Kant vs. Legendre on Symmetry: Mirror Images in Philosophy and Mathematics

In 1768, Kant published a short essay in which he inquired into the possibility of determining the directionality of space. Kant's central argument invokes the strategy that if one were to demonstrate directionality, then the relational view of space that Leibniz propounded would be refuted. This paper has been considered a major turning point in Kant's philosophical development towards his critical philosophy of transcendental idealism. I demonstrate that in this study, Kant came very close to the modern concept of symmetry. His novel construction of incongruent counterpart (inkongruentes Gegenstück) contains elements essential to the modern notion of symmetry. However, Kant does not consider the incongruent counterparts, which he designates as ‘Right’ and ‘Left’, symmetric; rather, he holds the French encyclopaedist view that symmetry is a kind of balance. This study convinced Kant that the solution to the problem of the nature of space lies not in mathematics but in metaphysics. He was wrong in this conclusion, at least with respect to symmetry. The solution was found within the framework of mathematics, that is, solid geometry. In 1794, Legendre recast the traditional encyclopaedist concept of symmetry by calling a certain property of polyhedra symmetrical. The view of Kant is contrasted with that of Legendre by comparing their usages of mirror image as an aid for understanding. While in both cases mirror images are not considered illusions—perhaps for the first time in the history of mirror reflections—the differences are substantial, highlighting the limitation of Kant's position and the great potential of Legendre's new concept of symmetry.