EVERT WILLEM BETH'S SCIENTIFIC PHILOSOPHY Miriam FRANCHELLA Universita degli Studi, Milano 1. Introduction The Dutch man Evert Willem Beth (1908-1964) is famous for his contributions to logic. It is also known that he was a philosopher and became interested in logic quite late in his life. Still, the aspects of his philosophical reflections and the details of its development are almost unknown. He himself gave some hints for ordering his pro­ duction. 1 He divided it into four periods: the neo-kantian, the antikantian, the antiirrationalist, the logical one. Within this frame­ work, it is possible to individuate a core around which Beth devel­ oped his reflections: it is the interplay betwe~n philosophy and the sciences. In effect, his philosophical reflections were always linked to the sciences in two ways: firstly, because he steadily checked his philosophical ideas against the current situation in foundational re­ search; second, because he occupied himself with the general ques­ tion of the relationships between philosophy and the sciences. We are going to analyse these aspects in detail. 2. The role of intuition: thinking ofKant At the very beginning of his work Beth was convinced that intuitionism was the only acceptable foundational school. He con­ sidered logicism as "out" at that period. The other two positions - formalism and intuitionism - appealed both to intuition: 1. "Een terugblik" ("A retrospective view"), De Gids 123 (1960), 320-330; also in E.W. Beth, Door wetenshap tot wijsheid (Assen, van Gorcum 1964), 112-113. 222 intuitionism directly, in the construction of mathematics, while for­ malism only in the metamathematical analysis. Beth observed on the one side that intuitionism was more direct in its development, and on the other side that these two foundational proposals led to state that mathematical judgements are synthetical, in the sense that they cannot be formed without a previous process of "verification" in (mental) reality. They represented for him a way to defend Kant, whose philosophy was the only reference point for him at that time at the university: in his rememberings2, he talks of "exclusive Mar­ burg orientation", which came to him in particular from his supervi­ sor l.C. Franken. It has been just his neokantian education that had caused to him this evaluation of scientific results which he would never abandon. Within this context, in 1935 he presented his disser­ tation "Rede en Anschouwing" ("Reason and Intuition") at the Fac­ ulty of Humanities of the Utrecht University. The subject had been offered by a question prize, which the Faculty had proposed: whether the necessity of space as a priori form of intuition no longer holds due to the possibility of founding geometry on a merely logi­ cal basis. The reference was Couturat's criticism of the Kantian the­ ory of spacial 'intuition in name oflogicism. Beth began by referring both Kant's theory (in a standard way) and Couturat's criticism of it. Then he proved that there is no substantial difference between ge­ ometry and the rest of mathematics. His reasoning was the follow­ ing. He tried a logical analysis of geometry, that is, he tried to find what charachterised the difference of geometry with respect to other parts of mathematics. He found no logical reasons and grasped the fact that other grounds produced the label "geometry" for a certain part of mathematics. Firstly, there were historical reasons: all sub­ jects related to Euclide's Elements have been called "geometry". Second, the use of geometrical methods - and relative terms - in other parts of mathematics in order to simplify their treatment led them to be labelled as geometry. Third, Klein introduced a system­ atic reason: geometry is the science of the properties of bodies that remain unchanged after displacement and reflection. Finally, Beth quoted esthetical reasons: this means on one hand (with a positive value) that geometry uses elegant ways with nice pictures; on the 2. Ibid. .