“Was it worth it?”

The other day, the Sunday New York Times ran an especially absurd headline (and I specify Sunday since that edition takes more the long view, rather than being rushed by breaking news, and hence copy editors -- if they any longer exist -- have more time to cull such absurdities):

As Self-Immolations Approach 100, Some Tibetans Are Asking, Is It Worth It?

It reads like a Monty Python skit -- You picture a chorus of peasants bellowing in unison, “O yes, totally!”   What renders it so darkly comical is that those who actually paid the ultimate sacrifice  are not available to be polled.

Yet let us dwell on this.  We are reminded of the notorious remark by Secretary of State Madeleine Albright during a 60 Minutes interview in 1996, in answer to a question pointing to reports of half a million Iraqi children having died as a result of U.S. sanctions (“that's more children than died in Hiroshima”):  She said: "We think the price is worth it,"  and was roundly skewered for this.   But the sly interviewer had put the words into her mouth, by the formulation of her question (“Is the price worth it?”).  The ‘question’ thus formulated  is akin to that interrogatory classic, “Have you stopped beating your wife?”

 

It is a paradox:  so stated, the defense of the Iraq sanctions seems heartless and outrageous;  yet this does not entail that the policy itself was evil or misguided.  (If you think it does, then you should be equally incensed at our harsh sanctions against Iran, which have been in place for years, with few people paying attention -- indeed, with the Republicans pretending that the current administration has been negligently doing nothing.)

So what are we to make of this?  There is a real issue here, beyond the matter of insidious reporters and gormless headline-writers.   The dilemma arises ever and everywhere, in political life.  A measure is proposed that will advance the nation’s goals or benefit society as a whole -- but always, somebody’s ox is going to be gored.    Every policy has its costs and benefits, so that “Is it worth it?” is -- logically, and rhetoric aside -- perfectly valid.  Yet the moment it is framed in that way, if you support the policy, you seem unfeeling towards the owner of the gored ox.   And if those gored fall into the class of Designated Tearjerkers, no rational solution is democratically possible.

Thus, some rational policy questions would include the following.   To respond to them publically is politically impossible;  I feel sure I shall be accounted heartless, merely for posing them (with no bias whatsoever as to possible answers):

Is it “worth it” to provide millions in taxpayer-funded lifetime medical care for (i) a brain-dead patient; (2) a homeless lunatic;  (3) a lifer in Federal prison;  (4) your unemployed or elderly neighbor next door.   Who, relative to your own neighbors, might be you yourself.

Is it “worth it” to require extremely costly and time-consuming airport security for every flight, on the off-chance that you might forestall a terrorist from blowing up the plane.  (Consider two classes of cases:  (i) You yourself are on that plane.  (ii) Donald Trump, your boss, and your mother-in-law are on that plane;  you yourself took the train.)

A woman wishes to abort her fetus.  Is it “worth it”?

The problem is, there is no accepted eudaemonic calculus , no characteristica universalis, that -- even supposing that the public were uniformly so selfless as to bow to its conclusions, whatever they might turn out to be -- no set of weighted values that applies (“Calculemus!”) uniformly and across-the-board, to all (or even any)  such cases.   The problem is a moral one, but in some respects even an algebraic problem.

Lexical footnote: 

The antiseptic-technocratic equivalent of the demotic “Was it worth it”  is the notion of acceptable risk :  generally calculated by those who are not themselves in immediate danger.

Adventures in Lineland

It is to be hoped that you have all had occasion, at some point in your childhood or thereafter, to meet the marvelous mathematical allegory Flatland, by Edwin Abbott, followed (in adolescence or later) by the masterfully pedagogical The Shape of Space, by Jeffrey Weeks, which takes you further into higher dimensions.

Here we do the opposite:  We take as our starting-point, a starting-line, either the continuum or some more manageable countable dense subset -- seemingly too exiguous to be interesting.  But remember:  Mind pervadeth all spaces of every sort;  and so we examine what it might be like, to live in this one.   A Leibnizian-Wolframian fantasy, fleshed-out.

[Note:  A prior exercise in Lineland physics was that of Ernst Ising in the 1920’s.   The one-dimensional Ising model attempted to model ferromagnetism, using the simplest possible assumptions.  It proved disappointing -- no phase transitions.   A two-dimensional model produced much more interesting behavior; a three-dimensional model  has resisted exact solution.]

ADVENTURES IN LINELAND

            At first blush, Lineland would seem a constrained sort of place.  As, if its population includes Tom, Dick, Harry, and Mary, arrayed along the line in that order, then Tom can never enjoy the immediate propinquity of Mary; and Harry is stuck forever next door to Dick, even if they don’t get along.

Yet Lineland  can actually be experientially rich, though the creatures cannot move or switch places.  And this, despite a certain apparent simplicity of its citizens. Its residents are monads (the term is due to Leibniz): physically, cells.  Their instantaneous expressivity is maximally limited: the only signal they can emit is “On” or “Off”; but internally, they have enormous storage.  The (public) state of all Lineland is identical with the pattern of On & Off along the line; what are the thoughts of the individual monads, we cannot know (they are 'noumena', 'Dinge an sich' -- Kant this time).  These publically inspectable states  evolve stepwise through quantized time, as in cellular automata. The physics is such that an “On” state of my neighbor  n cells down  emits a signal of strength ½^n (that is, falling off another half-strength with each cell) Hence the perception of a given monad  at any instant  consists of two real numbers (indeed, rational numbers, since n is always finite), one for the world to his right  and one for the left. The numbers may be represented as binary expansions, exactly corresponding to the pattern of On’s & Off’s. Thus, if my right-hand neighbor is Off, and the next beyond him On, the next two Off, and the rest On, the signal-strength is .0100111111….  The sensitivity and storage-capacity of the perceiver determines how many of these digits will actually be perceived by a given monad at each reception.  

            As time advances, the pattern unfolds: we, from above, out of time, see the whole thing spread out like a carpet. 

            The states of Lineland evolve according to some single given rule, of the sort familiar from cellular automata: thus predictably, so long as the world is merely “kicking over”.  However – and this is key --  at least some of the monads have (a very elementary sort of) free will. 

            Now there is, of course (lest you think this concession too grand), only one thing they can do with it: namely, refrain from turning “On” when the ground rule says they should (or vice versa).  These rare but bold interventions are then perceptible ‘from above’  as a switch in the pattern – which then propagates automatically for all time (in all the surprising ways that cellular automata can toss up, albeit deterministically), long after the rogue monad has fallen back into step.

This free will, though it is defined as the ability to resist the dictation of the master pattern, and though unambiguously displayed  only when that ability is exercised to produce a contradiction to that dictation, need not be exercised always as contrarian (like the youth who, to demonstrate their freedom from some convention, unanimously and invariably adhere to some other convention).  Any given monad might indeed have a rule (the rule being fixed, but its adoption free), that (to take a random example), whenever you have laid down five Zeros in succession, you output a One – whether or not this would be in accordance with the master program.

 

            Further, the sentient monads can ascertain  whích of their fellows have free will and whén they exercise it.  This, even though the capacity of any monad be finite.  For, suppose the reception capability sufficiently capacious to perceive the signal-strength  N-cells-out  on either side, and to retain a record of these states for T instants.  And suppose that the ground rule allows for determination  upon next output of a given cell  from no farther than K cells away.  Then the ambient band N on either side  produces an output in the next band N minus K on either side  that is completely determinate, providing it follows the ground rule.  (In the side-band, influences from outside the perceivable 2N bleed in, so the evolution there is anybody’s guess.)  If, within this narrower band, any cell does not manifest the predicted next state, it has exercized its free will on that step. 

            Different monads have different personalities.  Some exercize their free will sparingly, some often; and a few contrarians (who might as well be dead) invariably do exactly the opposite of what the ground rule tells them.  Monads exercise their will in a variety of entirely different styles.  Some are absolutely random; others, absolutely determinate; others exercise it when they darn well feel like it (whether this last option is anything more than some blend of stretches of pattern and stretches of randomness, is not immediately clear).  Those that are determinate (forever, or for a stretch) may be so in an infinite variety of ways.  One may defect from the ground rule  only at even instants; another, only at multiples of three; another, at prime numbers.  Others are determinate but not predeterminate:  Lineland’s loveslaves, these defect at t = T+1 if and only if their beloved neighbor  M cells to the right  defected at t = T.   Of course, the monads may fall in and out of love, mimicking another’s pattern  only for a time. Other monads blend all these strategies (in a bewildering variety of proportions).  As: Defect when and only when t is a Fermat prime, unless some specified defection pattern of one’s neighbors occurs  (as, iff an even number of neighbors at positions -3, -7, -22 to the left  and 5, 9, 220, 5555 to the right  have defected at t; this pattern itself may be fixed, or may evolve – deterministically or otherwise), unless one happens to feel contrary that day and does otherwise.

Monads have a psychic lifespan.  Before their soul is instilled, and after they die, they never exercise their will, but turn on and off as predicted by the ground rule.  Lifespans are a connected subset of the timeline, and may (exceptionally) be infinite, either into the future or into the past, or both.

            Internally to the monad, this lifespan is (we conjecture) entirely determinate: the soul is either present, or it is not.  But from outside, it is in principle difficult to tell  when life ends or begins (cf. our own inglorious extremal stretches, of blastula and senility).  For, suppose that a given monad has been inanimate for all time (that is, always following the ground rule), then suddenly at a given instant (call it t = 1) defects for the first time, thereafter defecting precisely at Fibonnaci numbers.  It hews to this pattern for a quintillioan iterations, then falls forever silent.  We may say that it has died – by definition, never defecting is “as good as dead” – but we cannot say when, just as we cannot say actually when it was born, even approximately. For its free-will pattern may have been: Follow the ground-rule for a quintillion iterations; then Fibonnaci for a quintillion; then the ground-rule for three quintillion; then blink on and off alternately for as long as you remain alive.  It’s actual lifespan may be anything from one quintillion to five quintillion iterations (it definitely died before it could implement its plan to blink alternately during its golden years of retirement).

            Reincarnation seems to be possible.  This consists in a personality (i.e. a temporal pattern of defection) reappearing after it has been extinguished for a time; it may reappear at the same cell, or in a different one.  If (as is usually the case) the personality in question lived only a finite time, then its pattern is largely indeterminate, and the reincarnation therefore only approximate or probable.   Still, if a monad, during its recorded life, emits a quintillion-long Fibonacci pattern followed by a sextillion-long pattern of Fermat primes followed by a digital representation of “Yankee Doodle” before falling forever silent; and then another monad (having been forever silent itself), exhibits exactly the same pattern before it expires, then we may certainly say that, while they lived, they exemplified the same spirit.  (Of course, had they both lived longer, they might have in time diverged – we’ll never know.)

            Such reincarnation can, of course, be multiple, and (if in different monads) temporally overlapping.

            Monads can also get married and have babies.  As, a monad defined by the keystream “Emit Fibonacci”, and one defined by the keystream “Emit Fermat Primes” ,may – at any offset (say, staring at the 17^th Fibonacci number for Mom, and the 19^th Fermat prime for Pop) – blend their instructions:  Junior (who may be born at any specific delay later) emits the mod-two sum of his parents’ keystreams.  Sometimes the results are rather beautiful, as when a digital recording of the violin part of a sonata (Mom) weds and procreates with the piano part (Dad).

            Polygamy, we regret to report, is permitted: a child may have any finite number of parents (its keystream being the mod 2 sum of all its progenitors).  There is even incest: a child may be the mod 2 sum  of two identical keystreams at some nonzero offset.  Par-polyploidal self-cloning is, however, fortunately self-stultifying: should some self-important monad attempt to bud off an offspring parthenogenetically with the mod-2 sum of 2k copies of its own genome (with no offset), the result is identically zero: the child is stillborn.

            Life in Lineland, I hope I have shown you, is a perpetual festival, a riot of laughs. Perhaps that world strikes you nevertheless as more monochrome than our own.  But then consider.  How many different personality types does our own world show – how many (however intricately varied) personal individual quiddities?  So far, only finitely many, at most as many as the number of people who have lived.  -- But I mean, in principle.   Fifty billion? umpety jillion? infinitely many?  Well, in Lineland, there is a one-to-one correspondence between personalities and real numbers: thus, there are uncountably many.  Nor are these, though admittedly numerous, as blandly indistinguishable as the real line seen “from afar”, all unnumbered.  There are, as we have seen, a variety of styles – actually, of classes of styles – or should we say, collections of classes of styles… -- in personality, each one of which has an unbounded number of variations.  And any one variation – any one little perfect little round little self-sufficient monad – contains any amount of evolving variety – sometimes an unbounded amount; sometimes infinite in both directions.  And despite the fact that all one monad can learn about any other monad at any given instant is whether that monad is Off or On, over time it can know an unbounded amount, up to the capacity of its own memory, about an unbounded number of fellow monads. 

            Furthermore, despite the solipsistic flavor of the basic metaphor, life in Lineland is boundlessly social.  Your own actions are affected (though not determined, owing to your ready reserve of free will) by the states of your neighbors.  At any given instant, K of them on either flank  are inputs to your next step, and mK on either side  to your action m steps later.  This can lead to cooperation, but also, alas, to conflict.  As, Mike the Monad likes to be flanked by neighbors (up to some depth; or, as many as possible) that are On as often as possible.  So he cleverly varies his output in ways that will tend to effect this (of course, the effect is not always immediate, but may take time to make itself felt).  If his immediate neighbors are inanimate, he can eventually build up a pretty cosy little neighborhood for himself, all On and atwinkle like Christmas lights, most of the time. And if his next animate neighbor, Marvin, twenty doors down, is like-minded, they can together do even better at keeping alive a nice On column between them (Marvin fending off contrary propagations from the right, Mike from the left). If, however, they have opposite desires for the real estate between them, then life is an endless battle of once-twice-three-shoot.

Such, then, is life  among the merry little monads.  And, as it may be, among ourselves.

~

For a comparable Minimalist  mathematical fable, check this out:

http://worldofdrjustice.blogspot.com/2010/12/constructivist-angelology.html

Realism: What

[We choose a classic Lockean title-style for this chapter, and begin with a middle-of-the-road definition.]

Realism Defined

First, a plain-man's characterization of what we might call minimalist/core/plain-vanilla Realism:

Outside our heads  there is freestanding reality.  Only madmen and a scattering of constructivist philosophers doubt its existence.

-- Edward O. Wilson, Consilience (1998)

This view may also be referred to as Platonism (though cf. a special restriction of this term  to mathematics, outlined below):

One issue that has traditionally divided philosophers  is whether ther are abstract objects.  Nominalists have held that there are not;  realists (in a special sense of the word) or Platonists (as they have been called  to avoid the troubles of ‘realist’), have held that there are.

-- W.V.O. Quine, Word and Object (1960), p. 233

Now for the more careful distinctions of professional philosophers:

A. E. Taylor, Elements of Metaphysics (1903; page references to the University Paperback reprint), p. 67:

By Realism is meant the doctrine that the fundamental character of that which really is, as distinguished from that which is only imagined to be, is to be found in its independence of all relation to the experience of a subject.  What exists at all, the realist holds, exists equally  whether it is experienced or not.

Within this, he distinguishes (p. 68) two subtypes:

Agnostic Realism, while asserting the ultimate dependence of our experience upon a reality which exists independently of experience, denies that we have any knowledge of the nature of this independent reality.

Sic:  not “full knowledge”: any knowledge.  Which is absurd.

Contrast (p. 69):

Dogmatic Realism, of which Leibnitz and … Herbart are the most important representatives … while maintaining that real being is independent of experience, at the same time  holds that it is possible to have positive knowledge  not only of its existence, but of its nature.

Sic:  postive knowledge, but not necessesarily full knowledge.  And in this form I heartily subscribe, despite the invidious label conferred by its opponent Mr. Taylor.   As William James put it, in The Principles of Psychology (1890), vol. II, p. 634:

Reality exists as a plenum.  … But we can neither experience nor think this plenum.  What we experience, what comes before us, is a chaos of fragmentary impressions  interrupting each other;  what we think is an abstract system of hypothetical data and laws.

Full knowledge -- plenary knowledge of Reality in all its fullness -- not only in some rarefied ding-an-sich sense, but in the ordinary sense of the sciences -- is doubtless impossible in the case of anything so wildly complex as an acorn or a rock;  but in the case of something simple, like Hilbert Space (infinite-dimensional, it is true, but tamed by a norm that is based upon an inner product), we can perhaps come close to exhausting most of what can be known of it.

Taylor fancies he has refuted Realism in all its varieties, summarizing his triumph thus:

Produce any instance you please, we said to the realist, … and we will undertake to show that it derives its reality for you  from the very fact that it is not ultimately separable from the experience of a subject.

Note the crucial bait-and-switch!   He whisks Reality -- an ontological category, and a big deal -- under the thimble, and takes out -- “Reality for you”, a miserable psychological gewgaw of no general interest whatever.   This latter concept is, we readily concede, drenched in irreducible subjectivism;  no need to bother to argue the point.  And bearing, it may be, little relation to Reality, whether theoretically or empirically.

For:  Reality, in reality-for-X, is an incomplete symbol.

Thus for instance:   It is not possible to survey the roster of extant men, and finally lay your finger upon one specimen, the average man.  The Average Man is not a real, but an ideal, to which various actual men may approximate to one degree or another.  And Real-for-Joe-Blow is not a real, but a figment, which may reflect more or less of an actual reality, with greater or lesser distortion.

~

~  Posthumous Endorsement ~

"Were I alive today, and in the mood for a mystery,

this is what I would be reading: "

I Don't Do Divorce Cases

Murphy on the Mount.

(I am John Locke, and I approved this message.)

~         ~

~

A more carefully phrased importation of subjectivism into the debate  is provided by Michael Dummett, in “Truth” (1959), repr. in Truth and other enigmas (1978), p. 23f:

The claim … should be rejected by a realist, who might, and I think ought to, agree to the following weaker principle:  that a statement cannot be true unless it is in principle capable of being known to be true. […]  The anti-realist interprets ‘capable of being known’ to mean ‘capable of being known by us’, whereas the realist interprets it to mean ‘capable of being known by some hypothetical being  whose intellectual capacities and powers of observation  may exceed our own.’ … The issue between realism and anti-realism … is one of the most fundamental of all the problems of philosophy.

Thus here, clearly, we are once again confronted -- like it or not -- with the question of theism.

So:  if you are a full-bore, double-barrelled, two-seed-in-the-spirit, dyed-in-the-wool copper-bottomed no-holds-barred Cantorian Realist … (we pause for the roars of approval to subside) … even in merely so much as mathematical Platonism:  must you therefore, of logical necessity, be a theist in the sense of the Abrahamic tradition?

Perhaps not; certainly, we have not shown so.  (Nor do we so much as aim to show more that that.   At this point, Jews Muslims and Christians are all singing kumbaya together in one big tent.)   You could, in principle, construct for yourself an ontological halfway-house, with a non-omnipotent, though omniscient, Knower.   And who might this Wiser Being be?  Why -- none other than Babar, the Elephant King!

*

In these essays, we are concerned mainly with the form of Realism known (in a mathematical context) as Platonism, defined as

the theory that mathematics describes a realm or system of real and independently existing objects, whose nature is known to us through proof, but which are entities  over and above the proofs by whch we discover them.

-- Roger Scruton, Modern Philosophy (1994), p. 384

(This is fine, but don’t let’s squabble about “objects”; “patterns” or “truths” would do just as well.  We’re not interested in reifying anything, but merely in claiming that we’re not just making it all up arbitrarily.)

And:

… the subject matter of mathematics as realistically (i.e. platonistically) construed.

-- Colin McGinn, “Truth and use”; in: Mark Platts, ed.  Reference, Truth and Reality (1980), p. 35

 But for the record, there are other sorts:

Do we wish to say that there is a moral reality, which underpins our moral judgements  and guarantees their truth?  Some philosophers have argued for such a view (‘moral realism’).

-- Roger Scruton, Modern Philosophy (1994), p. 98

Scruton himself is far from being a moral relativist or nihilist, but seems not to feel forced to accept such moral realism, at least in the form of what we might call a moral Correspondence Theory (as opposed to the real existence of God and his laws).    Indeed, he compares (we shall call it) Aesthetic Realism:

It is obvious that St Paul’s Cathedral is beautiful, and the new Lloyd’s building  repulsive;  but is there some ‘aesthetic reality’ that makes these judgements true?

You readily see how problematic this is:  Saint Paul’s to a Wahhabi or an Iconoclast would lack appeal; and presumably the people who designed and paid for the Lloyd’s building  felt it had a certain something.   I still think a kind of case might be made out (which I’ll not pursue), that it’s not just all a matter of personal preference, and that de gustibus certe dispundandum est.   The argument would proceed by analogy with the much tighter case in mathematics.  Many things are obvious to the expert which are not obvious at all to the general public, and which indeed could never be made obvious to them no matter how hard you tried (sheaf theory, for starters).  And more tellingly,  the Man on the Clapham Omnibus will pronounce some things obvious (particularly in the area of probability, e.g. the Monty Hall problem) that are demonstrably mistaken (though some will understand the demonstration, and some will not).    We do not conclude from this that mathematical truth is just a matter of taste (“we” here referring of course to men of good sense, and excluding the Postmodernists).   Some are more qualified than others to grasp certain truths.    Quite possibly the same is true in the field of aesthetics, as regards, say, the Goldberg Variations, on the one hand, and “Who Let the Dogs Out”, on the other.   One would have to make out the case that those who exalt the latter and contemn the former have certain other things wrong with them as well.

[Update]   I have just come across an intriguing analysis relevant to the case of the “Lloyd’s building”, and that buttresses the Aesthetic Realist conjecture above, to the effect that it’s not all just a matter of mutually incomparable ‘tastes’, but that there are objectively different levels of aesthetic competence, owing to the importance of informed appreciation.  The analysis appears in a delightful essay, “The Gherkin”, by John D. Barrow, in his book 100 Essential Things You Didn’t Know You Didn’t Know (2008;  the book is better than its title).  The “gherkin” in question is a new building in the City of London, a.k.a. “the Pine Cone” for its pocked appearance:   “Prince Charles sees it as symptomatic of a rash of carbuncular towers on the face of London”.   Yet it won the Stirling Prize for architecture.  What gives?

It turns out the building is, from an engineering standpoint, quite ingeniously designed, and that principally with a view towards eco-friendliness.  Thus it narrows somewhat at the bottom, to defeat the wind-tunnel effect that annoys pedestrians; and tapers again towards the top, which “opens up more of the sky and reduces the dominating effect of the structure because you can’t see the top from close-by on the ground.”   Even the pocks have their point:  “They bring light and natural ventilation deep into the heart of the building”, saving greatly on energy costs.  And more details in this vein.

I’ve never seen it, but it sounds … beautiful …

~

Realism in psychology:

According to his friend and biographer, Freud “had a high and serious respect for the reality of psychological facts.  They were as real and concrete to him  as metals are to a metallurgist.” (Ernest Jones, Freud: Years of Maturity (1955), p. 432)

[continued here]

Further juicy quotations:

As a Platonist, he saw everything on earth as broken arcs, which merely suggested the perfect rounds above.

-- Louis Auchincloss, The Rector of Justin (1964), p. 92

Realists [with a capital R] are not the same thing as ‘realists’ in daily life, who are men who expect neither themselves nor others to be any better than they ought to be, and generally much worse.

-- Ernest Gellner, “The crisis in the humanities” (1964), collected in The Devil in Modern Philosophy (1974), p. 15

To mathematicians who study them, moduli schemes are just as real as the regular objects in the world.

-- David Mumford, Forward to Mircea Pitici, ed., The Best Writing on Mathematics 2012, p. xi

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.