[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.
~
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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