Analysis Situs, the Foundations of Mathematics, and a Geometry of Space

This chapter focuses on Gottfried Wilhelm Leibniz’s writings on analysis situs—a collection of mathematical and philosophical investigations into the foundations, development, and formalization of geometry, examining how they contributed to the development of modern geometry. It first explains the concept of situation (situs) as the first source of Leibniz’s theory of a relational space and its use in metaphysics, then considers Leibniz’s development of a new formalism for geometry (a characteristica geometrica propria) and his use of (Euclidean) metric relations between figures or points as the main subject of his analysis situs. It also discusses Leibniz’s definition of space as an order of situations, his grand ambition to establish the principles of Euclidean geometry based on the abstract concept of a metric space, and how his writings on analysis situs became intertwined with metaphysics and the theory of knowledge. Finally, the chapter reviews Leibniz’s definition of space in relation to monads and bodies.