The direction-theory of parallels: geometry and philosophy in the age of Kant (Q6623920)

The author tells the story of the direction theory of parallels which gained popularity between the 1770s and 1870s as a tool for attempts to prove Euclid's parallel postulate. The author emphasises that the debates were intertwined with considerations of mathematical epistemology.\par Euclid had defined parallel lines in the plane as straight lines that do not meet. In the direction theory parallel lines are defined as straight lines that have the same direction. Sameness of direction is a transitive relation, a feature that does not exist in non-Euclidean geometries.\N\NThe author focuses on the early period beginning with the first, but not influential formulation by N.\ Kauffmann (Mercator), in his 1678 reworking of Euclid's \textit{Elements}. Several authors were influenced by G.W. Leibniz's ideas about an \textit{analysis situs}, e.g., W.J.K.\ Karsten and J.A.\ Segner, when they introduced direction theories in their theories of parallel lines in the last decades of the 18th century. C.T.\ Hindenburg was also influenced by Leibniz \textit{analysis situs}. According to Hindenburg, such a theory should necessarily contain a theory of parallels, but it ``had to be freed from the constraints of Leibniz's epistemology and developed through constructive, ruler-and-compass constructions'' (p.\ 523). Hindenburg's \textit{Neues System der Parallellinien} was published in 1781, the same year as the first edition of I.\ Kant's \textit{Critique of Pure Reason}. ``In the following 10 years, the philosophical landscape of Germany was completely transformed, and the debate on the analyticity and the syntheticity of mathematics came the forefront'' (p.\ 524). These topics influenced the debates on the direction theory between the Kantians (e.g., J.F.\ Schultz) and their adversaries (e.g., J.C.\ Schwab).\N\NThe direction theory became the mainstream in post-Kantian philosophy in the early 19th century. The author mentions, among others, statements by K.C.\ Langsdorf, J.G.\ Fichte, J.F.\ Fries, and G.W.F.\ Hegel. It is surprising that the discovery of non-Euclidean geometries did not damage the amenity of this theory and the attempts to prove the parallel postulate. The incompatibility of the theory with non-Euclidean geometry was not even an issue in the seminal refutations by C.\ Dodgson (1879) and G. Frege (1884).\N\NFor the entire collection see [Zbl 1537.01004].