sustainable in commutative algebra anymore, and the discord between Grothendieck and Bourbaki may well have been rooted in this rejection.

I can not recall where, but I read more than once more detailed descriptions on what Bourbaki did not accept from Grothendieck. The conservativeness of Bourbaki who did not accept the usage of category theory (not only "neglect") and non-acceptance of a very general approach of Grothendieck to the notion of "manifold" he envisioned for the future Bourbaki works were some of the main points of departure. The remark that as a proponent of "structures" Bourbaki had to include categories is anyway a bit lacking an argument. First of all, because of the size problems one can not take big categories on equal footing with, say groups, and considering only small categories would be strange and lacking most interesting examples. On the other hand, Grothendieck judged the lack stemming in conservativeness rather than in consistency of the structure-oriented style. Indeed, according to Dieudonne, Bourbaki felt comfortable only in including to the books already (meta)stable, "dead" mathematics and not the structures in the unstable "living" phase of development. This was the intended scope and self-conscious (according to Dieudonne) limitation of the work. One can accept this and still cry for an exception for so economic tool as the category theory (if taken in conservative and very basic sense), especially in the vision of the wish for generality, Bourbaki followed otherwise. Zoran Skoda

For those who like compliments: the triples website has had (for a while now) a link to a lecture by Voevodsky given at the American Academy of Arts and Sciences in October 2002, in which he describes "categories [as] one of the most important ideas of 20th century mathematics". The video of the talk may be found at http://claymath.msri.org/voevodsky2002.mov (the compliment isn't the only reason for watching!). And there are a few other categorical links on our site at http://www.math.mcgill.ca/triples/ Suggestions are always welcome. -= rags =- -- <rags@math.mcgill.ca> <www.math.mcgill.ca/rags>

Dear Colin, Zoran, Robert, Eduardo and All, I find the present discussion on Bourbaki and category theory very important. I recall asking the question to Samuel Eilenberg 25 years ago and more recently to Pierre Cartier. If my recollection is right, Bourbaki had essentially two options: rewrite the whole treaty using categories, or just introduce them in the book on homological algebra, The second option won, essentially because of the enormity of the task of rewriting everything. Other factors may have contributed on a smaller scale, like some unresolved foundational questions. In any cases, it was the beginning of end for Bourbaki. Bourbaki was a great humanistic and scientific enterprise. Advanced mathematics was made available to a large number of students, possibly over the head of their bad teachers. It defended the unity and rationality of science in an age of growing irrationalism (it was conceived in the mid thirties). I have personally learned a lot of mathematics by reading Bourbaki. Everything was proved, and the proofs were logically very clear. It was a like a continuation of Euclid Elements two thousand years later! But after a while, I stopped reading it. I had realised that something important was missing: the motivation. The historical notes were very sketchy and not integrated to the text. I remember my feeling of frustration in reading the books of functional analysis, because the applications to partial differential equations were not described. Everything was presented in a deductive order, from top to down. We all know that learning is very much an inductive process, from the particular to the general. This is true also of mathematical research. Bourbaki is dead but I hope that the humanistic philosophy behind the enterprise is not. Unfortunately, we presently live in an era of growing irrationalism. Science still needs to be defended against religion. Civilisation maybe at a turning point with the problem of climate change. Millions of people need and want to learn science and mathematics. Should we not try to give Bourbaki a second life? It will have to be different this time. Possibly with a new name. Obviously, internet is the medium of choice. What do you think? Andre -------- Message d'origine-------- De: cat-dist@mta.ca de la part de Colin McLarty Date: ven. 12/09/2008 14:46 À: categories@mta.ca Objet : categories: Re: Bourbaki and Categories From: zoran skoda <zskoda@gmail.com> Date: Friday, September 12, 2008 2:06 pm wrote, among other things

main points of departure. The remark that as a proponent of "structures" Bourbaki had to include categories is anyway a bit lacking an argument. First of all, because of the size problems one can not take big categories on equal footing with, say groups, and considering only small categories would be strange and lacking most interesting examples.

The claim is not that Bourbaki should have studied categories as structures. It is that Bourbaki was doomed to fail in trying to use their structure theory. Leo Corry shows in his book "Modern Algebra and the Rise of Mathematical Structures" (Birkhäuser 1996) that they did fail. And they should have seen this coming, because their theory had been "superseded by that of category and functor, which includes it under a more general and convenient form" (Dieudonné "The Work of Nicholas Bourbaki" 1970). best, Colin

Andre Joyal's message was inspiring. I think a new Bourbaki type of effort (this time with motivation and category theory) is called for. I would like a text on categorical algebra oriented toward those who have studied algebra and know enough category theory to use adjunctions, monads and the like. The same goes for categorical logic and model theory. I read Andre's closing remarks as a commentary on the emerging crisis in overcoming misconceptions about and outright hostility toward science and mathemtics. I have a current project, using my own meager funds and time, to advance science teaching using a book available through the National Academies Press, "Science, Evolution, and Creationism". From the title, I think you can see my motivation. In a similar (although less aprocryphal) vein, when I worked with computer scientists and applied mathematicians in industry ( and also when I've submitted papers to certain neural network journals) I encountered misconceptions about and outright hostility toward category theory. For example, in the dynamic systems community there seems to be a widespread myth that "category theory was tried and failed". I have followed this up to some extent and haven't found any basis for it. I am often told that the best way to counter skepticism is with a working application. Having tried that, and tried again, I've come to the conclusion that Yes, you need applications, but applications cannot by themselves counter a refusal to give a theory credit for being consistent with the data. You need a good, clear presentation of the theory couched in a language oriented toward the intended audience. As with biology teaching that shows clearly the importance of the theory of evolution, maybe mathematics teaching that incorporates category theory needs to begin in 6th Grade (in schools in the USA) if not sooner. Maybe a new Bourbaki project could have an extension into this level of instruction. Best Regards, Mike Please excuse my deviating from mathematics

Dear all, I would add some information on Bourbaki/categories/France. Charles Ehresmann has been an active member of Bourbaki from 1936 up to the end of the war, when he began to no more regularly participate and wanted to resign (it was not accepted but replaced by an age limit for active participation). What Andre says: >Bourbaki had essentially two options: rewrite the whole treaty using > categories, or just introduce them in the book on homological algebra, >The second option won, essentially because of the enormity of the task > of rewriting everything. is more easily understood if we take into account that communication between France and the USA were entirely broken during the war, so that mathematical ideas could not circulate and categories were only heard of after the war, at a time where the more general parts of the treatise were written or at least prepared (the successive versions process was very slow). Charles said to me that he did not recall to have read Eilenberg & Mac Lane's paper before the fifties, or at least not seen its interest. Naturally he had sooner made a large use of groupoids in is foundation of differential geometry, and he had even defined the general "composition of jets" and given its properties, but without linking it to the notion of a category. He exposed it in a course in Rio de Janeiro in the early fifties, and one of his students (Constantino de Barros who later came to Paris to prepare a thesis with him) suggested that there was a connection with categories. Charles' first large use of categories is in his seminal paper "Gattungen von lokalen Strukturen" (1957, reprinted in "Charles Ehresmann: Oeuvres completes et commentees" Part I). It is around this date that the word "category" began to circulate in France. In 1957, Choquet (with whom I prepared my thesis) suggested that I learnt more on the notion of category which he did not know but seemed to have many applications (it was the reason for which I first went to see Charles!). It should be noted that Choquet was less conservative than many French mathematicians. In 1959, he defended the development of probabilities by inviting Loomis to give a course (I remember Henri Cartan saying then to Paul-Andre Meyer that he should not study this domain for it would be bad for his career!). And later on, he defended Logic which was very badly considered. A final remark: the "disdain" for categories (not to be confused with 'ignorance') came only later on, since Charles was given the "Prix Petit d'Ormoy" by the French Academy in 1965, essentially for his recent work on categories... Andree C. Ehresmann

I agree with Andre. Encapsulating a group of mathematicians inside a single named entity fosters a kind of collaborative spirit in which good ideas are not kept for personal use later but are shared amongst the community. When ideas are shared in real time, good mathematics can be produced faster. Anyone who wants to join the collective can do so, and the collective produces highly useful material. Of course such an enterprise is orthogonal to name-recognition, and maybe to getting tenure! But there is certainly something good about it, as there is about wikipedia and the open source movement. I also agree that the internet could be used in a better way to transfer knowledge of mathematics. Math papers are written linearly, in the bottom-up (Euclid/Bourbaki) style, to some extent. Whereas words on paper are in this sense one-dimensional, computers offer many more dimensions for knowledge transfer. Even more interesting to me would be a kind of zoom-feature on proofs. Proofs are in the eye of the beholder: for example it has been debated as to whether Perelman's 70 pages was a full proof of geometrization. Given a proof with a statement which one does not understand, a mathematician may find himself reproving something that was obvious to (or wrongly assumed to be obvious by) another mathematician. The community could benefit if a mathematician who proves such a statement then uploaded the proof, even in rough form, to some kind of math wiki. If it were well-organized, this math wiki could revolutionize how mathematics is done. In fact, choosing the "right way" to organize such a site may itself be a problem which could produce interesting mathematics. Whatever the case may be, I am all for the idea of a new Bourbaki- style enterprise in some form or another. I think it may first require interested parties to get together at some physical location. David On Sep 13, 2008, at 10:17 AM, Andre Joyal wrote:

Dear Colin, Zoran, Robert, Eduardo and All,

I find the present discussion on Bourbaki and category theory very important. I recall asking the question to Samuel Eilenberg 25 years ago and more recently to Pierre Cartier. If my recollection is right, Bourbaki had essentially two options: rewrite the whole treaty using categories, or just introduce them in the book on homological algebra, The second option won, essentially because of the enormity of the task of rewriting everything. Other factors may have contributed on a smaller scale, like some unresolved foundational questions. In any cases, it was the beginning of end for Bourbaki.

Bourbaki was a great humanistic and scientific enterprise. Advanced mathematics was made available to a large number of students, possibly over the head of their bad teachers. It defended the unity and rationality of science in an age of growing irrationalism (it was conceived in the mid thirties).

I have personally learned a lot of mathematics by reading Bourbaki. Everything was proved, and the proofs were logically very clear. It was a like a continuation of Euclid Elements two thousand years later! But after a while, I stopped reading it. I had realised that something important was missing: the motivation. The historical notes were very sketchy and not integrated to the text. I remember my feeling of frustration in reading the books of functional analysis, because the applications to partial differential equations were not described. Everything was presented in a deductive order, from top to down. We all know that learning is very much an inductive process, from the particular to the general. This is true also of mathematical research.

Bourbaki is dead but I hope that the humanistic philosophy behind the enterprise is not. Unfortunately, we presently live in an era of growing irrationalism. Science still needs to be defended against religion. Civilisation maybe at a turning point with the problem of climate change. Millions of people need and want to learn science and mathematics.

Should we not try to give Bourbaki a second life? It will have to be different this time. Possibly with a new name. Obviously, internet is the medium of choice. What do you think?

Andre

Dear Andree, Could you please explain this better?: The only Bourbaki member I new personally was Sammy Eilenberg. As many of us, I knew him very well and I would say that he was more skeptical about the Bourbaki Tractate then one can conclude from Andre's message. Having in mind not just this but the content of Bourbaki's "Homological algebra" and what we see today from the followers of that Bourbaki group, I protest against Andre's "two options" and I insist that Bourbaki group simply did not see the importance of category theory (in spite of being brilliant mathematicians, as I said in my previous message). I hope Andre will forgive me and even agree with me. However, there were three great category-theorists in that group (plus there is this mysterious story about Chevalley's book of category theory lost in the train), and "did not see" cannot be said about them of course. On the other hand I have never heard of any joint work of Charles Ehresmann with any of the two others, Eilenberg and Grothendieck (and nothing jointly from them). I think apart from the time issues you describe, the relationship between Bourbaki Tractate and category theory should have been determined by their separate or joint influence and therefore also by their communication with each other (if any). Is this true, and could you please give details? Respectfully, and with best regards- George

Bourbaki redone as Bourwiki (thanks, Dusko!) with the benefit of category theoretic insights will hopefully clarify some segments of mathematics. What troubles me in this discussion however is its assumed scope of "some." I get the sense that there are people who want it to be mandated as "all." Perhaps it should be. Just now I looked through an issue of American Mathematical Monthly that came to hand to get a sense of the likely alignment of Bourwiki with what the mathematical community generally regards as the scope of its subject. Actually I do this periodically, and I don't see much change between the issue I picked up just now and any of the other issues I've looked at in the past with just this question in mind. If the subject Bourwiki is proposing to serve is mathematics, then perhaps it is time that the American Mathematical Monthly, along with the Putnam Mathematical Competition, the International Mathematics Olympiad, and the Journal of the AMS, abandon their pretense of being about mathematics and come up with a suitable name for their subject. Not only do categories, functors, natural transformations, adjunctions, and monads go unused in these 20th century icons of mathematics, they go unacknowledged. Clearly they have not gotten with the modern mathematical program and fall somewhere between a throwback to a golden age and a backwater of mathematics. When they die off like the dinosaurs they are, real mathematics will be able to advance unfettered into the 21st century and beyond. Judging from the talks at BLAST in Denver last month (B = Boolean algebras, L = lattices, A = (universal) algebra, S = set theory, T = topology), at least the algebraic community is moving very slightly in this direction. Things will hopefully improve yet further when algebraic geometry gets over its snit with equational model theory. Meanwhile if you need a witness for seven degrees of separation, look no further than AMM and CT. (I confess to being an unreconstructed graph theorist and algebraist myself. I may have to preemptively volunteer myself for re-education before it becomes involuntary.) Vaughan Pratt

I don't know what to say about the suggestion that a circle and a line make a sketch of which Euclidean plane geometry is a model. I would think you would need a point too, since intersections are crucial. Maybe complex projective geometry since then two lines intersect in one point (unless they coincide), a line and a circle in two (unless they are tangent or equal) and every pair of circles in four (ditto). Maybe the exceptions could be handled in some sketch. At any rate, it wold e interesting to try to sketch this in detail. At any rate, I never thought about this before. Michael On Tue, 16 Sep 2008, Andre.Rodin@ens.fr wrote:

When one defines, say, a group à la Borbaki, i.e. structurally, it usually goes without saying that the defined structure is defined up to isomorphism. The notion of isomorphism plays in this case the role similar to that of equality in the (naive) arithmetic. In most structural constexts the distinction between the "same" structure and isomorphic structures is mathematically trivial just like the distinction between the "same" number and equal numbers. It may be not specially discussed in this case exactly because it is very basic. The notion of admissible map, say, that of group homomorphism, on the contrary, requires a definition, which may be non-trivial. The idea to do mathematics up to isomorphism is not Bourbaki's invention; it goes back at least to Hilbert's "Grundlagen der Geometrie" of 1899. In this sense the modern axiomatic method is structuralist. In his often-quoted letter to Frege Hilbert explicitely says that a theory is "merely a framework" while domains of their objects are multiple and transform into each other by "univocal and reversible one-one transformations". Those who trace the history of mathematical structuralism back to Hilbert are quite right, in my view. I have in mind two issues related to CT, which suggest that CT goes in a *different* direction - in spite of the fact that MacLane and many other workers in CT had (and still have) structuralist motivations. The first is Functorial Semantics, which brings a *category* of models, not just one model up to isomorphism. From the structuralist viewpoint the presence of non-isomorphic models (i.e. non-categoricity) is a shortcoming of a given theory. From the perspective of Functorial Semantics it is a "natural" feature of mathematical theories to be dealt with rather than to be remedied. The second thing I have in mind is Sketch theory. I cannot see that Hilbert's basic structuralist intuition applies in this case. In my understanding things work in Sketch theory more like in Euclid. Think about circle and straight line as a sketch of the theory of the first four books of Euclid's "Elements". I would particularly appreciate, Michael, your comment on this point since I learnt a lot of Sketch theory from your works. I have also a comment about the idea to rewrite Bourbaki's "Elements" from a new categorical viewpoint. Bourbaki took Euclid's "Elements" as a model for his work just like did Hilbert writing his "Gundlagen". In my view, this is this long-term Euclidean tradition of "working foundations", which is worth to be saved and further developed, in particular in a categorical setting. I'm less sure that Bourbaki's example should be followed in a more specific sense. Bourbaki tries to cover too much - and doesn't try to distinguish between what belongs to foundations and what doesn't. As a result the work is too long and not particularly usefull for (early) beginners. I realise that today's mathematics unlike mathematics of Euclid's time is vast, so it is more difficult to present its basics in a concentrated form. But consider Hilbert's "Grundlagen". It covers very little - actually near to nothing - of geometry of its time. But at the same time it provided a very powerful model of how to do mathematics in a new way, which greatly influenced mathematics education and mathematical research in 20th century. In my view, Euclid's "Elements" and Hilbert's "Grundlagen" are better examples to be followed.

best, andrei

le 15/09/08 12:59, Michael Barr à barr@math.mcgill.ca a écrit :

...

I don't know about this. I took several courses in the late 1950s that seem to have been influenced by the structuralist ideas (certainly categories weren't mentioned; I never heard the word until Dave Harrison arrived in 1959) and each of them started by defining an appropriate notion of "admissible map". I do not recall any special point being made of isomorphism and I think in general it was used for what we now call a bimorphism (1-1 and onto) even in cases, such as topological groups, when they were not isomorphisms.

To be sure Bourbaki was not mentioned either, but this structuralist influence seemed strong.

Michael

David Spivak wrote:

I agree with Andre. Encapsulating a group of mathematicians inside a single named entity fosters a kind of collaborative spirit in which good ideas are not kept for personal use later but are shared amongst the community. When ideas are shared in real time, good mathematics can be produced faster. Anyone who wants to join the collective can do so, and the collective produces highly useful material. Of course such an enterprise is orthogonal to name-recognition, and maybe to getting tenure!

A partial solution: those who already have sufficient name recognition should proclaim the value contributed by those without it and especially in support of tenure for them. jim

Certainly! I shall try. Thank you for encouraging! This issue seems me also interesting from a different viewpoint. Even if the New Maths was a pedagogical failure it is still remarkable that the traditional school mathematics can be wholly spelled out in the Bourbaki-like terms. I wonder if CT allows for anything of this sort. andrei

Still, it might b interesting and even instructive to try to build a sketch whose objects are (interpreted as) sets of points sets of lines, sets of circles and intersection is an operation. I guess incidence would have to a relation. It might work.

Michael

Dear Steve,

Sketches are not mathematical objects in their own right, in the same sense that groups or spaces are.

Of course, they are not.

They are presentations (for theories), and have status similar to other sorts of presentations (for groups, rings, etc.)

I think about a sketch as an alternative to a string of formulae, which represents (axioms of) a theory. I didn't try to compare sketch theory with group theory. I tried to compare sketch theory with the general setting, in which group theory is developed a la Bourbaki (along with many other theories). A classical account of this general setting (which differs at certain points with Bourbaki's version) is Tarski's model theory. The notion of presentation in my understanding implies that what a given presentation is a presentation *of* is somehow given in advance. I try to think of a sketch as a means to build a theory, not to present a ready-made theory. Perhaps *representation* is a better word for it than *presentation*. best, andrei

Dear George, I thank you for your message. You wrote:

I insist that Bourbaki group simply did not see the importance of category theory.

It is difficult to know. The Boubaki group had shielded itself in secrecy, like a free mason cell. You are surely aware of the interview of Pierre Cartier in the Mathematical Intelligencer No1 1998. Everyone interested in the history of Bourbaki should read it. http://ega-math.narod.ru/Bbaki/Cartier.htm Let me stress a few passages:

The fourth generation was more or less a group of students of Grothendieck. But at that time Grothendieck had already left Bourbaki. He belonged to Bourbaki for about ten years but he left in anger. The personalities were very strong at the time. I remember there were clashes very often. There was also, as usual, a fight of generations, like in any family. I think a small group like that repeated more or less the psychological features of a family. So we had clashes between generations, clashes between brothers, and so on. But they did not distract Bourbaki from his main goal, even though they were quite brutal occasionally. At least the goal was clear. There were a few people who could not take the burden of this psychological style, for instance Grothendieck left and also Lang dropped out.

It is amazing that category theory was more or less the brainchild of Bourbaki. The two founders were Eilenberg and MacLane. MacLane was never a member of Bourbaki, but Eilenberg was, and MacLane was close in spirit. The first textbook on homo-logical algebra was Cartan-Eilenberg, which was published when both were very active in Bourbaki. Let us also mention Grothendieck, who developed categories to a very large extent. I have been using categories in a conscious or unconscious way in much of my work, and so had most of the Bourbaki members. But because the way of thinking was too dogmatic, or at least the presentation in the books was too dogmatic, Bourbaki could not accommodate a change of emphasis, once the publication process was started.

In accordance with Hilbert's views, set theory was thought by Bourbaki to provide that badly needed general framework. If you need some logical foundations, categories are a more flexible tool than set theory. The point is that categories offer both a general philosophical foundation—that is the encyclopedic, or taxonomic part—and a very efficient mathematical tool, to be used in mathematical situations. That set theory and structures are, by contrast, more rigid can be seen by reading the final chapter in Bourbaki set theory, with a monstrous endeavor to formulate categories without categories.

The interview ends with the following passage. The bold faces are from me.

When I began in mathematics the main task of a mathematician was to bring order and make a synthesis of existing material, to create what Thomas Kuhn called normal science. Mathematics, in the forties and fifties, was undergoing what Kuhn calls a solidification period. In a given science there are times when you have to take all the existing material and create a unified terminology, unified standards, and train people in a unified style. The purpose of mathematics, in the fifties and sixties, was that, to create a new era of normal science. Now we are again at the beginning of a new revolution. Mathematics is undergoing major changes. We don't know exactly where it will go. It is not yet time to make a synthesis of all these things—MAYBE IN TWENTY OR THIRTY CENTURY IT WILL BE TIME FOR A NEW BOURBAKI. I consider myself very fortunate to have had two lives, a life of normal science and a life of scientific revolution.

Is it the time for a new Bourbaki? Best regards, André -------- Message d'origine-------- De: cat-dist@mta.ca de la part de George Janelidze Date: lun. 15/09/2008 20:03 À: categories@mta.ca Objet : categories: Re: Bourbaki and Categories Dear Andree, Could you please explain this better?: The only Bourbaki member I new personally was Sammy Eilenberg. As many of us, I knew him very well and I would say that he was more skeptical about the Bourbaki Tractate then one can conclude from Andre's message. Having in mind not just this but the content of Bourbaki's "Homological algebra" and what we see today from the followers of that Bourbaki group, I protest against Andre's "two options" and I insist that Bourbaki group simply did not see the importance of category theory (in spite of being brilliant mathematicians, as I said in my previous message). I hope Andre will forgive me and even agree with me. However, there were three great category-theorists in that group (plus there is this mysterious story about Chevalley's book of category theory lost in the train), and "did not see" cannot be said about them of course. On the other hand I have never heard of any joint work of Charles Ehresmann with any of the two others, Eilenberg and Grothendieck (and nothing jointly from them). I think apart from the time issues you describe, the relationship between Bourbaki Tractate and category theory should have been determined by their separate or joint influence and therefore also by their communication with each other (if any). Is this true, and could you please give details? Respectfully, and with best regards- George

The three books I have authored or co-authored all contain a chapter of about 40 pages that is an introduction to category theory. They are largely identical (the one for Acyclic Models has a an added section on categories of fractions, needed for that book). The one in TTT is already freely available and, with Charles's permission, I would happily post that in whatever place you would like. Michael On Wed, 17 Sep 2008, R Brown wrote:

There are already some pretty good categorical entries on wiki; I have modified some of the entries on groups, actions, equivalence relations, to include references to groupoids, which has resulted in hits. But we should also consider planetmath.org (entries are contributed under the terms of the GNU Free Documentation License (FDL)) which allows for group work and is not so open as wiki to general modification. It needs a group of you wonderful energetic people to engage with reviewing what is on wiki and planetmath and making sure they express what is fealt to be desirable!

...

George Janedlize writes

Could you please explain this better... the Bourbaki group simply did not see the importance of category theory... However, there were three great category-theorists in that group... I have never heard of any joint work of Charles Ehresmann with any of the two others, Eilenberg and Grothendieck... ...the relation between Bourbaki Tractate and category theory should have been determined by their separate or joint influence and therefore also by their communication with each other (if any).

I'll try to explain why there is no contradiction. 1. Charles only participated actively to the Bourbaki group from 1935 to the mid forties, at a time he did not know category theory. In 1935 he had written a first version for the volume "Theorie des ensembles" where he introduced the notions of local structures and associated pseudogroups of transformations (not so far from groupoids!), but this version was not accepted and he did not like the final version published much later. After the war, he only participated irregularly because he felt that he was no more able to make himself heard, the decisions being taken by "those who spoke the more loudly" (as he said to me). 2. Around 1950 it was decided that active participation ended at 45 (the age Charles had then), lessening the influence of those (Eilenberg, Cartan, Chevalley and Dieudonne) who could have stressed the importance of categories. I don't know exactly when Grothendieck became a member, but it was much later, and I think he did not remain for long. Later on, disdain for category theory had developed in France... 3. As for the communication between Charles and the other category-theorists, he had no contact with Grothendieck who was much younger. He was friendly with Eilenberg but did not see him often. Before the war he lived in Paris and regularly met Henri Cartan, Dieudonne, and more specially, Chevalley (both had regular exchanges with the philosophers Cavailles and Lautman). But their communication almost ceased after the war when he developed all his activity in Strasbourg (up to 1955) and was out of France for a long part of the year. Anyway, before our joint work (from the mid sixties up to his death), Charles worked essentially alone and published no joint work at all, except 6 Notes on Topology or Geometry with some of his students. When he began to specialize in category theory in the sixties, it was not well understood by other French mathematicians, and his influence dwindled up to a real opposition in the seventies. Andree

Dear Michael, Still some content in Steve's claim could be imagined. A working mathematician (WM) works with Borubaki's structures like groups or vector spaces and leaves all worries about what his proofs actually mean for a working math logician. For such a WM, sketches (as any other syntactical machineries) are indeed technical minutiae rather than mathematical objects. It'd be perhaps a reasonable view unless a bunch of strong semantic results (Tarski, Mal'cev,Robinson) that our WM values so much, which are provided by bringing syntax onto the stage. Zinovy On Thu, Sep 18, 2008 at 10:31 AM, Michael Barr <barr@math.mcgill.ca> wrote:

Of course sketches are mathematical objects in their own right. Of course, the functor that assigns to each sketch the corresponding theory is not full or faithful. But the definition is precise, the notion of model is also precise, so I have no idea what, if any, content there is in the claim. Incidentally, you might with equal justice claim that triples are not mathematical objects since two distinct triples can have isomorphic categories of Eilenberg-Moore algebras. In fact there are triples (or theories) on Set that have infinitary operations, yet whose category of models is isomorphic to Set.

Michael

John Baez wrote:

On Wed, Sep 17, 2008 at 01:13:21PM -0400, Andre Joyal wrote:

...

Is it the time for a new Bourbaki?

Do we imagine this new Bourbaki as just systematizing and presenting what we know already, or struggling to create brand new mathematics?

I can imagine a new Bourbaki who tries to explain all of mathematics in the language of categories. But I can also imagine a new Bourbaki who tries to explain all of mathematics in the language of infinity-categories. It may be a bit too late for the first one, and a bit too early for the second one.

Perhaps we need both!

Best, jb

Is it necessary to have a global point of view to appreciate Bourbaki? I found them quite valuable locally - i.e.. just afew of the chapters by themselves earned my appreciaiton. jim

On Fri, Sep 19, 2008, John Baez wrote:

Do we imagine this new Bourbaki as just systematizing and presenting what we know already, or struggling to create brand new mathematics?

This is an important question. We need to have a clear view of the goal of such an enterprise.

...

From my point of view, the goal should be "educational": to help students and researchers to learn mathematics and cross the boundary between fields. Mathematics is vast, and every mathematician is a permanent student. The traditional way to learn is to read the litterature and to discuss with a master. I was told that Grothendieck had learned algebraic geometry by discussing with Serre. But few peoples have this chance. Obviously, Internet is opening new avenues for learning. Many peoples (and myself) have learned a lot by reading your bulletin "This Week's Finds in Mathematical Physics". You have a real talent to explain a subject by exposing the heuristic! A discussion forum like the "Categories list" is also very helpful. Wikipedia is a useful place to gather informations about a subject. But the Bourbaki Tractate was offering something more: a unified presentation of mathematics, including the proofs.

The Bourbaki Tractate was the result of a sustained collaboration of many generations of mathematicians from different fields. Conflicts are inevitable and mathematics evolve quickly. A unified, final presentation seems impossible. On Thur, Sep 18, 2008, Ronnie Brown wrote:

Something initially less ambitious like this might actually get done. Being electronic, it would be seen as a `current', rather than `final account', and so would better reflect the way mathematics develops, in which a slight shift of emphasis, or notation (like --> for a function), can have profound consequences.

A partially unified evolving presentation of mathematics seems possible. Today's littérature is already offering something like that! But I find it cahotic and complicated. But mathematics is naturally organised and simple! The complexity of the litterature is often artificial. Many proofs are complicated, simply because the author ignores the abstract argument that could simplify everything. Some statements are left unproved, and peremptory declared obvious when they are not. The reader who cant see the obvious thing is terrorised. He may as well quit mathematics. What can we do? Let me submit a few ideas for disccusion. Mathematics is naturally self-organised. The proof of most theorems can be broken in small steps of the form A_1,..,A_n --->B, where A_1,..,A_n is the list of hypothesis and B is the conclusion. Each step may have a simple proof. A complete proof maybe obtained by working backward from the statement of the theorem to the axioms ot to known theorems. Everyone who knows a nice proof of a meaningful implication A_1,..,A_n --->B should write a paper about it and put it in a special section of the arXiv (the NB section ?). On Mon, Sep 19, 2008, Michael Spivack wrote:

Even more interesting to me would be a kind of zoom-feature on proofs. Proofs are in the eye of the beholder: for example it has been debated as to whether Perelman's 70 pages was a full proof of geometrization. Given a proof with a statement which one does not understand, a mathematician may find himself reproving something that was obvious to (or wrongly assumed to be obvious by) another mathematician. The community could benefit if a mathematician who proves such a statement then uploaded the proof, even in rough form, to some kind of math wiki. If it were well-organized, this math wiki could revolutionize how mathematics is done. In fact, choosing the "right way" to organize such a site may itself be a problem which could produce interesting mathematics.

On Tues, Sep 16, 2008, Andrej Bauer wrote:

My opinion is that we have not yet found the right way to do "hive-science", but when we do, it will be a revolution. (A good start would be to get out of the hold that the evil publishers have on us.)

A special database for mathematics should be created (but I dont know how). Papers in the NB section of the arXves could be selected, modified and organised with a system of references, to give a partially unified presentation of mathematics. The same database could support different presentations realised by different competing teams. Each team could work like a mathematical journal, with an editor in chief and an editorial board. What do you think? Andre

On Fri, Sep 19, 2008 at 10:16:31PM -0400, jim stasheff wrote:

John Baez wrote:

...

I can imagine a new Bourbaki who tries to explain all of mathematics in the language of categories. But I can also imagine a new Bourbaki who tries to explain all of mathematics in the language of infinity-categories.

Is it necessary to have a global point of view to appreciate Bourbaki? I found them quite valuable locally - i.e.. just a few of the chapters by themselves earned my appreciation.

It's easy to appreciate their books locally - but I think they sought a global systematic viewpoint while writing them. It's possible that a "neo-Bourbaki" should take a less systematic approach. Mathematics may be too much in a state of foundational flux for a systematic approach to be successful right now. Maybe the best we can hope for is something a bit more like Wikipedia, where different people contribute different portions of text, and they don't cohere in a polished whole. But presumably anyone calling for a new Bourbaki wants something different from Wikipedia. There's "Scholarpedia": http://www.scholarpedia.org/ but it doesn't seem to be doing anything with math yet, and if it ever does, I bet it'll take a "midde-of-the-road" approach instead of pushing a specific intellectual agenda. I would like to see lots of people try lots of different things. Best, jb

I would just add or emphasize if implicit here one thought: a view into the often very human process of interaction viz. several of the anecdotes about `life with Bott' at his recent memorial conference jim Meredith Gregory wrote:

All,

i have been utterly delighted by this conversation. What i can't help but think about, however, is that with the internet we have a different sort of opportunity. Let me try to describe it.

- What is missing in most mathematical presentations is a view into the often very human and very messy process of getting to the presentation. What young mathematicians need -- in my view -- is a view of mathematicians doing mathematics. They need to see very top-down orientations rubbing elbows with very bottoms-up orientations. They need to see highly inventive, unifying viewpoints come up against skeptical viewpoints armed with vast arrays of counter-examples. They need to see people desperately trying to organize while others are desperately trying to de-construct. This is where the life of mathematics is. This is how people bring mathematics to life. - With the internet we have the opportunity to record not just the final artifact, tractate or wiki, but the process. Ever since Andre Joyal mentioned a 2nd life for Bourbaki i can't stop thinking about a Bourbaki colloquium run in Second Life <http://secondlife.com/> -- so that whatever the outcome of a given process is in terms of artifact, people can go back and look at the process, itself. They can see how people argued and counter-argued. There is getting to be a precendent for this, from Harvard<http://www.joystiq.com/2006/09/12/harvard-class-invades-second-life/>to Intel <http://softwarecommunity.intel.com/articles/eng/1283.htm>, to run serious technical conversation in Second Life.

Perhaps this idea is too far out, but i would urge those who seriously consider a second life for Bourbaki to remember to record the living part as well as the outcome. After all, looking over the last many emails to categories so much of it is an attempt to recover process -- how things got to be where they are.

Best wishes,

--greg