FOM: social construction?

[Lincoln Wallen]

I have just seen, and enjoyed very much, the posting by Charlie Silver
where he thinks his way to the point of suggesting that we act "like"
anthropologists when learning how to "do" a new area of mathematics.

I hope it was clear from my posting a few minutes ago in response to
Tait's message that it is exactly this view [expressed by Silver] that
I think has a lot of merit.  I put it differently: certain
anthropological schools have adopted the view, *not* that people (like
mathematicians) act like anthropologists when engaging in their chosen
activity, but *the other way around*: that there is no position
"outside" the activity one is studying from which to gain a more
direct understanding of the structure and truth of the activity.
Hence it is the anthropologists who must act ethnically and seek to
understand *as the participants themselves understand*, not in some
other way.  (Note: this approach causes consternation in mainstream
sociology as it removes the vantage point from which sociology seeks
to view the world.  Hence the profound difference of this thinking
with the seemingly similar "sociology of science" literature.  Again,
it is the latter many strong negative opinions expressed on FOM have
been effectively attacking.)

The anthropologist-mathematician is distinguished from his
mathematical counterpart only in the sense that he seeks also to
*articulate* these ethno-methods.  It is at this point that we must
address one of the most difficult scientific aspects of such
investigations: how do we articulate these ethno-methods?  That is,
how can one separate the particulars of the individual observations to
present a more general statement about the ethno-methods?  And,
moreover, what relationship does/should such general statements hold
to the circumstances, the phenomena, from which they were generated.
(NB. this is the same dilemma when writing books about language use it
seems to me, and the relationship we wish to hold between theories of
linguistic behaviour such as grammars, and use of language itself in,
say, dialogue.)

I won't bore you with the moves that have been made to address this
question, suffice it to say that the jury is still well and truly out,
and that unless you recognise the active role that people play in
deciding in practice whether or not a given "rule" is being followed,
you may well miss my point here.  (Cf. my posting about the practices
of counting pebbles and recognising squares on a Jurassic beach.) Such
theories are not natural scientific in any easy to grasp sense
precisely because the tool for measurement is still directly human.

But in the case of mathematics one could say that there has in fact
been fair progress.  For example, is it not reasonable to view
Gentzen's Natural deduction as a good attempt to describe certain
regularities in (written) mathematical arguments?  (Or other formal
systems for that matter.)  The question is: just what sort of a theory
is ND?  It is obvious that it is an idealisation.  It is also obvious
that its relationship with actual arguments is a judged relationship.
That is to say a fair degree of *mathematical* skill is required to
correctly identify exactly how an ordinary mathematician in writing
out an argument, is in fact producing argument which conforms to the
ND rules.  Failure to be able to give such an account may be
interpreted in different ways: error, misunderstanding of the reader,
etc..  Distinguishing between these alternatives requiring additional
communication (NB. I am not saying all these alternatives are equally
correct in every situation; it is for the participants to resolve
which account is the right one.  Cf. Wittgenstein's rule following
discussions and the interminable debates subsequently.) 

What is interesting is that if we are to raise ND to the status of
playing a large part in an ethno-method, we must account for how
mathematicians *are* able to recognise the structure ND gives an
account of in *actual* spoken and written mathematical discourse.
This is a much tougher question and if you begin to investigate it (as
I am actually doing, ethnomethodologically) it is not at all clear
that ND provides a sufficiently rich set of concepts to give an
adequate account of what people actually do and recognise.  This is
again similar to the problem formal linguists have in accounting for
conversation.  It is not that the grammars are incorrect, they are
sufficient for certain purposes, but too simple for others: the
idealisation is too crude.  We should not be frightened of this
surely.  It happens all the time!

Presumably the ability to recognise a mathematical development as
being conducted within set theory is also a skilful judgement.  The
question of category theory vs. set theory is partly a question of
accounts, of *reading into* a naturally occuring piece of mathematics
conformance with a scheme of organising principles.  The author will
no doubt attempt to lead you to a certain type of reading and the way
in which that is achieved is important.

The "enduring" nature of the product of mathematical activity is
likewise an achievement (as I have said before).  It is by no means a
denial of that fact to seek an account of how it is achieved.  Indeed:
the way in which that term "enduring" is being used can surely only be
understood in the context of an account of how what is properly
mathematical is actually produced.  (Cf. the reproducibility of
phsyical experiments rests on what we consider an experiment to be,
and an identification of what we consider invariant.  The distillation
of these notions is properly recognised as being the foundation of physical
science.  Why should it be any different for mathematics?)

There are too many directions to go in from here so I will stop.

Lincoln Wallen