FOM: Rhetoric in mathematics

[Corfield, David [CES]]

The best way of expressing  Hersh's thesis that mathematics is a social
activity is to call it a rhetorical activity. Lakatos's key insight was
precisely
this, derived from his knowledge of Arpad Szabo's work in the history of
Greek mathematics, and his study of Hegel's dialectics. Where he went
wrong, as I have outlined in a couple of works, is by restricting the
rhetorical
process to the level of mathematical propositions. With what amounts to
a
stabilisation of rigour, 'I don't believe your proof' is less commonly
heard.
these days. Even when problems in a proof are found, there is great
agreement
 about what needs to be mended.

Given that even in a very weak system the vast majority of theorems are
completely uninteresting, arguments occur over what's important.
The major arguments taking place in mathematics today concern:
whether a newly defined concept is worth its salt, whether a result has
been properly understood, whether this field is worth developing,
whether this more elegant  technique adds anything to what was
understood anyway.

These are all open-ended questions. What's interesting if you look at
these arguments is the range of rhetorical moves, many of which have
appeared on this mailing list: the concept appears in many fields; a big
name
has endorsed it; there's no great distance from what's gone before, yet
it's sufficiently
novel; this path is forced upon you; there are (reasonably) direct
applications
in science or computing. Frequent references will be made to the terms
'natural'
'inevitable' and 'unifying'.

Counter-arguments do not appear in print as often as one might like.
One of the most common appears to be silence.
Try reading a piece of math with the idea in your mind that the author's
saying
 'For God's sake, read me'.


To focus on the argumentative side of mathematics is not to deny that
something is providing resistance to mathematicians.
You can't just do what you want. Hamilton tried to create a
3-dimensional
number system resembling the complex numbers but had to go to four
dimensions. Others were happier with the three dimensional system of
vectors,
'hermaphrodite monsters' as Tait called them. What is interesting here
is the
dispute between the sides as to the right way to generalise the complex
numbers,

Lakatos was interested in the passage from the informal to the formal,
and
how questions could be raised about whether the informal had all been
captured. I argued in my papers that formalisation helps produce a more
refined informal thought. There's something informal about Witten's
integrating over an infinite dimensional space of connections, but it
could only have been thought up on the back of earlier formal work.

Arguments via motivation are open-fisted as we should have learned from
the
contributions to this list. As Thurston put it 'one man's motivation
is another's intimidation'. You can see how varieties of motivation can
affect even
apparently solidified notions of number. As a simple test to distinguish
Pratt's
 sanders from gluers, the following question can be put:

Does the fact that the fundamental group of the circle is isomorphic to
the integers
affect your concept of number?

The sand people would presumably say 'No', and  take the integers as
deriving from
the natural numbers. They can then point us  to counting stones or our
heartbeats
as the fundamental motivation.(The latter might help us avoid the
thought of their
finiteness.)

The gluers can point to the act of wrapping rubber bands around a
finger, or dogs
chained to trees.

Why should we want to restrict motivation to one variety?


David Corfield

School of Cultural Studies,
Leeds Metropolitan University,
U.K.