youtube: http://www.youtube.com/watch?v=xzSzItt6h-s The "documentary" is full of errors: http://www.realclimate.org/index.php/archives/2007/03/swindled/ http://www.climateofdenial.net/?q=node/3 André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

and if it makes one youngster pursue a career in mathematics instead of crime, that may justify its existence.

Best, jb

at least one or more episodes of Numb3ers made that point jim [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Aren't artistic types alleged to be more prone to mental illness than say sales clerks, real estate agents, auto mechanics, farmers, lumberjacks, etc? More generally, creative people? On that basis would one expect a higher prevalence of mental disorders among theoretical physicists (Boltzmann) than experimental ones (Rutherford), or among mathematicians than engineers, or among top chefs than short order cooks? Everyone seems to be a mental health expert today, just as everyone is an expert on evolution and global warming (but not quantum mechanics or ecology or anesthesiology, funny how that works). I'd be uncomfortable with any innuendos of this kind about theoreticians vs. practitioners, or creatives vs servants, or prima donnas vs. choristers, without some solid independent evaluation of this question by professionals with a substantial track record in mental health. Has any such evaluation been made? Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Vaughan Pratt wrote:

Aren't artistic types alleged to be more prone to mental illness than say sales clerks, real estate agents, auto mechanics, farmers, lumberjacks, etc? More generally, creative people? On that basis would one expect a higher prevalence of mental disorders among theoretical physicists (Boltzmann) than experimental ones (Rutherford), or among mathematicians than engineers, or among top chefs than short order cooks?

Everyone seems to be a mental health expert today, just as everyone is an expert on evolution and global warming (but not quantum mechanics or ecology or anesthesiology, funny how that works). I'd be uncomfortable with any innuendos of this kind about theoreticians vs. practitioners, or creatives vs servants, or prima donnas vs. choristers, without some solid independent evaluation of this question by professionals with a substantial track record in mental health. Has any such evaluation been made?

Vaughan Pratt

I seem to recall some such work on Asperger's see book by Ioan James and a benefides neuroscientist jim [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

[Note from moderator: while interesting, this thread has strayed well off-topic; to be posted further submissions must be relevant to categories.] Jim Stasheff wrote:

I seem to recall some such work on Asperger's see book by Ioan James and a benefides neuroscientist

Ioan James held the Savilian Chair of Geometry at Oxford during 1970-95, and seems to be the only author. Where does the neuroscientist come into it? Autism is a pretty disabling condition. The 20 individuals James writes about seem at worst to be only mildly autistic and surely would be better described as having Asperger syndrome if even that. In 1944 Hans Asperger described a behavioral pattern that, according to the Wikipedia article on him, included "a lack of empathy, little ability to form friendships, one-sided conversation, intense absorption in a special interest, and clumsy movements." Anyone who's gone overtime in a conference talk at the expense of the next speaker could be judged as having at least three of the first four of these, and the audience would then be riveted on the speaker's movements to assess their clumsiness. And how does one judge "ability to form friendships?" By the number of the speaker's coauthors? Some people are by nature private, others compartmentalize their time into private and social periods so that they can get some work done in their private time. How is someone to be judged when their social time is only say 10% or 20% of their private time? Wikipedia has the following article: http://en.wikipedia.org/wiki/People_speculated_to_have_been_autistic whose introduction reads --- Famous historical people have been speculated to have been autistic by journalists, academics and autism professionals. Such speculation is controversial and little of it is undisputed. For example, several autism researchers speculate that Wolfgang Amadeus Mozart had autism and other diagnoses, while other researchers say there is not sufficient evidence to draw conclusions that he had any diagnoses --- The body of the article includes "Speculation about their diagnoses is based on reported behaviors rather than any clinical observation of the individual. Fred Volkmar, a psychiatrist and autism expert and director of the Yale Child Study Center says, 'There is unfortunately a sort of cottage industry of finding that everyone has Asperger's.' " Here's the Amazon product description of James' book, and two customer reviews. --- This fascinating collection identifies famous figures from the past, whose behaviour suggests they may have had autism, a disorder that was not defined until the mid - 20th century. James looks at the lives of 20 individuals - scientists, artists, politicians and philosophers - examining in detail their interests, successes, indifferences and shortcomings. Among the profiles are those of mathematician and philosopher Bertrand Russell, who wondered in his autobiography how he managed to hurt the people around him quite without meaning to; biologist Alfred Kinsey, who excelled in academia but was ill at ease in social situations; and the writer Patricia Highsmith, who had very definite likes (fountain pens and absence of noise) and dislikes (television and four-course meals). From Albert Einstein to Philip of Spain, these intriguing individuals all showed clear evidence of autistic traits. This book will be of interest to general readers and anyone with a personal or professional interest in autism. --- --- FIRST REVIEW: "Offers opportunity for personal insight" This book is not what I expected, yet I appreciate its contribution to the literature on this topic. Twenty biographical accounts are arranged chronologically. Each offers basic facts about the life and accomplishments of the individual, and includes accounts of their behaviors, their own views of their challenges and accomplishments as well as the perceptions of their contemporaries. Taken together, these twenty biographies offer me insights into human diversity and the importance of accepting ourselves and others for our strengths as well as our idiosyncrasies. As a teacher it helps me both understand and appreciate unique children. --- --- SECOND REVIEW: "Not another book about famous people who might have had Asperger Syndrome!" I have to say, this genre is getting rather tired, and there is a lot of overlap between all the books on this subject, of which there are many. If you already own one of the other books on this subject, you don't need to buy this one, and if you don't already own a book on this subject, this one is as good as any. What else can I say? It's 20 mini biographies of people the author has researched and considers to show signs of having had Asperger Syndrome. Interesting enough, and probably succeeds in the stated aim of raising the self esteem of people with AS. --- Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

I think we should say it publicly. Gays get to have gay pride, why shouldn't categorists get to have category-theory pride? Perhaps we've just been in the closet too long. I think it's the right thing to do to explain to people that this stuff is interesting and worthwhile to us. Worst of all would be to let the fear of shame keep us from saying what we hold as true. If I consider category theory to be good, beautiful, valid math, then I shouldn't be shy about saying as much. If someone else doesn't consider it to be "real math," he or she can challenge me -- I'm up for that discussion. The worst they can do is not give me a job, but this is not an issue because I don't belong at a place that doesn't respect category theory. Andre is right -- category theory is probably the most important mathematical developpement of the 20th century. David On Mon, Dec 7, 2009 at 6:13 AM, Joyal, André <joyal.andre@uqam.ca> wrote:

Category theory is a powerful mathematical language. It is extremely good for organising, unifying and suggesting new directions of research. It is probably the most important mathematical developpement of the 20th century.

But we cant say that publically.

André Joyal

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Dear Andre, I think category theorists have done an excellent job at publicizing the secret. I am very much struck at category meetings what a variety of backgrounds the participants come from, lots from computer science of course, and now increasingly many physicists. It seems to me this is exactly because category theory has the qualities you describe. It enables the pure category theorists, the computer scientists, the physicists to meet and talk together with a high degree of mutual understanding. I don't think of myself as a pure category theorist, but I can't imagine trying to do what I do without it. All the best, Steve. Joyal wrote:

Category theory is a powerful mathematical language. It is extremely good for organising, unifying and suggesting new directions of research. It is probably the most important mathematical developpement of the 20th century.

But we cant say that publically.

André Joyal

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

I have read that Dirac had no empathy, not even for his family. I think the story goes the same for many other famous mathematicians/scientists. Why is it being so promoted that being a good mathematician and a good human being is impossible? Is it really true? -Mehrnoosh On Tue, Dec 8, 2009 at 5:23 AM, Robert Seely <rags@math.mcgill.ca> wrote:

Well, we might not say that, but Voevodsky did. Link on the triples page: http://www.math.mcgill.ca/triples/ (or directly http://claymath.msri.org/voevodsky2002.mov) where he says "Categories: one of the most important ideas of 20th century mathematics".

BTW - the Farmelo book, The Strangest Man, is one I recommend to my students - it's well worth looking at. But one thing that struck me was how *little* Farmelo plays the "strange man" theme - Dirac was indeed strange, but that's not what makes him worth reading about, nor was it what made him a great theoretician. Farmelo doesn't (IMO) make the same mistake so many documentary producers do ...

-= rags =-

On Mon, 7 Dec 2009, Joyal, André wrote:

Category theory is a powerful mathematical language.

...

It is extremely good for organising, unifying and suggesting new directions of research. It is probably the most important mathematical developpement of the 20th century.

But we cant say that publically.

André Joyal

-- Mehrnoosh Sadrzadeh EPSRC Postdoctoral Research Fellow Oxford University Computing Laboratory Research Fellow of Wolfson College http://web.comlab.ox.ac.uk/people/Mehrnoosh.Sadrzadeh/ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

I'm not too sure what the context was, but Andre' Joyal said on 7 December,

Category theory is a powerful mathematical language. It is extremely good for organising, unifying and suggesting new directions of research.

I completely agree.

It is probably the most important mathematical developpement of the 20th century.

It is too early to tell. [Comment attributed to Zhou Enlai (Chinese Communist leader 1949-76) when asked his opinion of the French Revolution.]

But we cant say that publically.

I think we should be wary of slapping ourselves on the back too much. The fact is that category theory alienated the rest of the mathematical world. Since the damage had been done in the 1970s, well before my time, I have never managed to work out how this happenned, or who was responsible. Probably it was the result of haughty claims about being the "most important mathematical development", and about being the foundations of mathematics before any serious technical work was done to justify this. Of course the ignorance and arrogance of mathematicians outside our subject had a lot to do with it too. Indeed, I believe that there is nothing wrong with pre-1980 category theory that cannot be attributed to the fact that it was done by pure mathematicians, and nor is there anything wrong with the post-1980 subject that is not the result of its having been done by computer scientists. However, discussion on that is not going to get us very far. What is more relevant and able to be fixed is the point in Andre's title, that category theory is a WELL KEPT SECRET. Secrecy, like charity, begins at home. For example, the notion of ARITHMETIC UNIVERSE was one of the most insightful developments of 1970s categorical logic. It captures exactly what is taught as "discrete mathematics" to computer science students (and is relevant to combinatorial mathematics), namely products, equalisers, stable disjoint sums, stable effective quotients of equivalence relations and FINITE powersets. It is the least structure that is capable of constructing the free internal gadget of the same kind, so the original idea was to prove Godel's incompleteness theorem categorically. Recently I was looking though the archives of the "Foundations of Mathematics" (FOM) mailing list at cs.nyu.edu/pipermail/fom/ and, amongst all of the personal abuse directed at Colin McLarty and Steve Awodey, came across an interesting argument against category theory, namely that the notion of elementary topos was merely an aping of the axioms of set theory. Arithmetic universes answer that objection extremely well. The work on arithmetic universes was done THIRTY SIX YEARS AGO, and many people since then have been nagging the author to write it up, indeed I myself have been doing so for half of that time now. I don't want anybody to read this as a personal attack -- it is simply an example of a general phenomenon, albeit an important example because of the importance of the material. Anybody in my generation or younger can cite lots of examples of "well known" "folklore" results that were supposedly discovered in the 1970s but have never been written up. The worst thing is that any younger person who is so impertinent as to write out a proof of one of these results has their paper rejected. To give another example, the theory of continuous lattices is crucial as background for my work on Abstract Stone Duality. I asked exactly the people who should have written it whether there was an introduction to continuous lattices suitable for analysts. There isn't, so I had to write my own. In this, I stated without proof that the evaluation map Sigma^X x X --> Sigma is continuous (when the topology Sigma^X is itself given the Scott topology) iff X is locally compact, and in this case Sigma^X is itself locally compact and obeys the adjunction Yx(-) -| Sigma^(-). The referee quite reasonably asked for a reference to a proof, but, so far as I can gather, no such proof exists in the literature. Two more examples: when is some Australian going to write "2-categories for the working categorist"? Where is the textbook on universal algebra based on monads? So, to answer Andre's question about why category theory is such a well kept secret -- it is because category theorists KEEP it as a secret. Each of us can help to leak this secret by doing two things: PUBLISH (= make freely available on the Web) all of the papers that you PRIVATISED by handing them over to commercial journals. WRITE textbook or encyclopedia accounts of your work for resources like the "n-cat lab", ncatlab.org/nlab/show/HomePage Paul Taylor [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

There are several comments I could make to this posting, but I will confine myself to two. First, I was around all through the 70s (and most of the 60s) and I have no idea what categorists did to earn the opprobrium described below. A colleague of mine commented one day maybe 25 years ago, that it seemed in the 60s that category theory would be important, but it hasn't turned out that way. I am not sure what didn't turn out that way, but that seemed to have been the general opinion. Second, a couple of papers by Linton and Manes in the Zurich Triples Book (LNM #80) makes very explicit the connection between triples and universal algebraic theories. Although doubtless out of print, a dozen of us put a lot of effort into retyping it in tex and republishing it as a TAC reprint. If someone wants to go ahead and replace every instance of "triple" by "monad" go ahead. Also Beck's tripleableness theorem is in Beck's thesis, another TAC reprint also retyped by volunteers. Incidentally (although Paul is well aware of this) every paper of mine later than 1985 and every earlier paper of which I had an electronic trace, is available on my personal ftp site. Also incidentally my wife and I retyped Grothendieck's Tohoku paper and are waiting only for proof-reading by the Van Osdols to post it (hint, hint, since I know Don reads this group). Michael On Thu, 10 Dec 2009, Paul Taylor wrote:

I'm not too sure what the context was, but Andre' Joyal said on 7 December,

...

Category theory is a powerful mathematical language. It is extremely good for organising, unifying and suggesting new directions of research.

I completely agree.

...

It is probably the most important mathematical developpement of the 20th century.

It is too early to tell.

[Comment attributed to Zhou Enlai (Chinese Communist leader 1949-76) when asked his opinion of the French Revolution.]

...

But we cant say that publically.

I think we should be wary of slapping ourselves on the back too much.

The fact is that category theory alienated the rest of the mathematical world. Since the damage had been done in the 1970s, well before my time, I have never managed to work out how this happenned, or who was responsible.

Probably it was the result of haughty claims about being the "most important mathematical development", and about being the foundations of mathematics before any serious technical work was done to justify this. Of course the ignorance and arrogance of mathematicians outside our subject had a lot to do with it too.

... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Michael Barr wrote:

First, I was around all through the 70s (and most of the 60s) and I have no idea what categorists did to earn the opprobrium described below.

My experience with CT may give some insight here. When I joined the MIT faculty in 1972 it was already 8 years since I'd taken Max Kelly's category theory class, and the only bit of it that I had retained by that time was that categories weren't required to have an underlying set functor (and I couldn't in 1972 even phrase that much of my recollection in the language of CT I used just now). Now skip the next two paragraphs (which are only there to preserve chronological order) unless you want all the boring details. In between I spent a year taking mathematics honours (fourth year), in which Max taught us topology (he had planned to teach us algebraic topology but realized we weren't prepared for it) and we got many other courses from John Mack (number theory), T.G. Room (axiomatic geometry), Bruce Barnes (group theory), etc. I then spent a year doing physics honours (a second fourth year, I did maths honours first because I wanted to be a theoretical physicist and had sensed that without maths honours the physics honours year would be insufficient grounding for a theoretical physicist). But then the next year I noticed that computer science was an as-yet-untapped gold mine of important yet easily solved problems, whence my career move from physics into CS. (A pity in some respects since I've always been good at solving hard problems once they engage my interest and the problems in physics had by then become quite hard and therefore should have been right up my alley.) So I became a CS grad student at Sydney then Berkeley then Stanford, and then Don Knuth's postdoc in 1971-72, and then spent a few months at IBM Yorktown Heights in a visiting faculty position in 1972. In 1972 the only person in the whole of 545 Technology Square (a 9-story building on the "other side of the tracks" from the main body of MIT) who talked about categories was Mitch Wand. Mike Fischer was his advisor. I lived out Mike's way and commuted to work with him much of the time, a half hour ride each way, so we got to discuss many things, but I don't recall category theory ever coming up. We mostly talked about algorithms and program verification and programming language design and group theory and other technical things, along with the vegetable gardens we were growing in our back yards as a joint project that also involved Albert Meyer. We both were totally oblivious to politics, which never came up. And it never occurred to either of us that we should discuss CT. While still a grad student Mitch gave a graduate course on CT. I didn't attend any of it, being rather busy as a junior faculty and having no occasion to, but I would occasionally hear feedback from those who did. The general feeling seemed to be that this was "mathematics made difficult," a way of obfuscating the obvious. I had no reason to defend CT at the time and simply accepted these reports as putting CT in the same ballpark that Rene Thom's chaos theory was later put by some of its detractors. In 1979, finding logic problems becoming more challenging, I (re)discovered algebra by way of universal algebra. I learned UA from Rasiowa and Sikorski, which I found to my surprise I could speed-read (must have been the excellent Sydney algebra courses), and successfully applied it to the logic problem I'd previously been stuck on. In 1983 I realized that category theory was the algebra of functions. I tried very very hard to understand Chapter 1 of CWM, which seemed far more obscure than universal algebra. Speed-reading that chapter was out of the question for me. Eventually I gave up and moved on to Chapter 2 and beyond, and after that it was just as easy as universal algebra. ---------- So I would say that the opprobrium could well have originated from the impression that CT was obfuscation, which Chapter 1 of CWM did nothing to dispel. Two four-syllable words beginning with the same letter, one leading to the other. ---------- So what do I think today? Well, I would rank three related concepts as being of fundamental but not equal importance, in the following order, most important first. 1. 2-categories 2. Dense functors 3. Natural transformations The algebra of 2-categories is barely algebra, it is really the associativity intrinsic to geometry. If you cut a string, even one with colored ink marks on it, in two places you can't tell after the fact in which order the cuts were made. If you cut a painting by Picasso vertically then horizontally into four pieces, the same holds even though the painting has depreciated. These are respectively associativity and middle-interchange. I hardly recognize these as algebra, they're geometry as far as I'm concerned, but they're the algebra of 2-categories. They suck, e pur si muove. Dense functors are important because they expose what is "natural" about natural transformations as an instance of 2-cells. To see how this works see http://boole.stanford.edu/pub/yon.pdf , "The Yoneda lemma as a foundational tool for algebra." I imagine Steve Lack et al have some equivalent way of describing this viewpoint which I'm still waiting to hear about (my 1962-1965 classmate Ross Street promised he'd get back to me on this but that was a while back). Meanwhile I've received enthusiastic feedback about it from Ronnie Brown and also a response from William Boshuk ("very enjoyable pamphlet"), though that's it so far. I've felt for at least 15 years that the notion of natural transformation as traditionally defined is a complicated concept. This I believe whether one thinks of them as a category theorist or (in their manifestation as homomorphisms) as an algebraist. Either way the idea is subtle. This subtlety of the concept is why I don't rank it higher. My ranking makes it ironic that transformations that are called "natural" should end up third. But then that's just my ranking, YMMV as they say. Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

On 10 Dec 2009, at 14:49, Paul Taylor wrote:

In this, I stated without proof that the evaluation map Sigma^X x X --> Sigma is continuous (when the topology Sigma^X is itself given the Scott topology) iff X is locally compact, and in this case Sigma^X is itself locally compact and obeys the adjunction Yx(-) -| Sigma^(-). The referee quite reasonably asked for a reference to a proof, but, so far as I can gather, no such proof exists in the literature.

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On Thu, Dec 10, 2009 at 9:49 AM, Paul Taylor <pt09@paultaylor.eu> wrote:

The fact is that category theory alienated the rest of the mathematical world.   Since the damage had been done in the 1970s, well before my time, I have never managed to work out how this happenned, or who was responsible.

Had it been really *done*? It may be just in "the nature of things" when a community A provides abstract models for community B. Something similar appears in relations between physicists and mathematicians, or between physicists/computer scientists and engineers. When a mathematician is building a math model for some physical theory, his main driving force is a question "What do they *really* do?" As the work is progressing, the question develops into a thesis "they don't actually understand what they do", and with this attitude, the mathematician finally brings something structurally neat to physicists. In a typical good case, the reaction would be like Manin recently formulated in his interview "we always knew that but thank you for attention". In a bad case, ... you know. Isn't it similar to the math vs.category theory case? For some people structural clarity and elegance is a matter of life and death, for others it's a dispensable luxury (it's a rephrasing of Edsger Dijkstra if I remember right). zd [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

On Mon, Dec 7, 2009 at 9:13 AM, Joyal, André <joyal.andre@uqam.ca> wrote:

Category theory is a powerful mathematical language. It is extremely good for organising, unifying and suggesting new directions of research.

one important point is missing: design and design patterns. Design from scratch is for geniuses while ordinary people design by adapting and developing preexisting patterns. Category theory created a powerful system of design patterns for math and beyond (computer science, physics, engineering). It seems it changed the very nature of mathematical design. Z.

It is probably the most important mathematical developpement of the 20th century.

But we cant say that publically.

André Joyal

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