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.  (Poincaré 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.

"Great Question"

Two flatulent practices relating to the posing of questions  have characterized political discourse in recent years, to the disbenefit of the polity.

(1)  Particularly associated with Donald Rumsfeld, though the maneuver antedates him, is posing ‘questions’ to oneself and immediately answering them.  Thus, let P and Q be propositions:  instead of saying simply

P.  Not-Q

the rhetorician draws it out into a monologue masquerading as a dialogue:

Is P true?  Sure it is!  Is Q true?   Absolutely not!

The advantages to the speaker are:

(a)  He gets to choose the ‘questions’, irrespective of whether anyone is interested in them or whether anyone would phrase the question in that manner.   As a result, they are invariably softball ‘questions’.

(b)  He gets to dwell in a bubble of his own voice.  Reporters, voters, maiden aunts, all fade into a kind of Meinongian subsistence;  and the Dialectic collapses to a trivial instantiation in the palaver of a single man.

Disadvantages to the audience include:

(c)  There is usually a substantive concern whether some proposition P be true, or what to do about it, or the like.   Yet by this slight-of-tongue, the bloviator poses instead some verbally related but substantively etiolated variant P-prime, and addresses that instead, thereby providing no real information or analysis, but leaving the impression that he has addressed P.

(d)  You have to listen to twice as much verbiage as for a straightforward “P” -- ten pounds of rhetoric in a five-pound bag.  And in time it gets freaking annoying.

(2)   In politically pre-sanitized settings like “town-halls” and related social Spectacles, an array of voters, carefully selected to represent each of the pressure groups to be pandered to, is permitted to pose a simple-minded, vaguely worded question, thus pushing a given button on the candidate’s library of prerecorded on-message speechlets.   But in a new development, the politician moves to snuggle-up to said Potemkin voter, with an enthusastic

   “Great question !!”

before launching in to her canned response.

The practice, repeated ad nauseam  (cf. SNL’s parody of Hillary Clinton’s use of this and similar maneuvers, in their satire of the Presidential ‘debates’), is distressing  even apart from its smarminess.    For the sort of low-info voter who most needs to be buttered-up and flattered, is least in position to actually pose a “great question”, or even a probing or pointed one.    Nor will, say, a scientist giving carefully-prepared, expert testimony on a complex subject  be patronized from the Congressional panel with a “Great question!”   The domains of “Great question!” and great questions, are virtually mutually exclusive.

Note:  The catchphrase “Great question!” is characteristically contemporary;  but the maneuvre is of longer date.   Speaking to a group,

Nixon would react to the most elementary question  as if he was hearing it for the first time.  He would ponder a bit, congratulate the questioner on his originality,  then give the answer he had given  hundreds of times before,  complete with some figures and a quotation,  as the citizen listened  in utter awe.
-- Stephen Ambrose, Nixon: the Triumph of a Politician 1962-1972 (1989), p. 139

~

And now, by an unhallowed congressus of these two tricks -- by Beelzebub out of Medusa -- comes the Great Rhetorical Question.  One came up this morning on NPR.

A Republican senator was touting Trump’s intention to

(i)  Increase spending (e.g. on infrastructure and defense)

and

(ii) Decrease taxes,

while

(iii) Holding down the deficit.

Now, this nation has been down that road many times before, both in speechifying and in practice;  so apart from the apparent logical incompatibility among those three desiderata, there is a track-record here, for anyone who cares to examine it.   That, however, is beyond the capacity of most pop journalists practicing today, as well as the attention-span of the average voter.    The substitute for critical inquiry, is a mere statement of undifferentiated unwillingness to be convinced -- the lazy-man’s lo-cal substitute for genuine skepticism (anatomized here).   In this case, the reporter’s Clarence-Darrow move was simply to ask, again, how (i)-(iii) might cohere.   But the politician was ready for it:

“Yes -- How can we address the problem of doing the things that need to be done, while holding the line on the deficit, without addressing the question of Entitlements? -- Great question!”

That may or may not be a Great Question, but the reporter had not posed it;  the rhetorician was congratulating himself.   And in so doing, bait-and-switched the subject from the incompatibility of (i)-(iii) to the problem of those pesky Entitlements.  A perfectly legitimate subject of debate, but here snuck-in via a side door.

~

The considerations above  recall to mind  what, for Europeans, must be the original Potemkin-“question” scam:  to wit, the Socratic dialogue.

The scenario has been solemnly sanctified,  countless times  over many years decades centuries.  The method has been dignified with the title  maieutic (etymologically relating to widwifery), whereby the savant or solon, by adroit interrogation, draws forth from the naïve listener (like a tapeworm from the bowel) innate knowledge  such that all might marvel that he hath.

As a build-up, this is terrific, only… You actually read the texts, and what you see is Socrates holding forth, more or less commendably, and harvesting the plaudits of uncomprehending minions or yes-men.

Thus, a demonstration of the Urysohn Metrization Theorem, via the Socratic method.

Socrates:  Let us, therefore, consider any topological space whatsoever, so that it have a countable basis.  You agree?

Minion: Sho’ nuff, boss!

Socrates: And may we not further stipulate, that said space be regular?  -- Surely that is not too much to ask!

Minion: True dat, Socrates!!

Socrates:  Whence it follows that, our space being regular, for any point thereof, we may define a continuous function  positive at that point, but vanishing outside of a neighborhood of that point. -- You agree?

Minion: Word up dawg, Socrates !!!!

Socrates: Whereupon we may plainly see, that by defining an infinite series of functions from said space, to …

(Socrates drones on;  the minion dozes, but his snores are taken as assent.)

Socrates: Whereupon we may plainly see, that our original space may be smoothly embedded into the countably-infinite Cartesian-product of the cube, which we earlier (you do recall this, don’t you, minion?) proved to be metrizable.  This metric is inherited by our embedded space in the subspace topology, thus proving the theorem.

Minion:  U da man, Socrates !!!

Socrates (beaming upon the evidently innate insight of his pupil):  Great answer !!!

Note:   We have it on good authority from the memoirs of Oxford dons, that any tutor who attempts to apply the Socratic “method” to draw wisdom out of the maws of wool-gathering undergraduates, comes up empty.

The Urysohn Metrization Theorem: a Perceptualist approach

At least since Locke, and down to our own day, an especially gormless form of empiricism, which we may call perceptualism, has held sway.

This theory is too boring even to summarize, let alone polemicize against, let alone analyze;  but -- credit where due -- we offer a

Perceptualist Proof of the UMT

(1)  I see … an array of color-patches, in my visual field. …  They form  (or so it seems to me, at this particular instant) … a topological space.

(2) They -- the pattern -- I see at once (unless I am but dreaming) -- it’s:  Regular !!

(3)  I focus on a single point (“Andrew”), and the topology there.  It has, one imagines, were there world enough and time to actually count and Booleanize all the open sets (1, 2, 3, …. ) a countable basis.

I try another point (“Bertha”):  likewise, the basis of the topology there is (let us say  -- obviously we could never really know this, any more than we can truly know that we have hands) countable.

Might this be the general case ??

(4) And -- yes! -- Charlie and Denise and Edmund and … all are graced with a topology enjoying a countable basis!   The space as a whole thus has one [NDLR: Fallacious step, but what do you expect] -- it’s Second-Countable !!!

(5)  And now … hoving in from beyond the horizon, as though from nowhere, it’s … look … a metric!  A very metric!  Hurrah!  The space is metrizable !!!!!!!

(6)  Q.E.D.

Eliminative materialism (expanded update)

…those bright and breezy Americans  who call themselves Behaviourists.  They declare with some warmth that there is really nothing in their minds, and that they only think with their muscles.

-- G.K. Chesterton,  Robert Louis Stevenson (1927)

Biology is only the Fascist substitute for sociology.

-- Arthur Koestler, Scum of the Earth (1940)

… that faulty inference  from the plausible view of the cognitive sovereignty of sense, to the absurd conclusion that explanatory models of human conduct must be in terms of elements similar to ‘sensation’ or ‘stimulus’.  The inference is quite fallacious, though an entire movement in psychology (Behaviourism) is based on the failure to see this.

-- Ernest Gellner, The Psychoanalytic Movement (1985; 2^nd edn. 1993), p. 16

Abstract:   The psycholytic style of current neuroscience-invoking reductionism  is intellectually and philosophically unilluminating;   and the social and political effects to which it is often put  are morally debilitating.   Taking ever-more-microscopic tallies of the goings-on in neural tissues  is of interest only to a histologist.   The attempt to press such observations into service as solvents of long-standing noëtic questions  is unsuccessful, probably in principle so. 

As Freud put it in the first paragraph of his final summary of his own work, the Abriss der Psychoanalyse (1938), we know about the brain on one side, and consciousness on the other, but

Alles dazwischen ist uns unbekannt;  eine direkte Beziehung zwischen beiden Endpunkten  ist nicht gegeben.  Wenn sie bestünde, würde sie höchstens eine genaue Lokalisation der Bewusstseinsvorgänge liefern  und für deren Verständis  nichts leisten.

That is to say such correlations amount to a higher phrenology.

~ ~ ~ ~ ~

[Prefatory note:  An earlier nineteenth-century forerunner]

Köhlerglaube der Wissenschaftler

In the course of the Materialismusstreit …, the German Darwinist Karl Vogt  published a widely read essay entitled Köhlerglaube und Wissenschaft (1855), in which he tried  with cynical witticism  to dispense with the problem of the human soul [NB: and mind].  According to Vogt, thought is a secretion of the brain  just as the digestive juices are secretions of the stomach, bile a secretion of the liver, and urine a secretion of the kidneys.

-- Kurt Reinhardt, Germany: 2000 Years (1950, 2^nd edn. 1961), p. 594

By now, the man in the street has become acclimated to the notion of secretions, and is unlikely to be swayed by that account.  He will notice that, somehow, the liver never secretes anything but bile, the kidneys, urine;  whereas the brain ‘secretes’ such unexpected and unprecedented and mutually contradictory things as:  Catholicism, Communism, situation comedies, and the Urysohn Metrization Theorem.

But the underlying atheist-nihilist idea, that of wave-of-the-wand eliminativism (it does not even deserve the name of reductionism), whereby you eliminate the problem from consideration, rather than doing anything to actually solve it,  ever and again resurrects itself, morphing into a semblance of life, from some discarded afterbith.

~      ~      ~      ~      ~

[Herewith our original essay  from a couple of years ago]

We did not have to wait for the tribe of those laboratory nihilists known as neuroscientists, to behold an academic school dedicated to reducing free will and cogitation to the level of meat.  The behaviorists were there first.    And it is our great good fortune that we are spared the chore of actually reading and refuting their earlier and generally gratingly ill-written output, since their fifteen minutes have expired(*), the quietus marked by a famous and scathing review of B. F. Skinner’s Verbal Behavior, by Noam Chomsky, way back in 1959 (a good year -- aye, a vintage year).

 

Skinner himself, and others of his spawn, continued to “engage in writing behavior”.(**)   And like the dragon’s-teeth of Cadmus, the doctrine springs up ever anew, in this or that contorted form.

Maxfield Parrish:  Cadmus

[(*)  I spoke too soon.  Savor my groveling retraction here.]

[(**)  The phrase was made famous by a journalist who, to test the nature of asylum-based therapy in general  and involuntary commitment in particular, had himself commited, and immediately began acting perfectly normal, and telling the doctors he’d like to go home.  He kept a journal of his experiences.   This latter fact did not go unnoticed by the keenly observant psychiatric nurses, who wrote in his chart, “Patient engages in writing behavior.”  An immortal and dactylic line.]

Skinner, apart from his paid occupation of making life miserable for rats, is best known as the author of Beyond Freedom and Dignity (1971 -- not such a good year).  The title echoes that of Jenseits von Gut und Böse,  by that other evil troll, Nietzsche:   beyond, not in the sense of uplift, but of, “Get O-o-o-vah-rit!” -- A similar bit of titular legerdemain (sugar-coated in the title, rat poison on the inside) was perpetrated by Patricia Churchland, in Neurophilosophy: Toward a Unified Science of the Mind/Brain (1986).  What is really being proposed there is not Unity, but Anschluss, with the mind playing the hapless role of Austria.

Professor Skinner, hard at work in the lab

Again, prior to contemporary neuroscientists, there were philosophers who ground much the same axe.   Wikipedia (s.v. “Self-refuting idea”), re eliminative materialism, speaks of identity theorists like the philosophers Boring and Smart, who “claim that ideas exist materially as patterns of neural structure and activity.”  Now, that is one unilluminating and bone-headed idea.  By which I mean, of course, no more than that it is an unilluminating and bone-headed pattern of neural structure and activity.  Owing possibly to an excess of calcium at one of the synapses.

And why, indeed, stop short at the level of the cells, themselves already doubtless epiphenomenal?  Take it right down to quantum mechanics!   Let some gaggle of labcoats come up with the Schrödinger equation for the Bible, and another for “Paradise Lost”, and show us how certain passages -- excuse us, certain Hermitian forms -- of the former, lead directly to the eigenkets of the latter.  Put literary critics  plumb out of work.
(Something more than just a joke:  Cf. entomologist/ultrareductionist Edward Wilson donning the robe of a Milton scholar here.)

-- A further word about this scenario of ideas as “patterns of neural structures and activity".  It really is little more than ye olde idea of Ideas as Pictures-in-the-head, but dressed up in a modern labcoat (the sort worn by actors playing doctors on TV); and this is one subterranean source of  plausibility of a doctrine that  on logical (and indeed theological) grounds  is a tissue of absurdity.

~

What gives the “thought is a physical pattern” account some specious plausibility, is our modern picture of vision:  which, to be sure, is vastly more complex than the everyman’s pre-theoretical account, whereby the vision of an apple is like a little apple in my eye.  (Compare, indeed, the etymology of the word pupil.)  -- Actually, the completely pre-theoretical account doesn’t even involve vision at all;  it’s just, “Behold! an apple!”  (or, holophrastically:  Gavagai!)

In that account, we begin with the beheld object.  

  (a)  It is, as a ding-an-sich, as hopelessly unknowable in its totality as a black hole;  yet like those reticent celestial bodies -- the veiled houris of the stellar harem -- by their corona ye may know them (that, and a bit of math).

  (b)  From this object -- let us take it to be specifically a coffee-cup, since we have proved the existence of these -- photons and phonons and perhaps even an occasional Higgs boson, bounce and boing and go-or-do-not-go through visual slits, in some inextricable and inexplicable quantum fashion, to wind up

  (c ) landing on, and lighting up, this or that spot on the retina.  From there,

  (d) ganglia shiver and synapses thrill, and eventually

  (e) the visual cortex receives various rude jolts, after which

  (f) the rest of the brain somehow sorts it all out, until finally

  (g) the mind says, “Aha!  There it is!  Time for a spot of java!”

Unfortunately, this comparatively simple scenario won’t do for the generality of ideation.  It would seem that, when these reductionist gentlemen contemplate an “Idea”, they go no farther than an analogue of (f) or at best (g), for an object of thought as simple as a coffee-cup (or, in their own case, their own endlessly contemplated navel) -- the ‘idea’ of one, rather than the image of one.  -- One is reminded of Hadamard’s plaint, in The Psychology of Mathematical Invention, that previous researchers into thought-processes had restricted themselves to the elementary thumb-sucking mental-motions of the Beoetians, which simply do not display the richness of thought-processes of research mathematicians, so that his own researches had to start from scratch.
(Similarly, generative linguistics would not have gotten far, had it never considered sentences more complex than "John is fat.")

For consider:

What the cognitive eliminativists are proposing, is not that a thought consists of some pattern of little electrochemical twitches and irritations plus ideation (the thinker contemplating the canvas of his own neuroprocesses, as it were);  their explicandum cannot involve the explicans.  No, it is really just -- splat -- that.

Now consider the following ideas:

  (a)  A topological space T is regular.

  (b)  T has a countable basis for its topology.

To each of these corresponds -- in Henry’s head -- a cortical panorama (A, B) not unlike a TV test-pattern plus static, or a Jackson Pollock painting on a bad day.

And consider that, as we all know by now, (a) and (b) imply

  (c)  T is metrizable.

which in turn duly corresponds to some porridge of pixels (C) on Henry's cortex.  Fine, we grant that.
But how  by all that is holy  do (A) and (B) imply (C)?  The moreso as Henry’s cortical idiosyncrasies  differ from those of Hector (A’, B’,C’), let alone those of the silicon-based mathematicians of the planet Gnorf:  who, whatever their physiological differences from ourselves, must reach exactly the same conclusion as did Urysohn, since that conclusion is, in fact, a fact (laid up in heaven, where Churchlands do not penetrate, and the worm corrupteth not).

[An attempt from an entirely different direction, to reduce the complexly interrelated richness of mathematics to triviality, by translating it into a non-mathematical world of discourse  entirely unsuited to it (in this case, the mushy one of metaphor), is  Where mathematics comes from:  How the embodied mind brings mathematics into being (2000), by   George Lakoff and Rafael Núñez.  You can tell already by the subtitle, that the authors are no Platonists.   I did coursework with George, briefly at Harvard and later at Berkeley, where at one point I TA’d for him;  so I shall not criticize him here, lest I fall into the crime of magistricide.   Yet I must sadly concur with the verdict of the reviewer, Terence Langendoen, in the March 2002 issue of the refereed journal Language.]

~     ~      ~

Ethicists should likewise soon be out of a job.  The latest in the public press:

The End of Evil?
Neuroscientists suggest there is no such thing.
Are they right? 

A word:  I have no quarrel at all with the reductionist program as propounded and practiced by, say, Steven Weinberg (who embraces the term).  The problem comes in with the eliminativists who do not really mean to explain things, but to explain them away.

For remarks on the ontology of attributes, and their vulnerability to the eliminativist attack, click here.

~     ~      ~

What tends to get eliminated, in these proceedings, are things like consciousness and free will -- features of which we had grown rather fond.   Their very familiarity (as “folk theory”), for these gentlemen,  somehow tells against them.  As though a good rule of thumb were, If the ancient Greeks believed it, it must be wrong.  An eternal verity gets rechristened a “stagnant theory”. 

(Extraordinary idea, that, in any case: that to be any good, a theory must, like a fashion-slut, annually change the length of her skirts.  By that measure, the truths of numbers known to Diaphantos must be positively mildewed.)

Here  phlogiston plays ever and again the useful role of whipping-boy.  (Indeed, given the continued prominence of this ethereal etwas in philosophical debate, it must count as one of the most fruitful scientific hypotheses ever.  If it didn’t not-exist, someone would have to uninvent it.)  

The line of reasoning, reduced to its essentials, runs:  Well if that thing turned out not to exist, maybe freedom and God and all that lot   are just illusions too.

One problem with this angle of attack is that there's more to physics than phlogiston.  Also on the roster of theoretic posits  we find:  acceleration, force, electromagnetism, atoms, protons …  Phlogiston went out on a limb, which broke;  atoms & co. went out on a limb, and so far it seems to have held.   The eliminativists are being rather choosy in what to highlight.

Another problem with their reasoning  is that it’s nonsense.   Shooting down this or that hypothesis of theoretical physics, or even the whole lot of them, is not at all the same thing as denying what every mother has been telling her children since the beginning of time.   We are rather more certain of our own consciousness and free will, than we are of protons.

Let us examine another favorite butt of their mirthless hilarity.  Your great-great-grandsires believed in God, and that the sun set in the west.  The more fools they!   We New Atheists proclaim triumphantly that the earth circles the sun, and not the other way about.  Therefore, everything else you believe is false as well.

But in point of fact, you haven’t got much further along by taking the sun as the rest-point of your dynamical system, than when you stood firm on the earth.  For now you may be held to depict a system in which the entire galaxy whirls lumbrously around the axis of a single, peripheral, faint star.

In reality, none of these choices of a reference frame intend the absurdities ascribed to them.   Each has its use for some purpose or other.  Grandpa was well within his rights, both cosmologically and theologically.

~     ~      ~

~   Commercial break  ~            

http://www.youtube.com/watch?v=FwzOXzZWUik

We now return you to your regularly scheduled essay.

*     *     *

All biochemists and molecular biologists today are ‘mechanical materialists’.

J. Maynard Smith & E. Szathmáry, The Origins of Life (1999), p. 11

Probably no harm in that, so long as the molecularists  stick to their last, and don't go sounding off in the Sunday supplements about matters that concern them not;  indeed, it would be more concerning, were they all Cantorian Realists, or Presbyterians.   But the day when all musicologists, or mathematicians, or literary critics, take mechanical materialism for their lodestar and touchstone, is the day when we shall cease to learn anything of interest about music, math, or literature.

~     ~      ~

 

We earlier glanced at the program of consilience, which (speaking somewhat informally) depicts the sciences stacking up one atop the other, like proverbial pancakes, while the life-giving maple syrup of shared method and ontology, seeping from one to the next, unites them.   (That summary, while not perfect, is both more concise and somewhat clearer than what you may glean from scattered passages in Wilson’s book.)  Eliminative materialism takes all this a step further:  the whole stack is squashed flatter than flat (again, the individual pancake has passed into legend for this very quality) into an inedible paste: the which, however, is declared to be the sole reality, things like syrup and pats of butter being just figments of your imagination (which itself does not exist).  Understandably, this philosophy was quite popular among the guards at Belsen-Belsen.

That may sound harsh, but truly, the often creepy frisson of the behaviorist/neuroscientific mindset does recall certain experiments during that unlamented Reich which were better left undone.  Steven Pinker, The Blank Slate, p. 19, on the founder of American Behaviorism:

Watson presented a baby with a white rat, and then clanged a hammer against an iron bar, alledgedly making the baby associate fear with fur.

Thus, to all our weightier indictments against Nominalism in all its forms, we add this slogan:   

“Behaviorists Hate Bunnies.”

(Our own favorite behaviorist, jolly Dr. Quine, does suffer from a certain lagomorphobia,  in his obsession with "undetached rabbit parts", documented here.)

Wilson pens a highly entertaining parable of the termite ethical code, extoling:

…the deep, saprophytic, basidiomycetic penetralia of the soil …. the sanctity of the physiological caste system; the evil of personal rights (the colony is ALL!); … the aesthetic pleasure of eating feces from nestmates’ anuses  after the shedding of our skins; and the ecstasy of cannibialism … (it is more blessed to be eaten than to eat).

Edward O. Wilson, Consilience (1998), p.  148

This is jolly stuff, skillfully done; probably give Donald Trump a stiffie.  Yet one senses a sour note,  as the (“consilient”) goal of the eliminative materialists   is to provide a similar account of human morality, decency, caring, flights of fancy, love …. Ever again and inevitably, these proctoscopists return to the anus, the central life-ring (Lebenszirkel) of their Weltanschauung.

~     ~      ~

To be fair to the reductionist program, let us briefly consider a proposal narrated by Wilson, who is both a much more graceful writer  and a more congenial companion than Skinner or the Churchlands.

Wilson offers one worked-out example of  basically eliminative reduction (or ‘consilient’ explanation): from oneirology straight down to biochemistry.  Here is a sketch of the bad old days of Freud:

Mysticism and science meet in dreams.  … When we sleep, the ego releases its grip on the id…  Freud … -- to put it as kindly as possible -- guessed wrong.

Edward O. Wilson, Consilience (1998), p. 74

Wilson is in fact a kind and temperate man; that mild rebuke is as strongly-worded as anything in the book.  (He is even able to discuss the Postmodernists without resorting to scatological invective, a task far beyond my own capacities.)

But now (p. 75) for the good news:

The competing and more modern hypothesis of the basic nature of dreaming  is the activation-synthesis model of biology.  … Sleep descends upon the brain when chemical nerve cell transmitters … such as norepinephrine and serotonin, decline in amount… Simultaneously a transmitter of a second kind, acetylcholine, rises in amount.  …

Etc., etc.; we skip to the exciting climax on the next page:

The electrical membrane activity, still mediated by acetylcholine at the nerve junctions, moves from the pons (the P of PGO), a bulbous mass of nerve centers located at the top of the brain stem, upward to the lower center of the brain mass, where it enteres the geniculate nuclei (G) of the thalamus, which are major switching centers in the visual neuronal pathways.  The PGO waves then pass on to the occipital cortex (O), at the rear of the brain, where integration of visual information takes place.

Voilà ! Any questions?  No?  We're done then.

Oh dear -- did I type all that up aright?  Was it not perhaps the serotonin that flows into the thingummy, thereby (de)activating the thalamo-geniculo-pataphysico reaction?  Well, no matter;  for you will notice that we have so far learned absolutely nothing about actual dreams.(*)

(*  Don’t misunderstand:  this is sound, painstaking work, and it is excellent that someone is doing it.  No doubt it will prove crucial to some hyperspecialists in the approriate microcompartment of biochemistry.  For the general man of science and letters, on the other hand, it is precisely as fascinating as the mysteries of the pancreas.)

This lacuna Wilson hastens to remedy by rushing in with his own broad-brush description, eclipsing that of that bungler Freud:

We fly through the air, swim in the deep sea, walk on a distant planet, converse with a long-dead parent … In dreams we are insane.  We wander across our limitless dreamscapes as madmen.

Hm.   Doesn’t ring a bell, either with my own experience or that of friends with whom I have discussed such things.    In my own dreams, I tend to be at work, or in a classroom, or (oddly) at some sprawling party on the grounds of a great country house.   No flying through the air;  more like, irksome difficulties catching the right bus.  But let that pass.   No doubt spending your days as a reductionist  means you must compensate with florid fantasies at night.

But the real killer is that last line.   It is, I believe, wildly and crucially false.   The real mind-set of the typical dreamer was captured perfectly by Louis Carroll:  Alice retains her level-headedness, while all around her are acting fantastically, and she rather taxes them for their behavior.  (A similar depiction occurs in the long-running comic strip, "The Strange World of Mr Mum".  The eponymous everyman wanders bemused through the strangest situations.)  What the dreamer beholds  may resemble the visions of psychotics:  yet the dreamer is not psychotic. In our dreams we remain our real selves:  we retain, in particular, our unbanishable morality, not doing in dreams what we would not do or at least fantasize doing in waking life. (Morality: a word anathema to some of these gentlemen -- though here Wilson himself is sound enough -- yet it survives the oneiric lysis, as it shall survive their nihilism.)  We remain perfectly rational, only…. just not at our best.   For instance, in dreams I often work out etymologies, or expound mathematical ideas.  These are without exception in error -- hey, happens to the best of us -- but that is what I do, I don’t imagine I’m a prophet, or a giant cockroach, or that everyone is out to get me, or what not.

Freud may not have nailed it, but you’ll learn more about actual dreaming from Die Traumdeutung than from this ultrareductionist stuff.

[Update]  I am currently re-reading Die Traumdeutung;  some actual reflections might follow.
Preliminary musings here:
http://worldofdrjustice.blogspot.com/2012/10/appendix-to-appendix-on-oneiric.html
And now this;
http://worldofdrjustice.blogspot.com/2012/12/die-traumdeutung-still-further-updated.html

~     ~      ~

That only somewhat facetious polemic brings me to another point.   While specialists of this or that stripe  may want to drill down to the last detail, what most of us require -- expressing this somewhat provocatively -- is an explanation that is as shallow as possible, in that it digs down only as far as it needs to, to clear things up a bit.  (Compare Quine's "Maxim of shallow analysis:  Where it doesn't itch, don't scratch.")  And even if you wind up doing a deep-dive yourself, what you intellectually retain is likely to be an explanatory skeleton. 

Thus, take mathematics.  When a proof is truly “irreducibly reductionist” (to coin a phrase), like that of the Four-Color Theorem,  where intermediate guiding Ideas, of general application, do not emerge, but everything is tediously and computationally broken down into a bad-infinity of special cases, mathematicians are disappointed and frustrated.   What we want ultimately is an agencement of Leading Ideas.  We might dutifully check the details of the proof, but these we shall probably not retain.

(Keynes, whose Treatise on Probability  brims with Leading Ideas, charmingly prefaces one particularly symbol-ridden chapter with absolution for any reader who wishes to skip it.  “We do it, to show that the thing can be done;  the exercise is more for the benefit of the writer, than the reader.” [Quoting from memory, so you might not be able to Google that.])

So, suppose we wish to know what launches birds or butterflies on their migrations.  What we hope to hear about is an account in terms of concepts antecedently familiar or at least learnable without too much fuss -- things like average temperature, extremal (trigger) temperature, average length of daylight, or depletion of ambient foliage.  Or, how do they find their way.   We are prepared for an explanation in terms of magnetism, starlight, or even “an instinct, varying in its details from species to species, whose guiding factors are as yet unknown”.   In the first-named case, we are open to talk about iron molecules in the inner ear;  but if the account in inextricably entangled in the details of the cross-effects between the ornithoboson cycle and the L-dextro-retro-feniculation mechanism, it’s TMI, and we tune out.  A ‘molar’ elucidation suffices for the general public and for the general zoologist   (and indeed, even this much is often too much to ask in practice;  apparently major mysteries remain about many migration-patterns, including the celebrated case of the Monarch butterfly).  An avian histologist might want to know more about those iron particles and their interaction with the inner ear.  But at some point even he is satified to call it a day, content to say, of levels yet further down, The Thing Can Be Done.

[Update 26 April 2012]  And now, for pigeons, it has been:
http://www.nytimes.com/2012/04/27/science/study-sheds-light-on-how-pigeons-navigate-by-magnetic-field.html?hpw

[Update 14 July 2012]  And trout:
http://www.latimes.com/news/science/la-sci-magnetic-sensing-cells-20120714,0,828028,print.story
And sharks (a segment this morning on NPR).

Now, sometimes, perhaps usually, no such shallow-draught explanation will be forthcoming.  Thus, the saga of individual animals gets you only so far towards the understanding the appearance and disappearance of species; a lower-level unit, the gene, turns out to be quite handy; and to further understand the behavior of these, we occasionally have to get down among the DNA.

Consider further,  the symmetries of snowflakes.  The usual explanation is straightforwardly intuitive, local and deterministic.  But somehow I don’t buy it.   It just feels as though such endlessly varied detail, with symmetries nonetheless rigidly enforced, brewed in a whirling cauldron of stochastic accident, must have some energy-minimizing post-editing -- something global (or “holistic”, if you like that word) along the lines of surface tension, or of standing waves on tympani.  In which case you’ll be dragging in Bessel functions, or calculus of  variations, or quantum mechanics.   And in that case  snowflakes will be one of those phenomena that can barely be intuitively understood.  In this, it would join ranks with what may be a host of questions that Homo sapiens will never really understand.  -- This is the Mysterian stance.  Believe it or not, this is also the orthodox Judeo-Christian-Muslim position towards the ultimate unknowability of the mind of God -- certain TV preachers and football coaches to the contrary.

So:  Reduce as you can and as you must, but not for its own sake.  Practically, cognitively, there is a price to pay.   The naturalist tracking an ecosystem knows he may need to have recourse to natural selection, perhaps to geology, a bit of statistics -- but it’s not as though he can’t wait to get his hand on the string theory of the thing. 

Or consider the case of generative syntax, which over the years has dug ever deeper, and become increasingly abstract in its ontology of explanation, drifting further and further from categories familiar to Jespersen.  Yet it did this, not out of any fetishism for abstraction, nor reductionism for its own sake, but rather because earlier, more traditional and intuitive attempts just did not work.  Indeed, the principle figure behind this sweep of developments has also consistently been in the forefront of debunking popular but vacuous styles of reductionism.

~     ~      ~

Again, to be fair, let us quote a man whose mind we entirely admire, whose prose sparkles like dew-bedecked blooms at break of dawn.  None other than -- but you have already guessed -- the master himself, Mr. Willard (“Van”) Orman Quine:

Corresponding to every mental state … the dualist is bound to admit the existence of a bodily state  that obtains when and only when the mental one obtains..  The bodily state is trivially specifiable in the dualist’s own terms, simply as the state of accompanying a mind that is in that mental state.  Instead of ascribing the one state to the mind, then, we may equivalently ascribe the other to the body.  The mind goes by the board, and will not be missed.(*)

(*In fairness to Quine, let us hasten to add, that his core concern here is quite abstractly ontological, even quantificational, as befits a symbolic logician.   It may even be compared to the notion, not of dualism, but of duality, in mathematics, being related rather like the positive and the negative of a photograph, where either one will do.  --  Still, his chosen periscopic viewpoint leads to some no very edifying reflections: “Mental states, construed as states of nerves, are like diseases.”  And such reasonings spawned venomous afterbirths  with Dennett and the Churchlands.)

This nifty epiphany casts our Quine into something of a Little-Jack-Horner state (= excitation of synapse #5328b), for (coming to his senses) he straightly confesses that “this effortless physicalism smacks of trickery”.   Yet what he says is true enough. --  True… enough, for sake of argument; that no such isometric reduction has ever actually been made and perhaps never will, is beside the present point.  Let us grandly grant it done;  Nobels all round, lads!  What then?  Well, then, nothing.  We are back where we were.

For: every carpet has an underside.  We may even concede (perhaps over-generously) that both sides are interderivable:  given the one, you can work out the other.  Now, you and I prefer the upper side, where green birds are depicted in an azure sky, and the prince urges his huntsmen  forward in the chase.  Equally -- and it is their right, -- the proctoscopic philosophers prefer to ogle the underside, an unintuitive tangle of knots and loose ends.  You and I can observe why the green beings  appear on a background of blue;  whereas the proctoscopists can only stare miserably at their welter of circumanal shorthairs.

Such, indeed, is the case most favorable to the proctoscopists, where we are concerned only with a single static pattern.  Consider now -- well, our old friend, the Urysohn Metrization Theorem.  All the proctoscopists can do, in trying to wrangle with that, is to examine the brains (not, note:  the thought processes) of actual people trying to wrangle with it, since the theorem itself is incorporeal.  (Compare the Competence:Performance  distinction, in linguistics. Recall as well the unprofitability of obsession with the latter.)  And granting our epistemological vivisectionists  superhuman abilities  far beyond the offing of likelihood, they will be able to peer into everybody's brain-pan  and inspect the various chemical blurps, electric fizz, and meandering fat-globules  of the would-be topologists.  Yet worse, far worse for their case -- even granted their ability to somehow read off, from this mephitic brew, such messages as “SMITH - NOW - CONSIDERING - FIRST-COUNTABILITY.  UH-OH, SMITH SAD.   MEBBE FIRST-COUNTABILITY NOT ENOUGH. OH NOES!!”, -- they will be not a whit closer to understanding the theorem.  For their data-set consists of all the various blunderers of the earth, barging about, running into dead-ends, committing fallacies, thinking they’ve proved it whereas in reality (victims of a misfolded protein) they did not -- really one would like to see Monty Python work this one up into a skit.  (They did the complete works of Marcel Proust;  why not Urysohn?)  For the truth of the theorem resides, not in the cloaca which these gentlemen are pleasured to inspect, but in the realm of Invisibilia, the kingdom of the Lord of Hosts.  And the valid mathematical moves made by this or that fallible incarnated creature, in their groping attempts to approach the truth, gain their validity, not from thermodynamic or statistical considerations, but from our miraculous harmony with such abstract truth itself, which stems, we know not whence;  though some of us have an inkling.

That, technically, one might perform such fundamental (or rather fundament-level) analyses or rather catalyses,  shows only that it is possible, by the wrong sort of reduction, to reduce meaningful phenomena to the meaningless.  We may dub this excretory reduction, as opposed to explanatory reduction.  And the craft practiced by these proctoscopic coprophiles, we may dub stercochemistry.  (Not a typo, that:  sterco-, not stereo-.)  Indeed, in this perspective, we notice that the doctrine of -- eliminative materialism, is singularly well-named.

Envoi:

… grob geredet, wie gesagt:

womit noch keineswegs  der Wunsch  ausgedrückt ist,

auch grob gehört,

grob verstanden zu werden …

-- Nietzsche, Zur Genealogie der Moral

[Update 10 III 2012]   For a briefer, parallel argument, in a lighter vein, click here:
Eliminative Immaterialism

[Update 28 III 2012] Roger Scruton weighs in:
http://www.spectator.co.uk/essays/all/7714533/brain-drain.thtml

*

Für psychologisch tiefgreifende Krimis,

in pikanter amerikanischer Mundart,

und christlich gesinnt,

klicken Sie bitte hier:

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

*

[Update 13 April 2012]  This particular post keeps receiving a disproportionate number of pageviews.  Since the Comments are virtually nonexistent, I am at a loss to know why.

If you have been enjoying it for its sober philosophical content, and want more, click here:

http://worldofdrjustice.blogspot.com/search/label/Realism

If, on the other hand, you rather relish the invective, trouncing the nihilists, then you might further appreciate this:

http://worldofdrjustice.blogspot.com/2011/09/philosophy-porn.html

The Urysohn Metrization Theorem: a poetic account

The Urysohn Metrization Theorem: 

a poetic account

( This poem

treats of the theorem

for a General Audience. )

“Stay!  Who can tame

this wild rough wood, 
and bring it into measure?”

they asked the land-surveyor.

“Aye, that can I,”

the stout lad said.

“So the terrain be not too pocked or dimpled,

nor stretch beyond all horizon

unto uncountability.”

So saying, he hoisted his transit over his shoulder

and went off whistling,

towards the West.

[We have likewise explained    the mysteries of physics,

in terms so simple,   even a Manager might understand.

For that,   click this.]

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.

*     *     *

~ 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!