Eduardo J. Dubuc wrote:

I wish you (V.P.) were more clear. I can not see what is your point. Witty msages for the Illuminati don't serve any pourpose, except amusement to some.

Yes, sorry about that. A couple of others wrote privately with the same request. I have no problem with the notion of mathematical truth per se, which I imagine to be what all mathematicians seek, along with mathematical tools and a consensus thereon by their colleagues. What I had in mind by the "haunting" remark is that the implications of Goedel's incompleteness results don't immediately leap out at one, and there is a certain optimistic tendency to minimize those implications and continue to argue the issues as though Goedel's theorems weren't relevant. We can't *define* mathematical truth (Tarski may have been the first to enunciate that implication most clearly), yet we can often recognize it when we see it. Learning to do mathematics amounts to learning how to find and communicate those mathematical truths that are easily recognized as such by other mathematicians according to community standards. We imagine that mathematics on Arcturus must be like ours, but mathematics is an intrinsically cultural subject and I don't see why Arcturan mathematics should be like ours. Do Arcturans have logic? Do they have algebra? Do they draw a distinction between the two? Do they believe in either? Add category theory as a third framework and ask the same questions of it. Do they know about initial algebras and final coalgebras, and if so which came first for them? Do they know about monads and adjunctions, and if so which came first? That's surely too brief to be clear. I'd be happy to engage further in this sort of speculation on the practice of mathematics as a cultural issue. I have less to contribute on the intrinsic nature of mathematics itself for lack of insight into its scope. Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Colin McLarty wrote:

[responding to Manin thoughts] I myself am also confident that people will calm down and notice that axiomatic categorical foundations such as ETCS and CCAF work perfectly well, in formal terms, and relate much more directly to practice than any earlier foundations.

Thanks, Colin. There I was nicely calmed down and then you got me all worked up again. :) I prefer the Euclidean plane over sets as a suitable starting point for understanding mathematics. What advantage is there to making geometry rest on set theory as opposed to vice versa? What is wrong with starting from a geodesic space as a place where it is always determined, given two points, what is the next one, subject to some simple equational principles? This is a common basis for the second postulate of Book I of Euclid's *Elements*, Newton's first law of motion, Einstein's theory of general relativity that a falling body is merely following a geodesic in a space curved by a nearby mass, and the notion of Hamiltonian flow of a vector field for an energy function defined on the cotangent space of a manifold as an expression of the principle of least action. In this framework a *set* is simply a geodesic space where the next point after x and y is x. (So if I ask what is the next element in the sequence 3,4,... the answer is 3, not 5.) More on this at http://boole.stanford.edu/pub/consgeom.pdf . A geodesic space or geode, aka kei, is related to a quandle (see http://en.wikipedia.org/wiki/Quandle ), the difference being that for abelian groups, quandles are merely sets whereas flat geodes (those satisfying Euclid's 5th postulate) form a symmetric monoidal closed category fully and reflectively extending Set (properly of course). Moreover its subdirect irreducibles are those of Ab except for those of even order as per the last slide. Quandles are for knot theory, not geometry. The difference between sets and geodesic spaces in foundations is like the difference between scales and Fur Elise for piano students. Both are good ways to get started but the second is more interesting. (Apologies again to Eduardo for my impenetrable writing, in this case I can only counsel patience since these ideas seem to come with a certain viscosity that inhibits any royal road of the kind Eduardo would like.) Best, Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Selon Colin McLarty <colin.mclarty@case.edu>:

I myself am also confident that people will calm down and notice that axiomatic categorical foundations such as ETCS and CCAF work perfectly well, in formal terms, and relate much more directly to practice than any earlier foundations. One hundred and fifty years of explicitly foundational thought has made this progress possible. By now, that can hardly qualify as "extraordinary"!

I do NOT believe that ETCS and CCAF "work perfectly well". Each of these involve two foundational "layers", namely, the classical "bottom" and a categorical "superstructure". By the classical bottom I mean NOT an underlying Set theory but the "Elementary theory of categories" (ETC), i.e. a theory of categories using the usual First-Order Logic (FOL) and relying on the standard Hilbert-Tarski-style axiomatic method. I agree with John Mayberry and some other people who argue that this aximatic method alone assumes a basic notion of set or collection. Unlike Mayberry I don't think that this fact implies that the project of categorical foundations, as a alternative to and replacement for set-theoretic foundations, is futile. Recall that the axiomatic method we are talking about (which is, of cause, quite different from Euclid's method and other earlier versions of axiomatic method) emerged together with Set theory. In order to make categorical foundations into a viable alternative of set-theoretic foundations we still need to provide Category theory with a new axiomatic method rather than use the older axiomatic method as do ETCS and CCAF. Elements of this prospective axiomatic method are found in what I just called the "categorical superstructure" of ETCS and CCAF but as far as these theories are concerned the classical background (FOL+ETC) is indispensable. This is why I say that ETCS and CCAF do NOT work perfectly weel as categorical foundations. Building of "purely categorical" foundations remains an open problem. It is not a matter of a ideological purity but a matter of complete "rebuilding" (Manin's word) of foundations: in my view, such a rebuilding is healthy and refreshing in any circumstances (unless it clashes severely with practice). [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

...

...

I invite everyone to read the interesting interview of Yuri Manin published in the November issue of the Notices of the AMS:

Manin is always entertaining but not very careful about what he says.

Hm, Manin is never just entertaining: he wrote several papers concerning physics, linguistics, psychology, and his thinking is an example of how a true mathematical mind works in complex areas like the humanities, generates unexpected views, reveals deep connections etc. If the results are readable and enjoyable, it just shows the literary talent of the author... :) I also wouldn't say that Manin is not very careful about what he says. The parts of the interview about foundations and physics say, basically, this. After Bourbaki, a correct mathematical text should consist of two parts: (a) definition of the structure in question (structure in the sense of Bourbaki), (b) deductions about this structure in some logic (perhaps, non-classical). Manin says that texts generated by physicists do have (b) but not (a). These are deductions about something that has not been defined and hence, for a mathematician, that does not exist at all (the Eiffel Tower is in the air). This situation is not unique, of course: Manin mentions Cantor's set theory at the time of invention, and it was and is so for engineering theories. Software engineering should be of special interest for this list because modern software executes deductions about categorical structures. It is not in the interview explicitly, but the following model of a mathematical text would be probably close in spirit to what Manin says. Mathematical texts form a span: PM <--- MM --->FM with PM -- the universe of "physical" mathematical texts (physics, computer science, engineering etc), MM -- the mathematician's universe of mathematical texts; they are written in a special subset of the natural language (nowadays, in accordance with Bourbaki or category theory), FM -- the universe of formal (machine-readable) mathematical texts. A physicist is interested in the left foot, a logicist -- in the right one, but mathematics is about the entire span (well, for a true mathematician, P stands for Platonic rather than Physics). If you want: the logicist view is more normative because it insists on the right right leg, but Bourbaki concerned about the entire span and did not want to fix neither right nor the left legs (unless P is for Platonic). So, they proposed a reasonable structure for MM for which the left and right sides of the whole could be added (if needed). It's indeed more about practical foundations... After all, Eduardo said it best:

Well, the fact that he is not very careful is precisely what makes his saying meaningful, interesting, fresh and enjoyable. He does not place himself within any philosophical or political frame. He feels free to say what it crosses his mind just as it comes. beautiful !

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Vaughan Pratt wrote:

Eduardo J. Dubuc wrote: [regarding Manin]

...

He does not place himself within any philosophical or political frame. He feels free to say what it crosses his mind just as it comes. beautiful !

Right, but clearly we cannot extend the same freedom to the hoi polloi, who cannot be trusted not to abuse it.

Vaughan Pratt

of course, only a few can make of such freedom a meaningful discourse e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

I get to André Rodin's comments, and the redoubtable John Mayberry, below. 2009/11/12 Joyal, André <joyal.andre@uqam.ca>: Writes what I entirely agree with:

I am convinced that categorical logic, which was wholly invented by Lawvere, is the most important development of logic during the second half of the 20th century. I find the notion of elementary topos absolutely extraordinary, almost magical.

I only mean it is not extraordinary that with enough time the developments become generally known.

Like every important mathematical discovery, it stands out as a colorful gem on a bed of grey stones.

This imagery makes perfect sense to me, for many great examples such as the complex numbers as André described. But I don't know if I could convey it to many philosophers of mathematics even in the most general terms -- let alone convince them anything to do with category theory is an example. Most philosophers, so far as I know, still consider the complex numbers far-fetched and "impossible to visualize" (which I find incredible).

I find astonishing that ETCS should be closely related to topos theory via the notion of an elementary topos. It is also surprising that the internal logic of a topos should be formally identical to intuitionistic set theory. The construction by Hyland of the realizability topos is also extraordinary because of the connection with recursive function theory.

Yes. And I agree with what André said earlier that there is room here for possible further insights into what remain profound mysteries about the hierarchy of infinite cardinals. (I do not claim to currently have those insights!)

One may argue that there is nothing magical in mathematics, since mathematics is rational by nature. I disagree. We are far from understanding completely the natural world, and mathematics is not a pure construction of the rational mind. Mathematicians are probing in the depth of a highly structured unkown. If we are patient and lucky enough we may catch a gem. The gem has a structure of its own and we can learn from it. This is were the magic is.

I am not happy to call it "magic" -- I collected rocks as a teenager and once did catch a "gem" (a thick tuft of pink-grading-to-green byssolite hairs with bright pyrite crystals suspended in them, 4 feet down a gray rock crevice that I could barely crawl into) but I do not call that "magic" either. Perhaps this is mostly a difference over words. 2009/11/12 <Andre.Rodin@ens.fr>: writes

I do NOT believe that ETCS and CCAF "work perfectly well". Each of these involve two foundational "layers", namely, the classical "bottom" and a categorical "superstructure". By the classical bottom I mean NOT an underlying Set theory but the "Elementary theory of categories" (ETC), i.e. a theory of categories using the usual First-Order Logic (FOL) and relying on the standard Hilbert-Tarski-style axiomatic method. I agree with John Mayberry and some other people who argue that this aximatic method alone assumes a basic notion of set or collection.

Mayberry says two things about this. The first, which has taught me a lot, is his stress that no formalization can be the basis of our actual knowledge of mathematics. This applies to all formalized foundations. Mayberry's point is precisely the reason why I say that ETCS and CCAF " work perfectly well, in formal terms." It is a plain fact that these axioms work as well as the formal ZFC axioms --- while Mayberry is right that formalized axioms cannot be the real basis of our knowledge. I believe John has underrated the dialectical relation between formalization and "the real basis of our knowledge." I have often discussed this with him and I am not sure exactly what he thinks about it now. Formal investigation of ZFC has changed our actual beliefs about sets. Category theory has further changed our actual beliefs about mathematics, and formal investigation of ETCS and CCAF has been part of this. But the key point is that ETCS and CCAF are not only formal axioms, any more than ZFC is. All are formalizations *of* our real beliefs about sets and categories. These real beliefs do not "assume a basic notion of set or collection" but rather *include* or *express* a basic notion. The next thing John says is that our basic notion of collection is best captured by ZFC. (Or, rather, he used to say that prior to developing his finitary set theory as an alternative foundation.) I say ETCS formalizes almost the same idea of set, but better than ZFC. The ETCS formalization is rather like the ZFC one, but omitting a lot of irrelevancies about transfinitely iterated membership. Zermelo and then Fraenkel and Skolem found these in the first attempts at axiomatization and I don't say i could have done better in 1908 or 1922. I say Eilenberg and MacLane's work of 1945 enabled Lawvere to do better in 1963. But even before Bill did that he had already seen that our basic notion of collection is not so much like that. It is typified by, say, the continuum, or the collection of Euclidean motions of the plane, and such. Our basic notion of the continuum is not that the discrete collection of points on it is equinumerous with the powerset of the natural numbers, and it is equipped with a lattice of open subsets -- our "basic notion" of it is rather a somewhat open-ended notion of continuous translation. The basic notions are in fact not very articulate in themselves, and throughout the history of mathematics it has taken further ideas to articulate them. Bill saw how to articulate these and many more, quite directly, in categorical terms not assuming any prior set theory. That articulation works even if you do not take it as foundational. But it gets a natural foundational character in the framework of the category of categories -- thus CCAF, the axiomatic theory of the category of categories as a foundation. best, Colin Unlike Mayberry I don't think that this fact implies that

the project of categorical foundations, as a alternative to and replacement for set-theoretic foundations, is futile. Recall that the axiomatic method we are talking about (which is, of cause, quite different from Euclid's method and other earlier versions of axiomatic method) emerged together with Set theory. In order to make categorical foundations into a viable alternative of set-theoretic foundations we still need to provide Category theory with a new axiomatic method rather than use the older axiomatic method as do ETCS and CCAF. Elements of this prospective axiomatic method are found in what I just called the "categorical superstructure" of ETCS and CCAF but as far as these theories are concerned the classical background (FOL+ETC) is indispensable. This is why I say that ETCS and CCAF do NOT work perfectly weel as categorical foundations. Building of "purely categorical" foundations remains an open problem. It is not a matter of a ideological purity but a matter of complete "rebuilding" (Manin's word) of foundations: in my view, such a rebuilding is healthy and refreshing in any circumstances (unless it clashes severely with practice).

best, Andre

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2009/11/12 <Andre.Rodin@ens.fr>: writes

ETCS is the formal basis of CCAF.

This is simply false. On some versions ETCS is a part of CCAF but even then it is in no sense prior to other parts.

ETCS relies on a pre-formal notion of set or collection just like ZF or any other axiomatic theory built with Hilbert-Tarski axiomatic method.

Do you mean that every formalized axiom system uses arithmetical notions such as "finite string of symbols." This is why that formal axioms cannot be the real basis of our knowledge of math, but it has no more bearing on categorical axioms than any others. Or do you think that pre-formal notions of "set" or "collection" are all based on iterated membership and Zermelo's form of the axiom of extensionality, so that CCAF is less basic than ZFC? That is a common belief among logicians who have not read Zermelo's critique of Cantor (where Zermelo points out that Cantor did not hold these beliefs) and who know a great deal more of ZFC than of other mathematics. In fact, long before mathematicians could analyze the continuum into a discrete set of points plus a topology, they were well aware of collections like the collection of rigid motions of the plane -- and that "collection" is a category. It is not just a ZFC set of motions but comes with composition of motions and with an object that the motions act on.

Elements of a new properly categorical method of theory-building are present in the "basic theory" (BC) that follows ETC. (I mean, in particular, the "redefinition" of functor in BC as 2-->A, etc. The standard definition of functor given earlier in ETC never reappears in BC.)

The "standard" definition of functor appears as the definition of a small category in the category of sets.

However in CCAF these new features are not yet developed into an autonomous axiomatic method - or into a new way of formalisation of pre-formal concepts, if you like.

Well, yes, they are developed into one. That was Bill's achievement with CCAF. best, Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Sorry. I did misunderstand that. But I still do not understand it. What is a "formal basis" of a theory T? Is any subtheory of T? Or is it any conceptually significant subtheory? (In the latter case I would not call it a "formal" basis.) Is it supposed to be a general rule that if a theory T has a "formal basis" then T cannot be a satisfactory foundation? The Eilenberg-MacLane axioms are a subtheory of CCAF and also have a natural, conceptually central interpretation in CCAF. I consider this an insight, Bill's insight, and I do not see how it becomes any kind of objection to CCAF. best, Colin 2009/11/13 <Andre.Rodin@ens.fr>:

Selon Colin McLarty <colin.mclarty@case.edu>:

...

2009/11/12  <Andre.Rodin@ens.fr>:

writes

...

 ETCS is the formal basis of CCAF.

I did NOT write this. I wrote "ETC is the formal basis of CCAF", please check my message. By ETC I mean the Elementary Theory of Categories. (You might take my ETC for a typo perhaps.)

best Andrei

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2009/11/15 <Andre.Rodin@ens.fr>: Suggests a better take on CCAF than the one he has been taking. That would be a take based more on Bill's published work on CCAF, and less on the philosophical objection that Geoff Hellman used to make about CCAF. Geoff himself has given up this objection.

AR: True, the most general notion of “collection” one can imagine may cover “category” and what not. But, I claim, the preformal notion of collection *relevant to the axiomatic method in its modern form* is more specific, and does NOT cover the preformal notion of category. I’m talking about “systems of things” in the sense of Hilbert 1899 rather than sets in the sense of ZFC or of any other axiomatic theory of sets.

This is the Hilbert conception where axioms are not asserted as true but offered as implicit definition; and so they are not about any specific subject matter but may be applied to whatever satisfies them. Lawvere from 1963 on has always been clear that his first order axioms ETCS and CCAF can be taken this way for metamathematical study -- but that he does assert them as true specifically of actual sets and categories. (Now Bill is not talking about any idealist truth or objects. He takes a dialectical view. But that is another topic.)

In ETC (the Elementary Theory of Categories in the sense of Bill’s 1966 paper) categories are conceived as collections of things called “morphisms” provided with relations called “domain”, “codomain” and “composition” (I hope I nothing forgot).

This is one use of ETC, and indeed a use made daily in mathematics. But it is not the use in CCAF. The fragment of CCAF you are calling ETC is asserted of specific things. Bill says it deals with: "the category whose maps are ‘all’ possible functors, and whose objects are ‘all’ possible (identity functors of) categories. Of course such universality needs to be tempered somewhat." The requisite tempering is very like that familiar in set theory, and Bill describes it. (The quote is his dissertation p. 26 of the TAC reprint.)

Even if there are pragmatic reasons to build theories of sets like ETCS and other mathematical theories on the basis of ETC rather than use axiomatic theories of sets like ZFC for doing category theory and the rest of mathematics, this doesn’t change the above argument.

What does change it though, is the interpretation of ETC in CCAF. That interpretation does not use The "Hilbert conception." Actually, it is best regarded as a single interpretation with a parameter: interpret "object" in the ETC axioms as "functor from 1 to X" where X is a fixed free variable of identity functor type in CCAF, interpret "morphism" as "functor from 2 to X" and so on always with the same free variable X. Interpreting the ETC axioms in CCAF this way is not at all treating them in the Hilbert way. But even take the interpretation corresponding to any one object A of CCAF. That amounts to specifying X as A in the parametrized interpretation. This interpretation does not deal with "the collection of objects of A" and "the collection of morphism of A". It never refers to any such collections. It deals with categories A,1,2,3, and functors among them. If you want to push this line:

Every major historical shift in foundations of mathematics so far involved a major change of the notion of axiomatic method. (I can substantiate the claim if you'll ask.)

Then you would do better to notice the novelty of these parametrized and single-category interpretations of ETC in CCAF and take this as the kind of major change that you expect to see.

AR: I called ETC “formal basis” of BT (“Basic Theory of Categories” in the sense of Bill’s 1966’s paper) meaning the two-level structure of BT. BT is ETC plus some other axioms. Conceptually the order of introduction of these axioms matters. My point (or rather guess) is that BT involves a prototype of a new axiomatic method (different from one I described above), which, however, doesn’t work in the given form independently.

This different axiomatic method is explicit in CCAF, and does work independently there. Specifically what is supposed to "not work" about it? Is it supposed to be formally inadequate to interpreting mathematics? (That is a non-starter, and even Feferman only made vague hints that it was so and never tried to fill them in.) Is it not really comprehensible? (Bill comprehended it already around 1960, and so do many of us now. Feferman argues well that he does not comprehend it, but falsely concludes that no one can.) best, Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Selon Colin McLarty <colin.mclarty@case.edu>: CM: Andre.Rodin@ens.fr Suggests a better take on CCAF than the one he has been taking. That would be a take based more on Bill's published work on CCAF, and less on the philosophical objection that Geoff Hellman used to make about CCAF. Geoff himself has given up this objection. AR: I don't know about Hellman's objection about CCAF and would be grateful for the reference. Talking about CCAF I mean first of all Bill's 1966 paper (leaving aside the problem noticed by Isbell as irrelevant to my story), rather than later versions of CCAF. I don't quite understand what does it mean a "better take" but if this means my argument then this argument is based on its own (as, in my understanding, any philosophical argument should be) but not on works of other people. CM: This is the Hilbert conception where axioms are not asserted as true but offered as implicit definition; and so they are not about any specific subject matter but may be applied to whatever satisfies them. Lawvere from 1963 on has always been clear that his first order axioms ETCS and CCAF can be taken this way for metamathematical study -- but that he does assert them as true specifically of actual sets and categories. (Now Bill is not talking about any idealist truth or objects. He takes a dialectical view. But that is another topic.) AR: This is an interesting aspect of the issue, about which I didn't think earlier. It might have a bearing on what I'm saying but so far I cannot see that it does. I am saying this: the axiomatic method in its modern form - which has been pioneered by Hilbert (among other people including Dedikind, et al. ) and then further developed by Zermelo, Tarski et al.) - involves a preformal notion of set or collection. Whatever first-order theory is built by this method objects of such a theory form preformal sets. In particular, when this method is used for building ETC then primitive objects of this theory called "morphisms" form preformal sets called "categories". In THIS sense the preformal notion of set remains a foundation of ETC. As far as I can see this situation doesn't depend on whether one thinks about axioms of ETC (or any other first-order theory) as assertive or as implicit definitions. CM: But even take the interpretation corresponding to any one object A of CCAF. That amounts to specifying X as A in the parametrized interpretation. This interpretation does not deal with "the collection of objects of A" and "the collection of morphism of A". It never refers to any such collections. It deals with categories A,1,2,3, and functors among them. AR: Right. This is exactly the reason why I say that CCAF has two well-distinguishable foundational "layers". At the first layer (ETC) a category is a collection of morphisms; at the second layer (i.e., in the core fragment of CCAF called in 1966 paper "basic theory" ), as you rightly notice, a category is no longer a collection. My problem with this is actually twofold. (1) The second layer depends on the first but not the other way round. Formally speaking, this simply amounts to the fact that axioms of ETC are axioms of BT but not the other way round. In THIS sense, once again preformal sets remain a foundation of CCAF. (2) The joint between the two layers remains for me unclear. From a formal viewpoint this looks trivial: CCAF is ETC plus some other axioms. But this doesn't explain the switch from thinking about categories as collections to thinking about categories as identity functors. In Bill's 1966 paper this switch is described as a new terminological convention made in the middle of the paper (that cancels the earlier convention). This change of notation points to but doesn't really addresse the issue, as far as I can see. CM: you would do better to notice the novelty of these parametrized and single-category interpretations of ETC in CCAF and take this as the kind of major change that you expect to see. AR: I do see this as a great novelty. But I claim that this novel approach in the given setting (i.e. in CCAF) doesn't work *independently* of the older approach; moreover there is a sense in which the older approach remains basic while the new one is a "superstructure". CM: This different axiomatic method is explicit in CCAF, and does work independently there. Specifically what is supposed to "not work" about it? AR: To sum up. ETC is built with the older Hilbert-Tarski's method. CCAF as a whole involves a genuinely new idea of how to build mathematical theories , I agree with you on this point. But since ETC is indispensible in CCAF - and morever since ETC is a starting point of CCAF the new categorical axiomatic method in the context of CCAF does not work *independently* (I am not saying that it doesn't work at all.) This is why I say that CCAF is only a half-way to genuinely categorical foundations of mathematics (that is only natural in case of such a pioneering work as Bill's 1966's paper). For a possible development of CCAF into a better categorical foundation my hopes are for developing the diagrammatic reasoning of the second layer of CCAF into a genuine logico-mathematical synatax, which could serve independently of the usual first-order syntax. I'm particularly interested in this respect in recent work of Charles Wells, Zinovy Diskin, Dominique Duval, René Guitart and other people. Actually I would be quite interested to hear from these people what they think about a possible relevance of their work to foundations of mathematics and, more specifically, to CCAF. A more general point: in my understanding, a dialectical attitude to foundations amounts to looking at them as a subject of further rebuilding - rather than looking at them as what is accomplished in principle and needs only working out some further technical details. best, andrei [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Andre Joyal wrote:

I find the notion of elementary topos absolutly extraordinary, almost magical. Like every important mathematical discovery, it stands out as a colorful gem on a bed of grey stones. A classical example of gem is the field of complex numbers.

Historically these two gems emerged as entirely independent developments. However they are arguably facets of a single gem, the abelian-topos categories Peter Freyd wrote about on this list in November 1997, archived at http://blog.gmane.org/gmane.science.mathematics.categories/day=19971031 or on Karel Stokkerman's topic-indexed archive at http://www.mta.ca/~cat-dist/catlist/1999/atcat and http://www.mta.ca/~cat-dist/catlist/1999/prattsli The complex numbers live within the ring M(2,R) of 2x2 real matrices as a subring of M(2,R) that happens to form a field. (The general linear group GL(2,R) is a larger *skew* field in M(2,R), but is there a larger *field* than the complex numbers therein?) M(2,R) in turn forms a one-object full subcategory of the abelian category Vct_R, with matrix multiplication (hence complex number multiplication) realized as composition. So the complex numbers form a subcategory of an abelian category. As Peter's treatment makes clear, abelian categories are a quite minor variant on toposes. An abelian category (resp. topos) is an abelian-topos category all of whose objects X are of type A (resp. T), meaning that the first (resp. second) projection of Xx0, namely from Xx0 to X (resp. 0), is an iso. This difference is expressed as a very simple elementary (first-order) predicate whose intuitive meaning is clear: simply multiply X by zero and see whether it remains X or collapses to zero. So within the relatively small universe of abelian-topos categories (by comparison with *all* categories) we find lurking therein both the field of complex numbers and the toposes (and hence in particular the effective topos, yet another gem Andre mentioned). Over on the Foundations of Mathematics mailing list, FOMers would presumably connect complex numbers to set theory by observing that the complex numbers form a set which lives within the universe of sets axiomatized by the Zermelo-Fraenkel axioms. To me the path from complex numbers to toposes via matrices and abelian categories seems somehow more intimate. Simply calling the complex numbers a set seems dry as dust (literally). (Peter J., are abelian-topos categories in the Elephant? They seem an obvious candidate yet I couldn't find them in either the table of contents or the index.) Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]