Saturday, January 15, 2011

The Urysohn Metrization Theorem (concluded)

(The continuation to this.)

Is there any distinction between a metrizable space and a metric space?   Seen naively, it’s the difference between a barn that hasn’t been painted yet, and one that has.
            Mathematically, the difference is insignificant.  Notice how one of the statements of the theorem  quoted above  slurs over the distinction:

     A compact Hausdorff space that is second countable is a metric space.

There is no mathematically interesting category of metrizable spaces prior to actual imposition of some specific metric -- analogous, say, to entangled quantum particles prior to collapse of the wave-packet, which are very interesting indeed, both philosophically (EPR Theorem, Bell’s experiments) and practically (quantum computing, quantum cryptography).  (For a quick course in the Uncertainty Principle, click here.)  If there actually were an analogy, how neat it would be, since in both cases  the final step involves (in some sense) “measurement”.
            There is, though, a lesson here for our larger project of Cantorian Realism, and the ontology and epistemology of mathematical objects.   Thus, consider a space (given initially as a base set and a defined neighborhood-system) which, after fiddling awhile, we find to be regular and second-countable.  Aha, so it’s metrizable, though knowing this does not by itself hand us a workable metric;  we’ll have to see what works.   Here, clearly, the metaphor of the unpainted barn breaks down.   For if barns -- which we build -- were like mathematical objects -- which (it is our contention) we discover -- some of them would prove recalcitrant to painting -- purely and simply unpaintable;  much as the Long Line can never be metric, howsoever it twist and turn.   Further, some paintable barns would admit more than one hue of paint, though not indefinitely many.

For let us emphasize:  Being metrizable is not a property of a bare set, but of a topological space -- that is, a base set together with a roster of which subsets count as open -- this roster itself is referred to as the “topology”.  The question then is whether a metric can be defined on the base set that will induce that roster of open-sets.  We have already been given the open sets we’re ‘aiming for’;  if the metric fails to yield these, then it is not a metric for that topology.  If no metric yields the right open sets, then that space (with that topology) is not metrizable.

Example 1:  Take the real plane, R x R, and let the interior of circles (i.e., open discs) be a basis for the topology.  Now define a metric on this set such that d(x,y) = 1 for all pairs of distinct points in the set.  This metric induces a topology all right -- the discrete topology, in which every pointset is itself open -- but it is not the Euclidean topology;  no cigar.  (Note:  The discrete topology is that of Leibnizian monadology, where every man is an island unto himself.)  The space itself is metrizable, however;   just use the usual Euclidean metric.

Example 2:  Now take a countably-infinite product of the set of reals with itself, R x R x R …  (You can pronounce this “R to the omega”.)   Assign the usual product topology to this (in which all but finitely many of the projections of an open set onto the individual R’s  must be all of that R).  You can induce this topology via a modification of the uniform norm.   But now instead assign the box topology (in which there is no restriction on how many of the slices may be less than all of R).  No metric induces that topology.

            As Dauben reports, Cantor himself eventually discovered the strange gap between our meeting a mathematical object for the first time -- presumably full-blown, yet still partially inscrutable -- and any eventual fullness of understanding. “Cantor no longer assumed that every set is born well-ordered.”


            Though the superficial similarity of the quantum case and the U.M.T.  doesn’t hold up, there does appear to be a rather arresting analogy with post-Chomskyan linguistics.
            The traditional view of language learning was that it involved general learning-strategies:  learning to make relative clauses was not radically different from learning your colors or the names of the kings of England (I caricature somewhat):  and just as different peoples conceive the color-palette in apparently incompatible ways, and the order of the kings might have been different (or no kings at all), so languages could differ indefinitely.
            Chomsky then challenged all this in ways much deeper and more philosophical than appeared to most people at first.   Many were surprised when, after laboring for a while at the forefront of fashionable linguistics, he out of the blue published a study of the time of Descartes, far outside the intellectual horizons of most of his followers.   But indeed, his project coheres, and always has.  Following his thought over the years, and finally getting the point, is a bracing intellectual experience.
            What initially attracted people was the positive expressive power in the slogan “Generative Grammar”;  yet very soon, those at the heart of the enterprise began to emphasize the theme of constraint
            In Chomsky’s view, as language-learners we must contend with certain hard (as in: hard-wired), quasi-algebraic parameters, each with a small finite range of possible values (often just two).  By our exposure to the particular ambient language in which we find ourselves, we (unconsciously) flip the various switches to their contingent, discovered position.   Certain combinations of settings will have further structural consequences.
            If we were as happily wired for topology as we are for language, we would meet a space, play with it in our cribs, learn in time what is the setting for its Separation parameter (T1, Hausdorf, regular, normal…), its Countability parameter (first-countable, second-countable, or neither) -- and having found that it is regular and second-countable, we would know it to be metrizable.

            Chomsky’s approach has been said, including by his fans, to involve an “innateness hypothesis”, a term at which he sometimes bridled.   And indeed, I called this roster of pre-existent parameters simply “hard”, where the imagery could be that of crystals (a full complement of Platonic solids, say) rather than that of a wiring diagram.  The default assumption in our scientific culture is, of course, that they reside on some gene or other;   but their actual nature renders problematic (not impossible) their visibility to the usual processes of Natural Selection.   (And again, to the puzzlement of his friends, Professor Chomsky never leapt with one bound onto the Darwin bandwagon.)
Also, if these parameters were coded for separately, one might expect a richer panoply of language-related mutations than is in fact observed.  What is the linguistic equivalent of lactose intolerance?
(Click here for our satire on the subject, which led to this whole U.M.T. thread in the first place.)  We might leave it open, just where these structures do reside:  perhaps upon that same hillside where the qualities of being Abelian, distributive, semi-simple, etc., may be found.

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