Saturday, January 22, 2011


[This is a continuation of a thread begun here.]

            How, then, are the prime numbers – or Hilbert space – different from a unicorn?
            (That reads like a joke, to which the reply would be:  “They lack a horn.”
Or:  Hilbert and this unicorn  walk into a bar… Yet the question is meaningful enough.)
            Unicorns do not (alas) exist, or so I’m told; but people do talk about them, and some (such as myself) may even believe in them.  Descriptions and depictions do exist, which purport (wrongly; or tongue-in-cheek; or in some extended sense, accessible only to those who believe in snow-bunnies) to depict them – in all their monocerous, silver-shanked magnificence.  (There are worse purposes, to which silk or ink might be put.)
A picture-of-a-unicorn (so goes the rote) is as much an unfissionable entity as a hotdog, which does not split neatly into a hot and a dog:  what at first glance seem  components (<picture> + <unicorn>)  are in reality synsemantic (“incomplete symbols”).  A picture-of-a-unicorn differs from a picture-of-a-phoenix, but you cannot extract the quantifier and give it wide scope:  there exists no thing which eíther is a picture of.  When we say (straining the usage of “exists” just a bit) that “The unicorn exists in heraldry”, we mean, nothing ontological, but merely: If you wish to find a picture-of-a-unicorn, go to the heraldry section of your local library,  and not to the section on pets.
            Okay, so no unicorn; darn.  What about its Meinongian doppelgänger, the “Idea of a Unicorn”?  The modern consensus, to which I overtly subscribe (with certain arrière-pensées concerning the unicorns themselves, whose hooves, like flint, strike fire from the stones  whereon they gallop, never heeding the wind) declines to admit that either unicorns or the “idea” of a unicorn does exist, in any concrete or meaningful sense – that is, as a thing with some independence, an idea that is there, somewhere, even if no-one is presently thinking it; a thing, so to speak, with a front and a back.  (And here I do truly, and passionately, agree:  if the forests lie sad and silent, unroamed by unicorns, then away with such rubbish as the “unicorn-idea”!)   I may be thinking of unicorns (as indeed I usually am), or entertaining (second-order) beliefs about them (their mane is like silk, like sand), but this is more like a verb than a thing.  (Again: There exists no unicorn, whereof I am thinking; I am merely daydreamicorning, unicontemplating…)  People may well continue to entertain ideas about unicorns (and by now you can supply the punctilious punctuation: entertertain-[ideasaboutunicorns]) so long as the (foolish, folk) tradition is passed down:  but once let the Earth disappear in a puff of smoke (which, bear in mind, it may do at any instant), and all ideas of unicorns die with it. (And here, though at my most sentimental, I make no objection:  They lie, with Thor, in a common grave.) Travelers from another dimension, roaming the ruins of our world, will never encounter a trace.

            The case with integers is quite otherwise.

            Let all sentient beings perish in the Big Crunch (this is meant, not as an optative, but as a [contrafactual] hypothesis); then let the universe evolve anew.  There may or may not, in that cycle, evolve any rational beings at all; or they may evolve, but be such sobersides as to have (like your boss) no interest in fables and fairy tales; or they may develop some folklore of their own – in all likelihood, nothing resembling a unicorn (these new rational beings  being themselves, for one thing, perfect spheres;  mothers frighten their children with ellipsoids):  yet let any one of them turn his hand (or flipper) to enumerating the stars of the firmament, or the ways in which he loves his sweetheart:  and the precise same integers as before  will stream to his aid.

(Yet let us pause for a bit, for I am seized with sudden sadness.  In principle very glad, that our sturdy friends the integers  survived that  cosmo-catastrophic transition;
and yet I do lament,
and sorely miss,
the brightness and that brashness,
of those   proud
                                 steeds ….)


Let us look  at the same thing  from another angle.  (A nice diagnostic, for things that are really realin the round.)
People have been tempted, and were scolded by Russell for so doing,  to say that Hamlet is real  in Shakespeare’s imagination, the way (or at least rather like) Napoleon in ours.  Russell’s retort is classic:

       When you have taken account of all the feelings roused by Napoleon in writers and readers of history, you have not touched the actual man; but in the case of Hamlet  you have come to the end of him. If no-one thought about Hamlet, there would be nothing left of him; if no one had thought about Napoleon, he would have soon seen to it that some one did.

Spoken like an Englishman!

Stop ...  being ... silly ..... ..... ...

I still have a sneaking sympathy with the reality of the gloomy Dane, in the sense that there are propositions about him (/it/whatever) that are true or false in our world, never mind in Platonic paradise, or in that of the late Shakespeare.  If, on a test, you identify Hamlet as the prince of Macedonia, you will be marked down; nor is his ladylove named “Buffy”.  (In southern California, that answer might get you half-credit.)  So, Hamlet is not real (unlike “Hamlet”, a play by the bard of Avon), but…real-ish.  (Compare the concept, discussed elsewhere, of truthitude.)  And contrariwise, the constructs of our world are – mostly  exactly that:  constructs, and thus ideal or even -- fake-ish.  -- I won’t emphasize this latter point, as it has been  if anything  overemphasized of late  by post-modernists and deconstructionsists, not to mention earlier and more honorable  skeptical attacks.  The upshot is simply that the two levels of Things  tend to edge towards  meeting in the middle.  We can still, faced with the following examination-question,

            Choose the odd man out:   
            (A) unicorn   (B) dog    (C) robin

correctly pick (A).  But we must concede that “dog” is a concept held together with duct tape, consisting as it does of an ever-evolving medley of breeds, beginning with one insensibly different from the ancestral wolf, and fanning out into some that may be no longer interfertile, and differing the one from another, in both appearance and behavior, more than do some allied species among themselves.  And as for “robin”, the term in lay use designates polysemically (though not really homophonically, the way “pen” designates both a female goose and a writing implement) a variety of superficially similar species, depending on where you hail from.  In terms of being a “natural kind”, the unicorn may have more uniformity than the canine.

There is another trait, though, which distinguishes more sharply between unicorns and zoologically more respectable species, than the existence of detailed descriptions (some fictional, some factual) of each.  And that is, the ability to reason with them, inductively.  If, on your trip to Shropshire, you meet a particular bird, let us call him Hoppy, which on visual inspection prompts you to call it a “robin”, an ejaculation which in turn prompts vigorous assent from the circumambient peasants, then, even though Hoppy is not actually quite the same sort of bird as what you used to call a “robin” back in Illinois, nonetheless, you’ll be correct in predicting that Hoppy can fly, and probably that he has a palatal penchant for worms.   Whereas, should you chance upon a monocerous equine in your ramble through Sherwood forest, you can conclude almost nothing.  Whether it would, like the unicorns in books, meekly lie down at the sight or scent of a virgin, is anybody’s guess.  (I’m betting:  Yes.   I mean -- I would….)

So now, let’s try our hand at this one:

            Choose the odd man out:
            (A) unicorn  (B) dog   (C) compact Hausdorf space

The answer is still (A); you can reason with the latter two.  And indeed, (C) is in a sense more tangible, and more reliable, than is (B).  Should you ever wake up one morning in a compact Hausdorf space, you can be quite sure that, should you chance upon an infinite sequence of points running along the path of your morning stroll, then that sequence will infallibly converge to a point in the space. And this would remain true, were the space somehow embedded (this time preferably without your presence) as a subspace of Hell.  For though the Dark Prince (whom God defeat) may have power over reprobates, and even the power  from time to time  to tempt the righteous, yet he has no power over topology.

            This reminds us of the old conundrum:  How can there be an omnipotent Supreme Being, who yet has no power to refute the truths of arithmetic, nor indeed any necessary proposition, such as those of topology?  The simplest answer may be, that topology is part of God.  You don’t refute your own arms and legs.

Straightaway (for conscience pinches) let me hasten to explain what I mean and do not mean by “Topology is part of God”.  It’s a nifty epigram (assonance and all), but it can mislead.  What I mean is – well, all that was said above, which is hard to summarize, but in a nutshell: If you consider the Creation to be part of what characterizes the Creator, then (actually, a fortiori) you should consider the principles by which that creating is regulated (topology among them) to characterize Him.  A simple point.  What I do not mean is – anything a breathless journalist might make of all this.  In particular, it does not directly say anything about what matters most to most of us, day to day, when it comes to God:  Does He bid us do this, or that? Might I be damned? Can I be saved? What must I believe?  Need I believe anything? --  If anything, contemplation of the multitude of invisibilium distracts our attention from such questions:  The more our mind must focus on God as Creator (since, as we come to learn, there is so much more to the Creation than Levittown, so much more still than we could ever behold with our eyes), the fewer neurons are left over for contemplating God as Judge, God as Comforter, God as Redeemer.  Fact is, He is infinite, we’re finite. Only so much bandwidth down here.
            Nonetheless, I must insist on this point -- trivial though, in a moral or eschatological perspective, it may be; if only to help correct the imbalance that has gone before. The Bible (meaning: the sum of the Old and New Testaments) dispatches the Creating in a couple of paragraphs (curiously prescient paragraphs though they be).  All the rest of the text just takes the visible world as given, asking after nothing more; and busies itself with guides to conduct, awful warnings, things admirable but not to be tried at home, before finally culminating in Christ, who if anything (despite a philosophical-sounding, if vague, “In the beginning was the Word” in the odd-man-out of the Gospels, John; an apophthegm  in any case  never really developed) is more centered than ever on the relationship of God to Man, not God and the Plan.

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