Date: Sun, 2 Mar 1997 15:19:43 -0400 (AST) Subject: Intuitionism's Limits Date: Sun, 2 Mar 1997 15:07:12 +1030 (CST) From: William James Intuitionism's Limits: if C is a category sufficiently complex to demonstrate that some C-arrow f:a-->b is monic and B is a subcategory of C containing just f (and the requisite identity arrows), do we still know that f is monic? Should we? (Or, in other words, which view *should* dominate: Intuitionism, Realism, the category theoretic...?) What if C is something (semi?)fundamental like a category of all sets and functions, or a category of categories? I suppose the answer is that monicity is relative to a category, but what supports this as a claim? And doesn't it seem to contradict the reasonable realist claim that we can somehow know f in B to be monic? (Or am I missing something straightforward: that properties can be granted to f by its relationship to C via an inclusion functor?) This goes to the issue of the adequacy of category theory as a foundation in more than the simply technical sense. (I could be using the term "realism" incorrectly too: I take it to be a positon, in maths at least, that mathematical entities can have collections of properties beyond the constraints of a given theoretical context.) William James Date: Mon, 3 Mar 1997 10:36:38 -0400 (AST) Subject: Re: Intuitionism's Limits Date: Sun, 2 Mar 1997 15:52:17 -0500 (EST) From: Peter Freyd To William James, You must be using a non-standard (philosophical?) definition of "monic", since it is obvious using the standard (mathematical) definition that a monic remains a monic in any subcategory containing it (to get unnecessarily technical, because it's given by a universally quantified Horn sentence). Could you tell us your definition? (For the record: f: A -> B is monic iff for all x,x':X -> A it x f x' f is the case that X -> A -> B = X -> A -> B implies x = x'.) Date: Mon, 3 Mar 1997 10:38:09 -0400 (AST) Subject: Re: Intuitionism's Limits Date: Mon, 3 Mar 1997 18:40:30 +1030 (CST) From: William James > Intuitionism's Limits: if C is a category sufficiently complex to > demonstrate that some C-arrow f:a-->b is monic and B is a subcategory > of C containing just f (and the requisite identity arrows), do we > still know that f is monic? Should we? (Or, in other words, which > view *should* dominate: Intuitionism, Realism, the category > theoretic...?) What if C is something (semi?)fundamental like a > category of all sets and functions, or a category of categories? > Whoops! The question is trivialised by using monicity as the relevant property. Reconsider it in terms of say f as an isomorphism, or of f holding some property in C that B lacks the resources to demonstrate. I'm thinking aloud on this question: constructive maths should say that of f in B there is no demonstration forthcoming, so judgment will be withheld on whether or not f has the property; a category theorist might say that category theory does not dwell on elements and that, in context, B is no different from any isomorph of 2, so there positively is no further property of f to be had other than that which can be demonstrated in any isomorph of 2. This is more than Intuitionism will allow. Might I, then, go on to say that the philosophies of constructive mathematics and category theory really are different? William James (if I'm digging a hole, I want it to be big) Date: Mon, 3 Mar 1997 10:35:52 -0400 (AST) Subject: Re: Intuitionism's Limits Date: Sun, 2 Mar 1997 15:27:28 -0500 (EST) From: John Baez William James writes: > I suppose the answer is that monicity is relative to a category, > but what supports this as a claim? It seems to me that category theory takes the sensible viewpoint that mathematical entities (e.g. objects and morphisms) only become interesting through their relationship with other entities. Every arrow looks just like every other arrow if we consider it in isolation. Every arrow in a category C is an image of the what James Dolan calls the "walking arrow" --- the nonidentity morphism in the free category C0 on a single morphism --- under some functor F: C0 -> C. Studying an arrow in isolation is just like studying the walking arrow, which is completely dull. The fun begins only when we have a bunch of arrows and start composing them. This is one reason why I think n-category theory should be useful in physics problems like quantum gravity, where it only makes sense to speak of where or when an event occurs relative to other events, not with respect to some spacetime manifold of fixed geometry. For some of the technical apsects of how this might go, see: John Baez and James Dolan, Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995), 6073-6105. Louis Crane, Clock and category: is quantum gravity algebraic?, J. Math. Phys. 36 (1995), 6180-6195. These both appeared in a special issue on diffeomorphism-invariant physics. Date: Mon, 3 Mar 1997 10:37:27 -0400 (AST) Subject: RE: Intuitionism's Limits Date: Sun, 2 Mar 97 21:09 EST From: Fred E J Linton <0004142427@mcimail.com> If a and b are *two* objects, then, in the category consisting solely of those two objects, their respective identity maps, and one further map from a to b (and nothing more), that map is both monic and epic. Once embedded in another category, however, that map may easily fail to remain monic, may easily fail to remain epic, may remain one but not the other -- there's no telling. And if a = b instead, and f and the identity on a are the only *two* maps there are, then clearly f *may* be idempotent, hence neither monic nor epic; then again, f *may* be involutory, hence a true isomorphism. I think true realism requires that one pay strict attention to the definitions, refraining from free-associations with the vibrations of the terms defined. Cheers, -- Fred Date: Mon, 3 Mar 1997 13:14:36 -0400 (AST) Subject: Re: Intuitionism's (read "Philosophy's") Limits Date: Mon, 3 Mar 1997 10:45:55 -0500 (EST) From: Peter Freyd William James continues to write: Might I, then, go on to say that the philosophies of constructive mathematics and category theory really are different? Constructive mathematics is a philosophy. Category theory is not. The question doesn't even type-check. Of course they're different. Date: Tue, 4 Mar 1997 22:41:52 -0400 (AST) Subject: Re: Intuitionism's (read "Philosophy's") Limits Date: Tue, 4 Mar 1997 18:19:51 +1030 (CST) From: William James > William James continues to write: > > Might I, then, go on to say that the philosophies of constructive > mathematics and category theory really are different? > > Constructive mathematics is a philosophy. Category theory is not. > The question doesn't even type-check. > > Of course they're different. Does category theory, being mathematics, have no associated philosophy? (I grant you the original question would have been more recognisable given better use of language: "...philosophies of constructive mathematics and *of* category theory...") William James Date: Wed, 5 Mar 1997 11:14:25 -0400 (AST) Subject: Re: Intuitionism's (read "Philosophy's") Limits Date: Wed, 5 Mar 1997 10:54:19 +0000 From: Steve Vickers > Constructive mathematics is a philosophy. Category theory is not. > The question doesn't even type-check. > > Of course they're different. Philosophy is the love of wisdom; type-checking is not. Of course category theory has its philosophy. To me it's "all things are connected" - you cannot fully describe anything purely in itself but only by the way it connects with others. Category theory makes the connections explicit (as morphisms) and then characterizes things by their universal properties. The philosophy plays a real role in categorical practice: for instance, in the idea that isomorphism between objects is more important than equality, which is not something that can be meaningful just in terms of the formal mathematics. The philosophy also yields a criterion for evaluating the theory: Is categorical structure adequate for describing the connections that we actually find? The strength of the categorical view of "connection structure" is amply confirmed by the power of the universal properties it can express (compare it with, say, graph theory); but if it does fail us anywhere, how might it advance beyond its present formalization? (There is already a plausible answer here: topology has a different way of describing the connections between a point and its neighbours, and the categorical and topological approaches combine to make topos theory.) I hesitate to try to reduce the philosophy of constructive mathematics to a single pithy phrase, not least because there are different schools of constructivism with apparently different philosopies. I shall therefore duck the question of comparing "the philosopies of constructive mathematics and category theory", but I don't believe it's a meaningless one. Steve Vickers. Date: Wed, 5 Mar 1997 11:15:07 -0400 (AST) Subject: Re: Intuitionism's (read "Philosophy's") Limits Date: Wed, 5 Mar 1997 12:46:26 +0000 (GMT) From: Dusko Pavlovic According to William James : > > Does category theory, being mathematics, have no associated philosophy? I'm afraid, William, that this presumed association of mathematics and philosophy is actually a bit of a sad romance: while some philosophies do like to be associated with mathematics, mathematics (it doesn't even have a proper plural) mathematics, most of the time, can't care less. While philosophy spends a lot of time defining itself and its relationship with the world, mathematics tends to be a kind of work some people like to do, taking up the world whichever way it comes to them: as a model of a process, as a game of signs or pictures, as a funny language shared between them and theri colleagues... Most mathematicians just smirk not only on philosophy, but even on category theory, or anything else deeply concerned with its own identity. They just like to solve their problems, and sometimes solve other people's problems, thereby gaining everyone's respect and admiration. At least, that's the way I have seen it. Perhaps it helps with your questions a bit. -- Dusko Pavlovic Date: Wed, 5 Mar 1997 11:13:38 -0400 (AST) Subject: Re: Intuitionism's (read "Philosophy's") Limits Date: Wed, 05 Mar 1997 00:56 -0500 (EST) From: Fred E J Linton <0004142427@mcimail.com> Any philosophy category theory may have would have at its core, I think, the notion that mathematical objects are known *not* in isolation but in the context of their comrades. The group of rational integers, accompanied *only* by its identity map, and the Thom space of the tangent bundle of some exotic manifold, accompanied once again *only* by its identity map, are, as categories, indistinguishable. Plucked out of their original contexts, there is no longer any social setting where one can find any difference between them that really *makes* a difference. According to some other views of mathematics, the group of rational integers, that particular Thom space, the real number {pi}, and my current left shoe, all have unique mathematical personalities that let them be "obviously" distinguished one from another, without any reference even to what I would call their "natural ambient environments". >From my perspective, admittedly that of a categorist, these views result from a simple failure to recognize that what passes for the "intrinsic structure" of a mathematical object is in fact nothing more (nor less) than a clear understanding of its relations with its mates, of roughly similar character, in some category (that "went without saying") they all jointly inhabit -- even the phrase "roughly similar character" is justifiable *only* by virtue of the fact that they *do* all inhabit some same category. I hope I'm actually making myself clear, and not just preaching to the converted. -- Fred Date: Wed, 5 Mar 1997 17:24:29 -0400 (AST) Subject: From moderator The philosophy of the categories list is to allow wide latitude for discussion, but the current discussion of philosophy is getting rather far afield. Therefore further contributions are discouraged, and will not be accepted beyond 5pm GMT on Friday, March 7. Best wishes Bob Rosebrugh, Moderator categories Date: Wed, 5 Mar 1997 17:21:16 -0400 (AST) Subject: Re: Intuitionism's (read "Philosophy's") Limits / flame on Date: Wed, 05 Mar 1997 09:55:51 -0800 From: John C. Mitchell >Philosophy is the love of wisdom; type-checking is not. Well that's a bizarre statement. Not that I want to get into this discussion or anything, but can't a love of wisdom be consistent with type checking? John Mitchell Date: Wed, 5 Mar 1997 17:20:03 -0400 (AST) Subject: Re: Intuitionism's (read "Philosophy's") Limits Date: Wed, 5 Mar 1997 11:27:42 -0500 (EST) From: James Stasheff this seems to ignore the distinction between neighbors (aka comrades) and parts (elements) The group of rational integers, with its non-identity automrophisms can, i thought, be distinguished from the Thom space of the > tangent bundle of some exotic manifold with its non-identity automrophisms without comparison to other sets of `numbers' or less exotic manifolds .oooO Jim Stasheff jds@math.unc.edu (UNC) Math-UNC (919)-962-9607 \ ( Chapel Hill NC FAX:(919)-962-2568 \*) 27599-3250 http://www.math.unc.edu/Faculty/jds May 15 - August 15: 146 Woodland Dr Lansdale PA 19446 (215)822-6707 On Wed, 5 Mar 1997, categories wrote: > Date: Wed, 05 Mar 1997 00:56 -0500 (EST) > From: Fred E J Linton <0004142427@mcimail.com> > > Any philosophy category theory may have would have at its core, I think, > the notion that mathematical objects are known *not* in isolation but > in the context of their comrades. The group of rational integers, > accompanied *only* by its identity map, and the Thom space of the > tangent bundle of some exotic manifold, accompanied once again *only* > by its identity map, are, as categories, indistinguishable. > > Plucked out of their original contexts, there is no longer any social setting > where one can find any difference between them that really *makes* a > difference. > > According to some other views of mathematics, the group of rational integers, > that particular Thom space, the real number {pi}, and my current left shoe, > all have unique mathematical personalities that let them be "obviously" > distinguished one from another, without any reference even to what I would > call their "natural ambient environments". > > >From my perspective, admittedly that of a categorist, these views result > from a simple failure to recognize that what passes for the "intrinsic > structure" of a mathematical object is in fact nothing more (nor less) > than a clear understanding of its relations with its mates, of roughly > similar character, in some category (that "went without saying") they all > jointly inhabit -- even the phrase "roughly similar character" is justifiable > *only* by virtue of the fact that they *do* all inhabit some same category. > > I hope I'm actually making myself clear, and not just preaching to the converted. > > -- Fred > > > Date: Thu, 6 Mar 1997 13:29:55 -0400 (AST) Subject: Re: Intuitionism's Limits Date: Wed, 05 Mar 1997 22:36:19 -0800 From: Vaughan Pratt Date: Tue, 4 Mar 1997 18:19:51 +1030 (CST) From: William James (I grant you the original question would have been more recognisable given better use of language: "...philosophies of constructive mathematics and *of* category theory...") Your original question was "Which view should dominate?", where "the category theoretic view" was one of your options. (You had several questions but this one seemed the most central.) If you are asking whether the primary expression of structure should be in terms of relations between elements or transformations of objects, then I would answer this as follows. The analogous question for physics is whether energy and matter consist of particles or waves. The consensus in physics today is that both energy and matter can be viewed more or less equally accurately, if not equally insightfully, as either particles or waves. Which offers more insight depends on the circumstances. The corresponding position for mathematics would be that structure can be expressed more or less equally well in elementary or transformational terms, and that which approach gives more insight depends on the circumstances. The extent to which this is not the consensus in mathematics today is less a reflection on either approach than on the conceptual health of mathematics relative to that of physics. Vaughan Pratt Date: Fri, 7 Mar 1997 15:43:37 -0400 (AST) Subject: non philosophy Date: Fri, 7 Mar 1997 14:24:15 -0500 (EST) From: Peter Freyd William James answered his own question when he asked: Does category theory, being mathematics, have no associated philosophy? Yes, category theory is mathematics. Therefore its associated philosophy is whatever philosophy one chooses to associate with mathematics. As for the latter, Dusko provided a pretty good answer. I would modify it only to reflect that mathematics (which, as always in the absence of a qualifier, means pure mathematics) is a subject matter. Most mathematicians are sufficiently confident about their subject matter that they feel no need for a semantics, much less a stated philosophy. (Yes, it has been notoriously difficult to define intensionally, but that's not special to mathematics. What's the subject matter of physics? If either mathematicians or physicists were -- using Dusko's language -- to spend a lot of time defining their subject, there never would have been much mathematics or physics.) Steve thinks that the existence of a philosophy of category theory is an "of course". In one of the public meanings of the word "philosophy" he's certainly correct but not, I think, in the sense that would include something like constructivism. (The public meaning in question has even less than type-checking to do with either love or wisdom. Well, maybe it has something to do with love.) May I suggest that the applied mathematician may have a very different understanding of category theory from the mathematician. Steve says that category theory is "all things are connected". But that's an article of faith for almost any mathematician. He goes on to say, "you cannot fully describe anything purely in itself but only by the way it connects with others." This assertion about what "you cannot" do sounds like it could be a good way of describing *applied* mathematicians. To begin with, Eilenberg and Mac Lane defined categories in order to define functors and they defined functors in order to define natural transformations. Immediately it was noted that a new tool existed to pin down -- in a formal way -- how it is that some of the all things are connected. It should be noted that categories -- and more to the point, functors -- have always been considered tools for studying the subject matter of mathematics. Tools, not the subject matter itself. I am on record that the language of categories began to become respectable when Frank Adams was able to count the number of independent vector fields on each sphere using a construction that quantified over functors: it produced an n-ary transformation on the K-functor for every n-ary endofunctor on the category of finite dimensional vector spaces (which he assembled into what are now known as the Adams operations). One of the better successes since then has been the use of categories in finding connections between various foundational systems. Because some of these systems are constructivist it has apparently caused some to think that categories are intrinsically constructivist. Strange. There's another important aspect of category theory. Most categories, in the beginning at least, were categories that naturally arose from existing branches of mathematics. Some of these categories, though, had never been lived in before they were invented as categories. Joel Cohen named one of these the "Freyd Category" (named not after Peter but Jennifer): its an abelian category whose full subcategory of projectives is the stable-homotopy category; all the other objects have no easy description; the category can be described as the target of the universal homology theory. But a much better example is in Serre's dissertation. This work, hailed by many as the single most substantive dissertation ever written, contends with the two abelian categories that result when one starts with the category of abelian groups and identifies with the zero group all finite groups in one case, or all finitely generated groups in the second case. Most remarkably, Serre did all this without using category theory. (The fact that the first non-trivial construction of a category occured without benefit of category theory must be reckoned an embarrassment for category theorists.) But in recent applications I think a very different type of question is being asked: "Is it possible that there is a category in which ... can take place?" These questions are at the heart of many approaches to programming semantics. And they are at the heart of many of the uses of categories in theoretical physics. But the first serious example came, in fact, a long time ago. In the late 60's Lawvere's approach to differential geometry asked just this type of question. Elementary topoi made their first appearance as just a preliminary part of the answer. So what's this all have to do with philosophy? Not much, of course.