FOM: hume's certainty vs objective certainty

[Robert Tragesser]

        Hersh's recent postings to Charles Silver
have clarified much for me,  and I find myself
rather more sympathetic to his outlook;  or at 
least I think I understand it better in that
I can see more clearly what I think his errors
of omission are.
        Consider his quote from Hume about how
we become more certain of our "proofs" the more
we go over them and the more they are accepted
by others.
        Hersh draws from such considerations
the conviction that "truth" in mathematics
will never be more than a matter of "probability".
(Hume's sense).
        First,  that sort of uncertainty attaches
to anything we do whatsoever -- have we done xxx right?
        Second,  and more deeply,  Hume fails to
appreciate a distinction appreciated by Locke,  between
"objective" certainty and "subjective" certainty.  Locke's
story is essentially this (I reframe it,  take shortcuts):
        Mathematical problems are such that we have 
definite,  determinate criteria for what counts as
a solution to them.  Propose a solution,  I can discover
whether or not it is a solution by inspecting whether or not
the proposed solution satisfies the criterion.  It is
this objective satisfaction that Locke calls certainty
proper.  (And has nothing to do with how "certain" I feel
about whether the proposed solution does indeed satisfy
the criterion.)
        Then,  for Locke,  we are in the domain of
"probability" exactly and only when the criterion doesn't
provide necessary and sufficient conditions.  
        Locke argues that we do not have such criteria
(providing necessary and sufficient conditions) in natural
philosophy,  and that is what natural philosophical knowledge
belongs to probability rather than certainty.
        Note that we have such objective certainty in mathematics
for Locke even though the mathematical ideas are human (but
definitely not Hume-ian) creations.
        
        It is interesting to observe that when Hersh describes
say being a passenger on a ferry or winning the World Series as
social realities. . .he fails to mention that in both cases
successfully winning the world series and successfully being a passenger
on a ferry require that certain (but hard to specify) physical
conditions be satisfied -- being a passenger and winning the world
series are at once social-cultural and physical/material achievements.
(Rota describes this dependence by what he calls Fundierung.)
        The deterministic nec&suff cond. criterion for whether or not 
a proposed solution to a mathematical problem is a solution
is somewhat analogous to the physical/material conditions on
winning the world series or on being a passenger.  It lies beyond the
terms of social reality -- if such and such is by the criterion not
a solution to the problem,  all the kings horses and all the Pope's Bishops
et al can't make it a solution. . .and neither can "G-d" for that matter. .
.
                rbrt trgssr