March 2006 - Categories - categories.org.au

Foundations?
by F W Lawvere 03 Sep '10

03 Sep '10

Down with "Foundations"! Up with algebra! Dear Friends, Presumably I am among those who are being "vilified" as "lovers of categories as foundation". By avoiding any precise definition, such a formulation might appeal to the widespread justified boredom induced by the past hundred years of "foundations as justification". Whenever I used the word "foundation" in my writings over the past forty years, I have explicitly rejected that reactionary use of the term and instead used the definition implicit in the work of my teachers Truesdell and Eilenberg. Namely, an important component of mathematical practice is the careful study of historical and contemporary analysis, geometry, etc. to extract the essential recurring concepts and constructions; making those concepts and constructions (such as homomorphism, functional, adjoint functor, etc.) explicit provides powerful guidance for further unified development of all mathematical subjects, old and new. What is the primary tool for such summing up of the essence of ongoing mathematics? Algebra! Nodal points in the progress of this kind of research occur when, as is the case with the finite number of axioms for the metacategory of categories, all that we know so far can be expressed in a single sort of algebra. I am proud to have participated with Eilenberg, Mac Lane, Freyd, and many others, in bringing about the contemporary awareness of Algebra as Category Theory Had it not been for the century of excessive attention given to the alleged possibility that mathematics is inconsistent, with the accompanying degradation of the F-word, we would still be using it in the sense known to the general public: the search for what is "basic". We, who supposedly know the explicit algebra of homomorphisms, functionals, etc. are long remiss in our duty to find ways to utilize those concepts also in guiding high school calculus. Best wishes, Bill Bibliography: - The Category of Categories as a Foundation for Mathematics, La Jolla conference 1965, Springer (1966) - Adjointness in Foundations, Dialectica, (1969), to be reprinted in TAC - Foundations and Applications: Axiomatization and Education, Bulletin of Symbolic Logic, (2003) vol 9, pp 213-224 - Sets for Mathematics, w/ Bob Rosebrugh, Cambridge Univ. Press, (2003) ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************

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cracks and pots 93
by Eduardo Dubuc 05 Apr '06

05 Apr '06

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.

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Re: cracks and pots
by Robert J. MacG. Dawson 04 Apr '06

04 Apr '06

Marta Bunge wrote; This [inviting researchers in fashionable applied areas to speak at category theory meetings] may lead to narrow > developments of any subject that they approach with this objective in > mind, and that is dangerous for the future of category theory (of > mathematics, in general). That is my main concern. My posting tried to > call attention to what I think is a sad state of affairs in category > theory, when it need not be. It is not clear to me that the majority of theoretical physicists agree with the negative view of categorical string theory held by the cited blog writers; and in the absence of a consensus among the physicists, I for one (with an undergradate degree and some graduate courses in physics) do not feel qualified to take sides; if anything, errors should be on the side of trying out too many ideas, not too few. I have this image of differential geometers saying to each other, a century ago, "Don't you think somebody ought to tell that Einstein to stop trying to use differential geometry to explain gravity, before our whole field gets a bad name?" Of course, the pioneering knot theorists probably thought that Lord Kelvin ought to stop trying to explain atomic nuclei as knotted loops of ether, too. But I think Einstein did differential geometry more good than Kelvin did harm to knot theory. A mathematical technique powerful enough to show that a physical theory does *not* work has shown its own value. What has sometimes gone on, at least for a while, is that very abstract physical theories have continued to be studied after it had become obvious that their predictions were wildly at variance with observation, or that they would never make any predictions. Even then I don't think the reputation of the mathematical theory being abused suffers, though that of the neighboring theoretical physicists may. I don't think this is the case with string theory yet, though I could be wrong. Cheers, Robert Dawson

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Re: fundamental theorem of algebra
by John Baez 04 Apr '06

04 Apr '06

Dear Vaughan - You write: > As a case in point, just now I checked a relatively recent Brittanica > article on algebra (1987 ed.), which states flatly (p.260a) that "No > elementary algebraic proof of [the FTAlg] exists, and the result is not > proved here." (Not even "is known" but "exists"; an expository article > should not assume that the reader knows the jargon meaning of this term > as "exists in the literature".) The authors taking responsibility for > this claim were Garrett Birkhoff, Marshall Hall, Pierre Samuel, Peter > Hilton, and Paul Cohn. They go into detail to show that z^n = a has n > roots, starting with the geometry of addition and multiplication in the > Argand diagram, so it's not as if their exposition was at too elementary > a level to talk in terms of mapping circles, or that "algebraic" ruled > out simple geometric arguments. > > I submit their nonexistence claim as prima facie evidence for my claim > that the very few who knew this argument weren't even letting the likes > of Birkhoff, Hall, etc. in on it, let alone "the rest of us." I really doubt those authors were unaware of the topological proof of the fundamental theorem of calculus in 1987. After all, it's exercise H.5 in chapter 1 of Spanier's "Algebraic Topology", copyright 1966. This book used to be the canonical textbook on algebraic topology, and Peter Hilton is a darn good algebraic topologist. I think I learned this topological proof sometime in grad school, around 1986. So, I don't think it was any sort of secret by then. I don't know what counts as an "elementary algebraic proof", but people often say that there is no "purely algebraic proof" of the fundamental theorem of calculus. After all, this theorem is about the complex numbers, which are often defined in terms of the real numbers, which are often defined as a topological completion of the rational numbers. I hope this is what the Encyclopedia article was trying to say. There are some so-called "algebraic proofs": http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra that use a bare minimum of topology. These proofs tend to have a purely algebraic core, namely "if odd-degree polynomials and the polynomial x^2 + 1 have roots in some field, this field is algebraically closed". But, they use the intermediate value theorem for continuous functions f: [0,1] -> R to show that C meets these conditions. So, I wouldn't call them "purely algebraic". It's sort of ironic that the so-called "fundamental theorem of algebra" doesn't have a purely algebraic proof. Gauss is famous for having given a proof of the fundamental theorem of algebra in his dissertation back in 1799. On the St. Andrews math history website they write: Gauss's proof of 1799 is topological in nature and has some rather serious gaps. It does not meet our present day standards required for a rigorous proof. They don't say how the proof went. So, I decided to find out! I was hoping I could irritate you by showing that it was just the topological proof you claim is so new. There's a discussion of it here: Hans Willi Siegberg Some Historical Remarks Concerning Degree Theory, American Mathematical Monthly, 88 (1981), 125-139. (Available on JSTOR, or via Google Scholar.) As the title hints, Gauss' proof uses ideas closely related to the winding number. Unfortunately, it's slightly different than the proof you like. The idea is to take a polynomial of degree n, say P: C -> C break it into real and imaginary parts P = U + iV, see where they vanish: S = {z: U(z) = 0} T = {z: V(z) = 0} and show that the intersection of S and T is nonempty. Gauss argues that far from the origin, S and T are smooth curves. Because the leading term of the polynomial dominates the rest, each of these curves intersects any sufficiently large circle transversely at n points. If we go around the circle these intersection points alternate: first a point in S, then one in T, then one in S, and so on. Moreover, the curves I'm talking about can't just disappear as we follow them into the disk, since they separate the region where U (resp. V) is positive from the region where it's negative. They may become singular, or intersect, but they can't just end! "So", S and T must intersect somewhere. This is true, but it takes more topology to prove it rigorously than was available to Gauss. Gauss knew his proof wasn't completely rigorous, so he invented some other arguments. The "winding number" idea you like is lurking in Gauss' third proof, which he wrote up in 1816 - but he only gave this winding number proof explicitly in 1840. According to Siegberg, Indeed, in a lecture "Theorie der imaginaeren Groessen (1840), Gauss mentioned [see Fraenkel, 1922] that his third proof of the fundamental theorem of the algebra originated from his first one, and he gave the function-theoretic argument that the winding number W(P|S, 0) equals n [the degree of the polynomial], whereas the winding number of any map F: (B,S) -> (R^2, R^2 - {0}) vanishes if there is no zero of F in B [see Fraenkel, 1922]. However, this argument cannot be found explicitly in [Gauss, 1816]. So, I guess that except perhaps for Gauss, nobody knew the proof you're talking about until 1840. Best, jb

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WHY ...CONCERNED? III
by F W Lawvere 02 Apr '06

02 Apr '06

WHY ARE WE CONCERNED? III The second main misconception about category theory Part of the perception that category theory is "foundations" (in the pejorative sense of being remote from applications and development) is due to a preoccupation with huge size. Since such perceptions hold back the learning of category theory, and hence facilitate its misuse as a mystifying shield, they are among our concerns. We need to deal with the size preoccupation head on. Experience has shown that we cannot build up or construct mathematical concepts from nothing. On the contrary, centuries of experience become concentrated in concepts such as "there must be a group of all rotations" and we then place ourselves conceptually within that creation; we state succinctly the properties which that creation as a structure seems to have, and then develop rigorously the consequences of those properties taken as axioms. The notion of category arose in that way, and in turn serves as a powerful instrument for guiding further such developments. Placing ourselves conceptually within the metacategory of categories, we routinely make use of the leap which idealizes the category of all finite sets as an object. The question is, what more? Of course we make use of the experience of those who have labored to justify mathematics, and it is fortunate that ultimately our results are compatible with theirs. (Mac Lane's use of the term metacategory is not mysterious; it simply refers to the universe of discourse of any model, in the special context where the elements of such a model are themselves called categories and functors. In the spirit of algebra, we do not concentrate on the cumulative hierarchy which might have been used to present the metacategory, but rather on the mathematical category itself.) The supposed size problems of category theory are often concentrated in functor category formation. For any two categories that are objects of the metacategory, the category of functors from one to the other exists in the sense that it also is an object in the metacategory (it is unique by exponential adjointness). That existence statement is compatible with standard set theory, although it is often presumed to be incompatible. In the original 1945 exposition of category theory, it was the Goedel-Bernays account of the cumulative hierarchy (see posting II) that was cited as probably relevant (in case the problem of justifying category theory should come up). As a result, category theorists have been worried about supposed "illegitimacies" that might arise from violating the Goedel-Bernays rules (which in essence stemmed from von Neumann). These rules expressed an expediency which was a very effective trick at the time, identifying two kinds of membership relation and truncating the content at a plausible level. The Goedel-Bernays theory is well known to have the same logical strength as the Zermelo-Fraenkel system. An important advantage is that the greater expressive power of Goedel-Bernays permits it to be finitely axiomatizable, whereas Zermelo-Fraenkel is not; the greater expressive power concerns an element V of any model in which all small sets of the model can be embedded (just as another smaller element captures all finite sets). But the greater expressive power still allows mutual relative consistency: To every model of Goedel-Bernays, a model of Zermelo-Fraenkel can be constructed in a fairly straightforward manner: just take the small elements; in the converse direction there are two procedures (left and right adjoint?): given a model of Zermelo-Fraenkel, one can take all definable subsets of it, or just all subsets, and in either case a model of Goedel-Bernays apparently results. Because these mutual interpretations are hypothetical, relatively weak assumptions are required on the background category of sets taken as the recipient of models. In fact, with only slightly stronger assumptions on the background category one can construct, for any model of Zermelo-Fraenkel, a model of what set theorists use daily as BG+, which contains as elements not only V but W = V^V, V^W etc. Our practice is consistent with the minimal assumptions of professional set theorists: For any model of BG+ the presented metacategory of categories is both cartesian closed (in the usual elementary sense) and also has an object S of small sets. (Those facts strongly augment well-known properties, such as the existence of the first four finite ordinals and their adequacy in the metacategory relative to the sub-metacategory of discrete categories; of course these same ordinals also co-represent one of the "2-category" structures on the metacategory). The category S is itself cartesian closed, and the categories of structures of geometry and analysis are enriched in it. Of course functor categories may no longer enjoy the same enrichment, just as functor categories starting from finite sets may not have finite hom-sets; but that is no reason to avoid considering them, and functionals on them, etc. when such considerations serve mathematics. It is of special interest to note that the restrictive "law" (under which categorists have been chafing) was already repealed forty years ago by Goedel and Bernays themselves. In their correspondence of 1963, it appears that they had been informed that a student of Eilenberg was working on a project to base set theory and mathematics on category theory; their immediate response was that mathematics will have to consider finite types over the class of small sets. (The relative consistency was presumably obvious to them.) Even though most set-theorists have themselves maintained clarity on the distinction, the identification of two kinds of membership in a formalized theory may have fostered in the minds of others a confusion between smallness (of a class or set) and existence as an element of the (meta)universe. Certainly, the specific meaning of smallness needs to be clarified (although for some purposes it can be taken as a parameter). There is a way of specifying smallness that is directly related to fundamental space/quantity dualities (rather than to imagined "building up" by stronger and stronger closure properties). Just as Dedekind finite sets X are characterized by the condition that a natural map X --->Hom(Q^X, Q) is an isomorphism, so indications from the study of rings of continuous functions and other branches of analysis strongly suggest that all small sets X should satisfy the same sort of isomorphism, with the truth-value space Q being replaced by the real line (in both cases, Hom refers to the binary algebraic operations on the object Q). There is the possibility to assume that conversely all sets X satisfying that isomorphism are small i.e. that, like the Dedekind-finite sets, they belong to a single uniquely-determined category S. That possibility in itself would imply no commitment concerning the existence or non-existence of super-huge objects in the metacategory "beyond" S, S^S, etc. Such an axiom would be somewhat stronger than ZF, but much weaker than the standard discussions of contemporary set theorists. ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************

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WHY ARE WE CONCERNED? I
by F W Lawvere 01 Apr '06

01 Apr '06

WHY ARE WE CONCERNED? I When Saunders Mac Lane penned his hard-hitting 1997 Synthese article, he was defending mathematics from an attack many of us hoped would just go away. But Saunders was aware of the seriousness of the threat, which indeed is still here with greater determination. Although the title of that article was "Despite physicists, proof is essential in mathematics", he was not opposing physics, nor even that immediate handful who, assuming the mantle of "mathematical physicists", gave themselves license to insult generations of scrupulously serious physicists and to demand that mathematics adopt a culture that considers conjecture as nearly-established truth. In essence it was an attack on science itself, as the highest form of knowing, that Saunders was opposing. The increased determination of that attack is expressed in two ways. To equip and organize the attack, finance capital has set up several institutions, some of which rather openly proclaim their goal of submitting science to the service of medieval obscurantism. Others say that they support mathematical research, but encourage a barrage of "popular" writings to shock and awe the public into continuing in the belief that they will never understand mathematics and hence never be able to actively participate in science. The contempt for Mac Lane's fight, recently expressed in articles supposedly memorializing him, takes the form of the claim that category theory itself is a "cool" instrument for deepening obscurantism. Not only Harvard's "When is one thing equal to another thing?" and the Cambridge "morality" muddle, but also a 2003 article aimed at teachers of undergraduates, quite explicitly support that claim. In the MAA Monthly, a Clay Fellow states as fact that category theory "is mathematics with the substance removed". Mastering the technique of disinformation whereby the readers are first told that now finally they will be informed, the article suggests that some raising of the level of understanding of the relationship between space and intensively variable quantity is going to be achieved. Then the author short-circuits any such understanding via the simplifying assumption that omits the distinction between covariant and contravariant functors as "unwieldy". As final display of the mastery of expositional technique, the categorical object which has, for nearly twenty pages, been heralded as simple, is revealed in the final pages in the most complicated and unexplained form possible. (Totally passed over is the issue that had led Grothendieck to the considerations allegedly being treated: not only the category of affine schemes, but also the category of all its presheaves, where the author implicitly wants us to work, fails to have the geometrically correct colimits needed to define projective space.) Another level of attack was launched when Cornell University was given very large sums of money to develop methods of teaching geometry without mentioning any geometrical concepts. No proof of the desirability of such a draconian excising of content needed to be given, beyond some phrases from the Dalai Lama. "Dumbing down" is an attack not only on school children and on undergraduates, but also one taking measured aim at colleagues in adjacent fields and at the general public. The general public is thirsty for genuinely informational articles to replace the science fiction gruel served constantly by journals like the Scientific American and the New York Times "Science" section. Those journals have never published anything resembling a mathematical proof and hence have rarely actually explained any scientific subject in a usable way. Nor have they even undertaken any program to raise the level of knowledge of calculus or linear algebra among their readers in a way which would make such explanations feasible. Instead, they provide games and amusements to divert the mathematically-interested public. In January of 2005 the Notices of the AMS announced that they had for a full ten years been strictly following a certain editorial policy. There had been a widespread demand for expository articles. To that demand, the response was a new definition of "expository": all precise definitions of mathematical concepts must be eliminated. Authors of expository articles were forced to compromise their presentation, or to withdraw their paper. Mathematicians, who were for several years becoming aware that these new expository articles are absolutely useless for developing a mathematical thought, were shocked to learn that a conscious policy had forced that situation. A peculiar sort of anti-authoritarianism seems to be the only justification offered for degrading the role of definition, theorem, and proof; certainly, serious expositors have never considered that the use of those three pillars of geometrical enlightenment excludes explanations and examples. Others have urged, however, that those instruments be eliminated even from lectures at meetings and from professional papers. That threat is part of the background for the concern expressed in the many messages to the categories list over the past weeks. Deeply concerned mathematicians ask me "How can we know?". Indeed, how can we know whether it is worthwhile to attend a certain meeting or a certain talk, and how can a scientific committee know whether a proposed talk is scientifically viable? If the "you don't want to know" culture of no proofs, no definitions, is accepted, we will truly have no way of knowing, and will be pressured to fall back on unsupported faith. ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************

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Re: WHY ARE WE CONCERNED?
by Peter McBURNEY 31 Mar '06

31 Mar '06

Peter Selinger wrote: > > "French lawmakers, for example, gave preliminary support this month to > a measure that would require the company to open the iPod to play > music purchased from any online music service; currently, songs > purchased from iTunes can be played only on iPods." > > New York Times, 2006/03/29, "Apple vs. Apple in Dispute Over Trademark" > > This is of course not a logical contradiction; but I would be very > surprised if it is what the writer really meant to say. Sadly, most > readers probably won't know the difference one way or the other. > I am not sure that mathematicians should cast stones here, given the decades it took for the ambiguities of First-Order Logic to be recognized, and then rectified with Independence-Friendly Logic. -- Peter McBurney University of Liverpool, UK.

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Fundamental theorem
by Michael Barr 31 Mar '06

31 Mar '06

The proof that Vaughan outlined is found in almost exactly the same form in Courant & Robbins, published around 60 years ago. Here is a proof I learned in grad school, based on three facts, one analytic and two algebraic. The analytic fact, which is an irreducible minimum, given that the reals cannot be defined algebraically (except as a real closed field of continuum transcendence degree, which misses the point) is the intermediate value theorem. In other words, order completeness. From this follows the fact that every odd order polynomial has a (real) root and that every real number--and with a bit of manipulation, every complex number--has a complex square root. The algebraic facts are the existence of a splitting field and the theorem on elementary symmetric functions, neither of which is quite trivial, nor very deep. The way you do it is by proving that every real polynomial of degree n = 2^k*m with m odd has a complex root, by induction on k. The case k = 0 is quite trivial, of course. So suppose that f is a real polynomial of even degree n and, in some splitting field has roots r_1,...,r_n. For each integer s form the polynomial f_s = \prod{i<j}(x - r_i - r_j - sr_ir_j) which has degree n(n-1)/2, which is less 2 divisible than n. The theorem on symmetric functions implies it is real and hence for some i and j dependent on s, r_i + r_j + sr_ir_j is in C. Since there are only finitely many pairs i and j and infinitely many integers, there are distinct s and t for which both r_i + r_j + sr_ir_j and r_i + r_j + tr_ir_j belong to C. Given that C has square roots, one easily discovers that r_i and r_j are both complex numbers. To go from real to complex polynomials, just multiply a complex polynomial by its conjugate; for a root r of the product, one of r and its conjugate is a root of the original. And the process of factoring completely is well-known. Incidentally, it is the case that if K is an algebraic extension of L and every polynomial with coefficients in L has at least one root in K, then K is an algebraic closure of L. The argument above pretty much does the characteristic 0; the prime case is trickier. This is not entirely elementary, but then neither is the winding number argument. Comparing them is difficult because the algebraic argument I have given could all be taught in a lower division course, while the winding number argument, intuitively appealing, really requires some sophisticated stuff to deal with rigorously. Which raises an interesting question. If we agree, as we seem to, that proof is the essence of what we mean by mathematics, then what does it mean to give an intuitively appealing non-proof and call it a proof? Michael

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May and Beck
by jim stasheff 31 Mar '06

31 Mar '06

I asked Peter May where he had commented on the relevance of Jon's work to his own: -------- Original Message -------- Subject: We'll talk (This refers to being together at the Mac Lane memorial fest - where others of you will be also) Date: Thu, 30 Mar 2006 08:11:07 -0600 From: Peter May <may(a)math.uchicago.edu> To: jds(a)math.upenn.edu No time to write, though. The paper was ``On H-spaces and infinite loop spaces''. The work of his I most liked concerned distributivity laws and monads, which I only learned about after rediscovering it for myself while doing multiplicative infinite loop space theory. See you next week, Peter

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Reminder: CMS Summer Meeting in Calgary 2006
by Robin Cockett 30 Mar '06

30 Mar '06

A gentle reminder: CMS 2006 meeting and other (categorical) events in Calgary : The summer Canadian Math Society meeting is in Calgary this year (June 3rd - June 5th) Steve Awodey is a plenary speaker for the meeting and there is a Category Theory session. (Note: Abstracts for speakers are due April 10th) IN ADDITION there are various other events of categorical interest occurring in Calgary round those dates: (i) June 2nd: Categories and Semigroup Workshop (University of Calgary) (ii) June 7th-9th: Foundational methods in Computer Science (Kananaskis Field Station, Alberta) LOCAL ARRANGEMENTS: If you wish to participate in any of these events you should let either myself (robin(a)cpsc.ucalgary.ca) or Pieter Hofstra (hofstrap(a)cpsc.ucalgary.ca) know so that we can keep you informed of local organizational details. WEBSITES: Please check the CMS site for registration details (http://www.cms.math.ca/Events/summer06) For the remaining events please check (http://pages.cpsc.ucalgary.ca/~robin/FMCS/FMCS_06/CatMeetings.html) Robin Cockett

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