FOM: Hersh, subjectivism, objectivity

[Stephen G Simpson]

Reuben Hersh professes himself to be, as it were, "shocked, shocked"
that anyone would link his anti-foundational, cultural constructivist
views to subjectivism and postmodernism.  However, it is precisely
this aspect of Hersh that has recently received quite a bit of play in
the popular press.  I urge the naive to read the long article in the
New York Times of 20 Dec 1997, pages A11-A12, titled "The Subjective
Underbelly of Hardheaded Math", subtitled "Multiculturalism takes aim
at numbers to make them friendlier".  Hersh may object to the New York
Times article, but he doesn't disavow his subjectivist views.

The focus of the article is Hersh's idea that mathematics is
inseparable from politics and culture.  This is presented in
opposition to what Hersh calls the "mainstream" view, that mathematics
is part of a quest for objective truth.  According to the article,
Hersh claims that acceptance of his anti-objectivity idea would bring
more students of different races and cultures into the study of
mathematics.  (The article doesn't explain why Hersh thinks that such
students are repelled by the "mainstream" view.)

Also discussed is Hersh's opinion that it is somehow objectively valid
to divide philosophers and mathematicians into "left-wing" and
"right-wing" camps, with "left-wing" being defined as "any politics
that is anti-elitist and increases political rights".  (Has Hersh ever
heard of the Soviet nomenklatura?)  Feyerabend and Lakatos are
mentioned prominently, as are feminist and Afrocentric mathematics.

The author of the article, Edward Rothstein, brings forth several
sensible, obvious, and well-deserved objections and counterarguments
to Hersh's and other postmodernist views.  He points out that, for
example, "neither Mr. Hersh nor any other champion of ethnomathematics
has provided examples of mathematical facts that are culturally
relative."  Rothstein ends the article as follows:

  In the midst of choosing up teams, Mr. Hersh has, perhaps
  unintentionally, demonstrated how powerful the political impulse is
  behind the belief that mathematics is culture-bound.  Meanwhile,
  mathematicians will work as they always have: rooted in a particular
  culture, place and time while striving for knowledge that leaves
  those variables far behind.

To give Hersh his due: Our mathematics has flourished and will
continue to flourish in the context of Western culture and
civilization.  This is obvious when we consider issues such as the
origin of mathematical notation, the accepted forms of mathematical
exposition, the choice of research problems, and the conviction that
mathematics is a wonderful subject which deserves to be pursued, both
for its own sake and for the sake of applications, in order to improve
the quality of human life by means of technology.  But none of this
means that mathematics isn't or can't be objective.

My own view is that, Hersh and other anti-objectivists
notwithstanding, mathematics is or should be an objective science.
Objectivity is a general epistemological concept referring to a proper
relationship between reality and human consciousness, wherein reality
comes first and our minds grasp it by means of an active, volitional
process.  As a seeker of objectivity in mathematics, I reject both
subjectivism (i.e. constructivism, the view that mathematics consists
of mental or social constructions) and intrinsicism (i.e. Platonism,
the view that mathematical truth exists independently of, and reveals
itself to, passive consciousness).

An apology: I'm not sure that we want to get into these general
philosophical issues here on the FOM list.  However, it seemed to me
that my remarks on Hersh would have been unintelligible outside that
context.

-- Steve

Name: Stephen G. Simpson
Position: Professor of Mathematics
Institution: Penn State University
Research interest: foundations of mathematics
More information: www.math.psu.edu/simpson/