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!

Hermeneutics/Epistemology

In Philosophy and the Mirror of Nature (1979), Richard Rorty makes much of a dichotomy between epistemology -- the traditional study of how, and whether, we know what we think we know (though elsewhere he redefines it as “the attempt to render all discourses commensurable  by translating them into a preferred set of terms” (p. 349), and elsewhere again (p. 353) as “roughly, a description of our study of the familiar”) -- and hermeneutics -- which is the kind of crap they teach at Yale.   Characterizing those who have qualms about Kuhn, he writes (p. 344):

If the study of science’s search for truth about the physical universe  is viewed hermeneutically, it will be viewed as the activity of spirit -- the faculty which makes -- rather than as the application of the mirroring faculties, those which find what nature has already made.

This dichotomy, then, shares some territory with what I’ve been labeling Realism vs. Nominalism.    However, the professor assures us, none of this matters:

In the view I want to recommend, nothing deep turns on the choice between these two phrases -- between the imagery of making and of finding.

Spoken like a hermeneuticist.

Still and all, the line between our two dichotomies must run not quite parallel, since he then says (p. 345) that this is “still not to say that the atoms, wave packages, etc., discovered by the physical scientists  are creations of the human spirit”, and (though himself leaning to the hermeneutical side of the line) even comes out manfully with:

To say, with Sartre, that man makes himself, and that he differs thereby from atoms and inkwells, is quite compatible with repudiating any suggestion  that part of his self-creation consists in ‘constituting’ atoms and inkwells.

(Imagine our relief!)  even though (p. 354)

its enemies assumed that anyone who overtly practiced hermeneutics  must be ‘antinaturalist’, and must lack a proper sense of the brute exteriority of the physical universe.

(Odd phrase, that -- “overtly practiced” -- as though hermeneuticists were given over to certain ... quite private practices as well...)

Edifying etymological footnote:
Hermeneutics is said to be named for the Greek god Hermes, described in the Homeric hymns as “shifty, cunning, a robber, a thief at the gates”, and by Wikipedia as “a liar, a thief, and a trickster”.  He is also the eponym of hermaphrodite.  Hm.

Babylonians again

If you ever took an astronomy class and paid attention, you know that there are two different ways of defining a month:  sidereal month and synodic month.  Frankly, these tax my retention.  But the real picture is even more complicated.

Shlomo Sternberg, Celestial Mechanics (1969), p. 26:

The return [of the moon] to maximum velocity  takes slightly longer than a sidereal month, and slightly less than a synodic month.  The period of return to maximum velocity is known as the anomalistic period or anomalistic month.

Thus far, a curiosity of astronomy;  one which, one might venture at hazard, was discovered by some persistent fellow with a telescope, prior to the discoveries of quasars or the Red Shift, but probably not all that long before.

But no.   The facts were known in the second century B.C.E. -- and this, in dizzying detail:

Hipparchus gives the relation that 251 synodic periods  are almost exactly equal to 269 anomalistic periods.

Now, at this point, things become seriously weird:

What is most striking is that we find this ratio 251/269 used in a theoretical way in Babylonian tables.

            This startlingly early anticipation does not mean that the ancient Babylonians had been informed of the facts by visiting Space Aliens, as the New Agers would no doubt have it (if they have heard of this story at all).  But it does suggest a preternaturally diligent interest in astronomical niceties.

            And indeed, preternatural is here the operative word.  For as Sternberg several times mentions, the desire for extraordinarily precise knowledge of the celestial facts  stemmed, among the ancient Hebrews, from necessities of religious observance.   Islam likewise has such requirements, a fact perhaps not unconnected with another blazingly insightful observation of the late 9^th/early 10^th century (p. 20):

The motion of the apogee (beyond the effect due to precession) was apparently first discovered by the Arabian astronomer al-Battani.

These, then, are instances in which religion was the midwife of science.  Other instances are familiar from the field of linguistics, where the preservation and explication of sacred Scripture (the Vedas, the Bible, the Koran) was the motivation.

Credo (concluded)

Bishop Berkeley famously quipped, in his (cogent) critical article on Newton’s fluxions (infinitessimals), that “he who can digest a second or third Fluxion,  need not, methinks, be squeamish about any Point in Divinity.”  Conversely,  might not a certain intellectual tolerance for points of Divinity, pre-dispose one towards seeking the reality behind such initially ill-presented abstractions as the infinitessimals, that they might be put upon a sound footing?  Indeed this has, since Berkeley’s time, been done, for the Calculus as inherited from Leibnez and Newton; and since that time, those very infinitessimals, once spirited away in the new Cauchy formulation, have shown up again unexpectedly at our door, no longer in rags, but smartly outfitted by Robinson.  And indeed these originally disreputable and now revived entities (very small ones, to be sure, but still entities) – which originally, I confess, struck me as rather a sterile abstraction, no doubt true enough in their way, but probably no part of the Palace – have latterly been suggested as just the broom for cleaning out the Augean stables of perturbation theory, of all massively practical things..  Is there nothing the mind of math may dream up, which may not prove someday as routinely serviceable as a hammer?

            Consider again the ill-fated squeamishness of Pythagorus, concerning what we now think of merely as the complement in R of Q.   Might his bias of nominalism been somehow connected to the limits of his theological horizons – Olympus and its rabble of godlings, with their petty jealousies and sordid couplings?  You are not likely to get much of an ontological leg-up from a shag-shanked goatgod; whereas God  the creator of all things visible and invisible…nay, Who dies and yet rises… a God who is One, and yet Three…

Thermodynamics without Tears

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Credo (continued)

[continues this]

I see no reason why we should have less confidence in mathematical intuition, than in sense-perception.

          -- Kurt Gödel

            To grant necessary ontological status  only to the non-negative integers, as does Kronecker and his thought-mates, resembles the attitude which prevailed among geometers from Euclid down to Gauss and Lobachevsky, which considered geometry as described by Euclid  to be the only possible geometry.  It was a word with no plural, like “universe”; that both these terms are now pluralizable (the latter in a conjectural, the former in a now very precise sense) represents a triumph of the human spirit, which now knocks tankards with the Invisible, in a toast to Him who made us all, the math along with the meat.  The old geometers believed this because they fetishized the visible: the flat, drab world (well, pied with beauty, true, but drab compared with the full panoply of Riemann surfaces and Finsler space) whose mensuration is approximated so closely by Euclidean geometry  -- at least, that stretch of turf that lies visibly close to hand – that the actual deviations that do exist cannot be detected by ordinary, plain-man means; and those who scoff at the invisible infinite, are very plain men indeed.  And it was this servitude to the locally visible – which is  in particular  to say, to the contingent – which caused the finest minds to fritter fruitlessly after a derivation of the Parallel Postulate from the rest of the Euclidean axioms, a fiasco that lasted literally for over two thousand years, from antiquity down to the nineteenth century. Indeed, it was not until we became familiar with the “invisible” worlds  revealed to us by Lobachevsky, Riemann, Klein and Poincaré  that we became fully clear on the status of Euclid’s axioms, and the distinction between axiomatics and model theory.

            So, to argue concretely:  If the number one is real, then so is a half, for I give you half this pie. And if “one” is real then so must be the square root of two, as being the measure of the hypotenuse of the isosceles right triangle, by the inexorable evidence of the Pythagorean Theorem. (Likewise the square root of 5, 13, 17, etc., and thus their products.) And the square root of two turns out not to be a ratio of natural numbers. Now, Pythagorus himself shrank from this conclusion, and stigmatized the postulant entity as irrational; yet now we take them for granted.   And if third roots or electricity are real, then so are the imaginary numbers required for their description; and if matter is real, and with it atoms, then so is quantum mechanics: which mean that compact operators on Hilbert Space are real:  you cannot see them or touch them, but you can almost hear them, buzzing all around us…  It is a slippery slope (we might almost say, a declivity whose slope is infinite) when we admit the reality of the visible world: it quickly (“quickly”, considered sub specie aeternitatis) drags in all the invisibles wíth it.

            It is true, the atmosphere around those higher turrets is rather rarefied.  Sometimes we become light-headed, and wonder if we are not after all just making some of this stuff up.  Yet no sooner do we begin to doubt our senses – or rather, to put too much trust in our senses, and too little in our carefully nurtured sense of the unseen – than Nature coughs up some concrete correspondence with our most arcane designs.  That same Riemann hypothesis now seems to be in some strange harmony with the energy-levels of atoms.  And as for connections on fibre bundles – meet the gauge fields of particle physics, your twin, separated at birth!

*

            The question remains, whether mathematical entities are, so to speak, real in general, or only real within a particular reality: much as a planet might pursue its course, in our universe but not another.  Now, the examples of mathematical reality adduced thus far, have all been given a clean bill of health by our actual, particular universe.  Hilbert space is as much a part of our daily reality as are porpoises – quite as vibrant, almost as much fun, and much less likely to go extinct.  But the Continuum Hypothesis… ahh.  That’s another matter.  One would really like to have a better handle on that one.  There are models for set theory in which it is true, and models in which it is false.  This tends to make us acutely uncomfortable, since, unlike the logically equivalent but intuitively more ethereal Axiom of Choice, it’s the sort of thing where you feel there ought to be a plain fact of the matter.  Nevertheless, its status may be ultimately no worse than that of the parallel postulate, which is quite placidly and understandably true in some geometries, false in others, and indeed true in our own universe at appropriately small scales (here I mean, of course, not what happens unobserved at infinity, but such local effects as the sum of the angles of a triangle), while false at others.

[concluded here]

Babylonian mathematics

In The Character of Physical Law (1965), the physicist Feynman presents an informal dichotomy between the “Greek” and the “Babylonian” approach to science.  The Greek: axiomatic, systematic.  The Babylonian: ad-hoc bricolage.  I satirized the latter in the parable of the humble woodchuck (“Constructivist Angelology”).

In this morning’s reading, I happened upon this:

Shlomo Sternberg, Celestial Mechanics (1969), p. 1:

In most popular accounts, the contributions of the Babylonians …  are consistently underestimated.  The Babylonians, in addition to having accumulated impressive observational data, based their tables and predictions on a form of Fourier analysis (using what are now known as “spline functions” instead of trigonometrical series).

!!! -- I stand corrected.

Party Hard Last Night?

 Commiserate with Murphy, over at his pad:

“Murphy Considers his Sins”