The Continuum: Mainstay or Menace? (erweitert)

The Continuum:  the original sin, from whose fecund loins
came all that is non-constructive in mathematics.

-- Anon.

Kronecker dismissed mathematical entities beyond the natural numbers as “Menschenwerk”.  An average practicing mathematician (who uses such entities all the time) may  agree with him to this extent:

(1)  Our intuitions about the natural numbers are clear and solid.   So long, indeed, as one deals only with some set of actual numbers (thus, a finite set), nothing especially surprising  or even all that interesting  turns up.  If we extend our horizon to the actual infinite of the set of all natural numbers, we meet some concepts that take getting used to (Hilbert's hotel):  but once we’ve done so, they seem natural enough.

(2) The rationals and negative integers  definitely, the algebraic numbers  probably, pretty much come along for the ride (that is, you can hardly exclude them once you’ve accepted N), and they still bring in no paradox – being, after all, of the same cardinality as the natural numbers themselves.  Though, a case could be made that these are not “entities” of the same standing as the integers, which in a sense we can hold in our hands (embodied in oranges, say), but rather abbreviations for operations on integers.  Thus, we cannot hold minus-two oranges in our hands; minus-two is not a thing, but a bookkeeping device. 

(3)  The real numbers, by contrast, are … a piece of work.  Maybe even Menschen-work, except that one could hardly imagine Menschen coming up with anything so intricate and even bizarre.  Their very cardinality baffles intuition  -- and the independence of the continuum hypothesis  shows that we are right to be baffled.  [Note:  The simple infinity of the integers already baffles *untutored* intuition;  but eventually you get the idea.  Click on the Label "Hilbert's Hotel" for further exemplification.  Whereas, the cardinality of the continuum is more like... Hilbert's Nightmare...] All sorts of queasy consequences arrive for simple quantification (cf. Quine re.  objectual vs. substitutional quantification).  The reals were invented (discovered?) for purposes of analysis, which in turn was developed largely for the sake of physics: but it now appears that physics (whether in its quantum cast, where Uncertainty provides a certain indissoluble granularity; or in the Wolframesque finite-automata approach) might not actually require, or afford, a continuum.

And yet standard mathematics speaks indeed ontologically of the reals, not merely pragmatically.  Thus for instance, Rudin’s standard text (Principles of Mathematical Analysis, 3^rd edn. 1976, p. 8):

We now state the existence theorem [emphasis in original] which is the core of this chapter.

Theorem. There exists an ordered field R which has the least-upper-bound property.

The author then mentions that the proof actually constructs the Reals out of the Rationals.  This is, of course, the most solid sort of proof of all – not one of those Cantorian diagonalization thingies that has you winding up assenting to the Infinite Woodchuck, without ever quite knowing how you got into such a fix.  It gives you an actual recipe for the construction of these extended numbers, as concrete and explicit as for baking a cake.  And yet… all kinds of things can be thus “constructed”, at will, including items which presumably are not part of the furniture of the universe, in the sense that angels actually sit on them.

~

A roaring vote of confidence in the continuum  is voiced by the noted mathematician René Thom:

“God created the integers and the rest is the work of man.”  This maxim spoken by the algebraist Kronecker  reveals more about his past as a banker who grew rich through monetary speculation  than about his philosophical insight.  There is hardly any doubt that, from a psychological and, for the writer, ontological point of view, the geometric continuum is the primordial entity.

-- “’Modern’ Mathematics: An Educational and Philosophic Error?”, in American Scientist (1971), repr. in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 74.

That is in-your-face Platonism, with which, quâ Realism, we have no quarrel.  But the psychological claim seems dubious:  Our intuition of the continuum is probably no more than a vague notion of a smear (and not very infinite at that, neither going out nor going down).   And as for the ontology … When we first meet the Real numbers mathematically (that was the very first thing we did in first-year calculus, with the opening chapter of Spivak’s text), we conceive them as the completion of the rationals.  And such they are indeed:  only, with respect to the metric provided by the absolute value.   With a p-adic valuation, you get a different completion of the rationals, the p-adic numbers.   Lastly, the surreal numbers augment the continuum in yet a different unexpected direction.  (I have less than no intuition about any of this.)

The physicist Schrödinger is less sure:

The idea of a continuous range, so familiar to mathematicians in our days, is something quite exorbitant, an enormous extrapolation of what is really accessible to us.

-- Erwin Schrõdinger, “Causality and Wave Mechanics”, repr. in translation in: James R. Newman, ed. World of Mathematics (1956), p. 1059

And from an Intuitionist (close kin to a physicist):

This could be done  by seeing the continuum as something that is infinitely becoming, instead of already being.

-- Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p. 333

(Compare our old friend the actio/actum distinction.)
Might be fine for physics, doesn’t work for math.  ‘See’ it however you like; that uncompleted-account doesn’t jibe well with Cantor-style constructions.

~

One might say:  The continuum feels unproblematic enough, so long you take it for granted, as just some kind of smooth dense slippery thing, like mud.  Yet so soon as you pause to enquire more nearly, you are back in Saint Augustine’s predicament with regard to Time: “Quid est tempus? Si nemo a me quaerat, scio …”

~

Even in a universe which (like Wolfram’s) abjures the continuum, the continuum might turn out to be mathematically indispensible for its treatment.   Cf. the indispensible role of “imaginary” numbers in electromagneticsm or quantum mechanics, even though all observables must be real-valued.

Twinned Eternities

In the poem portion of Nabokov's Pale Fire (1962), we read:

Outstare the stars. Infinite foretime and infinite aftertime.

The basic lexicon of Classical Arabic, as it happens (with its wealth of words), distinguishes these two, with a brief, monomorphemic word for each:

أبد  [abad] prospective eternity (with a terminus post quem but no end)

أزل [azal] retrospective eternity (with a terminus ante quem but no beginning)

Reflecting separately on these  raises interesting philosophical questions, e.g., regarding the indestructibility of the soul.  But as a merely intellectual matter, the picture will be clear to anyone who has got as far as geometry or calculus -- "half-infinite" directed intervals (such as the non-negative reals).

In Laughter in the Dark (English version 1938), Nabokov plays with the twinned hemi-eternities, in a passage full of satire:

“Haven’t I met your sister once?” queried Dorianna in her lovely bass voice.

“My sister is in Heaven,” answered Rex gravely.

“Oh, I’m sorry,” said Dorianna.

“Never was born,” he added.

Much is rustling beneath the sheen of this. First, Rex, po-faced, lays a logical snare for Dorianna, knowing that she would automatically assume that the now-absent soul is residing in the after-eternity (Arabic abad), which is all that most people ever think about, or even conceive to exist. An extra twist is provided by Dorianna’s automatic, flustered response at the unexpected reply – the conveyed sense is meant to be, “I’m sorry for your loss,” but simply stringing bits of syntax together yields, “I’m sorry that your sister is in Heaven.” (**) Rex then, indirectly pointing out the ambiguity in his initial statement, and implicitly rebuking Dorianna for failure to perceive ambiguity (again, in whatever sphere, most people don’t), reveals that this sororal Arlésienne is in fact at present domiciled in the ante-eternity (Arabic azal).

That passage, incidentally, rests as well on the paradox alluded to last time, that of the indestructibility of the soul. Intuitively, we think of a soul as the sort of thing (being brewed, after all, from the aethers of the Eternal One) that doesn’t simply wink out of existence. But by the same token, it shouldn’t be the sort of thing that simply winks in – in which case it must, as in the Platonic conception, be “stored up” somewhere (Rex specifies Paradise as the ‘green room’ or on-deck circle, though probably more from a faux-polite convenance than from any committed Platonism).

That observation does not exhaust the (depthless!) well of paradox subtending such insight. Follow it out, and you’ll encounter a retrospective version of the quantum-mechanical “many-worlds” scenario.

(**) P.G. Wodehouse plays on the same effect. In his world, earls are crotchety and idiosyncratic, and butlers are perforce imperturbable in the face of all.

            “Dash it, Beach, this egg is undercooked!”

            “Yes, sir.”

            “What – is the cook out sick again?”

            “Yes, sir.”

            “Well, I’ll be damned!”

            “Yes, sir.”

(Picture an infinitesimal lengthening of the glide initial: “Yyyes, sir.” The joke is, the double-meaning will be completely lost on the earl.)

~

It was the contention of “The Stokes Conjecture” (a chapter in my book, The Semantics of Form in Arabic)  that possession of such a neatly twinned pair of monomorphemes as azal and abad, as opposed to such roundabout (and nonce) phrasal confections as “infinite aftertime”, should track with a greater tendency of speakers of the language so outfitted, to be clear on the subject, for it to be cognitively relatively accessible, and thus to accrete around it  further developments both morphological (derivata) and semantic (metaphors).   Whether this be the case for Arabic letters and Islamic theology, I leave to my learnèd readers to make out.

~

The casual reference above, to the isomorphism of azal and abad, as constituents of infinite Time, to the half-rays  as subsets of the real line, conceals a textural disparity, in point of richness.   Not, as you might imagine, in favor of the human/experiential conception of Eternity, but the mathematical/intellectual science of the Real Line.

We really don’t know what to do with Eternity -- and literally, wouldn’t know what to do in it.    Like the silence of the infinite spatial reaches that so dismayed Pascal, we stand aghast at the prospect of doing anything “forever”, be it strumming harps or standing around on clouds swapping New Yorker captions.   The prospect of an infinite afterlife, for which we are supposed to yearn, is strictly baffling.

(Note that there is nothing heretical in that observation of human psychology;  notably, C.S. Lewis was converted to Christianity  before any sort of belief in or appreciation for  an infinite afterlife  was granted him.)
(For a mathematician's take on how the afterlife shall be spent, try this.
For an equine perspective,  this.)
(I riff upon the bafflement in the azal case, here.)

The Real Line, by contrast, is … infinitely diverting.   To begin with, even in a low-focus broad view (abstracting from the  so to speak  “quantum foam” of the infinitessimally inspected continuum), it harbors a great many other infinities within itself.  Thus, that infinite ray or half-line,  [0, ∞ ), is topologically equivalent to a mere half-open interval,  [0, 1):  in both cases, you can cover the space with a countably infinite sequence of disjoint intervals, no finite subset of which can do the job.  Thus, for the ray: [0,1), [1,2), [2,3) …. etc;  for the interval, [0, ½) [½,¾), [¾, 7/8) …

Further, the infinitude lies not only in thus stretching out forever, but in drilling down.   Any interval (even one that is closed, and thus compact) harbors infinitely many intervals as subsets;  and any such patch contains an uncountably infinite collection of points.  (We recall Dyson’s title, “Infinite in All Directions”, but with ‘directions’ differing qualitatively rather than simply like a compass needle.)

And those are just the appetizers.  The continuum is a regular zoo of exotic creatures -- the Cantor set, the Borel sets, projective sets, and a host of others studied in the discipline of “Descriptive Set Theory”.

~

As for those specifically twinned infinities, staring at each other from opposite sides of a mirror, they have a counterpart in modern physics, which takes reversibility of Charge, Parity, and Time (separately for some processes, in combination for others) as a kind of credo, like Liberté, Egalité, Fraternité.   Traditionally, time-reversed solutions were usually dismissed out of hand as unphysical;  other, more venturesome theorists, embraced them, telling fables of antiparticles as simply particles moving backwards in time, among other scenarios that chill the blood.

[ShoutOut:  Many thanks to Djinn ibn Sayârah  for help in formatting this for posting here.]

[Afternote]  My friend the Arabian theologian writes in :

Mark Twain wrote, regarding death:

“I do not fear death. I had been dead for billions and billions of years before I was born, and had not suffered the slightest inconvenience from it.”

That's kinda funny for as far as it goes, but no one who has experienced consciousness can be seriously flippant about it being snuffed out.

.

Minimalism vs. Nihilism

I’m currently reading an intermittently entertaining book by Jim Holt, with the simultaneously catchy and off-putting title  Why Does the World Exist?   (2012).    I actually gave my mother a copy for Christmas when it came out:  not having yet read it myself, but counting on that ever-companionable writer to make big ideas plain to the public, the notion being that she would read it, and then we would discuss it in our weekly phone-calls (we live on opposite seaboards:  she in a nursing-home on the west coast,  I in a sheltered workshop on the east).   Unfortunately, Alzheimer’s intervened, and we can no longer discuss anything substantive;  her days of wine and metaphysics are behind her.  Though, now, as I finally get around to reading it myself, it would seem that, in this particular case, she hasn’t missed much.  The problem is, his clear-headed addressal of that perennial topic of sophomore seminars -- Why is there Something, rather than Nothing? --  would have made an excellent laconic magazine-article, but Holt somehow wound up with a book contract for this ultimately (indeed, quickly) sterile subject, and has to pad things out.  So he takes various nugatory arguments by e.g. “metaphysical nihilists” more seriously than they warrant, and duly deflates them, but takes several pages doing so.

(Mr. Holt’s genteel approach to such gentlemen  is to humor them -- though, to his credit, ultimately to refute them.  Ours is rather to rip their throats out;  cf.  our diatribe contra Eliminative Materialism.)

The exercise calls to mind James Surowiecki’s canny column in this week’s New Yorker, “Punditonomics”, where he notes the (obvious) economic incentive to spew out short-shelflife twaddle if all you want is eyeballs.   A lot of that has been evoked by the disappearance of Malaysia Air Flight 370 (I actually work a few feet away from some Bureau integrees who are involved with that case, and poll them every so often.  BLUF:  UFOs are not involved.)   As Surowiecki reports,

The peak (or nadir) of the speculative frenzy  came when CNN anchor Don Lemon wondered aloud whether the plane might have been swallowed up by a mini black hole.

(Ceci renoue avec notre thème du néant.)  When a retired general who now shills as a CNN commentator, failed to cough out an equally (yet distinct!) OTT hypothesis, suggesting we wait and look at the facts,  the Host replied, revealingly,

“You know how cable news works, don’t you?  We got time to fill here!”

And thus, likewise, Mr Holt has some pages to fill, that were better left (in the spirit of Nothingness) elegantly blank.

On pp. 59-62, Holt gamely attempts a riff on nothingness as, “as Leibniz was the first to point out, the simplest of all possible realities,”  adding solicitiously, in case the reader has instinctually at this point  razzed in disgust, “Simplicity is greatly prized in science.”  (So, pipe down, you-all in the back row.)  In mathematics, by contrast, the technical term for this (the null group, or anything else) is:  trivial.

In the course of this, though, he does raise a somewhat interesting question:  “If our world turns out to have an infinite census of objects, why should it be, say, aleph-2 rather than aleph-29?  Only the Null World escapes this kind of arbitrariness.”  (The same observation applies to a finite ontology as well, of course.)  And the answer to that, we would suggest, as we have argued in a series of essays (Theologia Mathematica), is to take the invisible world seriously, quite on a par (as Gödel argued) with the visible (and  arguably  moreso).  Such infinities as exist, exist, and those that don’t, don’t, and there you have it.  (Of course, should there turn out to be some unexpected supremum like aleph-29, we will look for a principled understanding of that surprising fact.)   This line of thought has been applied more seriously to the problem of the cardinality of the continuum:  aleph-1 seems the only non-arbitrary value.

~

So what do you do when the puns on Nothingness run out?   If you’re a good author, like Jim Holt, you don’t simply repeat yourself, or inflate a chapter like a bicycle-tire:  You take the show on the road.   And so we are treated to a travelogue of several European and American capitals, meeting various Colorful Characters along the way.   It’s like a thematic tour, only instead of the theme being Chocolate or Wines,  it’s Big Ideas;  and instead of Mansions of the Rich and Famous,  it’s Cramped Apartments of the Brainy and Loquacious.

And thus we are whisked to Pittsburgh, where we make the acquaintance of the legendary Adolf Grünbaum, a Morris-Zapp-like personality, still robustly vigorous in advanced age.   And in addition to various choice philosophical obiter dicta, we are regaled with a memorable tale of the nighttime drive to the restaurant atop Mount Washington, the voluble crusty atheist at the wheel.  (B.L.U.F. : Never drive with the guy.)

[Afternote:   I have just begun reading the Collected Works of Adolf Grünbaum, the first volume of which  came out in 2013.  Evidently the wily old philosopher was delighted by the rather Zorba-the-Greek portrait he gets in Jim Holt’s book, for the dust jacket quotes Why Does the World Exist to the effect that “In the philosophical world, Grünbaum is a man of immense stature.  He is arguably the greatest living philosopher of science.”  On page one of the Introduction, the editor goes on to quote Holt at greater length along those lines.]

“…Mmmyess… Immmmense, that is quite the word ….”

Then before you know it -- like James Bond hopping from metropolis to metropolis -- we are in Paris, this time without the excuse of any actual philosopher to interview, the slender connective being the Café de Flore, where Sartre used to squat, and doodle about le néant (Why is there quelque-chose rather than rien du tout?, that is to say).   We never get any philosophy here, but the author does offer a vivid tableau of the nightlife of fashionable Eurotrash:

At a table in the back  I spotted Karl Lagerfeld, with his characteristic ponytail, dark glasses, and high white collar, in hushed conversation with one of his muses, who was wearing what looked like black lipstick.  Other than that, the place was pretty much empty: le Néant.

But then there was a noisy burst of activity.  A woman of a certain age … breezed through the front door, accompanied by a pair of what appeared to be Cuban gigolos  dressed in shell suits.  Giggling and grinding their teeth, this trio sat down with us  and began to jabber away.  The woman’s face was a sallow mask of leathery jollity, and she talked in a low croak  that put me in mind of Jeanne Moreau. … It seemed a good time to leave.

(Should you ask:  Why are there gigolos, rather than nothing -- the question is unanswerable.)

[Update 10 May 2014]  Jim Holt takes on Derek Parfit; wins by forfeit:
http://worldofdrjustice.blogspot.com/2014/05/have-theory-will-travel.html

Quine

Hovering, buzzing in the background, as I write these notes, is the solemn, rounded, currently extraterrestrial figure  of Willard Van Orman Quine.

I took Introduction to Logic from him  sophomore year  -- “Phil 140” -- one of the very few course designators I remember, along with  “Math 11” (Robin Hartshorne), “Math 55” (Andrew Gleason), and “Nat Sci 2” (George Wald).   All these went into shaping the man I am, quite as much as did the Y chromosome.

Simply as a stylist, he is almost my favorite writer -- right behind Chesterton.  Any paragraph at random, from either man, is guaranteed to delight, both in style and in substance.   For better or worse, his tight and chiselled, somewhat precious prose, infuses my own;  and he returned the favor, in a generous letter, praising The Semantics of Form in Arabic, which else must seek far and wide for any mention, let alone praise.    His style is mesmerizing -- I never find myself disagreeing, when I read his words;  though a paraphrase is never so compelling.

Yet on a core point of these essays, we seem to be at loggerheads.  Consider the following, from the celebrated “Two Dogmas of Empiricism” -- the original version in the Philosophical Review (1951);  a passage omitted (as Scott Soames points out) from the more accessible collection of essays, From a Logical Point of View:

Imagine, for the sake of analogy [“analogy” because his real game is the posit of physical objects, which he likewise deprecates], that we are given the rational numbers.  We develop an algebraic theory … but find it inconventiently complex, because certain functions,  such as square root,  lack values for some arguments.   Then it is discovered that the rules of our algebra can be much simplified by conceptually augmenting our ontology with […] irrational numbers.

So far, no quarrel at all.   But now we restore what we had suppressed in those square brackets:

… with some mythical entities, to be called irrational numbers.  All we continue to be really interested in, first and last, are rational numbers;  but we find that we can commonly get from one law about rational numbers to another  much more quickly and simply by pretending that the irrational numbers are there too.

            So:  The challenge to the Realist, is to demonstate, that the irrationals (so invidiously named) are indeed part of the fundamental furniture of the universe, and not mere spectral butlers, bustling about among the throning rationals, servile and ultimately dispensible.

            At present, I cannot meet this challenge.  In the first place, because I don’t understand much about the continuum, other than that it is a depthless well of mystery and paradox, and so don’t really know what to make of irrational numbers.  From Quine’s passage, you might imagine that they are harmless, simply a “rounding-out”, like adjoining an ideal point at infinity:  but they are much more than that.    With the rationals, we haven’t really left the comfortable, Kronecker-approved world of the integers:  the countable case.    Yet open the barred door, and the winds blow in.   “But to the rationals do the gods inherit;  beneath are all the fiends.”

            So in the meantime, while mulling it, here at least is one thought.  Even if you wished to spurn fractional rationals (on the grounds that there is an infinity of them in a thimble, and you don’t like infinities), and wished to stick only to the positive integers -- you would still run smack into the irrationals.   For, an isosceles right triangle with sides equal to unity has a hypotenuse measuring the square root of two.  --  OK OK, you say, I’ll buy irrationals, but not all of them: just algebraic numbers (the set of which is still countable).  -- And now you are on the slippery slope, right where the Realist wants you.  “I’ve got a couple of transcendentals [non-algebraic numbers ] I’d like you to meet, pi and his buddy e.  They’re right outside the door… and the window… and on the roof…. In fact, you can’t miss them.”

-- Egad, this just in!  Quine, replying to his critics, in Hahn & Schilpp, eds., The Philosophy of W. V. Quine (1986), p.  315:

            I admit the real numbers.

All is forgiven!  Van!  We are at one!