Review of "Proofs and refutations: the logic of mathematical discovery" by Imre Lakatos. Cambridge University Press 1976.

In computer science in general--but particularly in theoretical computer science--there are few aspects of mathematics that are more visible than formalism (e.g., consider that we have entire subdisciplines devoted to formal languages, abstract machines, and axiomatic definitions). In this entertaining and oftentimes beautiful book the author takes a deep look at what he calls the dogmatist view of mathematics (comprising logicism, formalism, and related philosophies) and finds it an inaccurate view of mathematics as it is learned and created. In Lakatos' own words, his "...aim is to elaborate the point that...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." Insofar as it contains contrary views, we think that a treatise on the philosophy of mathematics is of interest to this readership. But be warned that Lakatos is seldom subtle: there is something here to ruffle just about everyones' feathers. For those of us who are usually repelled by the puffery of philosophical writing, this book is a very comfortable one. First, the author is mathematically literate and he is as much influenced by mathematicians as he is by philosophers; so, many of his arguments have less the stilted flavor of syllogistic than a sort of free-flowing mathematical argument. Second, the style of presentation is perfectly matched to the subject: using the device of a dialogue between a teacher and his pupils, Lakatos provides a history of the development of an acceptable proof for Euler's formula V-E+F=2 (relating the vertices edges and faces of polyhedra)° There are copious footnotes which provide the historical, philosophical and mathematical commentary underlyin~ the dialogues. This complete and fascinating analyis of Euler's "testing" of the formula and the subsequent less-thansatisfactory attempts at proof by such luminaries as Cauchy forms the backdrop for the exposition of Lakatos' empirical approach to mathematical discovery and proof: counterexamples are examined to Field ideas for new proofs; proofs are disassembled and refuted to yield ideas for new counterexampies. Throughout, the technical footnotes help to support the Lakatosian view: