FOM: Imre Lakatos (Imre Lipschitz)

[Julio Gonzalez Cabillon]

(I hope this is not a duplicate, my first version disappeared).

Now, with more time, I expand my previous version. It seems that PSU.EDU
eats my mails. Anyhow...

               On Sat, 13 Dec 1997, Jeffery Zucker wrote:

| 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?

    Sorry, but I do not think that Lakatosian programme has had little
effect on mathematical pedagogy. What I think is that the introduction
of heuristic style, as Lakatos himself was aware of, would require not
only the rewriting of textbooks, but would make them so long that one
could never read them to the end. As a writer, I experienced this
problem myself deep inside in my own flesh.

    Lakatos' "Proof and Refutations: The Logic of Mathematical Discovery"
is certainly a wonderful book. As you say the main subject of this
masterpiece is the history of Descartes-Euler's theorem. The "problem"
with this theorem is that cannot be regarded as a paradigm for the history
of other mathematical results. In this sense some mathematicians think
that Lakatos' book "did a great disservice by treating it as typical".


| (2) What has been written in response to this work
| (for or against)?

    One of the outstanding mathematicians who has written on Lakatos
is Reuben Hersh. His books are really wonderful!

Cf. 
1. "The mathematical experience" / Philip J. Davis, Reuben Hersh; with an
introduction by Gian-Carlo Rota, Boston: Birkhaeuser, xix, pp. 440, 1981.
ISBN: 3-7643-3018-X.

2. "The mathematical experience" / Philip J. Davis, Reuben Hersh,
Elena Anne Marchisotto; with an introduction by Gian-Carlo Rota.
Boston: Birkhaeuser, xxi, pp. 487, 1995. ISBN: 0-8176-3739-7.

3. What is mathematics, really? / Reuben Hersh, New York, Oxford: Oxford
University Press, xxiv, pp. 343, 1997. ISBN: 0-1951-1368-3.

    On another line of argumentation, Brendan P. Larvor has addressed
recently "Lakatos as historian of mathematics", published in _Philosophia
Mathematica. Philosophy of Mathematics, its Learning, and its Applications.
Series III_, vol. 5, no. 1, pp. 42-64, 1997:

         "The goal of this paper is not to survey the history
         of mathematics with a view to testing Lakatos' theory,
         nor to carry out any part of a Lakatosian historiographic
         programme. Rather, it is to discuss the connection between
         the actual history and the philosophy. The easiest way in
         is to address the objection [to Lakatos' analysis in his
         famous book 'Proofs and refutations'] that the polyhedron
         theorem is a one-off. Nothing else in mathematics ever
         developed in anything like the same way. My reply (on
         Lakatos's behalf) comes in three parts.
         One could consistently maintain that Lakatos is right to
         hold philosophy accountable to history, and that Lakatos's
         own view of mathematics fails that test".


Further reading:

Ernest, Paul:
"The legacy of Lakatos: reconceptualising the philosophy of mathematics",
_Philosophia Mathematica. Philosophy of Mathematics, its Learning, and
its Applications. Series III_, vol. 5, no. 2-3, pp. 116-134, 1997.

Corfield, David:
"Assaying Lakatos's philosophy of mathematics", _Studies in History and
Philosophy of Science_, vol. 28, no. 1, pp. 99-121, 1997.

Glas, Eduard:
"Kuhn, Lakatos, and the image of mathematics", _Philosophia Mathematica.
Philosophy of Mathematics, its Learning, and its Applications. Series III_,
vol. 3, no. 3, pp. 225-247, 1995.

Freguglia, Paolo:
"Historiography and epistemology in Lakatos" (in Italian)
Epistemology of mathematics: 1989-1991 Seminars (in Italian),
pp. 67-76, Formazione e Aggiornamento in Matematica degli Insegnanti, 10,
CNR, Servizio Pubblicazioni, Piazzale Aldo Moro, 7, 00185 Rome, 1992.

Hernandez, Jesus:
"On the philosophy of mathematics of Imre Lakatos" (Spanish and English,
Spanish summary) III International Colloquium on Philosophy and History
of Mathematics, Ciudad de Mexico, 1992, _Mathesis_, vol. 8, no. 4,
pp. 459-477, 1992.

Koetsier, Teun:
"Lakatos' philosophy of mathematics: A historical approach"
Studies in the History and Philosophy of Mathematics, 3, Amsterdam:
North-Holland Publishing Co., xii, pp. 312, 1991. ISBN 0-444-88944-2. 

Dominicy, Marc:
"Falsification and falsifiabilization from Lakatos to Goodman" (in English),
_Revue Internationale de Philosophie_, vol. 37, no. 1-2, pp. 163-197, 1983.

Agassi, Joseph:
"Lakatos on proof and on mathematics" [in English] _Logique et Analyse.
Nouvelle Serie_, vol. 24, no. 95-96, pp. 437-439, 1981.

Currie, Gregory:
"Lakatos's philosophy of mathematics" _Synthese_, vol. 42, no. 2,
pp. 335-351, 1979.

Rott, Hans:
"Zur Wissenschaftsphilosophie von Imre Lakatos" [On the Philosophy of Science
of Imre Lakatos], _Philosophia Naturalis_, vol. 31, no. 1, pp. 25-62, 1994.

Hersh, Reuben:
"Introducing Imre Lakatos" _Mathematical Intelligencer_, vol. 1, no. 3,
pp. 148-151, 1978.

Ribes, Diego:
"The historical nature of the demarcation criterion of Lakatos",
(in Spanish), _Teorema_, vol. 7, no. 3-4, pp. 241-256, 1977.

Feyerabend; Cohen; Wartofsky (editors):
[*] "Essays in memory of Imre Lakatos", Boston Studies in the Philosophy
of Science. Vol. XXXIX. Synthese Library. Vol. 99. Dordrecht-Boston:
D. Reidel Publishing Company. XI, 767 pages, 1976.

Worrall, John:
"Imre Lakatos (1922-1974): Philosopher of Mathematics and Philosopher
of Science", pp. 1-8 in "Essays in Memory of Imre Lakatos" [*], 1976.

Feyerabend, Paul:
"Imre Lakatos", _The British Journal for the Philosophy of Science_
vol. 26, pp. 1-18. 1975.

Worrall, John:
"Imre Lakatos (1922-1974): philosopher of mathematics and philosopher
of science" (in English), _Zeitschrift fuer Allgemeine Wissenschaftstheorie_,
vol. 5, no. 2, pp. 211-217, 1974.
  
Best wishes from Montevideo,
                               Julio Gonzalez Cabillon