Friday, December 31, 2010


[Theologia Mathematica, ch. 2.   Continues this.]

There is a phrase within a clause within a prayer, of which I am particularly fond, and upon which I ponder ceaselessly:  “…the Father almighty, Maker of Heaven and Earth,

and of all things visible and invisible…”

… visibilium omnium et invisibilium.  Doubtless each line of the credo might profitably exfoliate into a tome,

but let us now, here, pause for a moment at this one.
            The phrase is no mere afterthought.  There are in fact a whale of a lot of invisibles out there, and it’s not just ghosts, or disembodied ectoplasm, or ethereal unstructured mush.  Nor do I mean “dark matter” or “dark energy”, though apparently there are gobs & gobs of that as well: more than of ordinary matter, which now rattles about in the cosmos like spare change.  (Or, to coin a phrase, like angry candy.) No, dark-whatever, doubtless all jolly stuff, and quaint in its way, but no more inherently fascinating than, say, rabbits.  It’s merely the library-paste and plasticine that happened to be lying about when God got around to making this particular universe on some particular day, possibly with leftovers from some earlier practice project; and now it hangs about, drifting moodily hither and yon, like so much unemployed blancmange.  Being invisible doesn’t make it significant or interesting.  Let us have no fetish about the invisible.  If for some reason the credo had said, “… and of all things probable and improbable,” and if the improbable had somehow been mostly ignored, yet contained most of what was of interest in the universe, then we’d be talking about the improbable; or the fantastical; or the ironical.  The entities  I shall be getting at here  are not significant because they’re invisible; I’d be even willing to concede that they’re significant despite being invisible, that visibility would be one further and delightful perfection, one which we may someday hope to glimpse.  In any event, what is meant here is the mathematical scaffolding, on which the sun and the moon and the quarks hang  like so much laundry.  That is, the plan of the thing, so much more permanent and pervasive than the things themselves.  I mean the symphonic score  from which our ephemeral melodies derive.

            Properly apprehended, it is a structure of – crystalline palaces, transparent and thus largely invisible to the untutored gaze, save as the light of insight  glances off them at an angle, and so catches the inner eye.  These ideal edifices are as hard and as chiseled and as real, as our own makeshift dungeons of stone: nay, more real, for these intricate perfections are the prototypes, whereof our own poor earthbound shantytown is but the fallen, partial, semi-crumbled, quasi-scrambled, half-forgotten misremembered afterimage.  They are, it is true, invisible: but in part (in increasing part) -- not unimaginable.  Through intense and lifelong study, we may – by luck, or grace, or mental sweat – eventually acquire a glimpse of their upper ramparts, from which turrets rise, from whence pennants flutter – flutter in a plenitude, an infinitude of dimensions, one upon the other like palace halls; so that our own most swirling ballet or crashing waves  are but as the slogging of an ant  trapped between the narrow glass walls of the ant-farm.
            These diaphanous entities, being (as we shall argue) a part of the Creation, display a different side of God, from what we customarily encounter.  Or rather, as it may be, many different sides: the mystery of the Trinity becomes the mystery of the Infinity.  For we must not think of “Math” as just some subject in school, or as a section at the bookstore, beyond “Gardening” and next to “Pets”.  For one thing, there is just so much of it,  acres of math like fields of wheat, with more unfolding with each passing day, and much which, when first met with, seems qualitatively, drastically diverse:  not like different species, say a wolf and a fox, but like different phyla – a microbe and a mastodon.  And even as science has discovered some of the commonalities between mastodons and microbes, in the process deepening our appreciation of each, so too does the steady, then accelerated, and finally springing advances of our collective understanding – as it might be, the Mathematical Overmind – deepen and widen and heighten and… beyonden  our sense of the unitary structure of All There Is.  The whole enterprise is so fantastic, with such unity-in-diversity (again, compare the Trinity) that whole new fields have evolved at a metalevel, just to keep tabs on it all:  Set Theory and Proof Theory, to police our reasonings, and Category Theory, to provide display cases for all the genera of the menagerie, in the museum of the mind. 

            At the bookstore-cum-giftshop in Hilbert’s celebrated Hotel, you will find aisles for:  History; Fiction (including Astrology and Economics); Physics ‘n’ Chemistry; Biology; Number Theory; Point-set Topology; Algebraic Topology; Algebraic K-Theory; Topological K-Theory; Real Analysis; Complex Analysis; The Riemann Hypothesis; Sheaf Theory; Topos Theory; The Poincaré Conjecture.;  and Miscellaneous (i.e., gardening, computers, self-help, sports, celebrities, stamp-collecting, and all the rest).  There is no separate section for Theology, since that overlaps all of them.

            Now, none of this is exactly new: it is paleo/retro-NeoPlatonism. A retread, you may say, and twice-refried.  Yet there is now much more concrete substance to the view, than was available to Plato or Plotinus. They might imagine the cube and the icosahedron, and Kepler might attempt to stuff the planetary orbits into such homely and visible (risible) boxes, but they never encountered a Riemann manifold – or rather, they did, because we live in one, but they couldn’t see it  so they couldn’t imagine it, any more than they imagined fibre bundles or E8
            The Atheist, viewing a world charged with the grandeur of God -- which is difficult to ignore, since it will flame out, like shining from (to coin a figure) shook foil -- sniffs and dismisses it as tinsel: seeing the shook foil but not the Shaker.  So too the Nominalist, beholding or rather failing to behold  the serried ranks of theorems rising like seraphim beyond sight, regards these as a mere medley of contingent things, simply frothed out of someone’s brain, and which might, like a limerick or a pop-tune, have frothed out into something quite different.  (This is if anything the more charitable of contemporary dismissals, vice the dismissal of math and science as being merely the Eurocentric patriarchal dogma of the racist sexist agist ablist ruling class…)


            The nominalist viewpoint was given epigrammatic utterance (1886) by Kronecker, thus:

“Die ganzen Zahlen hat der liebe Gott gemacht; alles andere ist Menschenwerk.”

(“Ganzen Zahlen” means integers, but he may have meant only the non-negative integers, or “natural numbers”.)
Kronecker, rueing the day that ever he denied the transcendence of higher mathematics

(Randbemerkung:  A delightful swipe at Kronecker, well below the belt, and delivered with punch by a pugnacious Platonist, can be savored here:  The Continuum. )

Stephen Kleene translates, “God made the integers, all the rest is the work of man”, and glosses:
(Introduction to Metamathematics, p. 19.)

The natural numbers – non-negative, non-zero whole numbers -- were, indeed, the only numbers recognized by the Pythagoreans.  One reads somewhere of a Pythagorean  casting himself into the sea in despair, upon encountering the proof of the non-rationality of the square root of 2; in fact, such self-drowning might just as well have been prompted by the sight of half an apple, since once you accept fractions, you are heading straight for E8.

            For in admitting the integers as being in no wise contingent – the work, indeed, of the Necessary Being – while desiring to dismiss the rest (“Menschenwerk” sounds even more like a kindergartner’s art project than the more Biblical “the work of man”), Kronecker has left the castle of mathematical agnosticism unguarded, by leaving open its postern gate.  Suppose we were – setting aside centuries of other riches – to begin by restricting ourselves to the laws of the natural numbers.  We would notice (as Euclid noticed) the primes, and require, for their adequate handling, great heaps of Number Theory.  Now, this already is no small thing.  You could fill a succession of lifetimes with nothing but Number Theory.  New discoveries emerge daily – some of them with such grave implications that they are actually classified, and at compartmented levels well beyond Top Secret, which does well enough for the design of an airplane or the movement of troops.  But the wealth is not just quantitative.  For to adequately handle nothing more than the primes, you need all of Number Theory, including even Analytic Number Theory, which brings in the continuum and the charmingly designated “imaginary” numbers (now much easier to imagine, though they remain as invisible as the number “23”), including indeed the Riemann Hypothesis, which already brings us to the frontier of knowledge with its outstanding unsolved problems.  Soon the whole of mathematics would come tumbling in through the unguarded door. 

[Note:  It’s never that simple.  I am aware that Kronecker himself was even more nominalistic that the famous quotation might suggest, as he did not accept the integers as a finished totality.  He went to some effort to derive results in a way that makes no use of such a totality.  Sort of neat if you can pull it off, like building a castle entirely out of toothpicks.  But if the idea of an actual infinity is problematic, that of the integers somehow running out of breath is even moreso.  I shall accept the natural numbers as given, and shall, for polemical purposes,  portray, as our foil, a sort of idealized Kronecker (who may even now be repenting of his nominalism, in some warm place) – the Kronecker of the quote – as accepting them as well.]

[As to the actual forked-radish of that name,  Joseph Dauben remarks (Georg Cantor, p. 66):
No-one could have been more opposed to Cantor’s ideas, nor have done more to damage his early career, than Leopold Kronecker.

Really, were it not for his key concession that the natural numbers are God-given, Kronecker might well be the villain of the piece.  As George Szpiro wrote of him (in Poincare’s Prize):  “Kronecker would not accept anything if it had not been invented by himself.”   Actually that can’t be quite accurate, since on that account he would not accept the integers… But anyhow, a perfect summary of the solipsist/nominalist epistemology.]

[The essay continues here.]

Wednesday, December 29, 2010

How Now, Round Cow

[Note:  It has been maintained that Origen held that “the resurrected body will be spherical”. Henry Chadwick, The Early Church (1993), p. 106]


It is an old joke and a good one, retold here with some improvements.

~ ~ ~

United General Dairy wished to enhance milk production, so they called in an engineer, a physicist, and a mathematician.

The Engineer said, “No problem. Gimme a week.”
Good as his word, he showed up a week later  with an elaborate 3-D CAD prototype of an Advanced Magnetic Milk Extractor, consisting of a reverse rotor hardwired to an alternator (see diagram 12-c) routing through an ANSI-standard bolometric squinch, relying on either hex-nut variable findipulators, or (depending on parts availability) …
It was all very clever, but the humble dairymen couldn’t figure out how to work the thing, and figured they’d spend all their time in tech support instead of milking cows, which is what they liked to do.  So they sent the engineer on his way.

The physicist frowned, pondered a bit, then said:  “Doable. Fund me for a month.”
Thirty days later he returned, visibly pleased with himself.  “This is so much more elegant than what that engineer came up with.  A simple cylinder, 500 miles long.  Behold, gentlemen:  The Relativistic Linear Cow Accelerator!  Insert cow at one end, she emerges at the other, with (provably) every last lactic atom extracted, and placed into appropriate containers.”

Physicist launches a cow

Management was impressed, but inquired as to the cost.
“Ohh,” said the physicist with an airy wave of his hand. “A billion, a trillion, something in that range.  Ask Congress.”
Calculating that a pint of milk would have to retail at over a million dollars, management bade the physicist adieu, and turned to the mathematician.

The mathematician, however, did not turn to them.  He was… thinking about something.
Eventually they managed to snag his attention, and explained the problem.  The mathematician slowly nodded.  “It’s really a most intriguing problem… with ramifications in unexpected directions…. Allow me a year’s sabbatical, and I might have something for you.”
Management shrugged, and basically forgot all about him, until, at the stroke of noon, one year later to the day, the mathematician burst in, his moon-face beaming.
“Gentlemen, I have it.  Consider a spherical cow….”

* * *

As with many a good joke – those that make you smile instead of smirk or snicker – there is a theological dimension to this. 
Just what it is, is difficult to put into words – unless you are Chesterton, for whom it was (so literally) child’s play; and who put it thus:

            I find that most round things are nice,
            Particularly Eternity and a baby.

This says it all, but a footnote for mortals.  For you see, the thing about spherical cows is, they are so  ----- cowishly round, so… profoundly round, so – so round all about:  yea,
take them from this end     or take them from that,
they are
         round all around. …..

And this, indeed, is worth considering,
well merits our contemplation,
and our meditation,
through many an eternity afternoon …. 


Bonus poem:  Symmetry viewed by a mooncalf  (Rilke):

Ach was ist das für ein schöner Ball !
Rot und rund wie ein Überall.
Gut, dass ihr ihn erschuft.
Ob der wohl kommt wenn man ruft?


Good heavens... I Googled "how now round cow", which I'd fancied a basically new tweak of the traditional "how now brown cow", just to see if the search engine was updating its indexing of this site -- turns out there are already tons of sites that use this phrase.   Nothing new under the sun.

So:   TWoDrJ still rules the "humble woodchuck" universe, but is at present an also-ran in the lovely rotund world of Round Cows.


William Thurston, Three-Dimensional Geometry and Topology (1997), p. 103:
Just like the circle and the two-sphere, the three-sphere is very round.  But there are some beautiful, classical aspects to its roundness  that are not easy to guess from its lower-dimensional sisters.

The easiest way for a human to visualize the three-sphere is as the one-point compactification of ordinary three-dimensional Euclidean space (basically, you adjoin a point at infinity and define open sets as sets containing this point and with compact complement).   But, the author cautions,
this picture suffers from a loss of symmetry: [this compactification] is not as round as it should be.

 Pleasingly, a Google search on “not as round as it should be” brings up the Thurston quote as the very first hit, ahead several pages of more humdrum physical uses.


Here is an actual unretouched photograph of a Spherical Cow:

Let the welkin resound with the rotundity of round!
Here you can behold a genuine round square in captivity.

Grandpa, how did it all begin?

For answers to this and all your other questions about Life, Love, and the Universe, see
"Murphy's Theory of Cosmology"   

 ($100 registration fee  waived for those accessing via this site.)

Tuesday, December 28, 2010

De casibus marmotarum illustrium

The cosmos watched with bated breath as TWoDrJ battled its way in the worldwide “Humble Woodchuck” sweepstakes,  from Google purgatory, to the third page of hits, to the second, to the first…   But just as we were about to send out an end-zone dance, the site inexplicably fell back.

[Scene:  A desolate battlefield.]
Eheu fugaces!   Où sont les marmottes d’antan?  After clambering to the top of the scrap-heap in the Search Sweepstakes,  the Humble Woodchuck has been unceremoniously cast down, apparently by a cabal of hackers (we suspect the PRC), all the way back to the third page of Google returns.

Fortunae rota volvitur;
descendo minoratus;
alter in altum tollitur;
nimis exaltatus.

But we did not rest at that, nor did sleep gain any purchase on our eyes, until we clawed our way back (woodchucks have very sharp claws, don’t underestimate), all the way to the First, Second, and Third hits on the very first page !!  Bronze, Silver, and Gold!   Hat trick!

As you can imagine, I’ve been fending off the media all night.  Sound-trucks blocking the driveway, the whole nine yards.

[Scene:  The Top of the World.]
CNN:  To what do you attribute this incredible surge of power, virtually unprecedented in the history of the Net?  Is it because of massive amounts of pageviews on your site?
Dr. J:   Er, not exactly;  not quite yet.  These things take time…
CNN:  Your own personal charm, perhaps?
Dr. J (modestly):  Well! ah, for the matter of that -- possibly plays a part.
CNN:  Or is it, perhaps, that you bribed the search engine???
Dr. J (flushing):  Preposterous! -- But no, the explanation lies rather in this:  that our prize woodchucks are  as humble as they come!  Humble as the day is long, b’dad!  Heck, they can out-humble any brand-X groundhog out there, better believe it!  You want humble?  We got humble!  Why, you just take Harry here --
CNN (fiddling with his earpiece):  What?  Come again?  Oops, sorry, you’re old news now, Doc.  Gotta hurry off to the Next Big Thing.
Dr. J :  But -- !  But -- !  Don’t I win anything?

Eheu fugaces ….

E8: a Riposte (concluded)

Let us examine a bit more closely  Synge’s picture of physics as bricolage,  where theories have the intellectual status of just-so stories, and are really little more than pragmatic techniques, or tools -- Newtonian mechanics and relativistic mechanics each useful in its own sphere, like screwdrivers and spoons, but of little interest in their own right.   Now, this is not to knock the status of a toolkit -- my respect for competent carpenters and electricians borders on reverence -- but fundamental physics is not like that.

            Synge presents the Newtonian view as having not been replaced or refuted by relativity;  it rules as before in its own realm.  Newton’s good for some things, Einstein for others, and Wiccan no doubt for others still.   But this view assumes a confusion.  For it is not the case that Newtonism and relativity are independently valid in their own way but incompatible;  rather, Newtonism is the limiting case of relativity, in a way very familiar in mathematics;  its continued use in everyday life is simply a calculational convenience, a shortcut.   To continue the tool metaphore:  Einstein and Newton are not like screwdriver and pliers, but like a hammer, and an old shoe used as a hammer, good enough for the task at hand.

            Furthermore, it is a good thing, not a bad thing, when initially separate paths converge.  If you only know one way to climb a thing,  perhaps it is only a Potemkin mountain -- a paper-maché façade, hollow behind the north slope.   It is quite a relief -- and an ontological ratification -- to meet another mountaineering party that has scaled up the other side.
            The reader may be familiar with the story of how Schrödinger and Heisenberg separately found Rome by different roads.  Let George Gamow tell it, in Thirty Years that Shook Physics (1966), p. 3:

The simultaneous appearance of Schrödinger’s and Heisenberg’s papers  in two different German magazines … astonished the world of theoretical physics.  These two papers looked as different as they could be, but led to exactly the same results concerning atomic structure and spectra.

We are, in hindsight, not overly surprised by this, since by now we most of us accept that there is something there at the quantum level, something real, something other than subjective, to be described.   It is describable by two quite different mathematical approaches, much as our peak may be scaled by walking up the north face  or rappelling up the southern cliffs.    Nor is such ‘duplication of effort’ a waste of time, for  in this instance, not only the factual success, but even the approaches themselves retained their usefulness -- for determining energy levels, Schrödinger’s wave mechanics was calculationally more convenient; and Heisenberg’s matrix methods had the edge when it came to calculated the intensities of the radiated frequencies.   Or, alternately, P.A.M. Dirac, The Principles of Quantum Mechanics (4th edn. 1958), p. viii:

Quantum mechanics … is known under one or other of the two names ‘Wave Mechanics’ and ‘Matrix Mechanics’, according to which physical things receive the emphasis in the treatment, the states of a system or its dynamical variables.

And (p. 115):

The Schrödinger form is the more useful one for practical problems, as it provides the simpler equations. … Heisenberg’s form for the equations of motion  is of value in providing an immediate analogy with classical mechanics.

Or again (R. F. Streater & A. S. Wightman, PCT, Spin & Statistics, and All That (1964), p. 4):

Throughout this book, states will be described in the Heisenberg picture of quantum mechanics.  The Schrödinger picture is much less convenient for the description of a relativistic theory, because it treats the time coordinate on a very different footing from the space coordinates.


P.A.M. Dirac, The Principles of Quantum Mechanics (4th edn. 1958), p. 311:

The Schrödinger picture is unsuited for dealing with quantum electrodynamics, because the vacuum fluctuations play such a dominant role in it. … They get bypassed when one uses the Heisenberg picture, and one is then able to concentrate on qualities that are of physical importance.

Dr. Matrix
Dr. Wave

Approaching an abstract but genuine reality from two different theoretical complexes  has its counterpart in different experiments, or different means of calculation, strengthen each other when they arrive at the same result.   Thus Einstein, in his annus mirabilis of 1905, when not inventing Relativity, found it worth his while  to “develop theoretically  three independent methods for finding Avogadro’s number.” (Abraham Pais, Subtle is the Lord (1982), p. 55.)   It was worth his while because, independently of our endeavors, this number is indeed there.

Summarizing:  For epistemology, the fact that two or more radically different approaches each manages to describe the phenomenon of interest, reassures us that we really do have our arms around this thing.   The lesson goes over, I would submit, in cases where what is being described is nothing so tangible as an atom (which Rutherford reportedly saw in front of his face as plainly as a spoon), but rather a four-manifold, or a simple Lie group.

~ ~ ~

I recently happened across the following curious passage:

The algebras G_2, […] E_8  are called exceptional.  In 1945, Chevalley remarked  that the existence of these algebras  is a brutal act of Providence  which we must accept blindly.  Perhaps this should be revised today  to assert that the source of these algebras  is the wisdom of the Deity  in allowing the Cayley numbers to exist.
-- Irving Kaplansky, “Lie Algebras”; in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 126

~ ~ ~

There’s one further type of brane in M-theory  that is really surprising.  This brane is the edge of spacetime. … The photons at the edge of spacetime participate in supersymmetric E8 gauge theory.
-- Steven Gubser, The Little Book of String Theory (2010), p. 95

Monday, December 27, 2010

E8: a Riposte (continued)

[A continuation of this.]

Synge waits until well into his second volume (J. L. Synge, Relativity:  The General Theory (1960), p.  104) to really let rip against Realism; and since he was himself very much a mathematical physicist, rather than an empirical experimenter, his testimony must be respected as coming from within the tent.  He distinguishes “Natural Observations” (NO) from “Mathematical Observations”, and opines:

Between NO and MO  there is a sharp and decisive break.  Only the simplest MO (counting) can be regarded as being NO also … Generally MO involve infinity (irrational numbers, differential calculus, and so on) and so lie outside physics and outside nature.

This is exactly the position of Kronecker (“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk”).

He then delivers himself of a curious passage, which Irrationalists would seize on with glee (fortunately none of them are reading this):

The peculiar fascination of theoretical physics  lies in the art of forcing meaningful truth out of the meaningless equation NO = MO, which is a symbolic form of the assertion that natural phenomena obey exact mathematical laws.  The true inequality NO =/= MO should not be spoken above a whisper, because it is extremely dangerous.   If believed, it would sever mathematics from physics, and reduce both to sterility through lack of mutual fecundation.  It is whispered here only as an apology to those readers who expect to see the mathematics of relativity [which he presents in great detail] tied to the physics of relativity  by a strong chain of clear thought.  It cannot be done.

These ring like the Night Thoughts of a relativistic physicist, on the eve of taking his own life.
            Despite his conspiritorial tone in that passage, Synge was by no means alone in his reservations. Here is another anti-Realist view from the world of physics:

A. D’Abro, The Rise of the New Physics (1939), vol. II, p. 728:
A hyperspace is obviously a mathematical fiction; and waves that can be represented only in a fictitious space  must themselves be unreal.

Now here indeed is a statement that has been overtaken by events.  In the view of string theory, this hyperspace, far from a mathematical fiction, is a physical fact, the one we live in;  indeed, we must beware lest those compactified but very real extra dimensions someday unfurl in our faces.  --  The point here being, not to make any point whatsoever about cosmic geometry, let alone to proclaim the truth of string theory:  but simply to counsel against that “obviously”, when dismissing the Realist picture.

[concluded here]

Sunday, December 26, 2010

E8: a riposte

[a continuation of this]

Writes J. L. Synge,  in Relativity:  The Special Theory (2nd edn. 1965), p.  163:

According to this hypothesis [viz. that of a unique mathematical structure for nature], the mathematical formulae of physics are discovered  not invented,  the Lorentz transformation, for example, being as much a part of physical reality as a table or a chair.

Hear, hear! The Realist raises his glass with a sigh, basking in the warm glow of these purling words.  -- But suddenly, the author surprises us with a basin of ice-water in the face:

But this hypothesis of a unique mathematical structure for nature  is actually very naïve.  It is the product of the eighteenth century, a period when mathematics was understood much less than it is today, and it is unacceptable to any physicist who has thought about mathematics, or any mathematician who has thought about physics. [Oh, snap!] When understood properly (i.e. as mathematicians understand them) these concepts exist in the human mind and not in nature;  it is a meaningless waste of time to debate whether the ratio of two measured lengths is rational or irrational, or whether matter is continuous or discontinuous, because the concepts of irrationality and continuity belong to a world of the intellect, a world of mathematics, and not to the real world in which phenomena occur and are measured by pieces of apparatus.

(… slow burn…)

And again, p. 207:

Is matter really discrete or continuous? … That question must be regarded as quite meaningless.  For ‘continuous’ is a mathematical word, not a physical word, and has only a very vague bearing on nature;  we must not try to attach physical meanings to mathematical concepts which involve infinite processes.
[…]  The above remarks merely underline the philosophical attitude [quoted above].  It is a theme that bears repetitition.

( Somehow in these last bits, I detect the tones of Dolores Umbridge…)

And p. 308:

We use  now this mathematical representation, now that,  seeking those representations which are convenient to work with  and which yield at least some correct physical predictions.

            Now, we quite agree that it doesn’t make sense to say whether a numerical physical measurement is rational or irrational.   And certainly, in physics, infinities are tricky.   The continuum may indeed not be what the doctor ordered for the texture of spacetime.  And we agree that it may be convenient to employ this or that formalism for this or that particular problem.  (We shall develop this point further in our discussion of the wave picture vs. the matrix methods in quantum mechanics:  but shall draw a very different moral from that of Professor Synge.)   And yet we hold to our Realist position:  which has, moreover, practical consequences, and is not simply a matter of private preference in ontology.

            The Realist claim is that mathematical objects are as real as physical objects.  That is not to claim that any given mathematical object -- be it the continuum, E8, or Klein bottles -- is actually instantiated in this particular physical cosmos, let alone that it applies everywhere and across the board.   Thus, take the continuum.  Our own spacetime may well be granular, not continuous;  the cosmos might be finite, both in diameter and duration.  It might even be downright cellular, as in Stephen Wolfram’s view.
Furthermore, the continuum itself is among the most mysterious of mathematical objects;  it is, after all, the eponym of the spectacularly counterintuitive status of the Continuum Hypothesis.  It sticks in the craw even of some mathematicians (though not the ones we prefer to share a beer with), let alone physicists or shopkeepers.

            However.  Considered purely as an object of group theory, E8 has calculable values along certain dimensions of assessment  (algebraic properties): Let us call the dimensions of assessment  "alpha, beta, gamma" … , and the values we calculate  "alpha-0, beta-0", etc.  Thus, for E8 (reading the values out of Wiki), alpha might be “What is its dimension?”, and alpha-0 turns out to be 248;  beta might be “What is its rank?”, and the answer “8”;  “What is its center?” -- “Trivial”; “What is the order of its Weyl group?” -- “696729600” (but you already knew that) …
            Now:  Suppose that spacetime does turn out to have six extra compactified dimensions, and that heterotic string theory is indeed the gospel truth.  Now, those in a position to know, inform us that  in that case  there are only two possible choices for the gauge group of our world.    Recall that earlier gauge groups we’ve met were things like the rotation of a wheel, things whose reality is familiar even prior to their employment as gauge groups:  but now we must choose between  SO(32) and E8 X E8.   The choice must be made on the basis of certain physical predictions.   Assessment-dimension lambda, for instance, might turn out to be physically measurable (specifying, say, the spin of a graviton, or the mass of a magnetic monopole, or what have you);  lo and behold, lambda-0 in E8 X E8 turns out to correspond what we have measured, whereas the different value that falls out of SO(32) does not;  and so forth for some other assessments.
            All right then:  The Realist thesis in this case says simply that adoption of E8 -- which is, in this view, pre-existent, and independent of physics just as it is independent of politics -- is a package deal.   Not every fact and feature about E8 will be physically interpretable, let alone measurable;  but for any that is -- say, omega -- the value as measured experimentally must turn out to be omega-0.   If it doesn’t, we have a problem.  E8 being pre-existent, and not invented by ourselves for our own convenience, it cannot be toyed and tinkered with, selecting some features and rejecting others in a Procrustean attempt to match experiments.   And Synge’s picture of using Newton or Einstein, merely as the mood moves us, as though one were an Allen wrench and the other a pair of pliers, won’t do at all.   If the mathematical structures we seem dimly to glimpse behind the veil of the world  were analogous, not to mountains, but to man-made tools, then we could always kludge one up for the occasion.   Then physics would be the theoretical equivalent of toenail clippers and pinking shears.

[continued here]


(Chastened by the stern but just admonishments of our Canadian colleague, regarding the tinsel and pinchbeck allure of Web celebrity based merely upon citizens’ obsessions with CUTE HEDGEHOGS and PLAYFUL PENGUINS,
we return now to the straight and narrow of Cantorian realism, leaving aside all reference to our animal friends, not excluding the lowly hedgehog (who knows but One Big Thing) or the HUMBLE WOODCHUCK.   You will find no mention of the HUMBLE WOODCHUCK in anything that follows;  and he that were so foolish as to search on the string

!! => “humble woodchuck” <= !!

in hopes of googling-up this essay, would surely search in vain.)


Before we proceed further, so that we have some acquaintances in common, let us consider our new friend,  E8.  This way, when I have occasion to refer to him again in future, we shall all give a knowing nod.

Now, E8 is, frankly, not such a big deal:  it’s just a single, exceptional simple Lie group (albeit not an exceptionally simple one), even if it does turn out to be the symmetry group of string theory, and thus of all the world.  For actually, all our major experiences of life – our loves, our sorrows, our proofs of the Riemann Hypothesis --  are carried out  quite independently of string theory (be that worthy enterprise  well-founded or no).  No need to get all misty and mystical about it, though journalists and publishers love to do so because it moves more newsprint and books.  (The “God Particle”, egad; it’s just a frigging Higgs boson.)  You could with as much reason wax mushtical over the mere numbers zero and one – the Nihil and the Ens, if you wish – who heroically alone shoulder all the burden of binary description, which encompasses so much.
            Nevertheless, in its own small way, E8 does have a certain fascination for us, the fascination of a small thing, perfected past admiration, as by a master craftsman, with all eternity wherein to work, and whose very existence seems a paradox, like a spinning top.  After all, word of this fait divers from the normally cleidoic mathematical world  did manage to make it into the print editions of both Le Monde and the New York Times,

thus elbowing out whatever might otherwise have occupied those column inches, be it an account of an auto accident, or a lost cat.  And this, despite the utter incapacity of the journalists to give us the least idea of what has been actually discovered.  It is as though they had sent a correspondent to cover a major speech, and then reported, “We could not make out a word he said.”  As the Times reporter put it (beneath the swooning headline  “The Scientific Promise of Perfect Symmetry”):

Eighteen mathematicians spent four years and 77 hours of supercomputer computation to describe this structure, with the results unveiled Monday at a talk at the Massachusetts Institute of Technology.
But it still is not easy to describe the description, at least not in words.
“It’s pretty abstract,” conceded Jeffrey D. Adams …
“You can’t really picture it,” Brian Conrey, executive director of the American Institute of Mathematics, said of E8--

and then offered his own endearingly goofy stab at a depiction: “It’s some sort of curvy, torus type of thing.”  (A torus, for those who are not aware of this, is a donut with a college education.)

Most previous symmetries have been simple enough in themselves -- like, translation (that is, just boogying along in a straight line), which is as simple as it gets -- but startling in physical consequences.  Thus:  symmetry in translation along the time axis -- and  voilà , Conservation of Energy!  Rotational symmetry -- conservation of angular momentum!  Likewise, to say that “the gauge symmetry of the electro-magnetic field is U(1)” is again to invoke the familiar wagon-wheel.  Even the PCT intricacies (Parity, Charge, Time) are based simply on the shuffling of easily visualisable bivalences (left vs. right, positive vs. negative, sooner vs. later);  the implications of these for physics, however, are beyond the reach of most of us.  With E8 we arrive at a symmetry group that is itself well-nigh incomprehensible -- certainly not surveyable without very extensive practice and training.  As for its ultimate physical implications -- anybody’s guess.   But whatever its ultimate fate in physics, the fact is, E8 is already there, in just the same way that the symmetry of a rotating wheel is there, and would still be there even if our cosmos happened not to contain anything physical that actually rotates.  We discovered E8;  we didn’t invent it.

[Update 4 Jan 2011:  The above provoked a playful and entertaining meditation by our Canadian Colleague:

[continued here]

Wednesday, December 22, 2010

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:
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.