Suum cuique

Concerning the Corpus Christi plays, which began in the thirteenth century:

The different scenes, whenever possible, were distributed in such a way  as to bear some relation to the occuption of the craft that performed it:  e.g. the task of producing Noah’s ark  was entrusted to the boat-builders, the adoration of the magi  to the goldsmiths.

-- Ward & Waller, eds. The Cambridge History of English Literature, vol. V: The Drama to 1642, Part One (1910), p. 45

Such nicety of fitness rests upon a folk notion of appropriateness, almost common sense, rather than being specially religious.  And indeed, we meet this meme in a quite plebeian guise, in one boy’s life, back in Eisenhower’s America:

https://worldofdrjustice.blogspot.com/2013/04/de-patre.html

Orthoëpy of “monotony” (Word-of-the-Day)

Please turn to page  58 of Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957) -- or, if you happen not to have your copy handy, here is the relevant quote:

Theorem.  If E is a finitely-additive, projection-valued set-function on the class of all measurable subsets of a measurable space, then E is monotone and subtractive.

.

Proof.  That E is subtractive is trivial;  monotony follows from 29.3.

Now of course -- you already knew that.   But the point here is rather the status of this interesting word monotony -- in particular, how to pronounce it in this case.

Familiar to everyone is monotony, the noun of monotonous, pronounced respectively and in parallel, mon-OT-on-ee and mon-OT-on-us.  But the adjective here, though cognate, is quite different:  monotone, pronounced MON-o-tone :  meaning:  increasing (bzw. decreasing) and never decreasing (bzw. increasing).  Alles klar?

The question then is:  How do we pronounce its associated noun? 

Now, I have never actually heard the term in speech, despite having sat through many a math-class.  The adjective is more usual than the noun, and the noun, when it does come up, is usually monotonicity  (mon-o-tone-ISS-it-ee) -- though strictly, this is the noun of monotonic (mon-o-TON-ik), not of monotone.  So how to pronounce this technical term “monotony” we have never heard in this meaning?    Every fibre of my being cries out, that we must pronounce it  MON-o-tone-ee.

The word is, thus, a homograph, with different pronunciations for its two (related) senses.

Compare and contrast  the situation with the words homology, pronounced ho-MOL-o-jee  in both the mathematical and the general sense;  versus homotopy, which has only a mathematical sense, and which is always pronounced HO-mo-top-ee.  Linguistically, this is an odd situation, since, semantically, homology and homotopy are very closely related, and one would expect their pronunciation-patterns  to march in step.

(See what you’ve been missing, by not majoring in math?  Don’t let your children make the same mistake !)

[Update, Juli 2012]  Someone just accessed this post with the searchstring

          alles klar pronunciation

Geht in Ordnung!
AH-less KLAHR.

The sense meant  is something between "Is that clear?"  and "Any questions?"
Intended  as gently  ironic...

The Urysohn Metrization Theorem: for real this time

Background:   Topology is familiarly, informally characterized as “rubber-sheet geometry”.  That is, unlike the Euclidean geometry that we learned in school, which applies to flat rigid surfaces,  you’re allowed to stretch and bend the space, so long as you don’t tear it or let it intersect itself.

But at some point, we might like -- without returning quite to the simplicities and rigidities of the Euclidean picture -- to make our space… a bit less rubbery.   As the godfather of Calabi-Yau manifolds puts it:

We start with some raw topological space, which is like a bare patch of land that’s been razed for construction.  On top of that, we’d like to build some kind of geometric structure that can later be decorated in various ways.

-- Shing-Tung Yau, The Shape of Inner Space (2010), p. 77

[Note from Jan 2011]  We were nonplussed to learn that this site comes up on the first page of Google search on “Urysohn Metrization Theorem”. 
[Note:  This has since changed, owing to a hack attack by the Nominalist Internationale.]
[Metanote:  It's back again, thanks to a counteroffensive by the Realist Underground.]
And ill at ease, since our post of that title is a satire on sociobiological/ultraDarwinistic  overreach, a satire of a sort practiced almost a century ago by G.K. Chesterton in his book The Everlasting Man.  Pity the unsuspecting physicist or math major who winds up there in hopes of learning the first thing about metrization, Urysohn or otherwise.   So we feel we owe it to these blameless Internauts to offer them at least a little something for their trouble.   Here, then, for the non-mathematician, or (God willing) the mathematician-to-be, is a thumbnail sketch of what led to this theorem in the first place. 

The intuitive content of the theorem is as follows.   If you have a space with enough structure to keep things apart which ought to be, and if the space itself is not too huge, then you can define a distance between any pair of elements.  The function that specifies this distance is called the “metric”, from the Greek word for 'measure'.

Thus, you mightn’t be able to do this if you lived in an oozy sort of world, where the minimal entities were like blobs with sometimes inextricably intertwined tentacles; nor if your world were scattered among separate universes.

(Footnote:  the idea is that you can come up with a nontrivial metric.  After all, any set whatever can be regarded as a (trivial) metric space, given the discrete topology.)

As for formal statements, these vary somewhat.  Here is a sampling.  (The following assemblage is an atavism from my days as a lexicographer at Merriam-Webster;  we worked from piles of attestation-slips, called "cites".)

John Kelley, General Topology (1955), p. 125:

Metrization Theorem (Urysohn)

A regular T1-space whose topology has a countable basis  is homeomorphic to a subspace of the [Hilbert] cube and hence metrizable.

James Dugundji, Topology (1965), p. 195, formulates it as

In 2-countable spaces, regularity is equivalent to metrizability.

and he labels this merely a “corollary” of

Theorem (Nagata and Smirnov) A topological space is metrizable if and only if it is regular and has a basis that can be decomposed into an at most countable collection of neighborhood-finite families.

The same can be said for the Bing metrization theorem, which likewise sharpens the sufficient condition into one both sufficient and necessary:  “a topological space X is metrizable if and only if it is regular and T0 and has a σ-discrete basis.” (Wiki)

Other formulations:

Stephen Willard, General Topology (1970), p. 166:

Urysohn’s metrization theorem.  The following are equivalent for a T1-space X:

(a)  X is regular and second countable

(b) X is separable and metrizable

(c )  X can be embedded as a subspace of the Hilbert cube.

James Munkres, Topology: a First Course (1975), p. 217:

Urysohn’s metrization theorem.  Every regular space with a countable basis is metrizable.

Michael Henle, A Combinatorial Introduction to Topology (1979), p.  283:

Metrization Theorem (Urysohn)

A compact Hausdorff space that is second countable is a metric space.

That one uses a stronger condition to reach the same conclusion, and is thus a weaker theorem; the same version appears here:

Boto von Querenburg, Mengentheoretische Topologie (3^rd edn.  2001):

Ein kompakter Hausdorff-Raum is genau dann metrisierbar, wenn er eine abzählbare Basis besitzt.

Something of an odd-man-out, possibly importing the stronger “normality” condition from the Urysohn Lemma, is this:

Seymour Lipschutz, General Topology (1965), p. 142:

Urysohn’s metrization theorem. Every second-countable normal T1-space is metrizable.

But cf. this:

George Simmons, Introduction to Topology and Modern Analysis (1963), p. 138, which offers a slightly stronger version, and names it differently:

Urysohn Imbedding Theorem.  If X is a second-countable normal space, then there exists a homeomorphism of X  onto a subspace of R-to-the-infinity, and X is therefore metrizable.

And indeed, the Lipschutz formulation is echoed much more recently in the October 2010 American Mathematical Monthly (“A Tale of Topology”, by Gerald Folland):

    Every second-countable normal space is metrizable.

If all that  already makes sense to you and seems obvious, you’re done.  If not, read on.

*

            The first order of business is to motivate the theorem.   What does it mean for a space to be metrizable, and why should we care?

            The space we’re best familiar with is the one we live in;  but the one we have studied most analytically, traditionally in high school geometry class, is the nice flat one, called the Euclidean plane.  This we studied  first by Euclid’s own methods, which date back over two thousand years, with axioms and proofs that justify each step -- the best possible mental exercise -- and lots of diagrams.  Later (if we stay the course) we take up a new approach, using analytic methods, which largely began with Descartes, in the seventeenth century.   Here we add a grid of axes, which measures exactly where each point is and how far apart they are, and prove things about figures: now not just triangles and circles and rectangles, but hyperbolas and cycloids and any shape we want, by means of equations.  You don't have much in the way of equations with Euclid;  for that, you need numbers -- given by the metric.

            And lo -- already, in these simple memories of high school, we have, in miniature, a picture of what has happened at the forefront of mathematical research over the past century or so.    For geometry,  in the sense with which you are all familiar, came to be generalized to a new subject, topology (originally called analysis situs -- both mean ‘the study of place’, as geometry means ‘the measuring of the earth’).   Whereas the Euclidean plane is rigid, we let these spaces get all stretchy and bendy.  In that case  we can no longer say what the circumference of a circle is, because by the time we wake up in the morning it may have stretched and drooped like one of Salvador Dali’s watches (in his painting, “The Persistence of Memory”):  but some things do remain true, such as the fact that that curve has an inside and an outside, meaning you can’t get there from here without crossing that curve.  (Note:  Such entirely general, almost naively simple-sounding statements are typical of topology.  The content of that one is called the Jordan Curve Theorem, and it's a real bear to prove.)

            Now topology was originally point-set topology, which mentally is rather like Euclid’s geometry:  you set up the ground rules for a space, then you ponder and visualize and reason things through, using pictures if you possibly can, and your own intuition.    Meanwhile, behind the scenes, a new view of topology was taking shape, somewhat analogous to what Descartes did for (or to) geometry:  instead of reasoning, half-intuitively, with spaces and shapes, you come up with an algrebra whose structures manage to reflect what is going on in those spaces in more detail, yielding numbers and equations and things you can calculate with.   It could have been called “analytic topology” by analogy with “analytic geometry”, but instead it is called algebraic topology.    Though very powerful, it is somewhat bloodless (at least for the beginner), and requires different habits of mind.   (Habits I alas lack.  Readers of my tales of woe will recall my bruising encounter with that subject;  the spot still smarts  in frosty weather yet.)

            So:  Cartesian geometry takes us, from shapes,  to the antecedently familiar realm of equations involving numbers.   Homology (a part of algebraic topology) takes us from more general shapes to the relatively tractable algebraic structures called Abelian groups.  

            The distance function in Cartesian geometry is what you get from the Pythagorean theorem.  The criteria for the general topological notion of a metric are a straightforward abstraction from this:  mainly, if you make a beeline from here to there, and another beeline from there to yonder, the distance traveled must be at least as much as had you simply gone straight to yonder from here.  As to what-all can meet the criteria -- ah, there lie surprises.

            Euclidean geometry is described as what you can do with a straight-edge and compass.   Sometimes people say “ruler” and compass, but that is a mistake:  we have no measurement-markings on our straight-edge; there are no pre-established units of measurement.   We can still determine that two different line-segments are the same length -- just take our trusty compass, measure the first segment with it, and now see if that compass-setting matches the endpoints of the second segment.  You might say that, in this world, length itself is not absolutely defined, whereas being-as-long-as is.   (This observation could be pursued in a syntactic direction -- that of incomplete symbols -- with interesting results.)

            Now, in the Cartesian approach -- analytic geometry -- we want to work with actual numbers, because that speeds things up.   So we turn the plane into a metric space -- “metric” just means ‘measurement’.   And the reason it is possible to do so is that the (pre-Cartesian) Euclidean plane was already rigid:  you do not change the length of something simply by moving it about; you can slide one triangle over to another one and see if they’re congruent.   

            Furthermore, the way we shall conveniently measure things  was already suggested to us by the celebrated truth of Euclidean geometry, expressed in the Pythagorean Theorem.   The earlier formulation of this was:  “the square on the hypotenuse is equal to the sum of the squares on the other two sides”:   meaning, the area of a square figure,  one of whose sides is the long side (the ‘diagonal’) of a right-angled triangle,  is equal to the sum of the areas of two other triangles likewise sticking off the shorter sides that are perpendicular to each other.    This was still a ‘point-set’ geometric view.   But now we start writing it in symbols, saying that, if x is the length of the one leg, and y is the length of the other, then the length of the diagonal is the square root of the sum of x-squared plus y-squared.  (This is known as the “Euclidean metric”.)  That’s algebra.  And the new viewpoint is reflected in the way you’ll here the theorem quoted nowadays: “the square of the hypotenuse is equal to the sum of the squares of the other two sides”:

            The rest  you are familiar with.  We briskly mark off a bunch of equal lengths along one direction, which we call the x-axis, and likewise along the y-axis that is perpendicular to it, and from this we get graph-paper, with its familiar grid.   And now our old friend the plane, which we first came to know as a tabula rasa -- the plane itself, and our own childish minds -- wears its Metric Space status on its sleeve, so to speak.

            This is all so familiar, that we are in danger of letting our memories do the thinking for us.   For in fact there are many different ways of deciding to set of a scheme of measurement on a flat surface.  We might stipulate that the ‘distance’ between two points shall be simply whichever is larger, the difference in the x-value or the difference in the y-value.  Or we might say instead that the distance shall be the square root of the difference of x-squared and y-squared, rather than their sum.   This is what Minkowski did, and it turns out to be the key to uniting space and time into a single Metric Space -- spacetime.   These and others are alternative possible metrizations of a plane.  (In one of them, it is possible to draw a round square -- the paradigm example of what philosophers tell us is impossible.   You can read about one here.)

            Of course, we don’t ourselves live inside of piece of paper (as Flatlanders do), we live in nice big rooms -- three dimensions rather than two.   We set up a third axis, the z-axis, like a tent-pole, to give us some breathing-space:  and now the grid shapes are little cubes instead of little squares.  Using this Euclidean metric, it turns out that an analog of the Pythagorean Theorem holds here as well:  we can consistently define the distance as the square root of the sum of x-squared plus y-squared plus z-squared.   Analytic geometry proceeds as before, with barely any change in methods (I originally wrote, "since the ones we used in the two-dimensional case were already so powerful"; but actually that puts the cart before the horse:  it is precisely such (unexpected) generalizability of a method that leads us to call that method 'powerful'.).    So now, instead of just circles and parabolas and so forth, we have a richer world of shapes, like cones and spheres and ellipsoids and helices, and on and on.  (Actually these were already known to the Greeks, though how they managed it with the pre-Cartesian methods they had, is something of a miracle.)  And it continues to be easy to prove things about these, since we still have basically the same metric, which is well adapted to equations and their numerical solutions.    Likewise in four dimensions, and on up as high as you like.
(Note:  People get all spooky when they hear things like 'fourth dimension', but these metric methods absolutely tame them. -- Children:  Study math.)

            Bottom line:  A metric space is a very convenient thing to work with.  You can pretty much know where you are and do what you want, even when the space gets hairy in other ways, like being infinite-dimensional, or very curvy.

            But.

            In the inexhaustible splendor of the Creation, there are many many different spaces, more numerous than the stars.    They sprang full-blown from the Creator’s brow, and it is up to us to discover their structure.  Unfortunately, when we first meet up with one of these, it may not be wearing a nice convenient metric on its sleeve.  It may be a very confusing, huge, menacing, squishy blob.  -- Recall that when we first met the plane, it too came without a pre-drawn grid.  But a particular grid was already implicit, because Euclid’s axioms imply the Pythagorean theorem.   (There are other metrics you could adopt where that theorem wouldn’t be true, but these would not conform to our everyday local experience, which is why Euclid chose the one he did, and why it took two millennia to generalize the naive notion of "distance" to the mathematical notion of "metric".)

            So:  Faced with such a blob, can we come up with a consistent measuring-scheme that will tame it, by turning it into a metric space?  That is, is it metrizable?  -- And the answer is:  Sometimes you can, and sometimes you can’t.   So, we want to come up with ways we can tell, whether the project is doable or hopeless.  It’s a bit like figuring out whether you can tame a given kind of animal.   Long ago, people figured out that you could do that with dogs, and later with horses, to a huge extent, so that instead of being wild beasts they are actually useful.  Cats, it turns out can be tamed to a lesser extent -- tamed to tolerate us, so long as we are not late with the cheeseburgers. They’re not useful, but they’re decorative.  (Meanwhile in Catland, the lecture reads:  “Peeps are redonkully e-z 2 tame;  goggies, not so much.”) So, you see a puppy and, no matter what breed it is, the mere fact that it is Canis familiaris tells you that you have an excellent chance of taming it:  though just how to do so may vary with the breed.  In a similar fashion, when we first meet an untamed space at the Space Store, we can know, by certain signs (which Urysohn specifies in his theorem) that the thing is in principle metrizable -- though it doesn’t give us a useful metric just for free;  for that we still have to do some work.

*

            If you and I are points in a metric space, the metric tells us how far apart we are:  that’s like analytic geometry.   But topology lets out the sails a bit.  In the most general topological space, you can’t say how far apart two points are -- but you can always say what bunks with what.  The bunks are known as “neighborhoods” or (roughly synonymous) “open sets”, and are given as part of the very definition of the space.   Thus, the most featureless space of all, justly (in this metaphor) called the “indiscrete” space, everybody bunks with everyone else, no privacy at all.  In the opposite, the “discrete” space (discrete, not discreet), each man is an island.  But most spaces, and every space of interest, is in between.

            To describe just where they fall on this in-between spectrum, we name various “separation” properties.   The simplest common one is that any two points can be separated (any two people can sleep in separate bunks).  In a metric space, this is easy.  If you and I are a certain distance apart, then if I draw (if we’re in a plane) a circle or (if we’re in a fatter space) a sphere  around me, with a radius less than that distance, then I’m in my own special bubble and you’re outside it.   Such elementary capacity of separation is also available in most non-metric spaces. This basic degree of separation is called T1.  If you and I can each draw such a neighborhood simultaneously, even better -- the space is “Hausdorf”.

            A more demanding requirement is that I can fix around myself a bubble (a neighborhood) which keeps me clear, not of just a single point, but of any collection of points called a “closed” set.  A set is closed if, for any point you can creep up on, in an infinite sequence, that point is in the set.  So for instance, on the number-line, take all the reciprocals of the natural numbers, ½, 1/3, ¼, etc:  these creep up on zero -- they get as close as ever you please, so the set of these reciprocals is not closed:  to close this set, you have to add zero.  Or, take everything inside a circle.  You can creep up to any point on the boundary, from within the circle, so the interior is not closed.  Add the boundary, now it’s closed.   -- So:  given a point, can that point stay clear (hide inside a bubble), not just from any other point (that’s easy), but from any other closed set that doesn’t contain that point -- no matter how pushy and encroaching?   Well, if that set is closed, it can’t keep creeping up on me indefinitely:  at some stage, it can’t come any closer, otherwise I’d be a limit point of that set, and since I’m not in that set, it wouldn’t be closed.  So at some point it keeps its distance -- I’m safe in my neighborhood (say, Beacon Hill), where the menacing set cannot encroach.  (In a metric space, this is easier to visualize:  it keeps its distance -- say, d.  So I draw a little bubble round me, of radius smaller than d, and I’m safe.(  -- Spaces that are like this are called regular.  Obviously, every regular space is Hausdorf, but not vice versa.

            (The next step up is:  Can any two closed sets -- not just one closed set and a point -- be kept apart by disjoint neighborhoods (neighborhoods that don’t intersect)?  If so, that space is called normal.  Normality is used in the Urysohn lemma, basically unrelated to the UMT.)

  

            So, being regular suffices to keep things separate enough to define distances between things.  But regularity by itself is not sufficient -- the space might be too ‘big’ to fit in a metric.  How big is too big?  Bigger than second-countable, the other premise of the U.M.T.  In that case, points can be just too far apart to have a finite distance.

*

None of these definitional and formal considerations  gets across the real power of the metric-space idea.   This arises when we begin to consider a more abstract sort of space, in which the “points” are not characterless, dimensionless ideal dots, but … functions.   (This is an example of the “Ladder ofAbstraction”.)   Andrew Gleason puts the matter well:

The assignment of a metric to a set of functions  gives this set an intuitively geometric character.  The success of the theory of metric spaces in analysis  can be attributed to the remarkable insight into the nature of functions  which has come from exploiting the geometric point of view.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p.  226

[ continued here …]

Categories for the Working Mom

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:

http://en.wikipedia.org/wiki/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 19^th 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.