The Urysohn Metrization Theorem: an Adaptationist Account (updated)

People in labcoats have been puzzling over the preponderance, over a wide range of far-flung and disparate societies, of belief in the Urysohn Metrization Theorem, to the effect that every regular topological space with a countable basis is metrizable.  How to account for this strange coincidence?

A rear-guard of Platonists and theists would persist in maintaining, that every such space is, as a matter of sheer fact, metrizable;  that the fact is “out there”, like a mountain, whether or not you or I are aware of it, and whether or not we can assemble some semblance of a demonstration to “climb” it -- to clarify the assertion, make it plausible, or to ‘prove’ it in some sense.

This, however, is not the method of modern science, which spurns the affordances of mere reason, and denies the evidence of our eyes, relying instead on various  techniques and equipment in well-funded laboratories.  Accordingly, herewith an account of how belief in the Metrization Theorem arose spontaneously, by the proven processes of Natural Selection.

You see, many many years ago, a number of tribes roamed the savannah.  Some went picturesquely naked, others were draped in animal skins.  And one of these tribes, fancying that the stronger separability criterion of normality was required, whereas the only spaces to be found in their ecosystem at the time were merely regular, despaired of ever metrizing anything; sickened, and died.   Another tribe failed to reckon with the necessity of a countable basis (mere first-countability being insufficient), and promptly went extinct.  Still another lowballed the separation condition, imagining that merely being Hausdorff was enough;  their metrizations went awry, and they were eaten by mastodons.

In this way, in the fullness of geological time, the only tribes remaining  possessed an innate belief  in the so-called Theorem (which is itself, of course, completely meaningless.)   Current estimates place the gene for this Theorem on Chromosome 14.

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~ Commercial break ~

http://www.linguasacrapublishing.com/justice.html

We now return you to your regularly scheduled essay.

*     *     *

[That was a philosophical satire.  For something a bit more substantive concerning the theorem in question, click here.]

~ ~ ~

Analytical appendix:

‘Twere a mug’s game, to cite specific instances of sociobiological overreaching -- just-so stories that purport to explain Love, Music, Art, what have you.  Chesterton already skewered these several generations back.   More worth noting are the (rare) cases where such thumb-sucking is found among mathematicians themselves.

Thus Reuben Hersch (apparently during a brief psychotic episode) wrote (“Some Proposals…” (1979); repr. in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 23):

Our mathematical ideas fit the world  for the same reason that our lungs are suited to the atmosphere of this planet.

Yet the author knows better.  Just a bit further up the page (with his customary lucidity) he wrote:

Consider the theorem  2^c < 2^(2^c), or any theorem in homological algebra.  No philosopher has yet explained in what sense such theorems should be regarded as referring to physical ‘possibilities’.

[Update 2 March 2013]  There is yet another, and deeper, level, to this Gedankensotie, which has only just now  become apparent.

At the very most superficial level, the piece is an attack of mathematical Realism.  Hopefully  none of my readers  understand it as such.

At the next, and still obvious, level, it is a defense of the same against Ultra-Darwinism (namely:  No conceivable considerations of mere individual survival can attach to these arcane topological considerations; ergo, the fact that the qualified international community is unanimous in embracing the truth of e.g. the UMT, is evidence for the transcendental truth of the latter.).  This is the spirit in which it was written, quite parallel to our essay “On the Existence of Penguins”.

But now (having re-read Freud, and been reminded of the role of the Wish-Fulfilment in dreams), I notice something quite further in this scenario of tribes perishing for misprizing the higher truths of topology:   namely, the wish that our evolution had been guided, not merely by such paltry contingencies as tricks of climate and the bite of the sabertooth, but by the very beckoning of such transcendant truths.  In pursuit of such ends,  gladly, I, and my tribe, would strive and maybe die!

Frontiers of ethnology

NPR just had a thinkpiece on race/nationalism/ingroup-outgroup etc.  Definitely a subject worth pondering.  But their idea of getting a “scientific” handle on the whole affair  was to interview some academic whose specialty is studying chimpanzees, but who was perfectly happy to pull further conclusions out of his fundament in the interest of fifteen seconds of fame.

In the classic phrase of the Jamies:  “I’m sorry teacher, but -- Zip your lip.”

Fortunately for the fair-and-balanced view, I have myself just completed a large and well-funded longitudinal-latitudinal-attitudinal study of the foraging and mating behavior of wild populations of sociobiologists and neuroscientists.   And can report that these untidy and ill-smelling beasts are remarkably focussed on their own faeces, which they regularly fling at normal decent working-people while hopping up and down and scratching their armpits.  Also Sprach die Wissenschaft.

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.

The Urysohn Metrization Theorem: an Adaptationist Account

People in labcoats have been puzzling over the preponderance, over a wide range of far-flung and disparate societies, of belief in the Urysohn Metrization Theorem, to the effect that every regular topological space with a countable basis is metrizable.  How to account for this strange coincidence?

A rear-guard of Platonists and theists would persist in maintaining, that every such space is, as a matter of sheer fact, metrizable;  that the fact is “out there”, like a mountain, whether or not you or I are aware of it, and whether or not we can assemble some semblance of a demonstration to “climb” it -- to clarify the assertion, make it plausible, or to ‘prove’ it in some sense.

This, however, is not the method of modern science, which spurns the affordances of mere reason, and denies the evidence of our eyes, relying instead on various  techniques and equipment in well-funded laboratories.  Accordingly, herewith an account of how belief in the Metrization Theorem arose spontaneously, by the proven processes of Natural Selection.

You see, many many years ago, a number of tribes roamed the savannah.  Some went picturesquely naked, others were draped in animal skins.  And one of these tribes, fancying that the stronger separability criterion of normality was required, whereas the only spaces to be found in their ecosystem at the time were merely regular, despaired of ever metrizing anything; sickened, and died.   Another tribe failed to reckon with the necessity of a countable basis (mere first-countability being insufficient), and promptly went extinct.  Still another lowballed the separation condition, imagining that merely being Hausdorff was enough;  their metrizations went awry, and they were eaten by mastodons.

In this way, in the fullness of geological time, the only tribes remaining  possessed an innate belief  in the so-called Theorem (which is itself, of course, completely meaningless.)   Current estimates place the gene for this Theorem on Chromosome 14.

*     *     *

~ Commercial break ~

http://www.linguasacrapublishing.com/justice.html

We now return you to your regularly scheduled essay.

*     *     *

[That was a philosophical satire.  For something a bit more substantive concerning the theorem in question, click here.]

~ ~ ~

Analytical appendix:

‘Twere a mug’s game, to cite specific instances of sociobiological overreaching -- just-so stories that purport to explain Love, Music, Art, what have you.  Chesterton already skewered these several generations back.   More worth noting are the (rare) cases where such thumb-sucking is found among mathematicians themselves.

Thus Reuben Hersch (apparently during a brief psychotic episode) wrote (“Some Proposals…” (1979); repr. in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 23):

Our mathematical ideas fit the world  for the same reason that our lungs are suited to the atmosphere of this planet.

Yet the author knows better.  Just a bit further up the page (with his customary lucidity) he wrote:

Consider the theorem  2^c < 2^(2^c), or any theorem in homological algebra.  No philosopher has yet explained in what sense such theorems should be regarded as referring to physical ‘possibilities’.