FOM: social construction?

[Lincoln Wallen]

   Date: Fri, 20 Mar 98 10:02:49 -0500
   From: <wtait at ix.netcom.com>

   Charles Silver wrote (3/20)
   >I believe Hersh is right that
   >we find out what math is by investigating what mathematicians would do in
   >certain circumstances.
   Maybe this is right if we are taking the approach of anthropologists, 
   investigating the workings of an alien culture. But if you replace ``we 
   find out what math is by'' by ``we learn math by'', I think you are 
   clearly wrong. E.g. do we learn English by investigating what English 
   speakers do in certain circumstances? This would not be a good 
   description even of how we learn a second language. But the case that is 
   analogous to learning math is learning our native language; and here it 
   is a non-starter.

   If the entire membership of AMS emailed in stating that there are only 
   finitely many prime numbers, I would be shocked that so many people had 
   gone mad all at once, not that an arithmetic truth had become false.

   Bill Tait 

Focus on the word "investigate" here.  In fact, an anthropologically
inspired view of langauge use and learning gives just this sort of
account of how we learn our native language: By observing the effects
our utterances have on the people around us we learn the meaning of
the utterances.  This type of "use and observation" seems good enough
to call "investigation" to me.

As a tutor of mathematics students one of the most important tasks is
to *react* to their mathematical utterances (marking, arguing etc) and
so provide a framework in which they can conduct similar
investigations.  In this way students learn to "speak" in the peculiar
way that enables them to do mathematics.

So I don't understand why this account is a "non-starter"?  Is it that
you think of an investigation as something an analyst does (hence the
reference to second language learning)?  If so, it is worth pointing
out that the methodological reason why (some types of modern)
anthropologist adopt the view that a particular cultural activity is
"alien" is precisely to try to seek the ways in which the participants
engaged in the moment-by-moment creation of that culture themselves
*investigate* the material (spoken, written, physical, etc) around
them.  It is an attempt to understand the practices of local
investigation, and a recognition that the primary "scientists" of an
activity are not the anthropologists, but the participants themselves.
The reconstructed task of the anthropologist is to articulate these so
called ethno-methods of (local) investigation.

In these terms I understand Silver to be agreeing with Hersh that we
need to articulate the ethno-methods used by mathematicians to create
and sustain mathematics.  Don't be put off by the "alien".  It simply
means treat what you see as strange, and seek the reason for its being
said right then, right there, rather than assuming some
extramathematical framework, dare I say, such as the philosophers and
mainstream sociologists try to provide us with.

Lincoln Wallen