> 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
> > http://iridia.ulb.ac.be/~marchal/
> 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.
If your definition of truth is limited to logical inference given a
certain set of axioms and inference rules, then what are we trying to
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