Der Fall Rorty (Explication de texte)

 Richard Rorty, Philosophy and the Mirror of Nature, p. 19:

If we try to clarify the orthodox notion of ‘the divine’, we seem to have either a merely negative conception, or else one explicated in terms of the notions of ‘infinity’ and ‘immateriality’. Since reference to infinity explains the obscure by the more obscure, we are left with immateriality.

            Rorty seems to have solved the Packing Problem:  This is ten pounds of bullsh*t in a five-pound bag.

            The first clause first, though it is the least exceptionable.  It neither says nor presupposes anything false; the bias is merely rhetorical.  We “try to clarify” (evidently doomed in our efforts, so obscure is the clarificandum) – as though clarification were not likewise a necessity for everything from rocks to snowflakes to the Continuum Hypothesis.  Bishop Berkeley tried, and failed, to clarify Newton’s fluxions; Cauchy succeeded.  Next!  -- And then those little tweezers, the squotes:  I use them myself, so I know what they’re used for.  Preferably not to denigrate divinity before we’ve even begun (“…of ‘the divine’ ….”).   

            The “merely negative conception” does point to a real current in theology (both Christian and Islamic), but a current so rich, that the “merely”, while essentially an emotion-word, almost rises to the status of a lie.  For the “negative conception”, the  via negativa, (discussed elsewhere in this series;  just click on the label) is nothing like nihilism, nothing like negativism, nothing even like skepticism.  It is a stance arrived at after a long – centuries-long – attempt to characterize, in positive terms, what is… well, it turns out, very difficult to characterize, at least in terms of the predicates we inherited from our hunter-gatherer and pastoral past.  We might compare it to the history of our contemplation of the Continuum Hypothesis.  In the early days, red-blooded mathematicians naturally set out on their stallions to prove it (positively) true; or, failing that, to prove it (positively) false.  The current position (after much intermediate agony)  is:  It is true in some models; false in others; and independent of the axioms as usually deployed in set theory.  This whole “independent of the axioms – neither true nor false, exactly, but not nonsense either” is a rich notion, almost too rich to digest,  which required millennia to arrive at.  It is indeed a kind of (unanticipated) negative result; but not “merely negative”.  And since the problem of God is a superset of the problem of the Continuum Hypothesis, we cannot well expect a simple etch-a-sketch portrait of the guy (as it were, posing along the boardwalk at Ocean City).  (Indeed, if you follow me closely, my adducing the C.H. for comparison, so far from being some typical theistical opportunistical move, if anything undermines our naïve conception of the Creator; at least, its initial effect on myself was sickening, as bad as the Beagle on Captain FitzGerald.  And there will be no post-initial effect for quite some time, until – God grant it – I gain greater insight into the thing.)

He goes on:  ”or else one explicated in terms of the notions of ‘infinity’ and ‘immateriality’.”

(Note again the sneer quotes, offered in lieu of argument by this ‘philosopher’.)

Being passably ignorant of theological history, I cannot say whether these two epithets, among the many that might be hazarded as regards the Godhead, are the two statistically most prominent ones, but let us suppose they are.  No theologian, and no widow lighting a candle, ever imagined that these were the crux of the thing.  The integers are infinite and immaterial; so is the Infinite Penguin; we worship neither.  How about “Creator”, “Redeemer” and “the ground of our morality”, yo?

  “Since reference to infinity explains the obscure by the more obscure…”    Good… Lord.  I wish it were so.   If the nature of the divine were actually clearer than the nature of infinity – I wouldn’t need to write this essay, it could be left to the kindergarten teacher, while the rest of us dance table-top, champagne in hand.   Infinity – by which I shall here always mean, the very teensiest flavor of same, aleph-nought – has been very well studied by now.  It is virtually suitable for the nursery.

            If I have hesitated to push the infinity-of-the-integers business too far (their infinity, as opposed to their Necessity), it is not because it is too obscure:  it is rather too simple, too almost shallow.  I can tell you a lot more about little-omega (that least, most modest infinity, with the ordinal type of the natural numbers) than I can about, say, something really difficult, like a duck.

            As for “immateriality”, that’s… immaterial.  Is He immaterial, or nonmaterial, or intangible, or  ethereal, or abstract, or funereal  -- wholly or partly so? Some say He once walked through Jerusalem, leaving footprints in the sand.  That is intriguing, if true (trust Rorty to latch on rather to the boring possibility); but as for immateriality, that is no part of our interest in, or devotion to, Him.  Smurfs are immaterial for the matter of that, and I don’t go to Smurfs on Sunday.  In fact, given a choice (say, by a dating service) between a Material entity and an Immaterial, I’d go with the Material every time.  It might tickle our idle curiosity to find out, whether He is – or rather, for that is nonsense, to what extent and in which ways He is ` ` ` immaterial ‘ ‘ ‘ (and here it is the philospher’s use, not the possible pis-aller use by the theologian, that I am punctuationally excoriating), -- how like a ghost, like the air, like the integers, like the non-algebraic numbers, like the angels, like aleph-one, like our late great-grandfather, like a poem once spoken but never written down, how like a prayer we would have uttered, but that the Reaper came too soon …. You’d need a lot of terminological tidying-up for the question to even make sense, and it isn’t worth doing.  Those who believe they have had some actual experience of God, have a number of tart things – be they true be they false things, but – a number of sharp and hard things, to say, about these experiences, which in no way resemble the mumblings of a sociophobic agnostic concerning the silences of a fog-enveloped all-encompassing blancmange. To ignore all this and focus on “immaterial”, is like never bothering to learn Relativity, yet loudly wondering (peevishly) about just what was Einstein’s favorite flavor of ice cream. (My understanding, incidentally, is that he preferred minkowskian, with sprinkles.)

            Nothing hangs on this red-herring of the “immaterial”.  Most folks who have had (as they imagine; again, righly or wrongly) any immediate experience of God, tend to emphasize the personal.   My own conception of the Creator is actually more immaterial than most, because I’m emphasizing the actual nature of the Creation, than which the Creator is logically-necessarily more complex: a Creation which has – I mean just plain in terms of Measure Theory -- far more of the mathematical (call that abstract or immaterial or whatever you like) than it does of rocks, or turds, or mxlnthnkxs. (These last are certain purely material items of Universe #138; they greatly outnumber our atoms, but there are far fewer of them than there are  integers.)  Were I someday actually to run into the chap, and were He to appear in the quite tangible aspect of Mr. Natural seated beneath a tree, I should, to be sure, feel mildly surprised, but philosophically neither cheated nor refuted.

(Key to that last paradox:

 While we dub the Lord the “Necessary Being”, that description, like that of “Father” or “Creator” or the “Lord of Hosts”,  does not exhaust Him.  It is impossible to conceive of Him as being nothing but the Necessary.  For in that case He  really would be just a giant math book. Hence the boring quality of the ontological argument (whether valid or not). We can conceivably deduce certain things about God along those lines, but nothing of His concreteness.)

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Rorty adds, later down the page:

If it makes any sense to speak of the existence of universals, it would seem that they must exist immaterially.

First:  It does indeed make sense to talk of universals, if it makes sense at all  to talk about integers, or modus ponens, or “all men are mortal”, or “the most wonderful mom there ever could be”   -- this “existence of” pre-modifier is something of a rogue.  It makes sense to talk of Hamlet, unicorns, democracy, love, and prime numbers; what is added by this “existence of”?  “Please pass the existence-of  salt.”  No, nonsense.

Next: These universals may “exist”, if you like, `immaterially’, or in any other fashion – I wouldn’t insist on the point, nothing hangs on this. For all I care, they exist in a pickle-jar; none of their properties (in universe after universe) are affected thereby.  As it happens, I personally tend to see Hilbert space as rather more concrete and well-defined, and mountains as rather more abstract and ill-defined, than has hitherto been customary. Likewise, Schubert’s piano sonata in B flat is – abstract /ideal/ immaterial/ call-it-what-you-may, it is still acoustically-ontologically tangible, and each recording or performance thereof is quite concrete.

And yet further a bit (Rorty goes on):

“…the immaterial – the mystery beyond the bounds of sense…”

Now this is a nice phrase, suggesting in particular  the existence of something beyond the confines of the lavatory, where nominalists spend so much time; so we receive it with respect.  Still, conscience oblige, we are compelled to observe, that the “bounds of sense” are a purely relative, species-bound, even individual- and moment-bound notion. A pickle-jar exists beyond the bounds of the blind bat, that doesn’t make it a mystery.  Hilbert space exists beyond the bounds of certain uninstructed individuals; whose bad?  The Blorks of the planet Fnoid can neither see nor touch a stationary object, but they can move it, and then sense it, using their solutions to the equations of motion, and this more accurately than with the eyes.   Cantor and Gödel had a sense for the infinite that compares favorably with many an ear for music or nose for wine. 

            And as for “mystery”, a term often used as a sort of hand-waving dismissal: many things, both physical and immaterial, are mysterious until you study them; others, unmysterious until you study them.  Rocks are much more mysterious after the discovery of atoms (my my, mostly empty space. And yet so solid).  Fractions, which every kindergartner now takes for granted, spooked the Egyptians, who for some reason expressed them, not in the simple form of today, but as an elaborately calculated sum of reciprocals.  Algebraic numbers, imaginary numbers, lose their mystery in a single intellectual wedding-night.  Transcendental numbers like e and pi, retain – or rather have gained, in mystery, but in a specific sort of mystery in each case: indeed, we have a solid handle at the hither end of it.  It is a mystery we never would have discovered, let alone elucidated, had we adhered to a Rortyan agnostic-proctological underview.

*

Edward O. Wilson, Consilience (1998), p. 190, re Rorty’s replacement of epistemology with hermeneutics:

Discourse among scholars, in short, can proceed without worrying about consilience.  About rigor too, it would seem.  Although this concession is welcomed by postmodernist scholars, it is a premature surrender that would drain much of the power and joy from scholarly inquiry.

Wilson counsels eschewing such a replacement, “except at cocktail time” -- a wise proviso, since indeed, prattling on about hermeneutics is much more likely than epistemological discourse  to get you laid.

 

Integers are our Friends

[Further reflections on a topic treated here, here, and here.]

It may be legitimately objected, that in beginning with a grubstake of mere integers, and proceeding stepwise to full mathematical Platonism, I am here sneaking past the goalposts via the fallacy of the sorites.  The classical example: We know that bald men and the hairy-headed equally exist, although we cannot specify, in that excruciating Gedankenexperiment in which each hair of the hirsute is plucked out (stop that!) one by one, at whích point precisely our unfortunate subject becomes glabrous.  Thus, suppose we agree to side with Kronecker and to grant ourselves the integers; and even grant, say, Arithmetic (that is, number theory using only elementary methods): Still, somewhere short of Topos Theory and Noncommutative Geometry – you’re not sure where, exactly, maybe you can’t even say specifically on which side of the divide Analytic Number Theory should fall, fair enough – somewhere this side that stuff, some right-thinking citizen needs to draw the line.  Noncommutative geometry – who ordered that?  You feel as though you’ve been sold a bill of goods.  Frchrssks, look at him:  that dude is bald.

Thus  the methodological objection.  There is also – especially these days, with our penchant for deconstructing and debunking – a psychological.  You may suggest that I have swallowed such a prodigious amount of abstract soup, merely because of some pre-existent hunger for it.  Now, I don’t believe that it was always pre-existent, in this particular case.  When I played cowboys and Indians, my little mind was on other things.  But, l’appetit vient en mangeant; and the integers were the appetizer.

            More concretely:  I have assumed less than may seem.  I have not so much as assumed any particular ontological status for the number “2”: I have merely taken Kronecker at his word, then attempted to refute, or at least to nuance, the second half of his epigram  (“… the rest is the work of Man”), the refutation being based simply upon the logical consequences of the first.  If, on the other hand, you were to begin by stubbornly denying that  one, two, and three (and I don’t mean “one, two, three, … infinity”, I mean: 1; 2; 3) formed an any more necessary part of the furniture of the universe, than Humpty Dumpty or Porky Pig, then I would be unable to convince you of anything by argument, having then no materials to work with.  We only got as far as I think we did, because of the perfectly enormous initial concession by the skeptic Kronecker.  You grant us the necessary reality of the natural numbers – their necessity bestowed, indeed, by the Necessary One – you have conceded a heck of a lot.  You have (it turns out) given away the ontological store.

            In fact, let us retrace our steps, and traverse some of the same terrain less hastily, and with less hunger for depth.  We have agreed to accept, as necessary, the natural numbers, and the simplest thing we can do is to count them – not worrying about primality, or odd-versus-even, or whether one number’s twice the size of another – not even necessarily ‘keeping track’: but just, ticking them off as they go by, like a bored doorman, waving the arriving spectators in to the stadium.  Now you will notice, in such a procedure, a tendency to nod off.  The numbers become dimmer and dimmer.  Has it been a thousand, or maybe twice that amount?  And should you ‘skip ahead’, and try to visualize, say, 17^(8371^545), you really can’t begin to imagine it.  The integers gradually wane, for all practical purposes, invisible.  Yet it is clear that these numbers are every bit as real, as the ones you noticed before you nodded off.  Furthermore, there are a whole heck of a lot more ‘invisible’ integers, than those that are (even with practice), visualizable:  to be precise, countably-infinitely-many more.  And if you are starting to stammer some objection about the possible non-necessity of numbers beyond Praxo (defined as the largest number that our species will ever actually need for anything; though  come to think of it  there is an interesting application of Praxo-plus-one …), then you are trumped, for we hold in our hands Dr. Kronecker’s get-out-of-nominalism-free card:  Every one of those dim distant integers  is as real as a rock, straight from the Maker’s quarry. By the time we are asked to swallow some new kind of quantity – say, a fraction, like “one-half” – we shall have swallowed a literal infinitude of whole ones.

[Update 10 IX 11] The title of this post, as well as this one,  is an example of the faux-naïf  -- a somewhat idiosyncratic concept which, like that of Minimalism (to which it bears some affinity) is slowly to be developed in the course of these posts.  In the meantime, let it remain a bit of a mystery.

Additionally, it has come to my attention that someone just found this post by searching on the words

            integers in our world

This is touching.  Though indeed, that search does not work very well, since the phrase in question did not appear in this post until this very moment.   To aid such sincere and innocent searchers in future, herewith some phrases for Google to match:

            =>  Integers at home and school

            =>  My favorite integers

            => The Campfire Book of Integers

            => Jonathan Livingston Integer

            => O Integer, my Integer !

            => Integers I have known

 

On What There Is (Whether or Not we can See it)

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

[Update III 2013]  The following is an early essay, and somewhat jejune.  The essence of mathematical Platonism concerns, not objects, but objectivity.   We wholly agree with the philosopher Putnam:

It is possible to be a Realist with respect to mathematical discourse, without committing oneself to the existence of 'mathematical objects'.  The queston of Realism, as Kreisel long ago put it, is the questiopn of the objectivity of mathematics ...

-- Hilary Putnam, “What is Mathematical Truth?”, repr. in Mathematics:  Matter and Method (1975, 1979)

~

As a proof-text for today’s sermon on the visible and the invisible, we  may cite so sober-pated an empiricist as Locke (Essay Concerning Human Understanding, II.xxiii.5; p. 270 of the Penguin edition) :

‘Tis plain, then, that the idea of corporeal substance in matter, is as remote from our conceptions, and apprehension, as that of spiritual substance, or spirit, and therefore  from our not having any notion of the substance of spirit, we can no more conclude its non-existence, than we can, for the same reason, deny the existence of body.

Locke’s reticence concerning the hypostasis of matter was prudent, since anything positive he might have said, would have been severely undermined, first  by the later atomic theory (and its successive proton/neutron and quark extensions, though for philosophical purposes these latter developments are minor); then by the mass-energy equivalence; and finally, most radically, by the quantum theory:  by which point our intuitions of just about anything  have gone by the board.

-->

~

~  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.)

~         ~

~

Locke goes on (p. 276):

It is for want of reflection, that we are apt to think, that our senses show us nothing but material things.  Every act of sensation, when duly considered, gives us an equal view of both parts of nature, the corporeal and the spiritual.

            We may concur with the great controversialist, and go him one better:  For in a way, the visible world has a more tenuous hold on reality than the invisible, in particular the mathematical.

            To say this, is no manner of skepticism as regards  the reality of what’s in front of our noses.  No, it’s there all right.  Not for us to second-guess what the Lord hath made and deemed good.  Doctor Johnson’s refutation of – not really idealism, more like nihilism – by giving a stone  a swift kick in the hindquarters, is final.  There is stuff all right; the problem is, are there things?

            For notice: We did not claim merely, nor did Kronecker merely grant, that there is some sort of number-stuff, some quantological porridge  – “there are numbers” like “there be dragons”, vague and unindividuated.  We posited rather (and also observe) an infinitude of neatly individuated entities, as different from one another as – well really, it is difficult to think of even a decent finite collection of physicals, that glitteringly differ among themselves so much as this.  A basket of apples, fine: some are knobbly this way and some are knobbly that; some are worm-eaten, some aren’t.  But to approximate the striking, almost shocking individuality of numbers – this one prime, this one a perfect cube, that one a taxicab number, and all the rest – you would need rather a basket of all manner of fruit, pineapple and breadfruit and durrian and pomegranate.  The seven brides for seven brothers are less distinct among themselves than the first seven integers (especially if we start with zero).

[Footnote: The tag “Taxicab number” springs from an incident in which G.H. Hardy, skeptical of Ramanujan’s apprently intimate acquaintance with the integers – despite a complete absence of formal schooling in the subject, he was on the same familiar terms with them, as Dr. Dolittle with animals – challenged him to find anything the least bit interesting about, oh, say, that integer there, on the number-plate of that cab, whatzit say – “1729 “.   Ho hum, not even prime.  -- Ah yes, said Ramanujan, with a familiar smile.  The smallest integer that can be written as the sum of cubes  in two different ways (1^3 + 12^3 vice 9^3 + 10^3). .  Wherupon he reached out and touseled its forelock, and fed it a hypercube of sugar.]

[Subfootnote: I was kidding, of course, about the integers differing among themselves more than most visible things.  Actual people differ more – but, note, not in their visible envelope.  The radical differences among people stem, precisely, from the realm of the invisible – from their minds, perhaps even their immortal souls.  Integers can’t compete with that.  They’re immortal, but they don’t have souls.]

            So then, what things are there?  The typical examples are:  This table, or this coffeepot.  But it is significant that these examples are usually things  that we (ourselves  created in the image of God, and thus rather already an irruption of the transcendent into the material universe) have crafted to our own ends:  this table, to hold our proofs of the Riemann hypothesis; this coffeepot, to pot our coffee.  Things get a lot more vague when you consider what we have nót remade: which is to say, most of the visible Creation  – wasteland, swamps (or rather: intermittently swampy territory, no license to individualize and pluralize just yet), and starry regions and intersteller detritus.  These things – or rather, this stuff exists all right, but where are the chiseled surfaces of the number “17”, where the English garden and terraced vistas of a really fine Banach space?  If Hilbert space were just a jumble of odd dimensions, mixed up anyhow, jutting out here and there like the spars of a shipwreck, we wouldn’t give a d*mn about it.

            All right, you concede, the physical universe, being all part of one quantum soup, stirred by the overarching Schroedinger equation, does not naturally individuate into midlevel objects.  Still (you contend), the elementary particles at any rate  are absolutely what they are,  and not another thing.  -- But unfortunately, once you get down to that level, new and worse problems arise.  Quite apart from the process, called “decay”, whereby particles spontaneously surrender their essence (to which one might sigh: “We all die…”), and even apart from the wave-particle duality, there are phenomena yet more puzzling. Granted an electron-neutrino is not vague like a swamp or a fog, but it does spend a certain amount of its time cross-dressing as a muon neutrino, so that we barely are authorized to assert, with the Bishop, that “everything is what it is, and not another thing”.  And as for bosons, suppose that they are staunchly now and forever bosons, still, the Bose-Einstein statistics require that no one boson can be separately and distinctly individuated from any other.  If the macroscopic world behaved like that, we would indeed retreat from our picture of the world as peopled by distinct individuals:  Tweedledum in practice and in principle indistinguishable from Tweedledee,  Hyde and Jeckyl randomly phasing in and out.

            Again, in a sense, the contingently-familiar furniture of the world  may be ontologically in worse case than the mathematical.  For, whatever’s familiar, we take for granted; we don’t look too closely into things.  Whereas every single mathematical discovery has been fought for  tooth and nail.  Gauss hid his discovery of non-Euclidean geometry, lest he be mauled by the Boe0tians; it wasn’t ready for prime time until it had been shored up from every angle, and soon even came in concrete models –Klein’s model and that of Poincaré, different visible photographic reductions of some robust pre-existent entity: which you can describe awry, or fail to describe at all, but which you cannot forever ignore.  It awaits you, like the lion.

            So: What value is the general testimony of the populace at large, swearing on a stack of People magazines, that the Real Things of this world are things like – bikini wax and lottery tickets and (oh, but I can’t go on, this is barely worth satire), -- whereas Gilbert space or whatever the hell it is  is just some cockeyed idea of a mad scientist?

            Well.  When it comes to things with which people are particularly familiar, we tend to credit their testimony  (“Tasty Twinkie, that”) and even their predictions-- “That’ll be Midge” or “He’s gonna go long” – particularly if the answer doesn’t really matter. (How about that, a quarterback sneak.  Well, whatever.)  But when it comes to intuitions about probability, or infinity, or angular momentum, or quantum phenomena, or the problem of induction, or the properties of the Cantor set, or the epistemological well-foundedness of what we hold (though typically loosely) in fact to be true, we are – not to offend any sensibilites, but – not to put too fine a point on it ---     but       ---

      ---     to-tal-ly f*cking retarded….

And by “we” I don’t mean:  with the evident exception of you and me and present company, and all of Rabbit’s friends and relations; I mean:  d*mn near everybody, with the possible exception of Feynman (when sober) and just possibly (though I have my doubts) Gauss.

            So, what’s going on in this visible so-called Reality, thing, here?  Ask an eyewitness; just don’t ask two of them, for they’ll tell you different things.  What’s Mary thinking?  No-one knows but Mary, and probably not even she.  What did Caesar say to Antony as they walked into the bar?  Wasn’t there; hard to recover. Nay, why did worm A, spurning the obvious attractions of worm B, chose rather to share its hermaphroditic slime with worm C?  Only another worm could tell you, if even (s)he(it). – Whereas:  Where lie the zeros of the Riemann zeta function?  This question is open to anyone who cares to investigate, regardless of race, creed, color, flavor, gender, nationality, chirality, sign of the zodiac, sexual orientation, membership vs. nonmembership in the National Association of Realtors, galaxy of residence for tax purposes, bodily composition (matter versus antimatter – a perfectly private question of your own personal space), -- height, weight, density, magnetic moment, Gaussian curvature, Euler characteristic (we absolutely do not discriminate on the basis of Euler characteristic  -- you wild ‘n’ wacky  K = -8 folks are totally welcome), …. human vice android vice klingon vice angelic biological status (archangels may participate, but don’t try to pull rank on the cherubim when it comes to the Riemann Hypothesis) … corporeal-status versus disembodied-cloud-of-intellect … existence within time or outside time or astride time, not a problem, yo, come one come all, we ride ‘em six to sixty, step right up, prove the R.H. and win a kewpie doll.