Eduardo, I think it is you who are suffering from superficial thinking, or at least superficial reading. Admittedly there was a deliberate superficiality in my topic--I describe 'faces' of category theory, aspects presented on the surface, to which those approaching the subject react, even as people in social contexts react to the face of those they meet. But the entire discussion has been about reactions to the public face of category theory, and about what and who should be that face. You evidently did not read my post carefully enough. It is not I, but the mathematical community as a whole that has a prejudice against 'categories as foundations'--and indeed against foundations, which most mathematicians try devoutly to ignore as my discussion of the attitude toward constructions of the real numbers illustrated. Both category theory and categorists have suffered as a result. In the 1980's, when I fell in love with category theory, in part because it did address big foundational issues, this prejudice resulted in the marginalization and ghettoization of category theory within mathematics. I began my post with the story of Moishe Flato's dismissal of category theory as 'a mere language' and his repentance from that view. I chose this because it was the most cheerful story I could tell to illustrate the prejudice against foundations, and category theory as such, and probably one most had not heard. Category theory is breaking out of its ghetto not by finding foundational applications in computer science--excellent though those are, both for the intellectual life of our community and job prospects for categorists--and certainly not by asserting its foundational role in mathematics, by showing its face as algebra to mathematics as a whole. Your last paragraph suggests, perhaps categorists are not. The attitude evinced by your reply to my post--dismissing the mathematical content of my remarks as "highly technical, sophisticated, difficult and impressively sounding words" (doesn't all mathematics sound that way until one masters the relevant concepts and definitions?), and adopting a 'blame the messenger' attitude to my report of anti-foundational prejudice among mathematicians--suggests that you are content to remain in the ghetto, and want to keep the rest of us there with you. Peevishly yours, D. Yetter On 30 Mar 2006, at 13:31, Eduardo Dubuc wrote:

Hi,

The 93 is because I have by now 92 msages in my cracks and pots file.

I apologize for the length of this posting. It is intended to be a (may be biased) partial account of the debate, and some comments.

Well, by now the "cracks and pots" debate is establishing itself as, in my opinion, an interesting and worth-wile event. Congratulations Marta !!

We are learning about:

a) Understand (for many of us) better what is mathematics, and what is physics, what is rigor and what is buccaneering, and also what is bullshit.

b) "Something is rotten in the state of category theory community"

Pay attention that The Bard does not say "category theory", but he says "category theory community"

I start from who has made the more refreshing, humorous, down to earth, honest and intelligent contributions to this debate:

**Vicent Schmitt: that theoretical physics, computer science, phylo., a mix of those, or whatever? , is used to justify poor "categorical" work is, in my view, an existing problem. More or less everyone is conscious of it (come on!...) but so far that has not been publicly debated.**

Yes Vincent!!, you point right to what it is at the center (or very near it) the problem raised in MartaÕs original "cracks and pots" posting!. And the "(come on!...)", beautiful !.

Now, talking about rigor, conjectures and proofs:

**Maclane : If a result has not yet been given valid proof, it isn't yet mathematics. This however does not deny the many preliminary stages of insight, experiment, speculation or conjecture, which can lead to mathematics. It states simply that a conjectured result is not yet a theorem **

It is relevant to compare this with Motl's distinction between physics and mathematics:

**Motl: In physics, we propose different conjectures about the real world, and it is important that we're not guaranteed that these conjectures will be true.

String theory itself is not just a conjecture, but rather a seemingly consistent mathematical framework. Once we accept string theory as an objectively existing mathematical structure, a structure that we treat as a part of "generalized physics" - which is of course what all string theorists are doing every day - we can ask a lot of questions about its properties.**

He does distinguish between "physics as conjecture" and mathematics with applications to physics. He call this mathematics "generalized physics"

But "conjecture" to be acceptable is not unrigourous neither buccaneering. he says:

**Motl: the statements about string theory are just conjectures, and they need to be proved or supported by evidence, otherwise they're irrelevant and "wrong", in the physical sense.**

He also says:

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

He is clearly saying that those "mathematically oriented people" are lacking rigor.

Many postings in this debate confound mathematical rigor with formalism, and push forward the idea that a formal and logically correct statement has automatically rigor. Even if it is foolish:

**V. Pratt: In axiomatic mathematics, everything that is not forbidden is permitted. **

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

It seems to me that he is equating here "mathematical standards of rigor" with "logically correct", and "the standards appropriate to that subject" (in this case, physics) with " buccaneering "

Nothing more wrong!! . In both cases, failing to convey what it should be considered "rigor in mathematics" and "rigor in physics"

But again Saunders and Lubos:

**MacLane: real proof is not simply a formalized document, but a sequence of ideas and insights**

** Motl: the primary physical motivation is to locate the right ideas and equations that describe the real world. Category theory has been used by many to achieve completely wrong physical conclusions - for example, by considering the "pompously foolish" quantization functor.**

He however seems to be pushing forward the same misconception of "rigor":

**Motl: It may be nice to be rigorous, but it's always more important to be correct: if the specific kind of rigor leads us to stupid conclusions in physics, we should avoid it.**

From the original Marta's "cracks and pots"

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

**J. Baez: I had never heard anyone before suggest that category theory could be discredited by applications to string theory. It completely surprised me. I'm used to the opposite complaint: that category theory is discredited by its *lack* of applications.**

Here it is a clear and rigorous answer:

(1) **W. Lawvere: The question is not whether mathematics should be applied. Most of us agree that it should. The concern is rather that our subject is sometimes being used as a mystifying smoke screen to protect pseudo-applications against the scrutiny of the general public and of the scientific colleagues in adjacent disciplines. We need to ensure that applications themselves be maximally effective, not clouded by misunderstanding.**

Now, an example of superficial conclusions:

** J. Baez: Indeed, the funny thing about string theory is that while leading to an abundant harvest of rigorous mathematical results, it has not yet correctly predicted a single result from a single experiment, even after more than 20 years of work on the part of many smart people.**

There is nothing funny about this. Lubos say:

** Motl: One of the fascinating features of string theory is that its objects and investigations, even though they've been partially disconnected from the daily exchanges with the experimentalists, remained extremely physical in character. All of the objects that we deal with are analogous to some objects in well-known working physical theories, to say the least.**

Bill has made a serious, well fundamented and non-bullshit contribution to "crack and pots" (he utilizes a different heading:" WHY ARE WE CONCERNED?")

In contrast to many passages of some contributors that it will be tiresome to reproduce here, and where one founds an overwhelming proliferation of

highly technical, sophisticated, difficult and impressively sounding words

such that it becomes impossible to see what they are saying, unless you are an expert, in which case you may find out that it is only superficial thinking (I am thinking specially in certain parts of Davis Yetter's postings).

** W. Lawvere: Professors may not consider the possibility of learning from undergraduate text books, and some may feel bored that I have once again repeated the above basic definitions and observations.**

If you have some real thoughts, you do not need impressive jargon.

See what an original and deep insight:

** W. Lawvere: As quantity includes zero, so structure includes the case of no structure, which Cantor considered one of his most profound and exciting discoveries**

Superficial thinking (which could be malicious, but very often is simply stupid) has manifested itself in these postings by pushing forward the idea that there are two different kinds of category theory:

"Categories as Foundation" and "Categories as Algebra", the first implicitly (but not explicitly said) the "bad one", and the second the "good" one.

** D. Yetter: All of these are part and parcel of a different face of category theory than one saw in the old days: category theory as algebra, rather than category theory as foundations.**

We have an excellent analysis of this fallacy in Bill's postings, which should be read carefully and slowly.

I imagine now to add something that Lawvere himself pointed out a long time ago: The laws of logic are a particular instance of the categorical concept of adjoint functors, a concept that grew out of mathematical experience.

There is any way some explanation to Yetter's prejudice against "categories as foundation". Often very poor category theory has been justified by people writing on foundations. Bill's quote (1) above also applies to this and related use of category theory in theoretical computer science.

Somebody else that does not need either noisy language sees better:

** Dusko: I am of course saying things very clear and familiar to many people on this list, but maybe they are worth saying nevertheless.**

** Dusko: but at the end of the day, I think, we'll all agree that the source of the unreasonable effectiveness of categorical algebra is its foundational content **

Then, he passes to consider Grothendiek's ("the greatest of the category theorists") work on Topos theory as work on foundations, which agrees with the analysis of foundations made by Lawvere.

I can not restrain myself to quote the following magnificent piece of meaningless hallucinogenic discourse:

**V. Pratt: In the millions of years of evolution of primate thinking, no productive mathematical mechanism has a higher probability of being stumbled on than mathematics founded on the Yoneda axiom. I know of no better explanation of how human thought could have evolved to its present form than evolution finding and exploiting the Yoneda principle**

Now, some serious business:

In recent years J.Baez and his followers have been occupying more and more space in the categorical community (this fact is at the starting point of the present debate).

I think this is so because they have some interesting category theory to show, but they are occupying more space than their mathematics deserves because they bring a refreshing air to a community until now dominated by an old guard that has not shown signs of necessary evolution, and that has not being able to attract very good and talented young mathematicians to the community. There is now not other exiting body of developments within the community. The old guard is being pushed out (prone or supine ?), but, alas, not by better mathematicians.

Category theory is in good shape (in particular pushed forward by the Russian school), and it is now passing over the category community. I have lost the information now, but recently it was in Europe an important congress that it had two subjects: one was a prestigious subject (that I do not remember now), the other was category theory. Not a single name (including Baez group) that we see in the category theory community meetings was there.

Best wishes to all e.d.

Dear Eduardo, Your message contains many important observations with which I very much agree, and a few others that need discussing. I may comment on it some time soon, publicly or privately. Right now, my purpose for replying to you is a different and more pressing one. I noticed that in your "cracks and pots 93" you implicitly refer to a lette= r which I had sent to categories in reply to Peter Selinger, and which I had also sent privately to various people, including you. The letter in questio= n never appeared!. It contained an attachment (to MacLane's article) which, according to Bob Rosebrugh, was difficult to include, hence the delay of it= s posting, and ultimately my replacing it by a brief message in response to Bill Lawvere, in which I included the URL, as suggested by Bob. It seems imperative now that I post the original message, without the attachment. Thus, portions of your letter may make more sense. It is below this message= =2E By the way, by my own count, there are only 75 messages posted in the "thread", but 183 more that were written privately to me in connection with it. Maybe you included some private messages in your count? Either way, thi= s means a *lot* of messages. Best, Marta ---------------------------------------------------------------------------= --------------------------------------------------------- Dear Peter, You were lucky to have been away on vacation, but perhaps quickly reading (how else) the mass of postings in the "cracks and pots" has caused you intellectual indigestion. Your reaction is therefore quite understandable. For your sake (and that of others in similar situations), I will sum up wha= t caused my postings, and be more explicit concerning

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

...

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

1. On March 13, I shared with you all a disturbing posting in Motl's blog, criticizing category theory in its applications to physics, and more particularly, John Baez. My concern was based on the possibility that any o= f this criticism might be justified because I could not failed to notice how John Baez had become more or less a prominent figure (as speaker/member of the scientific committee) in recent(ly announced) meetings in CT. Explicitly, I was thinking of Firenze, Ramifications of CT, Nov 13-19, 2003, Sydney, StreetFest, July 11-16, 2005, Union College, UC Mathematical Conference, December 3-4, 2005, Chicago, MacLane Memorial Conference (Unni Nambondiri Lectures), April 7,10,11, 2006, Halifax (near), CT'06, June 25-July 1, 2006. 2. On March 14, and in response to some, I asked more explicitly what caused organizers of meetings to bring to center stage one aspect of CT over others, particulaly one which seemed to me not to be in good standing after Motl's postings. Was it because it is indeed the case that CT is in disrepute, and if so its reputation needs to be restored, this being the best way to do it? Was it because it is funding for CT (notoriously lacking in the USA) that may be more easily secured that way? I wanted to know myself, but also possibly alert organizers of meetings to reflect on this issues, since their power and responsibility is indeed enormous in promotin= g a certain kind of research over another. 3. picked on one (March 17) to add some information that I had just come acros= s by reading Nature (on our coffee table, along with a dozen or so scientific journals), in an article which connected Lee Smolin of the Perimeter Institute with the Templeton Foundation, the latter a promoter of anything they can in the borderline of science and religion. In the Scientific American articles by Lee Smolin on Loop Quantum Gravity and the discretenes= s of the universe, a paper John Baez is quoted among the few references give= n at the end of the article. This, in turn, led me to research the Templeton Foundation itself, and with some help from a fellow categorist who seemed t= o know a lot about it, I easily located references to Templeton funding to th= e Goedel Centenary Symposium in Vienna, and to the A.Connes workshop on NCA a= t the Sir Isaac Newton Institute in Cambridge. I was, however, relieved not t= o find any direct connection between Templeton and Category Theory. Still, I meant to warn those unaware of this easy source of funding (with strings attached). In a subsequengt posting (March 27) I gave explicit references t= o these claims in response to some queries. 4. In short, I do not think that I can be blamed for not being explicit enough in matters that I could be explicit about. I still do not have all the answers to my questions. As I mentioned on March 27, I was mistaken in thinking of John Baez as a promoter of string theory when, in fact, he promotes a competitor thery, LQG. But the general question of categorical applications to physics remained. Why are they promoted now? As you, Peter, kindly offer as a possible explanation,

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

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

From: Eduardo Dubuc <edubuc@dm.uba.ar> To: cat-dist@mta.ca Subject: categories: cracks and pots 93 Date: Thu, 30 Mar 2006 15:31:17 -0300 (ART)

Hi,

The 93 is because I have by now 92 msages in my cracks and pots file.

I apologize for the length of this posting. It is intended to be a (may be biased) partial account of the debate, and some comments.

Well, by now the "cracks and pots" debate is establishing itself as, in my opinion, an interesting and worth-wile event. Congratulations Marta !!

We are learning about:

a) Understand (for many of us) better what is mathematics, and what is physics, what is rigor and what is buccaneering, and also what is bullshit.

b) "Something is rotten in the state of category theory community"

Pay attention that The Bard does not say "category theory", but he says "category theory community"

I start from who has made the more refreshing, humorous, down to earth, honest and intelligent contributions to this debate:

**Vicent Schmitt: that theoretical physics, computer science, phylo., a mix of those, or whatever? , is used to justify poor "categorical" work is, in my view, an existing problem. More or less everyone is conscious of it (come on!...) but so far that has not been publicly debated.**

Yes Vincent!!, you point right to what it is at the center (or very near it) the problem raised in Marta=D5s original "cracks and pots" posting!. A= nd the "(come on!...)", beautiful !.

Now, talking about rigor, conjectures and proofs:

**Maclane : If a result has not yet been given valid proof, it isn't yet mathematics. This however does not deny the many preliminary stages of insight, experiment, speculation or conjecture, which can lead to mathematics. It states simply that a conjectured result is not yet a theorem **

It is relevant to compare this with Motl's distinction between physics and mathematics:

**Motl: In physics, we propose different conjectures about the real world, and it is important that we're not guaranteed that these conjectures will be true.

String theory itself is not just a conjecture, but rather a seemingly consistent mathematical framework. Once we accept string theory as an objectively existing mathematical structure, a structure that we treat as a part of "generalized physics" - which is of course what all string theorists are doing every day - we can ask a lot of questions about its properties.**

He does distinguish between "physics as conjecture" and mathematics with applications to physics. He call this mathematics "generalized physics"

But "conjecture" to be acceptable is not unrigourous neither buccaneering. he says:

**Motl: the statements about string theory are just conjectures, and they need to be proved or supported by evidence, otherwise they're irrelevant and "wrong", in the physical sense.**

He also says:

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

He is clearly saying that those "mathematically oriented people" are lacking rigor.

Many postings in this debate confound mathematical rigor with formalism, and push forward the idea that a formal and logically correct statement has automatically rigor. Even if it is foolish:

**V. Pratt: In axiomatic mathematics, everything that is not forbidden is permitted. **

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

It seems to me that he is equating here "mathematical standards of rigor" with "logically correct", and "the standards appropriate to that subject" (in this case, physics) with " buccaneering "

Nothing more wrong!! . In both cases, failing to convey what it should be considered "rigor in mathematics" and "rigor in physics"

But again Saunders and Lubos:

**MacLane: real proof is not simply a formalized document, but a sequence of ideas and insights**

** Motl: the primary physical motivation is to locate the right ideas and equations that describe the real world. Category theory has been used by many to achieve completely wrong physical conclusions - for example, by considering the "pompously foolish" quantization functor.**

He however seems to be pushing forward the same misconception of "rigor":

**Motl: It may be nice to be rigorous, but it's always more important to be correct: if the specific kind of rigor leads us to stupid conclusions in physics, we should avoid it.**

...

From the original Marta's "cracks and pots"

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

**J. Baez: I had never heard anyone before suggest that category theory could be discredited by applications to string theory. It completely surprised me. I'm used to the opposite complaint: that category theory is discredited by its *lack* of applications.**

Here it is a clear and rigorous answer:

(1) **W. Lawvere: The question is not whether mathematics should be applied. Most of us agree that it should. The concern is rather that our subject is sometimes being used as a mystifying smoke screen to protect pseudo-applications against the scrutiny of the general public and of the scientific colleagues in adjacent disciplines. We need to ensure that applications themselves be maximally effective, not clouded by misunderstanding.**

Now, an example of superficial conclusions:

** J. Baez: Indeed, the funny thing about string theory is that while leading to an abundant harvest of rigorous mathematical results, it has not yet correctly predicted a single result from a single experiment, even after more than 20 years of work on the part of many smart people.**

There is nothing funny about this. Lubos say:

** Motl: One of the fascinating features of string theory is that its objects and investigations, even though they've been partially disconnected from the daily exchanges with the experimentalists, remained extremely physical in character. All of the objects that we deal with are analogous to some objects in well-known working physical theories, to say the least.**

Bill has made a serious, well fundamented and non-bullshit contribution to "crack and pots" (he utilizes a different heading:" WHY ARE WE CONCERNED?"= )

In contrast to many passages of some contributors that it will be tiresome to reproduce here, and where one founds an overwhelming proliferation of

highly technical, sophisticated, difficult and impressively sounding words

such that it becomes impossible to see what they are saying, unless you are an expert, in which case you may find out that it is only superficial thinking (I am thinking specially in certain parts of Davis Yetter's postings).

** W. Lawvere: Professors may not consider the possibility of learning from undergraduate text books, and some may feel bored that I have once again repeated the above basic definitions and observations.**

If you have some real thoughts, you do not need impressive jargon.

See what an original and deep insight:

** W. Lawvere: As quantity includes zero, so structure includes the case of no structure, which Cantor considered one of his most profound and exciting discoveries**

Superficial thinking (which could be malicious, but very often is simply stupid) has manifested itself in these postings by pushing forward the idea that there are two different kinds of category theory:

"Categories as Foundation" and "Categories as Algebra", the first implicitly (but not explicitly said) the "bad one", and the second the "good" one.

** D. Yetter: All of these are part and parcel of a different face of category theory than one saw in the old days: category theory as algebra, rather than category theory as foundations.**

We have an excellent analysis of this fallacy in Bill's postings, which should be read carefully and slowly.

I imagine now to add something that Lawvere himself pointed out a long time ago: The laws of logic are a particular instance of the categorical concept of adjoint functors, a concept that grew out of mathematical experience.

There is any way some explanation to Yetter's prejudice against "categories as foundation". Often very poor category theory has been justified by people writing on foundations. Bill's quote (1) above also applies to this and related use of category theory in theoretical computer science.

Somebody else that does not need either noisy language sees better:

** Dusko: I am of course saying things very clear and familiar to many people on this list, but maybe they are worth saying nevertheless.**

** Dusko: but at the end of the day, I think, we'll all agree that the source of the unreasonable effectiveness of categorical algebra is its foundational content **

Then, he passes to consider Grothendiek's ("the greatest of the category theorists") work on Topos theory as work on foundations, which agrees with the analysis of foundations made by Lawvere.

I can not restrain myself to quote the following magnificent piece of meaningless hallucinogenic discourse:

**V. Pratt: In the millions of years of evolution of primate thinking, no productive mathematical mechanism has a higher probability of being stumbled on than mathematics founded on the Yoneda axiom. I know of no better explanation of how human thought could have evolved to its present form than evolution finding and exploiting the Yoneda principle**

Now, some serious business:

In recent years J.Baez and his followers have been occupying more and more space in the categorical community (this fact is at the starting point of the present debate).

I think this is so because they have some interesting category theory to show, but they are occupying more space than their mathematics deserves because they bring a refreshing air to a community until now dominated by an old guard that has not shown signs of necessary evolution, and that has not being able to attract very good and talented young mathematicians to the community. There is now not other exiting body of developments within the community. The old guard is being pushed out (prone or supine ?), but, alas, not by better mathematicians.

Category theory is in good shape (in particular pushed forward by the Russian school), and it is now passing over the category community. I have lost the information now, but recently it was in Europe an important congress that it had two subjects: one was a prestigious subject (that I do not remember now), the other was category theory. Not a single name (including Baez group) that we see in the category theory community meetings was there.

Best wishes to all e.d.

Dear Eduardo, I promised to comment on your posting partly since you praise me in having started this discussion but also since you touch on something very relevant to the questions that I have been asking.

In recent years J.Baez and his followers have been occupying more and more space in the categorical community (this fact is at the starting point of the present debate).

You refer to the list of meetings where John Baez has played a prominent role as speaker in recent years -- and I forgot to mention Coimbra 1999 (School on Category Theory and Applications). I do not recall of anybody, with the exception perhaps also of Steve Awodey, to have been given so much attention in recent years. I was perhaps misled by "theological considerations", namely the Templeton Foundation and its efforts in trying to mix up science and religion. This is unfortunately true enough, but it has nothing to do with our present discussion. This is the conclusion that I have reached, and it is my obligation to say so publicly. You offer some interesting explanation for the state of affairs in category theory meetings.

I think this is so because they have some interesting category theory to show, but they are occupying more space than their mathematics deserves because they bring a refreshing air to a community until now dominated by an old guard that has not shown signs of necessary evolution, and that has not being able to attract very good and talented young mathematicians to the community. There is now not other exiting body of developments within the community. The old guard is being pushed out (prone or supine ?), but, alas, not by better mathematicians.

A general comment is that, whereas some of it may be true, there is much that should be justified, or questioned. For instance, that "they have sone interesting category theory to show", that "they bring a refreshing air", that ours is "a community until now dominated by an old guard that has shown no signs of necessary evolution", that our community "has not been able to attract very good and talented mathematicians", at present there is "no other exiting body of developments within the community". Would you care to be more explicit? At the request of others, and to the risk of attracting a lot of criticism, I have done so and have been grossly misunderstood. If I did not request it, others will do so. That way, it would be easier to respond to the above. Let us assume for the moment that the is a rift in our category theory community -- the "old guard" versus the "new guard". We could even put names to represent each one -- it may seem obvious to many that, whereas Marta Bunge "must have been influenced by Bill Lawvere (and others)", Eduardo Dubuc "must be influenced by Andre Joyal (and others)". I am still speculating. This imaginary discussion I want to refute, and I am sure that you would want to do the same, or maybe not. But, is there such a rift and, if so, why take one side against the other? I believe that *all* progressive forces within category theory (no matter where they come from) should join efforts rather than split over issues that very few people understand or care about. I am a great admirer of both Bill Lawvere and Andre Joyal (and of many others, eg Peter Freyd, Ross Street, Ieke Moerdijk, to name a few -- of course, beginning with Grothendieck) because, even if they may have different philosophies of mathematics (in particular of category theory), they all bring in novel ideas derived from great experience and insight.

The old guard is being pushed out (prone or supine ?), but, alas, not by better mathematicians.

This too, is cryptic, and too sweep a generalization. I believe that there is poor mathematics no matter which area of category theory, but there is also good mathematics both sides of the imaginary (?) rift. Instead of making divisions of the sort "the young against the old", which can be used to justify almost anything, let us make no divisions and try to preserve excellence and promise in whatever is being done, whether applied or not, whether fashionable or not, and filter out (as editors of journals, organizers of conferences) the sort of mathematics that is sure to discredit us. Who will then be the judges? Obviously good mathematicians, not biased, completely accountable for their choices, not short-sighted, independent, very well informed, anxious to preserve high standards. Is there anybody left after so many requirements? Well, Peter Johnstone is one such person, but there are several others, of course and it is not for me to make such a list. Once identified by general consensus (here is the hard part!), let them consistently make all the decisions (for a certain period of time, at least). That way, it will be unlikely that the same people be invited over and over again, and also unlikely that bad papers will be accepted to be presented at meetings or published in our journals. Sammy Eilenberg and Saunders MacLane may have represented different tendencies within category theory, and most of us feel having been influenced by one rather than by the other (some by both). Yet, there was never any open rift between them, which was wonderful. I decided to work on category theory after a brilliant lecture by Eilenberg at Haverford College in (I think) 1963. I thought -- this is what I want to do! And there (at Penn, where I was a grad student) was Peter Freyd giving a course in Algebraic Topology, and later one in Abelian Categories.The sheer beauty and depth of these new ideas (new then) was overwhelming, and I decided to ask Peter to let me become his student. As I said publicly at a recent celebration of Peter's 65th Birthday in Philadephia, this was to determine the rest of my (I think, most interesting) life. I met Bil Lawvere later, at a congress in Jerusalem in 1964. His new ideas were in a different direction, but equally fascinating, and I learnt a lot from him the following year at the Forschunsinstitut fur Matematik at the E.T.H. in Zurich, where I also met Anders Kock, Jim Lambek, and Fritz Ulmer, among others. I met "the others" (Myles Tierney, Jon Beck, Jean Benabou, Saunders MacLane, and others) at the first (1966) Oberwolfach meeting, where I was allowed to present the results of my thesis at the very end of it. I count as my advisors both Freyd and Lawvere (made official in the Genealogy Project). It was the confluence of ideas and people working in harmony that was so wonderful in those days. Why should they be gone forever? Let us work together to bring them back, if possible. Something good has to come out of this discussion, or else it may have been something worse than just a waste of time. What do you say, Eduardo? Best wishes, Marta

Dear Jim Your question "Should it not be sufficient to acknowledge good work whoever does it ? " would I think receive the obvious answer YES from all participants in the list. Yet it lies it the heart of most of the 200 messages, public and private, that I have received since Marta opened the discussion. For as I concluded in III, the problem is, How can we know ? We cannot acknowledge good work unless we have knowledge of it. This was not a problem for 40 years, or rather if there was temporary problem, it was usually due to the need to acquire prerequisites, which could be done. But in the past few years, a "new" barrier to finding out has been added to the mathematical culture (it was common in other parts of the culture already); disdain for communicating, starting with disdain for revealing definitions. That this trend is not due to a few individual culprits within our community is illustrated by the attempts to justify it "theoretically" by show business or business practice or by pop Oxbridge philosophy. Whether in exposition, in popularization, or in professional lectures, of course the practice of noncommunication typically begins with "'I am going to be your special communicator about this", etc. These additional barriers we are presented with make "How can we know" ( whether a work is good) much more difficult, and an accumulation of such difficulties has led to a lot of concern. Best regards Bill Quoting jim stasheff <jds@math.upenn.edu>:

Marta Bunge wrote (inter alia):

It was the confluence of ideas and people working in harmony that was so wonderful in those days. Why should they be gone forever? Let us work together to bring them back, if possible.

A consummation devoutly to be wished. In algebraic topology, throughout my career, there has been most of the time such harmony and a willingness to acknowledge each ohters work. Only briefly have there been periods of `turf claiming' and antagonism between individuals if not `camps'. (Of course funding was plentiful in my early days.)

Should it not be sufficient to acknowledge good work who ever produces it?

jim

[ balance of quotation omitted... ]

wlawvere@buffalo.edu wrote:

Dear Jim

Your question "Should it not be sufficient to acknowledge good work whoever does it ? " would I think receive the obvious answer YES from all participants in the list.

Yet it lies it the heart of most of the 200 messages, public and private, that I have received since Marta opened the discussion. For as I concluded in III, the problem is, How can we know ?

I have been raised to believe that the answer to this is peer-review, not public-trial-by-mailing-list. One problem with the latter method is that, since nobody wants to discuss particular cases, the discussion tends to center on generalities, innuendo and "I have heard it repeated many times"-type arguments, and not on evidence or actual facts that could be verified. Such evidence and facts, on the other hand, are available to editors and referees, who are also largely independent of public opinion. As I see it, this is exactly as it should be. -- Peter