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:
The subject matter of metamathematics is an abstraction of mathematics in which mathematical theories are replaced by formal systems, proofs by certain sequences of well-formed formulae, definitions by ‘abbreviatory devices’ which are ‘theoretically indispensable’ but ‘typographically dispensable.’ This abstraction was devised by Hilbert to provide a powerful technique for approaching some of the problems of the methodology of mathematics. At the same time there are problems which fall outside the range of metamathematical abstractions. Among these are all problems relating to informal … mathematics and to its growth, and all problems relating to the situational logic of mathematical problem-solving …
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:
Since informal mathematics is neither ‘tautological’ nor empirical, it must be meaningless, sheer nonsense. The dogmas of logical positivism have been detrimental to the history and philosophy of mathematics. The purpose of these [upcoming] essays is to approach some problems of the methodology of mathematics … in a sense akin to Pólya’s and Bernays’ ‘heuristic’ and Popper’s ‘logic of discovery’ or ‘situational logic.’
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”:
[G:] “I admit we were wrong in indicting Delta for surreptitious contractions of his concept of polyhedron: all his six definitions denoted the same good old concept of polyhedron he inherited from his forefathers …”
[D:] “Do you mean that all my definitions were logically equivalent?”
[G:] “That depends on your logical theory — according to mine they certainly are not.”
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.