Monday, January 3, 2011


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!

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