Truth and Provability

[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.

Kurt Gödel
("I'm looking throuuuugh you...!")

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?

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

~ Whatever can be meant,  can be expressed.  ~

(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.