FOM: social construction?

[Charles Silver]

> 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.

Bill Tait said:
> 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? 

	No, that's not what I'm saying.  We don't learn math by
anthropological investigations into what mathematicians do, but we do gain
insight about the nature of mathematics by noting things about the
activities of mathematicians.  Maybe it's beginning to look like I've
defected to the Hersh camp when I say this, but I don't think I have.  I
must admit, though, that I think his methodology of studying
mathematicians sociologically, or as you say, anthropologically, is a good
idea.  To my mind, it can establish general points about the subject
itself.  For example, an anthropologist can note that once a theorem has
been proved, it stands up for all time (unless a certain thing called a
"flaw" has been found).  That seems like an important point that
anthropologists can discover about the nature of mathematics (that
theorems are for all time).

	Actually, now that I think about it, it does seem as though it may
be *possible* to learn mathematics by conducting an anthropological
investigation.  For example, suppose two anthropologists are sitting at a
math colloquium, one of whom has studied mathematicians and the other has
not.  The speaker at this colloquium says that he will now prove something
by "mathematical induction."  At this point, the knowledgeable
anthropologist whispers to his colleague that mathematical induction
typically requires two cases, a basis case and an inductive case.  Then,
let's suppose the speaker skips over the basis case [I realize my example
is a bit ridiculous.  Please bear with me.].  The knowledgeable
anthropologist turns to his colleague and says, "Uh, oh.  I think he's
made what's called a 'mistake'."  Note that this anthropologist is
starting to learn some mathematics. 

	Now that I think more about it, I'm tempted to say that we were
all anthropologists when we started learning mathematics.  At least
partly, anyway.  We became aware that there were cetain acceptable ways to
do things that we learned from more bona fide mathematicians.  Of course,
we had to "get it" too.  There's the "seeing that a proof works" that's
different from just mimicking what others do.  But, aren't we a little
like anthropologists when we study someone else's work in order to learn a
new technique?  Wasn't forcing something like a whole new language to set
theorists in the early sixties? The big difference, though, is that the
anthropologist-on-his-way-to-becoming-a-mathematician must somehow "get
the idea" of a mathematical technique before he can really use it.  That
is, he can't do mathematics by strictly copying what he sees--or can he? 
I'm not sure about this.  Isn't there some minimal something that we call
"understanding" that must prevail?  On the other hand, there are
theorem-proving machines that don't really "understand" what they're
doing.  Isn't that so?  And, along these lines, I can prove things I don't
really understand, in the sense of really "seeing" the proof.  Sometimes I
just go through the steps that I know will work (because I've seen the
original proof). And, sometimes I can travel mindlessly (like an
anthropologist) down a standard path when working on a new problem, and to
my amazement a proof pops out, almost on its own.  Just by me playing
mathematical anthropologist.  Then, there are times when I (think I) 
"see" that something works, but I can't put together what's called in the
mathematical community "a proof." 

	Sorry for wandering all over the place in the above paragraphs. 
I'm not sure what's going on. Maybe someone can help straighten it all
out.
	
Charlie Silver
Smith College