FOM: social construction of mathematics?

Randall Holmes holmes at
Thu Mar 19 12:02:31 EST 1998

Comments on your reply follow.  Nice to hear from you!  (cc FOM list)

					--Randall Holmes

(You wrote)

Mathematics is the study of formal structure.  

        That is one view of what mathematics is.  You erroneously state
        it as a simple fact, not a controversial opinion.

(I reply)

	I certainly recognize that it is a controversial opinion.  But
this does not exclude it from being a simple fact, which it is (also).
Unfortunately, simple facts are not always obvious to everyone.  You
may regard it as a thesis of mine, if you like.

(You wrote, in response to my characterization of the kind of views
from which I believe yours spring) 

       Not my viewpoint.  In my view, the world has structure, which we try to discover.
        To describe and explain the world (and also for internal mathematical 
        motivations), we invent mathematics.

(I reply)

	That doesn't surprise me; note that I included "Hersh-like
viewpoints" in my phrasing.  I'm not necessarily referring to you
specifically.  Notice, though, that many who chime in enthusiastically
in support of your views clearly think that structure is something we
impose on the world.  The prevalence of such views has a great deal
to do with the popularity of the position you present.

	I agree that we invent mathematics to describe and explain the
world (and also for internal mathematical motivations); that doesn't say
what mathematics is about.

(You quote and say)

> If one does not acknowledge that there is real formal structure in the
> world, then one cannot understand the objective character of mathematics.

        This needs an argument to carry conviction.

(I reply)

	Of course it does.  I'm not arguing; I'm stating a position.

Globally, you criticize me repeatedly for not presenting arguments; I
am stating a position (which I am willing to defend subsequently), not
presenting an argument.  Also, not all criticisms given are criticisms
of you.

(You say)

        The view that mathematics is a social construction comes,
        on the contrary, from (1) being a mathematician and
        watching what mathematicians say and do (2) rejecting
        the other possibilities, that it is a transcendental
        immaterial "reality" (Platonism) or a mental process
        (Brouwer) or meaningless formal calculations (formalism.)

( I reply )

I don't see it as even a possibility.  I simply cannot see it as an
explanation.  Moreover, I believe that the only reason that such a proposal
can enjoy any support is because of the existence in the larger society
of the kind of views I describe in the posting (which are definitely attested
in postings on the list, whether you assent to them yourself or not).

The discipline of mathematics as a human activity is a social
construction.  To be able to say that the _subject matter_ is a social
construction and receive assent is extraordinary.

(I stand by my earlier statement)

> This conclusion is absurd, 

(end quote)

(You say)

        By "real world" do you mean the physical world, as I suppose?  
        Surely you know that most research and publication
        in pure math has nothing to do with any real or otherwise aspects
        of the real world.  Science isn't math, math isn't science.
        They overlap, they help each other.  Science is based ultimately
        on empirics, on some sort of physical observation or measurement.
        Math is not.  It is based on the joint thinking, mutual criticism,
        historically developed ideas  of people.  Not any old ideas,
        but the kind of ideas that can be checked, verified, confirmed, with
        virtual unanimity.

(end quote)

I agree with most of what you say here.  But ideas about social
constructions cannot be "the kind of ideas that can be checked,
verified, confirmed, with virtual unanimity."  Such ideas are ideas
about objective features of the world.  By the real world, I mean the
real world, which includes but is not necessary confined to the local
part of the physical world.

(you say)
        Everything is not "much clearer" if you confound, conflate
        and confuse math and science.  
(end quote)

I don't.  But I affirm that both of these subjects are about something.

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
not glimpse the wonders therein. |

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