Old post: Why are we concerned? F.W. Lawvere
This is an old post of Friday, March 31, 2006 taken from the Categories mailing list.
F.W. Lawvere: WHY ARE WE CONCERNED? I
Categories Mailing List
Sun, 26 Mar 2006 16:43:31 -0500 (EST)
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
************************************************************
F. William Lawvere: WHY ARE WE CONCERNED? II
Categories Mailing List
Tue, 28 Mar 2006 21:30:59 -0500 (EST)
WHY ARE WE CONCERNED? II
Misconceptions
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.
Some of the most important applications of our unifying efforts as categorists have been to the
teaching of algebraic topology
teaching of algebraic geometry
teaching of logic and set theory
teaching of differential geometry
These subjects all arose from the efforts to clarify and apply calculus; thus some of us have applied category theory to the teaching of calculus.
But it seems that we have not taught category theory itself well enough. Several recent writings reveal that basic misunderstandings about category theory are still prevalent, even among people who use it. Some of these concern the myth that category theory is the "insubstantial part" of mathematics and that it heralds an era when precise axioms are no longer needed. (Other myths revolve around the false belief that there are "size problems" if one tries to do category theory in a way harmonious with the standard practice of professional set theorists; see next posting.) The first of these misunderstandings is connected with taking seriously the
jest "sets without elements". The traditions of algebraic geometry and of category theory are completely compatible about elements, as I now show.
Contrary to Fregean rigidity, in mathematics we never use "properties" that are defined on the universe of "everything". There is the "universe of discourse" principle which is very important: for example, any given group, (or any given topological space, etc.) acts as a universe of discourse. As these examples suggest, a universe of discourse typically carries a structure which permits interesting properties and constructions on it. As the examples also show, there are typically many objects of a given mathematical category and also many categories, so transformation is an essential part of the content. As quantity includes zero, so structure includes the case of no structure, which Cantor considered one of his most profound and exciting discoveries. (His conjecture that the continuum hypothesis holds in that realm is probably true. [Bulletin of Symbolic Logic 9 (2003) 213-223].) Dedekind, Hausdorff, and most of 20th century mathematics followed the paradigm whereby structures have two aspects, a theory and an interpretation of it in such a featureless background. Because the background thus contributes minimal
distortion to the assumptions of the theory, the completeness theorems of first-order logic, the Nullstellensatz, and related results are available. The more geometric background categories which receive models are also viewed as structures (of an opposite kind) in abstract sets, for example the classifying topos for local rings as a background for algebraic groups. Such is "set theory" in the practice of mathematics; it is part of the essence from which organization emerges.
By contrast, the "set theory" studied by 20th century set theorists has a different aim and architecture. The aim is "justification" of mathematics, and the architecture is that of the cumulative hierarchy. The alleged need for justification arose in connection with the re-naming of Cantor's theorem as "Russell's paradox"; Cantor's theorem had shown that the system proposed by Frege was inconsistent, but there were those who dreamed nonetheless of restoring that rigidity. There was a bitter controversy between Cantor and Frege, and Zermelo swore allegiance to
Frege [Cantor G.: Abhandlungen mathematischen und philosophischen Inhalts, 1966, page 441, remarks of Zermelo on Cantor's 1884 review of Frege]. Von Neumann based himself on Zermelo and made explicit the cumulative hierarchy, which Bernays and Goedel used and which many subsequent set theorists presumed was the only architecture to be studied. The justificational aspect stems from the supposed construction of the hierarchy by a bizarre parody of ordinary iteration, parameterized by infinite ordinal numbers (Cantor's third discovery), entities which from the point of view of ordinary mathematics are even more in need of
justification than the analysis that supposedly needed it. (Indeed, in attempting to describe what these alleged infinite ordinals are and do, people often resort to stories about gods and demons.) Little or no progress has been made on this "justification" problem in a century, but work with the hierarchy has produced some knowledge about the possibilities for categories of sets. By adopting a standard definition of map and discarding the mock iteration (with its concomitant complicated structure), each model of the cumulative hierarchy yields a category of
abstract nearly featureless sets; most of the usual set-theoretical issues depend only on the mere category: measurable cardinals, Goedel-constructibility, the continuum hypothesis, etc.
Having thus briefly understood the two visions which are called set theory
(1) a category of Cantorian featureless sets which serves as the background recipient for the structures of algebra, geometry and analysis;
(2) the cumulative hierarchy with its rigid Fregean structure aiming to justify mathematics,
it is not surprising that the precise nature of the elementhood relations appropriate to each are quite different. While the Fregean image involves rigid inclusion and elementhood relations imagined to be given once and for all for mathematics as a whole, the usual mathematical practice instead considers inclusion and membership relations for subsets of a given universe of discourse (such as R^3). Thanks to Grothendieck's Tohoku observation, these mathematical local belonging relations are well globalized within the notion of category, whose primitives are domain, codomain, identity, and composition.
[The notion of category is a simple first-order theory of a semi-algebraic kind. It has myriads of interpretations, some in "classes", some "locally small" etc., but such undefined restrictions on interpretations have nothing to do with the notion of category per se. Many properties are best expressed within the first-order theory itself.]
Composition is a kind of non-commutative multiplication, hence there are two kinds of division problems. In any category, given any two morphisms a and b we can ask whether there exists a morphism p such that a = bp; if so, we may say that a belongs to b. This forces a and b to have as codomain the same object, which serves as their common universe of discourse. (The dual relation, f determines g, defined by "there exists m with mf = g", is probably equally important in mathematics.) There are two special cases of this belonging relation which are of special interest. First we say that b is a part (or subset in the case of a category of sets) of its codomain, if for all a belonging to b, the proof p of that belonging is unique; this is immediately seen to be equivalent to the usual notion of monomorphism. Then, if a and b are parts of the same object, we say a included in b iff a belongs to b. Any arbitrary morphism x with codomain X may be considered an element of X in the sense of Volterra (also known as a figure in X); we say that x is a member of b iff x belongs to b. Then clearly
a is included in b iff for all x, if x is a member of a, then x is a member of b.
The usual relationship between these two relations is thus maintained. Because in category theory the domain relation is as important as the codomain relation, we can be more precise about elements: very often it is appropriate to consider a special property of objects, and restrict the term element (or figure) to elements whose domain has that property, that is, to figures whose shape has the property. For example, in algebraic geometry the connected separable objects are appropriate domains for the figures known as "points"; in the algebraically closed case it suffices to consider elements with domain a terminal object 1 as points. On the other
hand, frequently it is of interest to choose a small class of figure shapes which generates in the sense of Grothendieck, i.e. so that the above equivalence between inclusion and universal implication of memberships holds even when the figures x are restricted to those of the prescribed shapes. A basic property of categories of Cantorian sets is that this holds with x restricted to those with terminal domain 1. In algebraic geometry, the figures whose domains have trivial cohomology are adequate. Note that if f is a morphism from A to B and if x is an element of A, then fx is an element of B of the same shape (of course in general figures are singular in that they distort their shape, for example, fx may be more singular than the figure x). Properties of x in A may be quite different from the properies of fx in B.
The mysterious distinction between x and singleton(x) in the hierarchical Frege architecture takes quite a different form in the categorical architecture where there is a natural transformation from the identity functor to the covariant power set functor; this natural transformation can be called singleton: singleton(x) is simply x considered as a special element of PX, rather than of the original X.
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. But if these basics were widely understood among algebraic geometers, perhaps misconceptions like "category theory is the insubstantial part of mathematics" would not have arisen. (As we know from experience, all of the substance of mathematics can be fully expressed in categories.) Perhaps the general term "A-points" for arbitrary rings A was confusing. "Spec(A)-shaped figures" is a more accurate rendering of Volterra's "elements"; that could be abbreviated to "A-figures", but points are in some sense special among figures. On the other hand, we often vary the background category, so that alternative terminology might involve passing from a category E to
E/spec(A), and restricting the notion of "point" in any category to mean figure of terminal shape; then the A-figures become, on pulling back to the new category, literally "moving points".
Whatever the particular chosen terminology, the important conclusion is to actively eliminate the mythology that spaces in categories have no elements, because as we see, this mythology obscures the simplicity of certain matters and thus provides a bogus basis for insulating one field of mathematics from another.
[The belonging relation is just the poset collapse of the categories E/X, whose actual maps serve as incidence relations, especially between figures in X. Thus every category E supports a certain geometrical imagery wherein all maps are geometrically continuous, in that they map figures to figures without tearing the incidence relations. Precise axioms about E are a key to further progress because they explicitly sum up and guide our experience with the objects and maps in E.]
************************************************************
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
************************************************************
F. William Lawvere: WHY ARE WE CONCERNED? III
Categories Mailing List
Date: Fri, 31 Mar 2006 09:15:28 -0500 (EST)
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
************************************************************
F.W. Lawvere: WHY ARE WE CONCERNED? I
Categories Mailing List
Sun, 26 Mar 2006 16:43:31 -0500 (EST)
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
************************************************************
F. William Lawvere: WHY ARE WE CONCERNED? II
Categories Mailing List
Tue, 28 Mar 2006 21:30:59 -0500 (EST)
WHY ARE WE CONCERNED? II
Misconceptions
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.
Some of the most important applications of our unifying efforts as categorists have been to the
teaching of algebraic topology
teaching of algebraic geometry
teaching of logic and set theory
teaching of differential geometry
These subjects all arose from the efforts to clarify and apply calculus; thus some of us have applied category theory to the teaching of calculus.
But it seems that we have not taught category theory itself well enough. Several recent writings reveal that basic misunderstandings about category theory are still prevalent, even among people who use it. Some of these concern the myth that category theory is the "insubstantial part" of mathematics and that it heralds an era when precise axioms are no longer needed. (Other myths revolve around the false belief that there are "size problems" if one tries to do category theory in a way harmonious with the standard practice of professional set theorists; see next posting.) The first of these misunderstandings is connected with taking seriously the
jest "sets without elements". The traditions of algebraic geometry and of category theory are completely compatible about elements, as I now show.
Contrary to Fregean rigidity, in mathematics we never use "properties" that are defined on the universe of "everything". There is the "universe of discourse" principle which is very important: for example, any given group, (or any given topological space, etc.) acts as a universe of discourse. As these examples suggest, a universe of discourse typically carries a structure which permits interesting properties and constructions on it. As the examples also show, there are typically many objects of a given mathematical category and also many categories, so transformation is an essential part of the content. As quantity includes zero, so structure includes the case of no structure, which Cantor considered one of his most profound and exciting discoveries. (His conjecture that the continuum hypothesis holds in that realm is probably true. [Bulletin of Symbolic Logic 9 (2003) 213-223].) Dedekind, Hausdorff, and most of 20th century mathematics followed the paradigm whereby structures have two aspects, a theory and an interpretation of it in such a featureless background. Because the background thus contributes minimal
distortion to the assumptions of the theory, the completeness theorems of first-order logic, the Nullstellensatz, and related results are available. The more geometric background categories which receive models are also viewed as structures (of an opposite kind) in abstract sets, for example the classifying topos for local rings as a background for algebraic groups. Such is "set theory" in the practice of mathematics; it is part of the essence from which organization emerges.
By contrast, the "set theory" studied by 20th century set theorists has a different aim and architecture. The aim is "justification" of mathematics, and the architecture is that of the cumulative hierarchy. The alleged need for justification arose in connection with the re-naming of Cantor's theorem as "Russell's paradox"; Cantor's theorem had shown that the system proposed by Frege was inconsistent, but there were those who dreamed nonetheless of restoring that rigidity. There was a bitter controversy between Cantor and Frege, and Zermelo swore allegiance to
Frege [Cantor G.: Abhandlungen mathematischen und philosophischen Inhalts, 1966, page 441, remarks of Zermelo on Cantor's 1884 review of Frege]. Von Neumann based himself on Zermelo and made explicit the cumulative hierarchy, which Bernays and Goedel used and which many subsequent set theorists presumed was the only architecture to be studied. The justificational aspect stems from the supposed construction of the hierarchy by a bizarre parody of ordinary iteration, parameterized by infinite ordinal numbers (Cantor's third discovery), entities which from the point of view of ordinary mathematics are even more in need of
justification than the analysis that supposedly needed it. (Indeed, in attempting to describe what these alleged infinite ordinals are and do, people often resort to stories about gods and demons.) Little or no progress has been made on this "justification" problem in a century, but work with the hierarchy has produced some knowledge about the possibilities for categories of sets. By adopting a standard definition of map and discarding the mock iteration (with its concomitant complicated structure), each model of the cumulative hierarchy yields a category of
abstract nearly featureless sets; most of the usual set-theoretical issues depend only on the mere category: measurable cardinals, Goedel-constructibility, the continuum hypothesis, etc.
Having thus briefly understood the two visions which are called set theory
(1) a category of Cantorian featureless sets which serves as the background recipient for the structures of algebra, geometry and analysis;
(2) the cumulative hierarchy with its rigid Fregean structure aiming to justify mathematics,
it is not surprising that the precise nature of the elementhood relations appropriate to each are quite different. While the Fregean image involves rigid inclusion and elementhood relations imagined to be given once and for all for mathematics as a whole, the usual mathematical practice instead considers inclusion and membership relations for subsets of a given universe of discourse (such as R^3). Thanks to Grothendieck's Tohoku observation, these mathematical local belonging relations are well globalized within the notion of category, whose primitives are domain, codomain, identity, and composition.
[The notion of category is a simple first-order theory of a semi-algebraic kind. It has myriads of interpretations, some in "classes", some "locally small" etc., but such undefined restrictions on interpretations have nothing to do with the notion of category per se. Many properties are best expressed within the first-order theory itself.]
Composition is a kind of non-commutative multiplication, hence there are two kinds of division problems. In any category, given any two morphisms a and b we can ask whether there exists a morphism p such that a = bp; if so, we may say that a belongs to b. This forces a and b to have as codomain the same object, which serves as their common universe of discourse. (The dual relation, f determines g, defined by "there exists m with mf = g", is probably equally important in mathematics.) There are two special cases of this belonging relation which are of special interest. First we say that b is a part (or subset in the case of a category of sets) of its codomain, if for all a belonging to b, the proof p of that belonging is unique; this is immediately seen to be equivalent to the usual notion of monomorphism. Then, if a and b are parts of the same object, we say a included in b iff a belongs to b. Any arbitrary morphism x with codomain X may be considered an element of X in the sense of Volterra (also known as a figure in X); we say that x is a member of b iff x belongs to b. Then clearly
a is included in b iff for all x, if x is a member of a, then x is a member of b.
The usual relationship between these two relations is thus maintained. Because in category theory the domain relation is as important as the codomain relation, we can be more precise about elements: very often it is appropriate to consider a special property of objects, and restrict the term element (or figure) to elements whose domain has that property, that is, to figures whose shape has the property. For example, in algebraic geometry the connected separable objects are appropriate domains for the figures known as "points"; in the algebraically closed case it suffices to consider elements with domain a terminal object 1 as points. On the other
hand, frequently it is of interest to choose a small class of figure shapes which generates in the sense of Grothendieck, i.e. so that the above equivalence between inclusion and universal implication of memberships holds even when the figures x are restricted to those of the prescribed shapes. A basic property of categories of Cantorian sets is that this holds with x restricted to those with terminal domain 1. In algebraic geometry, the figures whose domains have trivial cohomology are adequate. Note that if f is a morphism from A to B and if x is an element of A, then fx is an element of B of the same shape (of course in general figures are singular in that they distort their shape, for example, fx may be more singular than the figure x). Properties of x in A may be quite different from the properies of fx in B.
The mysterious distinction between x and singleton(x) in the hierarchical Frege architecture takes quite a different form in the categorical architecture where there is a natural transformation from the identity functor to the covariant power set functor; this natural transformation can be called singleton: singleton(x) is simply x considered as a special element of PX, rather than of the original X.
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. But if these basics were widely understood among algebraic geometers, perhaps misconceptions like "category theory is the insubstantial part of mathematics" would not have arisen. (As we know from experience, all of the substance of mathematics can be fully expressed in categories.) Perhaps the general term "A-points" for arbitrary rings A was confusing. "Spec(A)-shaped figures" is a more accurate rendering of Volterra's "elements"; that could be abbreviated to "A-figures", but points are in some sense special among figures. On the other hand, we often vary the background category, so that alternative terminology might involve passing from a category E to
E/spec(A), and restricting the notion of "point" in any category to mean figure of terminal shape; then the A-figures become, on pulling back to the new category, literally "moving points".
Whatever the particular chosen terminology, the important conclusion is to actively eliminate the mythology that spaces in categories have no elements, because as we see, this mythology obscures the simplicity of certain matters and thus provides a bogus basis for insulating one field of mathematics from another.
[The belonging relation is just the poset collapse of the categories E/X, whose actual maps serve as incidence relations, especially between figures in X. Thus every category E supports a certain geometrical imagery wherein all maps are geometrically continuous, in that they map figures to figures without tearing the incidence relations. Precise axioms about E are a key to further progress because they explicitly sum up and guide our experience with the objects and maps in E.]
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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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F. William Lawvere: WHY ARE WE CONCERNED? III
Categories Mailing List
Date: Fri, 31 Mar 2006 09:15:28 -0500 (EST)
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.
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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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Labels: category theory, mathematics, Old posts
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