Saturday, November 26, 2011

A Minimum Axiomatization for Reality (Part II)

[The following continues an essay begun here.  Read that first or the rest won't make sense.]

Calling our toy system of the physico-noöspheric world  “ZFC” (borrowing the label from bloodless, disembodied set-theory)  is of course  in the first instance  just a fun pun: the “choice” is rather different in either case.  Nevertheless,  the analogy may be more circumstantial than is afforded at first glance.  For, the indubitability of free-will  depends upon ourselves being actual incarnated instances of same; some extraterrestrial scientist, reviewing the labnotes on time-series behavior-sequences of H. sapiens terrensis, might not be driven so forcefully to this conclusion.  As for Choice in the set-theoretical sense, we ourselves live outside of Mathland – most of us, very far outside it – and are thus  at best  in the position of that transgalactic analyst.  But for that special few of us who actually did emigrate to Mathland, and gain status as a Landed Immigrant, that axiom has been more familiar.  Michael Potter again (p. 291):

The centrepiece of Zermelo’s original axiomatization  was the axiom of choice.

And p. 259:

Cantor made frequent use of the axiom of choice in his work on cardinal arithmetic.  Indeed there is no evidence to suggest that Cantor ever doubted the validity of the axiom  for a moment:  it was a principle which, in Zermelo’s words, he ‘unconsciously and instinctively used everywhere, and expressly stated nowhere’.

Such fideism will scarcely be found  even in a Saint Augustine.

Indeed, even from outside, lacking all insight into what goes on inside our human pretty-little-heads, that alien sage (after examining reams of printout)  might be led to the positing of free will in H. sapiens, simply to make any sense of the data at all. Compare (Potter p. 260):

Many branches of abstract mathematics  are very much streamlined by the assumption of the axiom of choice.  A good example  is general topology, which becomes decidedly disconcerting in its absence (Good and Tree 1995).

The article referenced, incidentally, is entitled “Continuing horrors of topology without choice.”  Replace “topology” with “philosophy” and the phrase remains apt.

[Continued here]

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