FOM: Mathematical certitude; response to Shipman

[Robert Tragesser]

        I'd like again to draw attention
to a fundamental difference between Locke
and Hume of mathematics.
        The "skepticism" that Hume uses
to rob mathematics of its "certainty" applies
to any human activity whatsoever where there
is an issue of right or wrong: well,  golly,
one can always have made a mistake.  it's
a kind of skeptical ether generated by
neurotic,  insecure minds.
        Locke sharply distinguished
such subjective certainty and subjective
probability from the objective certainty
and objective probability that pertains
to Scientia generally,  and mathematics
most particularly.
        What is characteristic of mathematical
problems (even if it is not universally true of
all problems posed in mathematics) is that there
is a generally not explicable "criterion" for
when a proposed solution to the problem is
a solution to the problem.  If we have an idea
(as for example and idea of the imagination)
for the solution,  we can determine _a priori-
whether or not it is a solution by seeing
if it satisfies the necessary and sufficient 
conditions supported by the criterion.
        Objective probability occurs -- as
in natural science -- when the criterion
supports necessary but not necessary and
sufficient conditions on the solution to the
problem.

        Also Hersh's notion of a "correct proof"
ought to be taken to task.  The proofs in the
opening chapter of Hilbert's FOG are correct,
even though Hilbert gave no logical principles
nor logical syntax. I'd also contend that the
proofs in Euclid's Book I are sound,  even if
formally deficient in the sense (pointed out e.g.,
by Leibniz) that he didn't manage to give
adequate defintions of line, angle,  and so had to
resort to postulates to take up the slack;  the
reasoning in E's demonstrations are sound on the
basis of the intended meanings,  even if
Euclid did not manage to capture the meanings
in an adequate defintion (so that all the
postulates could be demonstrated on the basis of
the definitions).
        Indeed,  Hersh in his seminal Advances in 
Mathematics paper in effect acknowledges the
presence of a cogent "insightful understanding"
when he attacks "formalists" as forgetting that
one must see that the formal rules are correct
and correctly applied.--A pity he didn't
follow this out,  but wandered off into the
skeptical bogs.

        robert tragesser


                robert tragesser