FOM: objectivity, postmodernism, multiculturalism, feminist etc.

[Stephen G Simpson]

Reuben Hersh says:
 > Since you rely on the New York Times to tell you what
 > I think,
 > ...

I'm not relying solely on the New York Times.  I've also read Hersh's
postings on the FOM mailing list, including his 22 Dec 1997 summary of
"What is Mathematics, Really?".

Based on what I've learned of Hersh's views, I think Rothstein's
discussion of them in his 20 Dec 1997 New York Times article is
reasonably accurate.  If Hersh thinks that Rothstein misrepresented
him, I would suggest that Hersh publish a line-by-line critique
detailing exactly where Rothstein went wrong.  I'd be very glad if
Hersh would use the FOM mailing list for this purpose.

 > ...
 >
 > 'A world of ideas exists, created by human beings, existing in their
 > shared consciousness.  These ideas have objective properties, in the
 > same sense that material objects have objective properties.
 > ...
 >
 > 	Is that enough about objectivity?

No, this isn't enough.  This is merely lip-service to objectivity.
This kind of "objectivity" is pure subjectivism.  We might as well
say: "Sherlock Holmes exists and has objective properties, in the same
sense that material objects have objective properties, because the
world of Sherlock Holmes was created by human beings and exists in
their shared consciousness."  I would sharply disagree with this
statement, for the same reason that I disagree with Hersh's view of
the nature of mathematics.  Objectivity is not "shared consciousness".
Objectivity is not "consensus".  The essence of objectivity is respect
for reality, including the recognition that reality is prior to
consciousness.

 > Then you repeat Rothsteins' canard that "feminist and Afro-American
 > mathematics are mentioned prominently."  The only sentences that
 > can be construed as such "prominent mention" are on page xv of the
 > preface.  May I please with your permission quote them?

First, I said Afrocentric, not Afro-American.  Secondly, it's clear
from the context that I meant "mentioned prominently in Rothstein's
article," not "mentioned prominently in Hersh's book."

Neither Rothstein nor I said that Hersh endorses Afrocentrism or
radical feminism in mathematics education.  However, I think that
Rothstein is correct in implying that Hersh's subjectivist ideas are
highly compatible with these anti-objectivity educational movements.
Hersh says:

 > "Some people think female mathematicians and mathematicians of
 > color see the nature of mathematics differently than do white male
 > mathematicians.  I'm not convinced such differences are present.
 > If they are, I'm unqualified to write about them."

This statement is too cautious and too weak.  The feminists and
Afrocentrists would simply add: "Of course Hersh is unaware of and
unqualified to write about such differences, because Hersh is a white
male."

Why doesn't Hersh go farther and say the following, as objectivity
demands?  "Gender and race are irrelevant to mathematics.  Female
mathematicians and mathematicians of color, qua female and colored, do
not see mathematics differently than white male mathematicians.  If
some women and people of color see mathematics differently, that is
due to the philosophical ideas held by those people, not to gender or
race."

 > You also quote Rothstein to the effect that I claim my so-called
 > anti-objectivist views would bring more students of different races
 > and cultures into mathematics.  My book contains not a single word
 > about students of different races and cultures.
 > ...
 > Where do you or Rothstein get your "different races and cultures?"
 > Do you call this honest, accurate, or responsible?

I quoted Rothstein accurately.  Rothstein says:

   Mr. Hersh argues that his view of mathematics as fundamentally
   social and multicultural would, if accepted in the classroom, bring
   more students of different races and cultures into the study of
   mathematics.  He calls this vision of mathematics "humanist" and
   argues that it fits with "left-wing anti-elitism" and its striving
   for "universal literacy, universal higher education, universal
   access to knowledge and culture."

If Hersh has never argued in this way, then obviously Rothstein's
statement is neither honest nor accurate nor responsible.  The
question is, has Hersh in fact argued in this way?  

[ My own view is that I'm all for universal numeracy, but I don't
think we need to sacrifice objectivity in order to achieve it. ]

 > I did say that in my opinion formalist and Platonist philosophies
 > can be educationally harmful, and that humanist philosophy might be
 > educationally beneficial.

Why does Hersh think that his "humanist" philosophy is educationally
beneficial?  Why does he think that other philosophies are
educationally harmful?  

[ My own view is that it's educationally beneficial to stress the
objective, reality-oriented nature of mathematics.  Morris Kline's
"Mathematics and the Physical World" is an example of this approach. ]

 > As for "postmodernism", I once tried to find out what it is, and
 > failed.  In the present connection, however, I think I can see what
 > it is.  Isn't it a stick to beat an author whose ideas you dislike
 > so much you can't bear to pick up his book?

That's not what postmodernism means to me.  I would characterize
postmodernism as follows: It is a trendy movement which now dominates
the humanities departments in many European and American universities.
It tries to reject and undercut the values of Western civilization and
culture, especially the Enlightenment, objectivity, reason, science,
technology, and human progress.

For a survey of contemporary postmodernism, I would suggest "Higher
Superstition" by Paul R. Gross and Norman Levitt, Johns Hopkins
University Press, 1994.  Levitt is a mathematician at Rutgers.  An
excellent companion volume is "The Flight from Science and Reason",
Johns Hopkins University Press, 1996, which is the proceedings of a
conference sponsored by the New York Academy of Sciences.

-- Steve

PS
I'm not saying that Hersh is a postmodernist.  I am saying that
Hersh's views are highly compatible with postmodernism, and will
inevitably be picked up by and exploited by postmodernists.
Similarly, Atiyah's anti-rigor views were picked up and exploited by
the postmodernist Horgan in his Scientific American articles.  Atiyah
complained that he had been misunderstood.  This was discussed here on
the FOM list.  Search for Horgan in the September 1997 archive,
fom.9709.