Thursday, December 1, 2011

A Minimum Axiomatization for Reality (Part V -- finis)

[A completion of the essay begun here.)

In fact, neither the thesis of axioms as foundational, nor the antithesis I have presented with the label “regressive strategy”, is the whole truth.  We stand before a dialectic, well described by Bertrand Russell, in Introduction to Mathematical Philosophy (1919; 2nd edn. 1920), p. 1, after distinguishing the (so to speak) synthetic from the analytic approach:

Early Greek geometers, passing from the empirical rules of Egyptian land-surveying  to the general propositions by which those rules were found to be justifiable, and thence to Euclid’s axioms … were engaged in mathematical philosophy …; but when once the axioms … had been reached, their deductive employment … belonged to mathematics ….

~ ~ ~
A couple of further oddities about axiomatics, not conforming to their traditional status as epistemological bedrock:

Shaughan Lavine, Understanding the Infinite (1994), p. 47:

[Cantor] did not work axiomatically.  He believed in the reality of his ordinal numbers and sets, and he saw himself as discovering their properties.  Therefore, no axioms were necessary.

Joseph Ullian, in Hahn & Schilpp, eds., The Philosophy of W. V. Quine (1986), p. 585:
Truth accrues to an axiom, if at all, from the success of the system in which it participates.

What an extraordinary phrase -- “truth accrues”.  And what a surprising thing for it to “accrue” to -- axioms, which one had rather imagined to be beyond that whole dimension of assessment:  they are (the common thought had run) foundational -- stipulated, not assessed.

~ ~ ~

A curious coda to all this.
We have argued that, to be adequate to our experience of the world, we must posit the existence of Free Will as an axiom:  this, since  without it  no aspect of our experience makes sense, and since (so materialists assure us) it cannot be itself derived from the rest of science.
(At this point, the materialists are content to deny Free Will altogether, and to lapse into a robot coma;  where we shall leave them.)
We have further argued that no such apodictic necessity adheres to the thesis of the existence of God;  there are logically possible alternatives, though they are all horrible.

All this, in the course of a sort of Gedankenexperiment or finger-exercise, whereby we set up a (distant, but beguiling) metaphorical connection between, on the one hand, the standard set-theoretical foundations known as ZFC, where the “C” refers to the Axiom of Choice (in what is intended to be a strictly mathematical sense, but which, in its explication, often gives rise to images of voluntarism), and, on the other hand, our own worldview as rational beings incarnated in this cosmos, where the (relatively uncontroversial) counterparts of the “ZF” basics are now the metaphysical underpinnings of the scientific enterprise itself (we discussed these here), and the opposite-number to the Axiom of Choice is now…. Choice itself -- Free Will.

It is a remarkable milestone in mathematical logic  that three fundamental postulates (none of them theorems in themselves, and indeed later shown to be unprovable in any ordinary sense) -- the Axiom of Choice, the Well-Ordering Principle, and Zorn’s Lemma -- each arrived at independently in the course of mathematical history (by which I mean, of course, the history of our own mathematizing, the truths of mathematics themselves being timesless)  have been shown to be logically inter-equivalent.  That is, any one of them can be logically derived (via sophisticated arguments) from either of the others.

The fact is surprising enough in itself;  more surprising still, in light of their psychological inequivalence.   A classic joke runs: 

The Axiom of Choice is obviously true;  the Well-Ordering Principle, obviously false;  and Zorn’s Lemma -- who can understand it?

It would be worth your while to obtain a Ph.D. in mathematics, simply to be able to get that joke (which contains deep truths).   Nothing else in the universe is nearly so funny.

With that excursus -- back to our original program.
Zorn’s Lemma (as we know now, as a “lemma” it is misnamed, not being provable without assuming one of the other two principles) may be stated thus:

Suppose a partially ordered set P has the property that every totally ordered subset has an upper bound in P. Then the set P contains at least one maximal element.

Now, the structure of this proposition is (mutatis naturally mutandis) isomorphic to the Ontological Argument for the existence of God.   Notoriously, that argument as Anselm stated it  is unconvincing in isolation (as philosophers pipe-puffingly put it, it “does not go through”).   As is Zorn’s “Lemma”.  But !  Given the assumption of the Axiom of Choice (which mathematicians have, in the course of their practice, come to feel largely indispensable), you do get Zorn’s Lemma.   The parallel being (in our metaphorical or possibly not-so-metaphorical thought-experiment) that, given the axiom of Free Will, you get … Anselm’s Lemma, as we shall call it in this form.
Not a proof;  just a thought.  But what a thought !

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