FOM: Lakatos

[Jeffery Zucker]

I recently read "Proofs and Refutations" by Imre Lakatos.  
Many subscribers to this list will be familiar with this 
classic work, but I'll summarise it briefly.

It grew out of Lakatos' 1961 Cambridge PhD thesis under 
R. Braithwaite, and was strongly influenced by Karl Popper. 
It was edited and published posthumously by John Worrall 
and Elie Zahar in 1976. 

I'll quote briefly from the author's Introduction:

"Its modest aim is to elaborate the point that informal,
quasi-empirical, mathematics does not grow through 
a monotonous increase of the number of indubitably
established theorems but through the incessant 
improvement of guesses by speculation and criticism,
by the logic of proofs and refutations"

The bulk of the book is in the form of a case study 
(in dialogue form), in which Euler's celebrated theorem for
polyhedra (V-E+F=2) is subjected to a series of successive 
proofs and refutations, the latter being dealt with (not only 
by revisions of the proofs, but also) by successive
re-formulations of the theorem, involving re-definitions of
the notion of "polygon". 

I was impressed, not only by the author's philosophical
brilliance (as it seems to me), but also by the 
dazzling erudition displayed in his historical footnotes. 

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

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