FOM: Objectivity&Hersh/Krieger/Rota/Simpson

[Robert S Tragesser]

Contra-Hersh: The Sense in Which Mathematics
is Essentially and Cruelly Objective
        
        There are (at least) three often
conflated senses of objectivity --
on one of them only mathematics is essentailly
and cruelly (but beautifully) objective:

(1) objective = intersubjective.
        Intersubjectivity-- full or
at least competent shared/shareable
understnading. Mathematics is not objective 
in this
sense.  Intersubjectivity is very hard
won and always highly limited (not only
relative to all humans,  but relative
to all those who make an effort to learn
mathematics).  Full Intersubjectivity is
very obviously not much or readily
achieved among the small class of
first rate mathematicians.
(2) objective -- objectively, indepen-
dently existing entities.
        This is probably not important
for mathematics.
(3)  objective = internal,  unavoidable
constraints on the solutions to problems
concerning. . .
        It is above all in this sense that
mathematics is objective.   It is this
that makes mathematics severe,  difficult,
and unfriendly to fuzzy-wuzzies.  There
is no way of softening mathematics in this
sense -- to do so is to open the gates
to all the circle-squarers who are out
there still.  If Fermat had given a
"short argument" for his Last Theorem,
and if,  as experts now say,  it was very
likely invalid,  it would be invalid not
just from the point of view of our
mathematical culture,  but from Fermat's--
that is,  we could point to the mistake
in Fermat's argument,  and he would acknowledge
it -- or else we are not discussing "the same"
problem.  (But it is indeed amazing how
problems move across historical-cultural
barriers -- the profound and unexplicated
sense in which mathematics is inter-subjective.)
        
Cultural-dependence?  Martin Krieger in 
"Proofs as cultural artefacts" lets us
see how cultural-historical tendencies can
influence and free choices of problems,
methods,  formulations. . .  But,  in
contrast to Hersh,  nothing he says robs
mathematics of its objectivity in sense 3 above.

        Rota (in Indiscrete Thoughts) has pointed
out the decontextualization is part of mathematical
development.  A beautiful case in point is the
history of probability "theory"--as Hacking et al
(including Krieger) have shown,  it was drenched
in cultural presuppositions.  But it has sub-
sequently been decontextualized,  it has become a
branch of pure mathematics (Kac's point):  as for
example,  Klain/Rota,  INTRODUCTION TO GEOMETRIC
PROBABILITY,  and Erdos/Spencer PROBABILISTIC
METHODS IN COMBINATORICS.
Ex.2: Dirichlet's Principle.  Montavesky has
argued that,  for Riemann,  Dirichlet's Principle
was a physical principle and,  as such,  unobjec-
tionable.  Weierstrass's complaint was that it
was inadequately treated as a mathematical
principle -- regarded as a mathematical problem
(the problem:"Dirichlet's Principle?") it was
unsolved,  and that Riemann had (would have had?)
no difficulty at all in acknowledging this.
        

        In sum:  mathematical problems have severe
built in constraints as to what can count as
a solution.  Those constraints can make the
problem very hard to solve.  To study a
problem and not grasp those constraints is
TO FAIL TO UNDERSTAND THE PROBLEM.  
        Mathematicians are not free to add or
subtract constraints (without thereby changing
the problem).
        It is just here that Hersh's consensualistic,
anti-objective philosophy of mathematics falls
apart -- the intrinsic constraints on the solution
to a problem make the difference between true and
false solutions,  and no amount of emotional
support or sweet,  generous consent will make a
fALSE SOLUTION INTO A TRUE SOLUTION.

        robert tragesser
Class of '43 Professor in the History & Philosophy of
Science and Mathematics at Conn C.
        Current Interests:  (1) chronobiology,  (2)the transition
from Aristotle to Galileo,  (3) philosophy and foundations
of mathematics and logic.