Monday, November 28, 2011

A Minimum Axiomatization for Reality (Part III)


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Improbable though it seems – I began this thought-stream rather as a lark, but suggestive analogies keep cropping up -- here is yet another similarity between the mathematical and the anthropological settings of Choice.  Here our author, Potter, resorts to theological language  (p. 250)  to do justice to the mathematical state of affairs

The arguments that are given by mathematicians for believing the axiom of choice  are often quite weak.  One common argument generalizes from the finite case, on the basis that there is no reason to suppose that infinite collections behave any differently.  The difficulty with this is that we have nothing except a wing and a prayer to support the view that they do not behave differently.
            -- Michael Potter, Set Theory and its Philosophy (2004)
 
 
The naivety of the reported reasoning (Sicut in finito, sic in infinito) seems extreme  -- two things could scarcely be more different, and mathematics overflows with processes that do not generalize straightforwardly to the infinite case.  (Consult discussion here.)  But we are in little better case, in our everyday implicit beliefs, when challenged to defend them.  Saint Augustine’s epigram (“Quid est tempus? Si nemo a me quaerat, scio …”) is classic.  We may offer a more homely case, our response to a nihilist who doubts the presence of this coffee-cup, and demands to know why on earth I would postulate the existence of such a thing, which Bishop Berkeley had surely already adequately refuted.

I:  Because I see it in front of me, and hear the ching of the spoon against the china, and feel its warmth and weight and solidity in my two hands, and the taste of the wholesome beverage that issues from its porcelain interior.
He:  Ha!  Your brain evidently floats in a pretty well-appointed vat.

The only really adequate answer to his sly challenge, is Axiom I, below.

Potter goes on:

Another variant  proceeds more cautiously, by generalizing  first from the finite to the countable case  in the constructive manner already outlined, and then generalizing to the uncountable case by appeal to the idea that an ideal being could achieve the choices required of him (or perhaps Him).

And so, at last, towards the very end of the logic book, He puts in an appearance.

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