The time has come (the walrus said) to talk of many things:
of shoes, and ships, and sealing wax; of cabbages, and kings.
-- “The Walrus and the Carpenter”
Logician Lewis Carroll’s implication was, of course, that any such conversation would be absurdly desultory. Nothing connects those topics. Yet let us briefly consider in what ways topics – categories of objects, natural classes of things – may be connected.
[Note: the following will be funnier, and perhaps more revealing, if you have a smattering of acquaintance with Category theory:
It will be more telling still if you personally know some mathematicians. If you do not, simply check into Hilbert’s Hotel, and scout out the lobby. ]
We shall, in the manner of mathematicians, and for our own amusement, print the names of our categories boldface. Thus, consider: Orange; Grapefruit; Banana. Each is a natural kind – indeed, a species. The first two group naturally as members of a larger category, Citrus; adding Banana requires instead considering these three as merely instances of Fruit. Now try to add Bowling-ball. It spoils everything. There are functionally defined sets that contain all these, but they have much less interesting structure. Yet, if we group only Orange, Grapefruit, and Bowling-Ball, we once again have something interesting, Balls. (Banana, having been rudely voted off the island, might wander off and hook up similarly with Football.)
The field of biology is rich with such systematically related natural kinds. Their study is called taxonomy or systematics. There are even morphisms of a sort within this theory, whereby, for instance, a man’s arm, a bat’s wing, and a whale’s flipper are said to be homologous; the bat’s wing and the butterfly’s wing are not homologous but merely analogous. You might say, they correspond functionally but not functorially.
Such systematics is possible because of Evolution. Mathematical objects, by contrast, exist eternal and unchanging in Platonic heaven; they do not come presystematized. It is not obvious to the novice, and was obvious to almost no-one prior to the 19th century, that there could be any interesting systematics involving all of them. Upon initial acquaintance, the class of Sets, of Abelian Groups, of Differentiable Manifolds, of Knots, and what have you, may seem as various as the catalogue of the walrus.
But just for fun, let’s take three categories of things, chosen pretty much at random, and see if there is anything at all to say about their relations.
Ducks; Refrigerators; Topologists.
(Indulge me here. No animals were harmed in the filming of this fantasy.)
Pretty clearly we’re not going to get very far if we get too fine-grained. We must take no notice of such things as: Having feathers; Having a handle; Having a tendency to stare off into space. But let us daydream a bit.
For Ducks (I have mallards in mind), we find, for example (free-associating):
(I) Modes of operation: Dabbling is the default. Flying may or may not be necessary. Waddling is worst-case: a waddling duck is not at his finest.
(II) Sexual dimorphism: Marked.
(III) Phonation/vocalization: the quack.
(IV) Growth, at two different logical levels:
(A) Individual: There is a well-defined life cycle, from the egg to the watery grave. Essentially isomorphic across individuals (no real correlation with (II), for example. The cycles are moreover connected (by procreation), and the whole thing has the overall topology of a directed set, of cofinality 2 (Adam Drake and Eve Duck).
(B) Group: At the flock or species level, the numbers may go up or down with time. There is nothing nearly so structured or interesting to say as in (A).
(V) Purpose:
(A) Individual: Each duck is so constituted internally as to be purposive, pursuing its own ends – to eat, to mate, to quack, to dabble, to swim, to fly. These are essentially identical across individuals (with a very slight behavioral proviso in the case of mating, which parallels the division in (II)).
(B) Group: There are two ways of looking at the collectivity; and now it makes a difference, as it essentially did not in (IV B).
(1) Flock: The purposes in (A) continue to make sense at the flock level, in a merely derivative way; additionally, some more flock-level behavior comes in, like migration.
(2) Species: Again, there is more than one way of considering this; and the result is radically different in kind from those in (A) and (B) – being, for one thing, quite unconscious.
(2a) Traditional view: Each species has a rung on the Great Chain of Being, and is part of God’s plan.
(2b) Post-Darwinian synthesis: The sort of things discussed in The Selfish Gene.
Okay now, Refrigerators. If you were considering them on their own, without reference to our project, you would of course come up with quite a different list of noteworthy features. For instance, unlike the case with Computer Chips or Unmanned Drones, there is no premium at all on miniaturization: The watermelon still has to fit in the fridge. This characteristic is shared by Refrigerators and Passenger Aircraft, but let’s not go there. Instead, let’s just compare the checklist of the Ducks.
(I) Modes of operation: On (default). Off may or may not be part of its cycle. Worst-case: Broken. – A faint analogy to Ducks I.
(II) Sexual dimorphism: Absent. (If you disagree with this assessment, you need help.)
(III) Phonation/vocalization: Well, it hums when it’s on, so you might call it that. But there is decidedly no homology with duck phonation, and barely even analogy.
(IV) Growth: None.
(V) Purpose: Well, they do have a purpose – to keep food cool – but it’s not they that have it. The purpose is externally imposed – we create refrigerators to serve our purpose. It is interesting to note this difference in itself; it wasn’t an idea that had occurred to me before in such sharp form. But that doesn’t mean there is any interesting cross-category morphism. At best, there’s a kind of analogy to Ducks V B 2a.
Okay, so far disappointing. Now let’s look at Topologists.
(I) Modes of operation: No discretely identifiable modes.
(II) Sexual dimorphism: Absent.
That is to say: There are both male and female topologists, considered as People; but this difference is entirely irrelevant quâ members of the present Category, Topologists. Men and women might actually tend to have somewhat different thinking styles in their chosen subject-matter, just as blind topologists might; but that is structurally as irrelevant to Topologists, as the fact that male topologists tend to weigh more, or that Contemplative Topologists tend to weigh next to nothing.
(III) Phonation/vocalization: It may be present or absent. Communication with other individuals in the category may be oral or written. A subclass, the order of Contemplative Topologists, live in trees and never speak to anyone. Some interesting correlations with Trappist Monks, but we won’t go there.
So, still no interesting similarities across our categories; and yet the nature of the contrast between Ducks II and Topologists II is rather engaging. And we probably wouldn’t have happened upon quite this thought, but for our quixotic experiment.
(IV) Growth:
(A) Individual: There is growth (in topological insight, as your career matures). It is no longer isomorphic across individuals, though there are some similarities (you almost always know more with time, and there tends to be a certain continuity in terms of your subspecialties, though occasionally some pioneer with make a radical break, or even found a new subspecialty). As for the topology of the whole set of growth-curves, there is a slight resemblance to Ducks IV A, except that now, instead of each individual bearing the offspring-of relation to precisely two other individuals in an all-or-nothing way, now each individual may bear an intellectual-offspring relation to indefinitely many individuals, and in varying degrees; morever, unlike the case in Ducks, both individuals P and Q may bear the relation to each other. There might even be a very few individuals in the order of Contemplative Topologists, who bear this relation to no-one, having acquired their initial insights directly from God (think Ramanujan).
(B) Group: There is an uninterestingly similar growth at the ‘flock’ level, as the profession thrives or withers. There is a kind of growth at the ‘species’ level: that is to say, in the field of Topology itself, pursuit of which defines membership in Topologists. No homology with Ducks IV B (pace Richard Dawkins and his stupid “memes”).
And finally:
(V) Purpose
(A) Individual: Entirely analogous to Ducks V A. That is to say: mutatis mutandis. The actual behaviors are entirely different, but at least analogous if not homologous. So, of course, once again, the fact that many topologists (considered as People) want to mate (though no-one in Contemplative Topologists does) is structurally irrelevant to this category. Purposes here include: Prove theorems; understand stuff; etc.
(B) Group: At the ‘flock’ level it is similar to Ducks V B 1 – again in an uninteresting way. At the ‘species’ level – topology itself – although the field is by now largely internally motivated – la topologie pour la topologie, having outgrown its role as handmaiden to analysis – there is nonetheless a certain degree of external purposing, as requirements arise in other sciences. Thus Poincaré’s topological approach to the three-body problem, which turned out to be definitive, and beyond physics as such. There is ebb and flow – topology sent another pseudopod in the direction of physics, by finally rigorously deriving the behavior of pendula – previous derivations had involved a fair amount of handwaving, as physics almost always does. But this time (as Ivar Ekeland remarks, in The Best of All Possible Worlds), the physicists didn’t much care. They knew empirically what pendula did, and don’t care for real rigor beyond a certain point.
There might even be a kind of vague analogue to Refrigerators V and Ducks V B 2a Namely: the sum total of possible topological truth is Out There, in Platonic heaven, undiscovered by ourselves. But the routes we shall wind up taking are in part conditioned by their existence, external to ourselves. The analogy would be to mountain-climbers inhabiting a misty land of limited visibility. Their general purpose remains the same, and internal: to climb mountains. But which mountains they will climb, and the kinds of mountains, and the kind of climbing techniques they will need to come up with, are largely externally determined. It could even, in principle, be externally purposed: God populates the planet with a graded series of ever-craggier peaks, arranged in concentric circles, to train the mountaineers and lure them on…
*
So, where are we. We have not really come up with any interesting “functors” (structurally sound relations) across these categories; although, oddly, what began almost as a satirical exercise did lead to noticing some features one otherwise might not have. There is some utility, to contemplating topologists sub specie anatum and vice versa, if only to pass the time while waiting for a bus. But such musings aside, it is obvious that, if there existed a body of theory that actually illuminated all these categories and more in new ways, bringing out previously unsuspected analogies among them, and in the process actually contributing concrete new content to Ornithology, Appliance Tech, and (Sociology, Philosophy of Science, who knows), that would be spectacular.
And that is what, for the objects of mathematics, Category Theory proposes to do. At the outset, this might seem an unlikely enterprise. Mathematical objects exist eternally, independently of ourselves, transcendentally; but our knowledge of same is partial, historical, contingent, and in various ways defective. Some of the structures-as-we-conceive-them are historically offshoots of others, some popped up in all sorts of unrelated ways. Why should there be a general Theory of Everything? The notion that some theory could peer down from above, like a Fatlander inspecting the innards of the denizens of Flatland, from a level more “meta” than metamathematics (which is mostly just proof theory), is astonishing.
And indeed, apparently we do not yet have such a theory. Saunders MacLane (one of the impressarios of Category Theory) puts it thus:
Categories and functors are everywhere in topology and in parts of algebra, but they do not as yet relate very well to most of analysis. … There is as yet no simple and adequate way of conceptually organizing all of mathematics.
[Mathematics: Form and Function (1986), p. 407]
In other words: So far there is no way at all. (It’s not as though we have a way that is adequate but not simple.)
*
One reason you might think that such a project could eventually succeed is that the big story of mathematics over the past fifty to a hundred years is in the growth in density of reticulation among the various fields, quite apart from the increased internal richness of each field in itself. What were initially far-flung enterprises turn out to be able to take in one another’s washing. It is not clear, however, that such mutual assistance pacts must be functorial, part of a single overarching theory, rather than being analogous (not homologous, of course) to, say, increased cooperation among nations, as they discover the value of low tariffs, outsourcing, academic exchange programs, etc., and are served by new connective technologies (jet planes, the Internet). The phenomenon is real, and hugely important – as important as, and in some ways even similar to, the invention of the Internet, since it has managed, against all odds, to halt and reverse what had otherwise seemed certain, namely increasing hyperspecialization and fragmentation into Fachidiotie. True, hyperspecialization is still often required: witness algebraic geometers, who speak a language known only to dolphins. Nevertheless, said magicians may from time to time come up with a result that turns out to be just what the number theorist needed. (Or the cryptographer: in which case said geometer mysteriously disappears.) So: real, and important, but perhaps only superficially similar to the kind of interconnections that are the province of Category Theory.
I say “perhaps”, because I have no idea, since I don’t understand Category Theory. Perhaps on the contrary all such fruitful cross-connections are at bottom functorial. And if this is the case (not saying it is, mind), then conceivably (I mention this merely as a logical possibility) this circumstance in turn derives from the (as it might be) fact, that all mathematical structures are siblings, owing their origin to: the one Father. (Praise Him.)
*
Category Theory was sired by homological algebra out of algebraic topology. As the name suggests, algebraic topology connects the (ab initio quite disparate) fields of point-set topology with algebra. But in fact such a connection, of geometry (the predecessor of topology) with algebra, has occurred, not once, but several times. Each time it has been fruitful in quite different ways; and this fact in itself suggests the potential fruitfulness of cross-categorial connections generally.
Geometry in its infancy was literally geo-metry – measuring out fields and furrows upon the earth. Such a state of the art did not lend itself to anything cross-categorial, being itself not even a true category yet, but just a bag of tricks. (See our fable of the humble woodchuck, in “Constructivist Angelology”.) Once it had become formalized – and indeed axiomatized – as Euclidean geometry, it had reached a stage of sufficient richness that it could be further enriched, not merely quantitatively, by proving still more theorems, but qualitatively, by its transfer into algebra. This was accomplished by Descartes and others, in the field of analytic geometry; and it qualifies as a Hegelian Aufhebung or sublation, the field being simultaneously preserved (all the theorems still hold) and transformed beyond recognition (the mental world of its algebraic reconceptualization being galaxies apart from that of Euclid).
And for a while, that was as far as you could go, until geometry itself took a qualitative leap forward, with the development of non-Euclidean geometries. Once that momentous step had been taken (and really, the history of mathematics is so sweepingly orchestral, that the annals of empires are by comparison but so much chaff) – once geometry had become geometries, it was in a position to receive yet another wholesale transmogrification thanks to algebra, this time by Felix Klein and his followers in the Erlangen program. Here group theory was used to enrich and systematize Geometry (for the discipline thus regimented once again deserved -- by its unity and depth -- the dignity of the singular, and we shall capitalize it as well) in ways previously undreamt-of.
Now one more turn of the dialectic. Once again, the field itself – the field that grew almost literally out of the fields, where Adam dolve and oxen plowed – underwent a qualitative self-transformative inner enrichment, as Geometry, with its shapes and metrics, gave birth to Topology, where now metric spaces are but a special case, as Euclidean space is but a special case of metric spaces. And once that triumph had arrived, the result was sufficiently sprawling that it once again needed to be systematized with new tools. Whence Algebraic Topology.
A parallel narrative, in miniature, could be spun for the evolution from such knots as Odysseus tied on his ships, to today’s heavily algebraized Knot Theory.
*
Almost have I managed to re-whet my appetite, and give CTh yet another try. Yet somehow all previous encounters have yielded nothing but dismay. Thus, the least enjoyable math book I have ever read is Sze-Tsen Hu’s Elements of Modern Algebra (1965) – and this despite the fact that one of the series editors (for Holden-Day) was Andrew Gleason, my favorite math teacher ever, and at antipodes from this bloodless book -- the great Gleason, always genial and brimming with toothsome examples. (It was in his undergraduate course on group theory that I became acquainted with the art of Escher.) This Hu is a heavy hitter in homotopy, and no doubt knows his stuff, but his book feels like if-it’s-Tuesday-this-must-be-graded-algebras. Structure after structure is introduced without motivation or exemplification, and related in all sorts of impeccably “natural” ways to other likewise unmotivated structures. Diagrams commute, commute, commute, with the joylessness of draftees doing jumping-jacks. If your eyes glazed over at semigroups, you’ll go legally blind at semigroupoids. Suicide begins to appear an attractive alternative. This book is the night in which all algebraic structures are grey.
Compare the remark by the delightful Klaus Jänich (Topologie, 1980, translated as Topology, 1984; p. 123 of the latter):
Only with some hesitation do I introduce yet another topological concept: paracompactness. There are so many such concepts! An A is called B if for every C there is a D such that E holds – this is quite boring in the beginning.
*
Now: these last remarks would be unimportant -- if a subject is boring, just ignore it, move on – save for one telling incident in undergraduate days. Namely, when my then-classmate, comrade (later Professor) Kudla, unbosomed himself of the opinion, that Category Theory was the best invention since sliced bread, and was the light at the heart of Algebraic Topology.
One would not have assumed so – even assuming the self-sufficient magnificence of CTh. For: compare the role of math in, say, the vocational training program you go through to become an electrician. It crops up, though not centrally, nor often. A guy could be really good with wiring, and have an intuitive feel for circuits, be manually dextrous, know how to get really great prices on parts, and have a sharp eye for things not up to Code, but be behindhand at mathematics, even find it irksome, and still be a terrific electrician. Of course, if he does happen to cotton on to math, he might find it useful, not only for electrician stuff, but for carpentry (the other day our son was discovering a trigonometric formula to handle some thorny problem involving beveled molding that must meet at a corner) and much else, though it’s not the main part of any of the vocations.
Thus, in Algebraic Topology, a functor (think: a function with a Ph.D.) takes you from Top to Group. The functor is itself necessarily at a higher level of abstraction than either its domain or its codomain. But, having ascended by its means, surely you can now throw away the ladder. That is to say: Once our friendly neighborhood functor has spit out the requisite homotopy groups or what have you, surely its work is done, and it is at liberty to fold up its tent, and (much like the Arab of proverb) steal silently away. We concentrate on the resultant algebraic structures, which (we hope) are easier to calculate with than are the topological spaces that spawned them (with the functors as midwife), and we forget the functor: just like that math stuff you met in Voc Tech, and promptly forgot.
*
A curious sidelight. Professor Kudla himself replied to my earlier trifle on the subject, thus:
As it happens, I am reading some Grothendieck style algebraic
geometry lately, which involves a lot more category theory than my usual, rather more classical, fare.
His lectures from 1959-61 are still difficult (for me) to absorb.
Now, notice: This, from a man, who while still in his teens, supped on Category Theory with an ice-cream spoon; and who has since had a long, uninterrupted career as a professional mathematician. Be further advised: A similar confession would be made by almost every other living mathematician – those, that is (a distinct minority), who have ever even bothered to attempt to wrap their minds around the inimitable, nay ineffable Grothendieck. Indeed, rumor has it, the only creature on this planet, ever to completely understand Grothendieck, is a certain Emperor penguin, who, as such, is unable to communicate his (or her; it’s hard to tell with penguins) understanding to ourselves. And this, mind you, relative to writings almost half a century old; and that, please be aware, in a field evolving with such rapidity, as to compress the rise, decline, and fall of the Roman empire, into a single summer afternoon.
All of which is merely to observe, that, compared with mathematics, in point of depth, breadth, and density, all other human endeavors are approximately homeomorphic to the sort of gunk that tends to build up beneath your toenails, unless you observe proper hygiene. Which, in the (alas) absence of more substantive guidance after all this is said and done, you are hereby advised to do.
[Question to folks in the business: Whence the abstruseness of Grothendieck? Does it inhere in the subject-matter? In his own quirky approach thereto? In his style of exposition? -- I have touched on similar matters in another essay, “On Scope and Difficulty”, reprinted here.]
Update: I have at last found a book on category theory that is truly pedagogical:
Lawvere & Schanuel, Conceptual Mathematics (1997).
Had I known of this book at the time, I would probably not have written the above.
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