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Proofs and Refutations: The Logic of Mathematical Discovery
Publisher: Cambridge University Press (1976)
Details: 186 pages, Paperback
Topics: Polyhedra, Philosophy of Mathematics, Classic Works
This book is in the MAA's basic library list.
MAA Review[Reviewed by Michael Berg, on 06/05/2012]
In his Author’s Introduction, Imre Lakatos presents us with the following pithy characterization of what is certainly one of the central concerns of modern philosophies of mathematics:
Lakatos then proceeds to elaborate the notion of Hilbert’s formalism, finally characterizing it as “a bulwark of logical positivism … [in which] a statement is meaningful only if it is either ‘tautological’ or empirical.” He then launches his own gambit:
Accordingly we learn already from Lakatos’ introduction to his book that his position is diametrically opposite to that of the formalist school which, again in his words, holds that “mathematics is identical with formalised mathematics.” By aligning himself with Pólya, for example, he indicates that his approach to the subject is centered on conveying what it is that a mathematician actually does, as opposed to formalized products (per se) that might subsequently appear as a result of mathematical work. Lakatos puts it this way: “[The present book’s] 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.”
What follows is a lengthy “essay” in the form of a “dialogue … [reflecting] the dialectic of the story [itself]: … a sort of rationally reconstructed or ‘distilled’ history.” The trajectory Lakatos considers starts off with an elaborate, polemical, and in places humorous discussion of how the V – E + F = 2 affair, originally due to Euler, gave rise over the years to a sequence of important and ultimately rather sophisticated geometrical generalizations, replete with controversy, give and take, and even a sort of Sturm und Drang. The tone of this discussion is exemplified by, for instance, the following interchange on p.86, pitting “Gamma” against “Delta”:
It turns out that the main player in the shadows here is Cauchy, and the discussion that started with what we would now call combinatorial topology begins to undergo a transition to analysis situs and, in point of fact, the book under review ends with nothing less than Carathéodory’s definition of a measurable set.
Proofs and Refutations is not everyone’s cup of tea: its discussion-cum-dialectic style is, shall we say, unusual. As a Platonist, I have serious objections to the dialectical position Lakatos appears to embrace already in his aforementioned introduction, suggesting that the history of mathematics can be described entirely in terms of the opposing poles of dogmatism and skepticism (in the author’s parlance). However, regarding Lakatos’ hugely important subtext, even Hilbert, the putative architect of formalism, arguably did not believe that mathematics could in truth be embedded in formal logic (and, in any case, Gödel settled that hash conclusively already in Hilbert’s lifetime) — and any one who reads Reid’s brilliant biography Hilbert cannot escape coming to this conclusion.
All this having been said, it is important and instructive to give Lakatos a proper hearing, even in this day, thirty-six years after the book’s original appearance. Additionally, as a compact study in history of mathematics proper, it is very much on target.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.
BLL** — The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.