Geometry, construction, and intuition in Kant and his successors (Q2754657)

In three sections, the author discusses several aspects of the role of intuition in \textit{Kant}'s theory of geometry. In the first section he presents the logical approach to Kantian intuition first articulated by W. E. \textit{Beth} and J. \textit{Hintikka}, according to which intuition ``serves to generate singular terms in the context of mathematical reasoning in inferences such as we would represent today existential instantiations'' (p. 186). He particularly defends this logical approach by focusing on the role of intuition in geometrical constructions in proof procedures employed in \textit{Euclid}'s ``Elements'', and illustrates it with the help of arguments posed by Kant in a dispute with \textit{Eberhard} in 1790.NEWLINENEWLINENEWLINEThe second section deals with Kant's distinction between geometrical constructions with the help of ruler and compass, mechanical constructions using more complicated means, and construction in intuition. Furthermore the relation between geometrical construction and motion in space (i.e., translation and rotation) is discussed.NEWLINENEWLINENEWLINEThe author sees ``the relevant formal structure of intuitive or perceptual space as fundamentally kinematical'' (p. 199). In this interpretation Kant's conception of spatial intuition moves nearer to the one of some of his critical followers. The author particularly discusses H. v. \textit{Helmholtz}'s program to found geometry on the formal structure of perceptual space, H. \textit{Poincaré}'s conventionalist interpretation of geometry, and H. \textit{Weyl}'s group theoretical solution of the generalized ``space-problem''.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00010].