[FOM] Simpson on Tymoczkoism

[David Corfield]

Neil Tennant wrote:

>There is another philosopher's distinction that might be relevant to
>Steve Simpson's exchange with David Corfield. This is Reichenbach's
>distinction between the context of discovery and the context of
>justification.

>In the 23-page sample of his book which I downloaded---David,
>why can't CUP cough up the whole of the first chapter for us?!---

I did ask for more of course, but there's a strict page limit policy.

>Corfield
>emphasizes the relevance of real/core mathematical practice to the
>philosophy of mathematics, and complains that philosophers pay too little
>attention to it. Perhaps his emphasis here is really on the context of
>discovery---involving the Lakatosian process of conjecture, failed
>attempts at proof, subsequent conceptual tinkering, and eventual success
>with proof---rather than the context of justification.

>In response to this, the philosopher of mathematics who emphasizes instead
>the context of justification will be waiting on the sidelines, as it were,
>for the mathematician (or mathematical community) to "clean up its act",
>and present us with the finished product. Then, and only then, does the
>mathematical foundationalist of the Simpsonian kind roll up his or her
>sleeves and get down to the business of anatomizing the conceptual
>structures and deductive resources underlying the results proved.

It's useful to bring up the two contexts idea here, but I believe
what Lakatos and I are doing has plenty to do with justication in
mathematics. We're interested in a different kind of justification.
F.o.m. activity is aimed at the justification involved in passing from
starting assumptions to certain conclusions, e.g., examining which
logical resources are required.

Lakatos and I (and others) are interested in how you justify a piece of
mathematics (a single proof, a journal paper, an expository paper,
a textbook, a research programme, a research tradition) as being 'good'.
Here is the mathematician Ian Stewart's paragraph-long attempt to do so:

"Mathematics is good if it enriches the subject, if it opens up new
vistas, if it solves old problems, if it fills gaps, fitting snugly and
satisfyingly into what is already known, or if it forges new links
between previously unconnected parts of the subject. It is bad if
it is trivial, overelaborate, or lacks any definable mathematical
purpose or direction. It is pure if its methods are pure - that is, if
it doesn't cheat and tackle one problem while pretending to tackle
another, and if there are no gaping holes in its logic. It is applied
if it leads to useful insights outside mathematics. By these criteria,
today's mathematics contains as high a proportion of good work
as at any other period, and as any other area; and much of it
manages to be both pure and applied at the same time."

Logic is mentioned, of course, but there is so much more. It is true that
we do look at mathematics in the making, but we are not confined
to this. To examine the near universal appraisal of today's mathematicians
that the work inspired by Riemann of tying together the algebra, analysis,
geometry, and topology of 2-dimensional spaces is excellent mathematics,
we don't just stay in the 19th century. Rather, we examine the position of
this work
in the mathematical network, and how it features in the current images of
mathematics. Read Atiyah's recent lecture
'Mathematics in the 20th Century', Bulletin of the London Mathematical
Society
34(1), 1-15, 2002, for one such image. According to Atiyah, the 20th
century saw us 'doing a Riemann' on 3- and higher dimensional spaces,
the 21st will probably see us doing  the same for infinite dimensional
spaces.

Of course, it is interesting to return to Riemann in the 19th century, to
see how
it was that his work generated a fundamental shift in what was conceived to
be good, and it's informative to find out the kinds of resistance to this
shift. Are
we just dealing with a change in mathematical aesthetics occurring
fortuitously?
Is Riemann still revered for merely contingent reasons?  Perhaps, the likes
of
Atiyah have bullied the math community to accept their image of
mathematics.

You could imagine a three way conversation:

A) The use of Riemann surface ideas in string theory and the mathematical
venture to unify
algebra, analysis, geometry, and topology in higher dimensions are fads. We
continue
to revere Riemann because of vested interests. Things may look totally
different in 100 years.
If they don't it'll be down to social inertia.

B) Local social rationalism: The best mathematical developments are those
accepted in open discussions conducted on the basis of the criteria of the
time. These criteria will change but should do so through constructive open
debate. So long as decisions are reached by careful scrutiny by an open
society of mathematicians, in a 100 years we may well justifiably arrive at
a very different way of dealing with 2-dimensional spaces, or not even
choose to look at such
spaces at all.

C) Riemann uncovered a piece of 'conceptual reality'. If we don't continue
to work to elaborate this 19th century work into the 21st century, we will
have lost the plot.

There are several more subtle shades of these positions.


>Put more succinctly: Why is the XXX-mathematics of today philosophically
>sexier than the XXX-mathematics of yesteryear?

Somehow, I don't think 'Towards a philosophy of XXX-mathematics' would sell.

Hopefully I've made clear my view that the XXX-math of yesteryear is
philosophically
sexy.

As for whether anything special is being turned up by today's
XXX-mathematicians,
I've said before that I think we should pay attention to recent conceptions
of space.
It never ceases to amaze me how metaphysicians don't think to see if
mathematicians
have done anything useful for them.

I'm also quite intrigued by the thought that there are non-algorithmic, but
suggestive
indications of how to generate important math from existing important math.
For
a process known as categorification see chap 10 of my book and references.
The
Russian mathematician Vladimir Arnold suggests other processes -
complexification,
symplectification, etc. - see 'Polymathematics' available at his web-site.
One almost
catches a glimpse of something resembling chemistry's periodic table.

David Corfield