Lakatos' Philosophy of Mathematics, Volume 3: A Historical Approach (Studies in the History and Philosophy of Mathematics) [Hardcover]

T. Koetsier (Author)

Book Description

Publication Date: December 4, 1991 | ISBN-10: 0444889442 | ISBN-13: 978-0444889447 | Edition: 1

In this book, which is both a philosophical and historiographical study, the author investigates the fallibility and the rationality of mathematics by means of rational reconstructions of developments in mathematics. The initial chapters are devoted to a critical discussion of Lakatos' philosophy of mathematics. In the remaining chapters several episodes in the history of mathematics are discussed, such as the appearance of deduction in Greek mathematics and the transition from Eighteenth-Century to Nineteenth-Century analysis. The author aims at developing a notion of mathematical rationality that agrees with the historical facts. A modified version of Lakatos' methodology is proposed. The resulting constructions show that mathematical knowledge is fallible, but that its fallibility is remarkably weak.

Editorial Reviews

Review

A.M. Coyne
...very clearly written and can be understood by advanced undergraduates in either mathematics or philosophy. The bibliography is extensive. It will interest all philosophers of mathematics and any mathematician interested in reflecting on the methodology of mathematics.
Zentralblatt für Mathematik

Product Details

• Hardcover: 312 pages
• Publisher: North Holland; 1 edition (December 4, 1991)
• Language: English
• ISBN-10: 0444889442
• ISBN-13: 978-0444889447
• Product Dimensions: 8.9 x 6 x 0.8 inches
• Shipping Weight: 1.5 pounds
• Average Customer Review: 2.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #9,073,915 in Books (See Top 100 in Books)

More About the Author

Discover books, learn about writers, read author blogs, and more.

Customer Reviews

Most Helpful Customer Reviews

2 of 2 people found the following review helpful:

2.0 out of 5 stars Useless, June 5, 2008

By 

Viktor Blasjo - See all my reviews
(REAL NAME)   

This review is from: Lakatos' Philosophy of Mathematics, Volume 3: A Historical Approach (Studies in the History and Philosophy of Mathematics) (Hardcover)

To the limited extent that this book has to do with Lakatos' philosophy of mathematics it is an unsympathetic and unimaginative summary of some of Lakatos' views interspersed with Koetsier's incompetent extrapolations and misrepresentations.

An example of Koetsier's unreasonable extrapolations is his claim that "Lakatos's views apparently imply that nothing in mathematics is self-evident. Self-evidence in mathematics is an illusion." (p. 24). This statement immediately follows a quotation from Lakatos, Papers, II, p. 42, where Lakatos comes nowhere close to saying anything of the sort.

Another utterly absurd observation is that "Lakatos's notion of 'problem-shift' is similar to Brouwer's notion of 'jump from goal to means'" (p. 64). The latter is a quotation from Brouwer's "Life, Art and Mysticism", which Koetsier quotes as if it was a scholarly work containing "notions" for analysing the development of science when in fact it is a rebellious student manifesto that has nothing at all to do with science or mathematics. The quotation in question in fact occurs in the context of Brouwer's condemnation of modern industrial society.

An example of Koetsier's foolish misrepresentations is his critique of Lakatos' "Cauchy and the Continuum." Here Koetsier misconstrues Lakatos' standpoint by overemphasising the Robinsonian non-standard analysis aspect. Lakatos explicitly concluded that the Robinsonian interpretation is not correct and that its role was that of "a powerful stimulus" (Papers, II, p. 57). Lakatos' argument instead rests ultimately on the claim that Cauchy rejected the canonical counterexample since it diverges at x=1/n (cf. pp. 85-86). Lakatos may very well be wrong, but Koetsier's discussion does not help us decide since it misses Lakatos' point entirely.

The bulk of the book is devoted to Koetsier's proposed improvement on Lakatos: a "methodology of mathematical research traditions" (MMRT). This is amateur philosophy at its worst, complete with ambiguities spewing out its ears and feeble attempts to support it by pretentious terminology that is both ill-defined and never actually used. The latter pillars are often erected on pseudo-Laktosian sand, with pompous distinctions, never employed in the case studies, between e.g. "heuristic progress" (producing conjectures) and "absolute progress" (proving conjectures). It is no wonder that this distinction is never used later, as it is virtually vacuous (no tradition has had only heuristic progress). Koetsier's entire theory in the end amounts to a list of properties that are deemed desirable in a mathematical theory (p. 170), and the dictum that mathematicians should assign a "tradition" a value "proportional to its expected progress", as defined, within a margin of error of fifty-eight thousand miles, by this extremely vague list.

Koetsier's case studies never actually use his MMRT theory in any substantial way (how could they since this theory is all fluff?). One case study is a long and rambling and previously published survey of the history of the theorem on the equality of mixed partial derivatives. Presumably having been forced to include it to meet his page quota, Koetsier is desperately grasping for a way to connect it to his MMRT theory. But this time he does not have the imagination to come up even with fluff. Instead he establishes the desired connection by simply maintaining that the mere existence of research traditions provide resounding proof for his theory: "The different eighteenth century version of the interchangeability theorem ... support the rational reconstruction on the basis of the MMRT ... in the sense that they show the unity of the formalist tradition." (p. 249).

The book is also packed with typos and clumsy formulations. The list on p. 170 referred to above, for example, is said to be "undoubtedly not incomplete", when obviously the opposite is meant. As usual the fat cats at Elsevier want to stick the profits from their ridiculously overpriced books in their own pockets instead of hiring descent proof readers.