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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