Bruno Marchal wrote:
> Le 25-oct.-06, à 13:57, 1Z a écrit :
> > Brent Meeker wrote:
> >
> >> It's even more than seeing where axioms and rules of inference lead.
> >> Given some axioms and rules of inference the only truths you can
> >> reach are those of the form "It is true that axioms => theorems".
> >
> > For formalists, all mathematical truths are of this form.
> And that is why the doctrine of formalism in mathematics (or just
> number theory) is dead since Godel has proved his incompleteness
> theorem.
> We definitely know today that number theoretical truth escapes all
> formal theories.
> Physicists can still dream today about a formal and complete theory of
> "everything-physical", but number scientist knows that the number realm
> is not completely formally unifiable.
> Bruno

Again, the kind of formalism that says
everything can be brought under a single
formal scheme (the Hilbertian
programme) is different from the kind
that says mathematical truths are dependent on axioms,
and different truths will be arrived at under different
axioms. Of course the key point  here
is "different truths". Tom is not entitled to assume that
all roads lead to Rome.

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