The Urysohn Metrization Theorem: an Apology

The other day, we learned to our distress, that, owing to an earlier post, a Google search on the phrase 

 => “ Urysohn Metrization Theorem “ <=

brings up our own Cantor-cum-woodchucks site  on the very first page.   This is embarrassing, since we have never said anything substantive  specifically about this well-known theorem.   Innocent youngsters searching on this phrase may be led to a page that only purports to deal with that celebrated result of point-set topology, but which actually is simply a satire on sociobiological overreaching, a genre practiced by G.K. Chesterton a good hundred years ago (in The Everlasting Man and elsewhere).
[We interrupt this post to bring you a sobering update, 11 I 11:  that first-page hit has been axed, probably by the Nominalist International.  For an an earlier such episode, click here.]

By way of atonement, we present a picture of the saintly man himself: 

"Golly, I sure do hope y'all enjoy my swell theorem!"

            To be sure -- the Urysohn Metrization Theorm was a spot-on choice, among many possible such choices, to represent something for which an adaptationist account must collapse of its own absurdity.    Nor can the U.M.T.  readily be smuggled into a “spandrel”, as an exaptation of some more basic skill.  It would be a stretch even to demonstrate the differential, for successful procreation, of the… conscious… ability to count up to twenty.   All sorts of complex non-conscious, instinctual behaviors can be of such value, as witness those mighty engineers, the spiders.   (Note:  I became fonder than ever of spider-kind, upon reading of the sheer variety of their engineering styles, in the pages of Richard Dawkins.   Let that be noted here, in case I am ever driven, nolens volens, to say something not-nice about the man.)  But conscious mastery is requisite, though not sufficient, for extension of our curious mathematical faculty  to such attainments as proving -- nay, even conceiving -- the Poincaré Conjecture. 

            So:  Instinctual unconscious knowledge  by no means suffices  for further conscious exfoliation of its possibilities.   Neither birds nor bats will ever invent the moon rocket.   But note further -- at this point  simply as a curiosity -- that our instinctual arithmetic armamentarium  is pretty paltry.   Some languages don’t even have words for integers above three or so.   (I am leaving a lot out at this point, in the logical sequence;  but were it filled in, it would only strengthen the case.)   I came across a passage recently in which one scholar wished to project all of mathematics from our instinctual ability to recognize groups of two objects, as such, and of three objects, as such.   But he misses a subtle logico-linguistic point.  That ability, whatever it may be, is not in itself evidence of the ability to count -- that is to say, to have mastered the successor-function, and have grasped the resulting structure.   The instinctual world of such peoples is populated by pairs, and by triads, and perhaps quartets and, for some, maybe even a quincunx:  but these no more form an intrinsically and extendably ordered sequence than do squares and triangles.   Pairs and triads populate the tribe’s ontology -- but qualitatively, not quantitatively:  much in the manner of tigers and lions.  Such ability (which scarcely extends beyond half a dozen or so, apart from supposed cases of idiot-savants) is not an instance of actual counting, but a qualitative (non-quantitative) substitute for it -- and which will help you get by, so long as your horizon does not extend much beyond the six-pack.

            Other adaptationist accounts of advanced (and by no means universal) human capabilities -- say, Mozart is to birdsong as wings-for-flying are to winglets-for-heat-exchange -- though really just just-so-stories, are still ingenious enough, and have a certain plausibility:  birdsong is, after all, demonstrably utilized in courtship, and likewise in our own species:  the swain beneath the balcony, with his lute.    You’ll have a harder time with Schoenburg -- dodecaphony ne’er won fair maid.   But mathematics is far more recalcitrant, because vastly more developed.  To say it’s all just a spandrel -- well, here the spandrel would be bigger than the entire cathedral.

*

            Actually I am quite sympathetic to the program of trying to discover as much as we can about the causal-temporal structure of the biosphere along Darwinian lines, as I am sympathetic to every serious attempt at explanation which employs -- always bounded by common sense -- a due reductionism.  Thus, parts of chemistry have indeed been illuminated by (if not quite “reduced to”) parts of physics:  quantum mechanics and the structure of the simpler atoms; statistical mechanics and aspects of thermodynamics.   The complexity of the biological Creation becomes only more beautiful when lit up by the insights of evolutionary science -- what had seemed random and unrelated, even absurd (what’s the whale doing with that tiny useless floating bonelet where a hipbone ought to be?), stands revealed as a splendidly articulated and unfolding (“e-volving”) panorama:  no longer mere complexity or complication, but intricacy, worthy of a watchmaker.

*

            To return to the Urysohn Metrization Theorem and to our heartfelt apology.   We feel a responsibility to those unsuspecting souls, astray in the Googlewoods, who in quest of enlightenment concerning the Urysohn Metrization Theorem, should wind up at our own humble cottage door.   Accordingly we have formed the intention to make good, and to come up with a narrative that may illuminate this result (the Urysohn Metrization Theorem, I mean), beginning with no more than the reader may recall from high school geometry, yet winding up with some intuitive feel for that theorem (namely, the Urysohn Metrization Theorem).   This represents a serious undertaking, since first we’ll have to understand the darn thing  (the Urysohn Metrization Theorem, that is).  But we are inspired by the splendid example of Sir Jeffrey Weeks (we have just now decided to knight him) in The Shape of Space.   If we can do, for the <<Urysohn Metrization Theorem>>, even a part of what he did in bringing the geometry and topology of three-manifolds within the grasp of the suckling undergraduate, we shall not have lived in vain.

Note:  Anyone who fancies that this post about the Urysohn Metrization Theorem, which bears the phrase “Urysohn Metrization Theorem” right in its title, is simply a ploy to keep our stats up (namely as regards a phrasal search on “Urysohn Metrization Theorem”, or perhaps “Urysohn’s Metrization Theorem”, or “The Metrization Theorem of Urysohn”, or even “Urysohn:  Whither the Metrization Theorem?”, “The Urysohn Metrization Theorem for Dummies”, “The Urysohn Theorem:  Metric or Menace?” and “The Urysohn Metrization Theorem explained for the Tired Business-Man”, usw.), is just plain cynical.

'Nother M&M

 [“Bite-sized essays for the busy businessman” ®]

A. D. Aleksandrov, in his splendid  essay on Non-Euclidean Geometry for the Soviet Mathematics overview (vol. II, p. 177, of the English translation published by the MIT Press), is quite concerned to defend mathematics from any charge of being a “just-so story”, or abstraction for its own sake; and this, for motives rather opposite to those that move mé. He, in the service of orthodox Marxism; I, in the service of orthodox Christianity.  But in this case it comes to the same thing:

Riemannian geometry has its applications as a method of abstract-geometrical description of physical phenomena.  This description is not at all arbitrary, and is not an idle play of the mathematical mind; it reflects the real mechanism of the phenomena in question, but reflects it in an abstract form.

I greatly look forward to meeting Aleksandrov in the next life.  (Eventually.  I reckon we’re both headed for an extended stint in purgatory, though for different reasons.)

Quine

Hovering, buzzing in the background, as I write these notes, is the solemn, rounded, currently extraterrestrial figure  of Willard Van Orman Quine.

I took Introduction to Logic from him  sophomore year  -- “Phil 140” -- one of the very few course designators I remember, along with  “Math 11” (Robin Hartshorne), “Math 55” (Andrew Gleason), and “Nat Sci 2” (George Wald).   All these went into shaping the man I am, quite as much as did the Y chromosome.

Simply as a stylist, he is almost my favorite writer -- right behind Chesterton.  Any paragraph at random, from either man, is guaranteed to delight, both in style and in substance.   For better or worse, his tight and chiselled, somewhat precious prose, infuses my own;  and he returned the favor, in a generous letter, praising The Semantics of Form in Arabic, which else must seek far and wide for any mention, let alone praise.    His style is mesmerizing -- I never find myself disagreeing, when I read his words;  though a paraphrase is never so compelling.

Yet on a core point of these essays, we seem to be at loggerheads.  Consider the following, from the celebrated “Two Dogmas of Empiricism” -- the original version in the Philosophical Review (1951);  a passage omitted (as Scott Soames points out) from the more accessible collection of essays, From a Logical Point of View:

Imagine, for the sake of analogy [“analogy” because his real game is the posit of physical objects, which he likewise deprecates], that we are given the rational numbers.  We develop an algebraic theory … but find it inconventiently complex, because certain functions,  such as square root,  lack values for some arguments.   Then it is discovered that the rules of our algebra can be much simplified by conceptually augmenting our ontology with […] irrational numbers.

So far, no quarrel at all.   But now we restore what we had suppressed in those square brackets:

… with some mythical entities, to be called irrational numbers.  All we continue to be really interested in, first and last, are rational numbers;  but we find that we can commonly get from one law about rational numbers to another  much more quickly and simply by pretending that the irrational numbers are there too.

            So:  The challenge to the Realist, is to demonstate, that the irrationals (so invidiously named) are indeed part of the fundamental furniture of the universe, and not mere spectral butlers, bustling about among the throning rationals, servile and ultimately dispensible.

            At present, I cannot meet this challenge.  In the first place, because I don’t understand much about the continuum, other than that it is a depthless well of mystery and paradox, and so don’t really know what to make of irrational numbers.  From Quine’s passage, you might imagine that they are harmless, simply a “rounding-out”, like adjoining an ideal point at infinity:  but they are much more than that.    With the rationals, we haven’t really left the comfortable, Kronecker-approved world of the integers:  the countable case.    Yet open the barred door, and the winds blow in.   “But to the rationals do the gods inherit;  beneath are all the fiends.”

            So in the meantime, while mulling it, here at least is one thought.  Even if you wished to spurn fractional rationals (on the grounds that there is an infinity of them in a thimble, and you don’t like infinities), and wished to stick only to the positive integers -- you would still run smack into the irrationals.   For, an isosceles right triangle with sides equal to unity has a hypotenuse measuring the square root of two.  --  OK OK, you say, I’ll buy irrationals, but not all of them: just algebraic numbers (the set of which is still countable).  -- And now you are on the slippery slope, right where the Realist wants you.  “I’ve got a couple of transcendentals [non-algebraic numbers ] I’d like you to meet, pi and his buddy e.  They’re right outside the door… and the window… and on the roof…. In fact, you can’t miss them.”

-- Egad, this just in!  Quine, replying to his critics, in Hahn & Schilpp, eds., The Philosophy of W. V. Quine (1986), p.  315:

            I admit the real numbers.

All is forgiven!  Van!  We are at one!