Tuesday, January 31, 2017

The Urysohn Metrization Theorem: a biochemical derivation


(I)  Various physicists, working with cesium-filled chambers and the like, have reported results that, at least as reported in the press, seem supraluminal.   Their technique is admirable; their results, at least as entertaining as that of the Euler disk; but they by no means dethrone the position of the speed of light in a vacuum, c, which still reigns at the center of physics.  Their surprising lab results may be compared with the paradoxical distinctions of group-velocity vs. phase velocity, in wave phenomena, or unphysical illusions such as in the thought-experiment of flipping a flashlight left to right-- millions of years later, on the arc of heaven as it were, an illumination mimics that trajectory with distances multiplied by a zillion, so that the 'signal' moves supraluminally, though nothing is actually moving left to right. A curiosity.  Einstein in heaven is not tearing his wild white hair.

(II) Various neuroscientists, sticking electrodes into hapless experimental subjects, detected electrical blips in the brain, as many as seven seconds before the fellow (responding to the prompting, “Coffee tea or milk?”) at last says, “Tea please.”

Neuroscientists -- an excitable lot -- froth over this.  “We feel we choose, but we don’t” enthuses one from the University College, London (whom I shall not name, so as not to defame; quotation from 1 Sept 2011 Nature).  Furthermore (adds one of his henchmen, in the same issue):

It’s possible that what are now correlations  could at some point become causal connections between brain mechanisms and behaviours.  If that were the case, then it would threaten free will, on any definition by any philosopher.

[“… any … any…” : sic; sic.]

The only reason such illation might have any color of probability, is that the neuroscientists (whose characters were formed in the laboratory, pithing frogs) regularly choose actions too trivial to reflect our real humanity.  In the case above -- I lied about the Stewardess Trilemma, really all the experiment was about was pushing some damned button, a button which moreover had no effect upon anything.  Boring beyond belief. No doubt the experimental subjects were in a coma throughout the experiment, or Dreaming of Babylon;  meanwhile the burblings of the subcortical tissues continued, much like the similar borborygmus of the bowels;  until eventually a finger twitched:  Ah what the heck, push now.
Not to knock it -- such results might even scale up.  Thus, to take the creature dear to the Churchlands, the sea-slug:  quite possibly its “choice” to slime-right or slime-left, at any given instant, owes its promptings largely to meteorological and other influences, with any accompanying mentation bearing but the faintest resemblance to the decision to enter the priesthood, or to see if cohomological algebra can yield insights into Yang-Mills theory.  But it doesn’t scale up to us -- at least, not to us at our best.  Prior, say, to coming up with the Urysohn Metrization conjecture, and then coming up with its proof, there was plenty of unconscious ruminating, and well more than seven seconds’ worth.  No mathematician denies it; all proclaim it.  (Poincar√© stepping onto the bus.)   But that doesn’t mean that the truth of the Urysohn Metrization Theorem, or even the validity or coherence of our proofs thereof, are caused by, or bear any structural resemblance to, the sort of subcerebral singultus that so fascinates the guys in the lab.  While button-pushing may be close enough to a reflex action, that you could imagine it might just fall out of biochemistry, without invoking mind or will at all, that really doesn't work to explain such mental palaces as the UMT.

After all:  Free will, like the Cosmos itself, is a gift from our Maker, not just something we picked up at WalMart.  Hence  there is no a-priori reason why it should be intuitively understandable, or even analytically explicable (without millennia of hard work, and even then may lie beyond our poor powers), any more than is the Cosmos.

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