FOM: Lakatos

[Moshe' Machover]

>I have two questions, with which members of this list
>may be able to help me:
>
>(1) In spite of Lakatos's strong (to me) arguments,
>his work seems to have had little, if any, discernible effect
>on mathematical pedagogy, including the style of textbooks.
>Why is this?
>
>(2) What has been written in response to this work
>(for or against)?
>
>BTW, Moshe Machover (whom the editors acknowledge for
>his help) may have some interesting insights on this.
>
>Jeff Zucker

Recently there has been a revival of interest in Lakatos' approach to the
history and philosophy of mathematics. Of special interest is the work of
David Corfield.

The following is an excerpt from a report I wrote recently on Corfield's work.

> [Begin quote]
A seemingly promising start in a new direction was made by the late Imre
Lakatos, in his Proofs and Refutations, which was concerned with deep and
important problems of mathematical methodology. But after his untimely
death 23 years ago, his project has remained almost totally isolated. There
have been several interesting critiques of Lakatos, but no-one has actually
carried his programme forwards to any substantial extent. Moreover, Lakatos
and his few followers were almost exclusively concerned with 18th and 19th
century maths. (The main `story' of Proofs and Refutations begins with
Euler and ends with Poincare'.)

Corfield's work shows both *why* Lakatos' project has got stuck, and how
it can be radically reformulated and re-launched as a philosophical project
relevant to contemporary maths. Corfield pinpoints the shortcomings of
Lakatos' view of mathematical theories, and the inadequacy of importing the
notion of *research programme*--which was one Lakatos' innovations in the
philosophy of natural science--to the philosophy of maths (as some have
attempted to do). Corfield exposes two main shortcomings. First, Lakatos
and his few immediate followers characterize the core a mathematical theory
or research programme in terms of its central established *propositions*
(theorems, axioms) and conjectured propositions. Also, they concentrated on
isolated conjectures or, at best, isolated areas of mathematical research.
In the papers under review (as well as in his PhD thesis, whose theme these
papers extend and amplify) Corfield gives an impressive outline of salient
features of modern mathematical research, displaying consummate skill and
extensive knowledge in a broad front of fields (including algebraic
geometry, algebraic number theory, algebraic topology, category theory and
topos theory, as well as mathematical logic and constructive mathematics).

He argues convincingly that what typifies a mathematical research
programme is not a bunch of mathematical propositions, but more general
meta-propositions, aims and methods. He also shows that the single notion
of research programme is by itself inadequate for coping with the immense
complexity of present-day maths, and the high degree of interaction between
its various sectors. He proposes supplementing the notion of research
programme with the higher-level notion of *tradition* and the lower-level
notion of *project*.
[End quote]

For further information about his own work and realted work by others,
please contact Corfield. He can be reached at the following e-mail address:

		a.j.hall at lmu.ac.uk






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