FOM: Goedel: truth and misinterpretations

[V. Sazonov]

At the very beginning of this discussion it was 
very precisely characterized by Thomas Forster 

<T.Forster at dpmms.cam.ac.uk>
Re: FOM: CH and 2nd-order validity
Date: Fri, 13 Oct 2000 17:45:59 +0100

as pro and contra the theology in f.o.m. 

I essentially agree with Vladimir Kanovei's comments on
misinterpretations of the well known main incompleteness
and undecidability results. But I would add some more 
comments. 

However mathematical truth is, really, nothing else than 
provability, mathematicians, in their everyday work, use the 
term "truth" also in some *different* sense. It is very 
convenient for them to have illusion of living in a mathematical 
world, say, in a universe of set theory. It is because 
mathematical intuition plays extremely important role. Here the 
formal logical rules of (classical) logic help very much in 
supporting this intuition (illusion of existence of mathematical 
objects and relation of them to reality in the case of applied 
mathematics). The rules of the formal games such as the law of 
excluded middle "A or not A" give a strong illusion that A is 
either true or false and that everything is happening AS IN 
an extremely simplified reality. Working mathematicians may 
even forget that this is only an illusion. But this would be a 
biggest mistake for philosophers of mathematics. Thus, there are 
two interrelated points of view: purely formalistic ("true" 
= "provable") and intuitive based on illusions. We need both 
of them simultaneously. But who, being in right mind, could 
assert coincidence of illusions with reality or truth? 
The term truth is here absolutely unsuitable as it has strong 
association with reality. But there is nothing dangerous in 
using the term truth in mathematical logic as well as, e.g. 
in using the terms ring or lattice in algebra. They are under 
a very strong control!  


Kanovei wrote:

> > true arithmetical sentence A such that ...
> True -- where ? in which sense ?

It is a very good question. The answer is - in our 
illusions, dreams, in an illusory sense. This illusory 
truth should not be taken too seriously if we do not 
want to go into useless philosophical labyrinths.  Let us 
faithfully discuss about illusions as about illusions 
and also how it is possible that they are so useful in 
science, in getting something quite real. When we are 
acting as philosophers, we very often abuse the terms 
"truth", "existence", "all", etc. In a sense these terms 
should be better forbidden as in a sense indecent, 
inappropriate, or at least used extremely carefully in 
discussions on f.o.m. I would not say that we have a good 
culture in using them. How it is possible at all to use 
these concepts both in mathematical logic as quite clear 
technical terms and in philosophy of mathematics so that 
a greatest mess arises. If the same term (like truth) 
is occupied both in philosophy and in mathematical logic, 
let us at least split them on two versions and then look 
for what will happen. 

Let us consider, for example, the so called truth of 
the statements Con(PA) and "Peano Arithmetic is 
consistent". Thus, we have ZFC |- Con(PA) and we understand 
not only the formal proof, but the intuitive meaning of 
all steps of this proof. We also can prove in ZFC that 
\omega|= Con(PA). This is a mathematical technical statement 
of truth. No philosophy is needed here. On the other hand, 
"Peano Arithmetic is consistent" means essentially that no 
human being can find a formal proof (of reasonable, non-imaginary 
length) in PA of 0=1. It is rather clear statement, however 
not so precise as above. What is "human being", etc.? It is 
greatest mistake to identify the formal statement Con(PA) 
with "Peano Arithmetic is consistent". But things like this 
are permanently happening here in FOM! 

>From the very well-known clear and *technical* sense of words 
we can assert that not all arithmetical truths are provable 
in PA (and in any reasonable extension of PA). Here we have 
precise mathematical definitions of terms "truth", "provable", 
etc. When philosophers interpret this I have no clear idea what 
do they mean by these terms. I know only "illusory truth" 
and "feasible proof" as the counterparts. Both these concepts 
are sufficiently clear, I think for all of us, but very imprecise. 
It is completely unclear which way we could conclude (without 
making all the necessary distinctions!) that from some 
philosophical point of view there are arithmetical truths 
(IN WHICH SENSE, PLEASE?) which are unprovable (IN WHICH SENSE, 
PLEASE?). 


Torkel Franzen wrote:

>   Well, whether you use the axioms of ZFC or some other axioms, how would
> you justify those axioms?  Not by proofs, surely?

Exactly by proofs - by participation in proofs, 
by their role in them. Also the truth in the reality 
(experimental, not philosophical truth) may play a 
crucial role. Also some aesthetics. 


Vladimir Sazonov