`As if' reasoning in Vaihinger and Pasch
This paper is about the interplay between some of the ideas of the mathematician Moritz Pasch (1843--1930) and the philosopher Hans Vaihinger (1852--1933). Pasch made fundamental contributions to the axiomatic foundations of geometry and arithmetic. (In particular, his introduction of the concept of the halfplanes determined by a line in a plane led to the first complete axiomatization of planar Euclidean geometry.) Vaihinger's `as if' philosophy emphasized the importance of fictional ideas in philosophy and mathematics. He also believed that the fundamental concepts of mathematics are logically incoherent. The author indicates that Pasch showed that Vaihinger's examples were wrong, but that Pasch partially excused Vaihinger's mistakes on the basis of the fact that the usual expositions (for example, of the techniques in the calculus for finding maxima and minima) are not clearly explained. Pasch believed in an empiricist approach to mathematics, viewing geometry, for example, as `a branch of natural science'. In his later years, he devised a system K of `combinatorial principles' that yielded Peano arithmetic and whose consistency he believed cannot be doubted. Moreover, his empiricist views were also alleged to show how to understand the applicability of mathematics and to advance mathematical exposition and teaching. Even more surprising is the fact that Pasch, almost as if he was following Vaihinger's ideas, invented intuitive pictures, so-called just so stories, to help motivate and explain the application of the mathematical theories. It might have been helpful to the reader if examples of those stories as well as axioms of Pasch's theory K had been given. At the end of the paper, the author insists that Pasch did not subscribe to a structuralist view of mathematics and suggests that things like the just so stories provide something essential to our understanding of mathematics.
scientific article; zbMATH DE number 5797478