Thursday, January 20, 2011


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

“Die ganzen Zahlen  hat der liebe Gott gemacht…”
    -- L. Kronecker, in a rare moment of candor

So, God made the integers, good job; but what *are* they, exactly?  -- The question may be ill-posed, as unanswerable as what Matter really 'is', or Time.  A mathematician will be less troubled by this, than a physicist, since mathematicians are accustomed to things being characterized only up to isomorphism.  Even so, a thoughtful mathematician is aware that there is an ontological problem:

“The” natural numbers  have a conceptual existence which is quite independent of set theory [dbj:  Certainly it is cognitively prior], and it is an act of faith on our part  to agree that the set of “the” natural numbers  endowed with the “count-one-more” function  is a configuration satisfying the postulates for a simple chain.
-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p.  153

The Intuitionist philosopher Michael Dummett points out that, even in this most basic and most intuitive object, unexpected depths and terrors lurk:

The notion of ‘natural number’, even as characterised by the formal system, is impredicative.
-- Michael Dummett, “The Philosophical Significance of Gödel’s Theorem” (1963), repr. in  Truth and other enigmas (1978), p. 199.

(Impredicativity is a petri-dish for paradox.)

Wishing to dish up something quotable  and a little less evasive than what St. Augustine said when asked the nature of Time, we coin this epigram:

Z  is the substrate  on which we grow the truths of number-theory. (*)


The best-known approach to 'reducing' numbers to something supposedly more basic, is that of set theory, in any of several varieties.
The world waited for Russell, for them to be declared “classes of classes”.   Thus ‘four’ is -- not merely may be analogized to, but is -- a great bulging bag of examples, one of these being the number of the Evangelists (or, in more modern terms, the Beatles).  -- So:  are we commited to an ontology of classes of classes?
            Maybe not.  Even Russell says that “classes…” (let alone classes of classes) “cannot be regarded as part of the ultimate furniture of the world.”  We can in fact, for the present, afford to be completely agnostic as to the best mathematical or philosophical characterization of integers, just as we can afford to be agnostic as to the best characterization of rocks.  For, nothing so far hangs on this characterization.  You can build castles out of rocks, and that didn’t change when rocks were discovered to be atoms surrounded by mostly empty space; they didn’t suddenly become porous, or bounce.  And seven will always be prime, whether it turns out to be best understood as a class of classes, a mess of masses, or the representative on earth of His Holiness Septimus the Seventh. In a sense, I’m not saying we need believe anything abstract or ethereal or (to begin with) even ‘mathematical’ about numbers:  I’m saying they’re already real as rocks.  But if you believe in rocks – if you don’t think you’re just a brain in a vat and are hallucinating them – then you must believe what you discover about the nature of rocks: some are hard, some are friable, etc.  And so it is with numbers: some are composite, some are prime, some are zeros of the Riemann zeta function.  Mathematically, these things just irresistably follow – follow without our pushing them along, like those now-unemployed angels pushing the planets-- follow irrespective of philosophy.  Philosophically, we are at liberty to contemplate the integers with the breezy, boozy indolence of the chimp: one banana good, two banana more good, three banana more gooder still.  What we are not at liberty to do, is to exclude them from our ontology.
            Nothing really hangs on what numbers themselves “are”, or whether the question even has an answer, or has meaning.  Everything hangs on their objective transcendent fixity as positions in a pre-existent pattern.

            Another parable, on the inflexible reality of the integers.  Even if no-one in your entire universe ever ever counts anything, the numbers are present, like the stone guest at the feast.  (Rather as electromagnetic radiation is present, even if everyone is blind.)
            Take for instance planet Xymol.  There, all reference to anything quantitative, or logical, has always been taboo.  The activity of ‘counting’, which (by report) exists in other galaxies, they hold to be a merely contingent and local (and rather disgusting) custom, much like cannibalism.  The inhabitants of Xymol are purely “artistic”, purely emotional and qualitative; numbers, they maintain, are non-existent, and they have no need of them.  And one of the emotional and qualitative things that little Timmi von Xymol loves to do with his cubical blocks, is to arrange them into perfect rectangles (non-trivial ones: each side greater than a unit length).  As Timmi’s guardian, you want him to be happy.  But despite your best intentions, you’d better be careful what you give him.  If you give him 24, or 93784668225, or RSA-576 blocks, he’ll be fine, since these are composite.  But if you give him 17, or Mersenne-41, then after a great deal of fretting and fussing, the poor lad will break down in tears. For these will ever be prime, on the peaks as in the valleys,  though you deny the existence of numbers.

            The question, “What are numbers?”, is  strictly in itself  not very interesting: neither mathematically, theologically, nor philosophically.  Though Carnap’s outlook is in general foreign to the views developed here, his deflationary remarks anent that question, in “Empiricism, semantics, and ontology” [1950; repr. in Benacerraf & Putnam 1983] may stand.
            It’s like – What are penguins, really?  Godlike birds?  – to be sure.  Fusiform instantiations of ornithological perfection? – freilich, freilich.  But what are they quantum-mechanically, cosmologically, mathematically?  -- Oh, away with you, let us simply admire them, as they slide on their tummies along the ice!
(*) Footnote, lest that splendid epigram be taken in a Nominalist sense.
The metaphor is actually fairly exact.  We don’t invent tomatoes:  we discovered them in nature, and now help them propagate their pre-existent kind.  The ‘truth’ of the tomato -- its genetic type -- existed prior to our discovery.   When we grow any individual tomato, it may be stunted, or waterlogged -- we simply do our best to approach the magnificent pre-existent Platonic Tomato.   And likewise with math:  our (imperfect) constructions and theorems are peasant-style approximations to the Type.

God grant that, in Paradise, my resurrected body may feast on a Platonic BLT.

[Further reflections here.]

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