FOM: Goedel: truth and misinterpretations
V. Sazonov
V.Sazonov at doc.mmu.ac.uk
Wed Oct 25 15:52:23 EDT 2000
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
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