Thursday, February 3, 2011

Our Friends the Integers

What, after all, is a natural number?  There are Frege’s version, Zermelo’s, and von Neumann’s, and countless further alternatives, all mutually incompatible, and equally correct. … There is no saying absolutely what numbers are;  there is only arithmetic.
W.V.O. Quine, “Ontological Relativity” (Journal of Philosophy, 1968)

Randbemerkung:  The repeated reference to the integers in these notes  might mislead the reader into imagining that I accord them the least importance, mathematically or ontologically (let alone theologically).  Not so.  I only recur to them on the rhetorical grounds that the arch-nominalist Kronecker gave them up for free.  Had he instead said:

            Die Mengen hat der liebe Gott gemacht; alles andere ist Menschenwerk

then we should have instead busied ourselves with the necessary Menschenwerk of building up the integers out of set theory in any of the usual ways, pointing out that this new epigram eventually commits you to the integers in any event.  And had he said (oh would that he had):

            Die offenen Mengen hat der Liebe Gott gemacht…. ,

then the touchstone would have been topology.  Yea, had he instead, like certain of our very ethereal contemporaries, posited rather Category Theory at the base, sets and numbers and all the rest to be developed out of that – well, I might personally have balked, because I don’t understand Category Theory.  Still, a donkey does not understand the Goldbach Conjecture, so intellectually that is no objection. 

            The infinities of the Creation  -- and a fortiori, of the Creator – are --- whatever they might be, we may never know, but in any event, nothing particularly to do with whatever a handful of our own species happens, at any particular instant, to latch onto.  The integers are one toenail in one foreleg (or is it hindleg) of the Infinite Elephant.  It matters not what an infinitessimal portion this may be, of that great beast (to Whom even Babar tips his hat), nor how ineptly we manhandle it:  what matters is that the Elephant is Real.  Praise Him!

            Anyhow, the integers are fine, but nothing special.  Frankly, in fact, the Riemann Hypothesis bores me to tears (mainly because I still cannot truly intuit its significance). The Poincaré Conjecture is much  more inciting – though, having been finally, after a century, proved in its entirety, it serves less well than the R.H. as an image of the blaue Blume.  Compare, indeed, the final chapter of Russell’s Introduction to Mathematical Philosophy, for a nice dissing of the integers.  Likewise Wittgenstein (Zettel 706): “Die Zahlen sind der Mathematik nicht fundamental.”

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