FOM: social construction?

[Randall Holmes]

I think that I can identify Hersh's error (as have some others).  It
is the confusion of use with mention; that is, Hersh confuses talk
about the objects of mathematics with talk about the symbols (and
the psychological phenomena in our heads) which refer to the objects
of mathematics.  

(quote from Hersh)

To your questions about the validity of mathematical theories
in the past, as mathematical theories, about mathematical 
objects, with mathematical hypotheses--it's not that they
weren't valid, they weren't there at all.  To claim that they
were, you have to imagine that human thoughts and ideas can
exist without humans to think them.  Do you say they are or were
already inherent in the pebbles or the planets?

(end quote)

The symbolic formulations of mathematical assertions which we might
apply were not there in the past.  This much is true.  Nor were any
human thoughts or ideas there in the past.  But the features of the
structure of the world which those assertions and thoughts refer to 
were there in the past.  There is no other way to make sense of the fact
that mathematical formulations do indeed turn out to be applicable to
situations of which no human being was aware at the time.

(quote from Hersh)


We make
theories to study them, whether in the past, present or
future.  The theories try to be true to exterior reality,
but they are not in or part of exterior reality, they are
our devices to describe exterior reality.  Without human
readers, our libraries are mere piles of ink and paper.
Mathematics is something we do, for several purposes, including
to do physics.  The physical world goes on without us.
Our marks on paper or magnetic chips are only our ways of
thinking together and talking to each other.

(end quote)

There is nothing here that I disagree with (as long as "theories" are
understood as human thoughts and symbols, rather than in a technical
metamathemtical sense).  In physical science, it is understood that the
theories are about things that are there (Hersh acknowledges elsewhere
in his post that the planets, which are objects of scientific theory, did
exist in the Jurassic :-) ); theories are true or false insofar as they
agree or disagree with what is really there.

Exactly the same considerations apply to mathematics.  Moreover, the
reference of physical theories to a physical world stands or falls with
the reference of all kinds of theories to the world.  There are people out
there (including a recent poster on this list) who say with all seriousness
that the planets themselves didn't exist in the Jurassic; "planet", after
all, is a human idea (so is "Jurassic", of course :-) )  The same arguments
which Hersh is trying to use to deconstruct mathematical reality can be
and are being used to deconstruct physical reality.

Hersh needs to defend the distinction between physical theories (which
surely are not meaningful unless they refer) and mathematical
theories.  It is clear from other statements which he has made that he
is concerned to defend a materialist, empiricist, "skeptical and
critical" philosophy.  It is quite difficult to see how mathematical
discourse can refer on a materialist view; that is why I am not a
materialist.  But the other adjectives all appear to me to apply to
Quine's positions (for example), and Quine is convinced of the reality
of mathematical objects (against his will) [reference for the last is
personal conversation, circa 1990].

My opinion is that Hersh has allowed himself to be led into an elementary
philosophical error by his distaste for the implications of the reality
of the subject matter of mathematics, which do indeed lend support to the
denial of naive materialism ( and maybe even the resumption of theology as
a serious intellectual pursuit, though that is a longer stretch :-) )

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes