Russell and the development of mathematics

Abstract

Russell and the development of mathematics by Gregory H. Moore George Temple. 100 Years ofMathematics. London: Duckworth, 1981. Pp. xvi, 316. £32.00. AT THE RISK of antagonizing certain historians of mathematics, one might categorically assert that the most difficult period about which to write the history of mathematics is that of the last hundred years. This difficulty is due both to the extreme technical complexity ofthe historical material and to its enormous quantity. While a historian of Greek mathematics must make the most offragments and a handful ofsurviving documents, the modern historian is confronted with a plethora of material , which, regrettably, often has gaps at the most inconvenient places and whose interconnections are frequently obscure at best. Thus the reviewer regards with considerable sympathy the task that George Temple has set himself. Temple, a former Sedleian Professor of Mathematics at Oxford, is best known for his contributions to that part of mathematics known as analysis, particularly generalized functions, and to mathematical physics. It is as a mathematician rather than as a historian that he describes his purpose in the work under review: This book has no pretensions to give an encyclopaedic history of the mathematics of [the last hundred years] .... Its object is to present the work of some of the mathematicians who have carried out this transformation of mathematics, to describe their ideals, their concepts, their methods and their achievements.... This book has been written therefore to appeal to those who desire a broad survey of the main currents of mathematical thought. (P. I) To carry out this programme, Temple divides his book, as Caesar divided Gaul, into three parts: number, space, and analysis. The topic of number receives four chapters-devoted respectively to real numbers, infinitesimals, transfinite numbers, and (somewhat anomalously, considering the previous chapter) finite and infinite numbers-while the topic of space is subdivided into chapters on vectors, measurement, algebraic geometry, and topology. Analysis, to which more than half of the book is devoted, is allotted seven chapters-on functions, the notions 89 90 Russell summer 1985 of derivative and integral, distributions, differential equations, the calculus of variations, potential theory, and mathematical logic. Russell's work is discussed at sporadic intervals, but it is pleasing to find mention of his logicism, his criticisms of definitions of real number, his work on the foundations of geometry, and his theory of types. In the introduction Temple acknowledges two ofhis predecessors, the book Elements d'histoire des mathematiques by Nicolas Bourbakil and the essay "A Half-Century of Mathematics" by Hermann Weyl,2 and says about them: "These essays set a high standard of exposition, but they do not pretend to cover ... all the main divisions of the subject. There is room for a history of mathematical ideas which will demand less mathematical expertise and offer a more detailed account of the motivation ofresearch" (p. I). Nevertheless, Temple's account demands almost as much mathematical expertise as Bourbaki's (and, indeed, much more than that of Weyl) but fails to provide a better account ofwhat motivated research. Temple has a tendency to conflate Russell's genuine accomplishments with his marginal contributions. In particular, when describing the construction of the real numbers put forward by the nineteenth-century mathematicians Weierstrass, Cantor, and Dedekind, Temple writes: "The task of submitting these theories to a minute logical analysis has been carried out with devastating results by Russell [in The Principles of Mathematics] and the best service which an historian can now render is to describe the fundamental principles ofanalysis which a benign interpretation can discover in the wrecks of these theories" (p. 13). On the contrary, Cantor's construction (using Cauchy sequences of rational numbers) and Dedekind's construction (using Dedekind cuts in the rationals) remain the standard constructions of the real numbers. The later mathematical improvements in these two constructions are of a secondary nature. It is in the chapter on mathematical logic (which Temple places idiosyncratically in the part ofthe book devoted to analysis) that Russell's work is discussed most fully. Here one finds a brieftreatment ofRussell's paradox, logicism, the theory of types, and Russell's critique of the Peano postulates for the natural numbers. From a...