[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 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.
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