Saturday, March 2, 2013

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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We now return you to your regularly scheduled essay.

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

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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!

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