[FOM] Re: Question on the Scope of Mathematics
a.hazen at philosophy.unimelb.edu.au
Sat Jul 31 01:09:15 EDT 2004
It has been pointed out (as a "sociological" fact) that
mathematicians, even when finding and convincing themselves of the
(probable) truth of propositions about undeniably mathematical
subject matter SOMETIMES proceed by giving rigorous proofs, and
sometimes... do other things.
Just as another "sociological" observation, not all "mathematical"
discourse consists of expounding (rigorous and non-rigorous) proofs.
One thing that is certainly a significant part of teaching
mathematics, and I think also of discussion among mathematical
professionals, is explanation: particularly, explanation of why
initially attractive proof strategies WON'T (or at least shouldn't be
expected to) work. And I think this sort of explanation is a useful
part of "mathematical life" (mathematical conversation?) even when it
does not contain anything proof-like, rigorous OR heuristic.
Trivial example: an elementary logic student, impressed by how
usable the tableau method is for classroom examples, is surprised
when told that First-Order Logic is undecidable: don't tableaux
constitute a decision procedure? And we explain that, yes, the
tableau method WOULD be a decision procedure IF we could set a bound
on the number of new instantial terms we might have to use, but that
if there is an AE formula on the branch, we may have to go on
forever. We haven't proven that FOL is undecidable, we haven't
proven that tableaux aren't a decision procedure, we haven't even
shown that there isn't some-- unobvious-- way of calculating a bound
on the number of instantial terms such that we can say a formula is
satisfiable if the tableau hasn't closed after that many new terms
are added. But I think we have said something useful, and something
that it is the business of "mathematicians QUA mathematicians" to say.
Generalizing wildly and irresponsibly from the example... I guess
I'd like to make the philosophical claim that mathematics includes
the giving of rigorous proofs, but it ALSO includes the
question-asking and preliminary discussion which -- when we are lucky
-- leads to something we CAN give a rigorous proof of. (I don't
think this necessarily commits me to denying the centrality of proof
to the whole family of activities making up mathematics, however.)
University of Melbourne
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