Sunday, December 28, 2014

A New Proof of the Existence of Coffee-Cups

But the math had not clarified one basic question:  what the hell did it all mean?  How could the world of atoms be nothing but a puff of probabilities, and yet conglomerations of those atoms could create something as strong and unbending as the  chair on which he sat?
-- David Kaiser, How the Hippies Saved Physics (2011), p.  54

[Update to an earlier essay.  For the original, with readers’ comments, click here.]

There comes a time in the life of every young person, when they doubt the existence of coffee-cups.  So here comes jolly old Doctor J, to prove it for you;  and more importantly, to reassure you that, even should you someday forget the details of the proof, you may continue  to place entire faith in the existence of these useful objects, asking no questions for conscience’ sake.
(We earlier sketched the idea of the proof, but there the argumentation was informal.)

So!  It’s a quiet Sunday, the desk and the mind are clear, let us pour out a generous helping of this fresh-ground, new-brewed, steaming darkling French roast, into the convenient receptacle which (to all appearances, at any rate) sits within easy arm’s reach upon the solid oaken desktop, to stimulate the grey cells, and set ourselves to this great task  of confounding the nominalists and solipsists.

[Excursus:  Historique du problème
Dr Johnson famously proved the existence of stones  by kicking one (a martyr to science -- he reckoned without his gout).  The Cambridge philosopher of banality, G. E. Moore, satisfied himself (I choose the verb with care) of the existence of physical reality, by noticing his own hands.  (The choice was not a happy one, being potentially seen as solipsistically self-centered, not to say onanistic;  further, the phenomenon of phantom limbs complicates the picture in irrelevant ways.)
Accordingly, we shall proceed by proving the existence of coffee-cups:  both because they form the prototypical physical object for philosophers (apart from mats, which are ignored unless they have cats on them), and because, for mathematicians, who in general care not a fig for the physical, do care crucially about this one concrete object, since coffee is their essential fuel.]

But before embarking upon the abstract part of this argument (since some folks are uncomfortable with abstractions -- though really, they need not be;  what follows makes a lot more sense than most of what gets uttered around the water-cooler), I shall offer what critics have called the objective correlative:  a description of the actual coffee-cup whose space-time coordinates are My-Here and My-Now.  (Rather than the ambiguous “here and now”;  for such coordinates must indeed be relativized to some observer, though not necessarily a mortal observer.)  After all, we wouldn’t want you to wonder whether we might not be just making the whole thing up.

From the standpoint of the chemist, the object in question would appear to consist in some sort of ceramic, whatever that may be, though I really couldn’t say -- anyhow, something sturdily non-porous.  In general outline, it forms a cylinder, sealed off at the bottom and evacuated at the top.  Somewhat spoiling this sleek Platonic profile, a handle protrudes from the side, a concession to the physical infirmities of hominoidal incarnation.
It is thus strictly speaking a ‘mug’ rather than a cup;  but still I say “coffee-cup” because, for philosophers (who don’t get out much), that is the prototypical object in the cosmos.  (At least this is true for hypercaffeinated Americans; more leisurely Oxonians look to "the tree in the Quad").  In similar fashion, the propotypical Contingent Truth is:  “The cat is on the mat.”  (That last one is not true at the moment, b.t.w.;  she seems to have wandered off.)

That is pretty much all you need to be  to call yourself a coffee-cup.  But these days, most cups are not quite so minimalist:  most of them sport some wacky slogan of office lore, or the logo of a sports team.  This particular specimen displays the image of a medieval knight, standing in contrapposto, his tabard emblazened with a cross pommée.  On his face is an expression difficult to read, some blend of troubled arrière-pensées and manly determination.  He is set largely within a blue circle, which his mailed headgear slightly overtops; at the bottom -- a detail I just noticed only now -- the end of his belt (bulging slightly at the tip) laps down and over, in a way that, hm,  might be misinterpreted.  And written boldly above him, the words:

Catholic  Attitude

Well.  So much for the objective correlative.  Now for the proof.

~ ~ ~

We must confess at the outset (well, it’s a bit late for that) that we cannot actually prove the existence of coffee-cups, by the lofty standards of proof first intuited by Euclid, and refined  in our own day  to a high sheen.   Outside indeed of mathematics, such a capability appears not to exist:  Whenever I try to read a physics article reporting recent research, it always appears at some point to be hand-waving, “You’ll just have to trust us on this.”   And within mathematics itself (trade secret; don’t whisper this to anyone) we seldom explicitly set out all the steps, even in the final published results.   As for the actual process of mathematical discovery, it normally bears no relation at all to Hilbert’s formalist program;  at its deepest and most mysterious, it is more like…. (and here we’ll have to whisper very softly) … Revelation…

Anyhow, we obviously can’t actually prove the existence of coffee-cups, because logically, strictly speaking, they might not exist:  We might all be just brains in a vat, hallucinating the whole thing.   So to clear the air, we frankly state our Auxiliary Assumption:

(AUX)  We’re not just brains in a vat.

If you personally reject that assumption -- well, happy marinating.  For the rest of us -- to proceed.

[Excursus:  Among the merits of d’Abro’s history The Rise of the New Physics (1939), is that he repeatedly points out the (sometimes unacknowledged) auxiliary assumptions that were required in the development of even the most successful programs in the physical sciences, by the most celebrated researchers, during physics’ Golden Age.
Almost nothing we do in everyday life  is free of unspoken assumptions, sheer force of habit, and random whims.  So, simply by listing such propositions as (AUX), we are making some modest progress.]

The principal premises of the proof, and the only ones that would occur to most people, are evidential.   This is where the hard work of science is done.  Fortunately, in the case of perceptions of ordinary middle-size objects, Nature does it for us, that we need not continually trouble our little heads:  all sorts of cross-connections and epistemological assumptions are hard-wired into our brains.  (These brains, b.t.w., reside in a cranium, vice a vat.)  We shall nevertheless lay out some of the perceptions and observations that serve to buttress the conclusion to objectual existence -- not so much to convince you yet further that your mug is real, but rather, virtually the contrary:  by such explicit exposure of our evidential grounds, to make plain their essential poverty, absent certain grounding metaphysical principles, along the lines of (AUX) though more substantial -- or rather again, more abstract and as it might be insubstantial, since (AUX), though it has the look of a metaphysical assumption, might in some circumstances be actually demonstrable, and thus empirical, as happens in the movie “The Matrix”.

(PE) Perceptual Evidence

(1) I seem to see before me a colour-patch (Note to the lay reader: These colour-patches compose the whole of the world for the positivist empiricists, who never quite manage to convince themselves that coffee-cups are real, and therefore cannot drink the coffee, and die of thirst), roughly cylindrical in outline -- though the exact projection upon the retina depends on the viewing-angle in complex ways (blah blah blah; insert usual empiricist verbiage here).
(2) When I reach out with my hand (for reassurance as to the existence of your own hands, consult the works of G. E. Moore), groping in the general direction of the above-named coloured patch, I abruptly encounter what seems to be a solid object -- a bit too abruptly, it turns out, as some sort of dark hot liquid is now pouring into my lap.
(3) (Insert more such evidence here -- crucially, cross-modally, involving the sound of the mug as you strike it with your pen; the smell and the taste of the coffee within).
(4)  When I pour a small amount of coffee into this apparent container, it does not quantum-tunnel out:  thus recalling, by uniformity and analogy, such similar objectual posits as the Water-Glass.
(5)  Jones here -- a stout fellow of sound mind -- affirms that, egads, he too perceives a coloured patch in what, calculating the parallax by triangulations, dum-de-dum, doing the math, appears to correspond to the same space-time locus that I myself have identified.
(6)  (etc. etc. etc.)

(C ) Conclusion

Coffee-cups really do exist.

(They do, but this one's just an image of such a cup)

Now:   The thoughtful reader will already have noticed how grotesquely speckled with gaps  such reasoning is -- why, it shouldn’t convince a child.  For, though it pretends to consist of but simple reports of perceptions -- “observation sentences” in the lingo -- rather than any abstract reasoning that might be open to critique, it actually smuggles in a great deal of unbuttressed assumptions.  Thus -- what ties (1) and (2) into any sort of connection with each other?  Why,  the unexamined assumption that the visual coordinates of (1) map smoothly and without controversy to the kinesthetic coordinates of (2):  a fact by no means obvious  -- intellectually, that is;  of course, the identity is more or less hard-wired, though it may take baby a certain amount of groping and spoon-dropping to get the respective ordinates and abscissae to finally match up.  Nor is the correlation in any sense necessary -- indeed, it can be easily overturned in simple experiments involving funny spectacles.
And as for that “Jones” there -- you are assuming the existence of Other Minds, about which great vats of ink have been spilled!  (There is, in fact, one sense in which philosophers are brains-in-a-vat.)  And indeed you need vastly more than that -- for the Other Minds projected by mere analogy, might be only Other Monads, and incommunicable among themselves.  All right, wire them one to another -- still not enough:  you are assuming that you each mean ‘the same thing’ when you utter the same (or: “similar”) syllables:  a circumstance demonstrably false in every political summit or marital argument.  All right, plow ahead and assume that:  you’re still not there.  With all the intelligence and good-will in the world, your private knowledge might not be transmissable intact: witness the celebrated case of determining whether or not your extragalactic pen-pal resides in a world made of matter or of anti-matter, or whether he is right- or left-handed.  -- And here my grey-cells throw in the towel; but were you (younger, and keener) to pursue this line of speculation further, things would probably only get worse.

No, for the various clauses of (PE) to have any cohesion and probative force at all, we require a vast apparatus of non-evidential, metaphysical assumptions, almost never made plain.  The above syllogism,  (PE) => (C ), is thus in reality an enthymeme .  Suppressed is what we might dub the enthymematic assumption, which we now state here:

(EA -- full version)  Background metaphysical premises
(Insert the entire body of philosophy here, along with the whole of science.  Do not omit to mention the Assumption of Cosmic Uniformity (spatial; temporal; spatio-temporal), the Problem of Induction, the Reliability of Deduction, along with some still-unnamed rules of thumb concerning our right to ignore pesky quantum-mechanical paradoxes when speaking of mid-level objects, etc. etc.)

Now, that is rather a tall order, and even were it somehow to be accomplished for this particular case (which might or might not "go over" towards founding the existence of pickles, say), no individual mind could survey the results and verify their correctness and inter-consistency.   And yet, this is the heavy machinery that we need to conclude to so modest a proposition as the existence of coffee-cups.  So let us superadd, as a concession to our feeble powers of ratiocination and memory, the following Concise Version:

(EA -- concise version)  God is Good.

This one works for me;  but if you’re an epistemological stickler, you can probably get by with this weaker assumption, as stated by Einstein, and applied, not only in his life, but in the practice of his physics:

(EA -- weakened version)  Boshaft ist er Nicht.

Which is to say, anglicê:  at least He is not plain mean.   By which he meant:  The cosmos is not just some vast joke, jerry-rigged to baffle our senses.  Overall, it ultimately makes sense -- even if what we perceive here below, is but the thread-gnarly underside of the grand patterned carpet.

(Note:  The basic point is more logical than theological;  still, the realms do tend to intertwine.)

Historical footnote:

Descartes derived the principle of inertia from two premises:
* the homogeneity of the straight line, and
* the immutability of God, of which the constant quantity of motion in the world is an expression.
-- Charles Gillispie, The Edge of Objectivity (1960), p. 90

Note incidentally that the first of these Cartesian principles is at least as fraught as the second (the problems and paradoxes of The Continuum).

~ For a theo-philosophical fantasia, try this:
Murphy and the Magic Pawnshop
~ ~ ~

To add some punch to our insistence on the Importance of the Enthymeme, with all its tacit metaphysical assumptions, let us instance another, less familiar syllogism.

(P) Evidential Premises:
1. Something funny’s going on here.
2. What’s that smell?
3.  I can’t find my car keys -- probably somebody swiped them.
4.  What’s that noise?!
5.  So where does Jones get the money for a fancy car like that?
6. No way I’m letting that bastard merge into my lane.
7. Why are all those people staring at me?
8. Ouch!
9. It’s a conspiracy.

(C ) Conclusion:  The world is ruled by giant lizards.

That conclusion is an actual doctrine, firmly held by some people:  non-institutionalized people, who have the right to vote (and who seemingly exercise that right with disproportionate frequency).  I’d provide a link to the relevant Web sites, but fear lest these poison your computer.

What, though, must be the missing enthymeme, which licenses this deduction?  Apparently something like this:

(E)  The cosmos is ruled by Satan.

Compared with E., C. begins to look almost reassuring.

~  ~  ~

But enough of such stuffy studies!   The sun, renewed, has broken through the clouds, the bird-sounds rise in chorus, and a breeze bestirs itself among the leaves.   Let us then fare forth, into the wide world, whose splendor bears the imprint of its Maker, as plainly as had He signed it, John-Hancock-style, with His celestial pen.

~  ~  ~

[Afterthought]  But what then, of the java, that flows from the mug whose reality we have just proved???  Why, it flows to fuel the brains of mathematicians! 

[Appendix, Jan 2015]  Having at length satisfied ourselves as to the reality, or at least reliability, of coffee-cups, would should not  on that account sink back into an attitude of Moorean complacency (“I’m all right, Jack;  I’ve got hands”).  For our commitment to these  suggests yet further commitments, which we had not realized were there to assess.  Such as :  Realism with regard to quantum state vectors.

The question of ‘reality’ must be addressed in quantum mechanics -- especially if you takes the view that the quantum formalism applies universally to the whole of physics -- for then, if there is no quantum reality, there can be no reality at any level.
-- Roger Penrose,  The Road to Reality (2004), p. 508


The question of the objective existence of the objects of mathematics … is an exact replica of the question of the objective existence of the outer world.
-- Kurt Gödel, “What is Cantor’s continuum problem?”, in American Mathematical Monthly, 1947.

In for a penny, in for a pound.


This notion of ‘enthymeme’ deserves further comment, and here is not really the place;  yet I can find no other home for it, among these essays.  Thus:

This is characteristic of ancient informal logic -- that is, of the logic of proof or of thought-experiment …. ; we regard it as enthymematic  only through hindsight:  it was only later that an increase in content  became a sign, not of the power, but of the weakness, of an inference.
-- Imre Lakatos, Proofs and Refutations (1976), p. 81

For more from this pen, try this:

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