FOM: Pratt on absolute truth

[John Mayberry]

	There are two bones of serious contention here that I want to 
take up.
	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. 


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John Mayberry
J.P.Mayberry at bristol.ac.uk
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