Wednesday, April 19, 2017

Truth and Provability (expanded)

[A footnote to this.]

Trying to suss out the nature of Truth by staring straight at it  is like attempting heliology by staring at the Sun.   In both cases, more assimilable enlightenment  comes from the corona.

Thus the related but distinct notion of Provability. Gödel was the first to neatly delineate the two notions:  the Propositional Calculus is deductively complete (i.e., all truths may be derived via the defining rules of the system), whereas anything as robust as the integers is deductively incomplete :  one can, within that more expressive system, formulate statements that are true but (intra-systemically) unprovable .  Previously, the Formalists (Hilbert et al.) had seen provability as an analytic explanation of what is meant by ‘truth’ itself.
Pre-scientifically, and indeed theologically, we are not surprised that the two notions should not be equivalent (though of course we had no notion of the precision afforded by Gödel’s results).  Some things, existing from before we were born, and lasting ever after,  just are true;  why should they be logically derivable, or even humanly comprehensible?

~     ~     ~

There is a perhaps related notion within the philosophy of language:

(Intended-meaning : truth  ::  expression : provability)

Yet the very existence of our word ineffable, suggests that we at least entertain the possibility that this may not be so.

In the words of a Neothomist philosopher:

La communicabilité de la pensée  est un fait immense, incontestable  et elle n'est possible que par le langage;  mais tout suggère que, dans le langage, la pensée reste  par nature  essentiellement autre que son moyen de communication.
-- Etienne Gilson, Linguistique et philosophie (1969), p.  39

Taking this in a maximalist sense  would imply, not merely that certain thoughts are ineffable, but that no thought is quite equivalent to its verbal expression.

A further discussion of this topic may be consulted here:
The "idea" idea.


It is clear that we need a notion of truth independent (or partially independent) of provability (however vexed and vague that notion must necessarily be, without such buttress), else how to assess the validity of what purportedly is proved.   In the Awful Warning against dividing by zero, delivered to pupils in their tender years, the young scholars meet a Falsidical Paradox:  an impressive display of mathematical handwaving, purporting to show -- there it is on the blackboard, plain as day --  that zero is equal to one.  Yet even a lad in short-pants surely resists such demonstration, if only with an incoherent “Uh, no, it’s not.”

Compare the smoke and mirrors with which philosophers and neuroscientists demonstrate that consciousness is an illusion, and free will  a will-o’-the-wisp.  Equally we reply: “Uh, no, it’s not.”


Relativist conceptions of truth  are familiar.  As the Harvard philosopher wittily put it, mimicking the cant of the post-‘60’s undergraduates:

“You may not be coming from where I’m coming from, but I know relativism isn’t true for me.”
-- Hilary Putnam,  Reason, Truth, and History (1981), p. 119

Less familiar is a relativist conception of provability.   Here, Ernest Gellner comments on the position of fellow-philosopher Michael Oakeshott:

What is proof? -- he asks.  There is no such thing as proof in general, he answers himself.  There is only proof  persuasive for this, that, or the other kind of man.   Cogency of proof  is relative to what you are.  he notices that this does not seem to apply to mathematics, and brazenly comments that just this has always made him suspicious of mathematics.
-- Ernest Gellner, Contemporary Thought and Politics (1978), p. 180

Actually Oakeshott  put his case too weakly:  varying standards of proof are relevant in mathematics -- indeed, it is only within mathematics  that such scruples have structure and are in point.   In pre-Cauchy/Weierstrass analysis, proof was a bit of a kludge.   Later on, Constructivist qualms  came into play.  And in our own day, we distinguish between theorems whose proof requires the (disputed) Axiom of Choice, from those that can dispense with it.

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