[FOM] Re: Question on the Scope of Mathematics

A.P. Hazen 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.)
Allen Hazen
Philosophy Department
University of Melbourne

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