Beth, Evert Willem (1908-1964)

Abstract

During a conference entitled “Un logicien consciencieux. La philosophie de Evert Willem Beth,” held in Nancy in 1998 and published in this journal, volume 3, cahier 4 [Barth, Heinzmann et al. 1998-1999], the participants regretted that a reasoned biography of Beth was not available1. In view of the efforts made since then in this direction, in particular by Paul van Ulsen in his PhD thesis in Dutch [2000], this article attempts to provide an overview of Beth’s work for a broader community of philosophers of science. However, my focus is not so much on Beth’s widely discussed and important contributions to mathematical logic, where his most creative merits reside. My aim is to emphasize Beth’s philosophical and cultural background. Both are also discussed in the first two chapters of Ulsen’s thesis, which globally focuses on Beth’s work in logic. Of course, philosophical and scientific aspects cannot be systematically separated, but Beth’s influence on the philosophy of mathematics between the 1950s and 1960s seems to me large enough to highlight it here. Beth is one of the very few philosophers who also presented highly technically relevant results in logic, or one of the few logicians who also wrote philosophical papers, not only in his youth or old age, but throughout his career. In his 741-page magnum opus The Foundations of Mathematics [1959], he combines the treatment of philosophical and logico-mathematical problems under the perspective that the traditional deductive procedure is an essential and not merely incidental element in mathematical proof so that “logic must play the principal role in a philosophical examination of mathematical thought” [1959, XI]. By 1937, Beth was “convinced that the only possibility of founding mathematics in a truly satisfactory way is to apply intuitive and formal methods simultaneously in such a way that they can control and verify each other” (“L’évidence intuitive dans les mathématiques modernes” [1937, 165]). In 1956, he still justified the emphasis on the intuitive element of proof by “certain complications that the development of these [formalist] tendencies would later give rise to” (“Poincaré et la philosophie” [1956b, 238, transl. G. H.]).