FOM: Pratt on absolute truth
J.P.Mayberry at bristol.ac.uk
Fri Feb 6 11:21:36 EST 1998
There are two bones of serious contention here that I want to
First, absoluteness. I think it is a mistake to talk about
*absolute* truth when we should be talking about *truth*, plain and
unvarnished. Truth is not subject to degrees. Absolute *certainty* is
another matter. But in discussing certainty we stray into psychology
and epistemology, subjects which no doubt impinge on mathematics and
its foundations, but aren't really central to it.
We all know that reflexive uses of the notion of truth can lead
to paradoxes. But in ordinary circumstances, to claim that "A" is true
is just to claim that A: to claim that "There are infinitely many
primes" is true is to claim that there are infinitely many primes. My
certainty, absolute or otherwise, concerning this proposition has
no bearing on the question of its truth. Absoluteness here attaches, or
fails to attach, to the certainty of my conviction, not to the truth
of the proposition that is the object of that conviction. In all human
discourse on whatever subject, even just to believe that A, however
tentatively and with whatever reservations, is to believe (tentatively
and with reservations) that A is *true*. Take away *that* sense of the
word "true" and you are effectively struck dumb.
Second, formal proof. A formal proof, insofar as it is formal,
is not really a proof, and, insofar as it is a proof, it is not merely
formal. A formal proof - let us say for the sake of discussion - is an
assemblage or configuration of signs constructed in conformity with
certain syntactic rules. Whether such a configuration (or a description
thereof) is (or describes) a genuine formal proof in accordance with
the rules is a geometrico-combinatorial question. But the configuration
becomes a real proof, and not merely a formal one, when we cease to
regard it as merely a configuration and *read* it as a sequence of
propositions advancing to a conclusion.
Now we can give mathematically rigorous definitions of "formal
proof", "interpretation of the language L", "truth in the structure
M",... But we must already possess the aboriginal notions of
*proof*,*interpretation*, *truth*,... in order to judge whether or not
our mathematical definitions do the jobs they were designed for, and,
more generally, to assess the significance of these rigorously defined
simulacra of our ordinary notions.
So I say that that these precisely defined notions - formal
proof, truth in a structure - cannot be taken as primary or
foundational. They are parasitic on the corresponding ordinary notions.
This is hard to say, and I'm not sure I'm saying it very well, but we
must already have mastered the notions of proof, interpretation, etc.
before we undertake to give mathematical accounts of the corresponding
mathematical entities - formal proofs, etc. - and mastery does not
require that we be able to give informative, watertight verbal
definitions of them.
This is why I disagree with Vaughan Pratt when he says
Our problem with using your notion of "what you have proved" is that
until you can say precisely what hedging to add, you can't even say
what you have proved. Formal proof systems do not have this problem,
they are perfectly clear about what has been proved.
On the contrary, there are all kinds of presuppositions underlying the
definition of formal proof - all kinds of "hedges" are required.
Formalizing arguments doesn't guarantee them, or even help us to
understand them. Nor does it render all their presuppositions visible,
for there are the presuppositions that underlie the logical formalism
itself and justify our calling it a system of logical *proof*. A proof
formalized yields a formal proof. But we must unformalize it again to
reconstitute it into a *proof*, that is to say, a real argument, not a
mere configuration of signs.
One final observation. Samuel Johnson's remarks about spoon
counting are not in the least illogical. They are a witty way of
calling attention to the principle that one ought to take into
account the practical consequences of one's theoretical beliefs.
J.P.Mayberry at bristol.ac.uk
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