Sets and Maps as a Foundation for Mathematics

Abstract

According to Plutarch, the great philosopher Plato said: άεì ό θεòς γεωμετρει, God eternally geometrizes. The sentence was remembered in 19th-century Germany, and it was subject to changes that reflect the changing conceptions of mathematical rigor and pure mathematics. During the first half of the century, one of the greatest German mathematicians said, άεϬ ò θεòς άριθμητíζει, ‘God eternally arithmetizes;’³ geometry had lost its privileged foundational position to arithmetic. Gauss was of the opinion that, while space has an outside reality and we cannot prescribe its laws completely a priori, number is merely a product of our spirit or mind [Geist; Gauss 1863/1929, vol. 8, 201]. Dedekind essentially agreed, and his most important foundational work, Was sind und was sollen die Zahlen? [1888], bears the motto: άεì ó ???νθρωπος άριθμητíςει, ‘man always arithmetizes.’ It seems that, in Dedekind’s view, numbers are not made by God, but by men;¹ mathematics has nothing to do with a world of essences or a Platonic heaven, it is a free creation of the human mind [Dedekind 1888, 335, 360].