FOM: social construction of mathematics?

[Randall Holmes]

(this is from Randall Holmes)

This is a comment on the views expressed by Hersh and others concerning
the "social construction" of mathematics.

To make any sense of such views, they need to be placed in a larger
context.  Mathematics is the study of formal structure.  One may
attempt to answer Hersh or Hersh-like interlocutors by pointing out
all the structure which exists in the world and asking if it depends
on social consensus.  One is not going to get much satisfaction this
way, because the viewpoint that spawns these kinds of views is one which
holds that we impose structure on the world rather than find it there.

If one does not acknowledge that there is real formal structure in the
world, then one cannot understand the objective character of mathematics.
Formal structure (whether it exists Platonically or inheres in a more
Aristotelean manner in objects, independently of us in either case) is
what mathematics is about.

If structure is supposed to be imposed by minds, then there is the
further problem of why it is that structure in general (including
scientific laws and other things, not just mathematics) is
intersubjective.  We know as individuals that we didn't really create
all of it.  Thus the retreat to "society" or "culture" or (cleverly)
"language" as the place where structure is created (by mental
activity, but collaboratively, bit by bit).  This view leads
inevitably to the idea that mathematics is a social construction.
This conclusion is absurd, but the underlying assumption also has the
power of confusing the faculties of its adherents enough that they
have trouble seeing this.

It is impossible to argue with Hersh or any others like him (others have
posted similarly) without addressing the basic assumption.  The structure
of the world was not created by human individuals or human societies.
Scientific laws are not sociological phenomena; neither are mathematical
theorems.  Both science and mathematics are human activities, but they
are activities devoted to the study of real aspects of the real world.

Unfortunately, the assumption in question is so deeply rooted that it makes
conversation with its adherents very difficult; each side keeps missing the
point of what the other side is saying.  I am deeply convinced that a
necessary consequence of the assumption that I am challenging here is that
the world is entirely unintelligible; it is an intellectual poison which
has already caused an enormous amount of damage.

If the offending assumption is abandoned, the objective character of
mathematics becomes obvious and unproblematic.  There are still problems
with the philosophy of mathematics (resolving the Plato vs. Aristotle
question mentioned in passing above, for example) but everything becomes
much clearer.

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