Space and the Continuum from Kant to Poincaré
Anna Franchini

Abstract:
The present thesis explores Kant’s transcendental philosophy, focusing on the cognitive processes that bring to the formation of the concept of space. In light of the interpretation proposed by Pinosio and van Lambalgen, we provide an in-depth analysis of the fundamental passages of the Critique of Pure Reason expounding the synthetic activity that produces the consciousness of space as a formal intuition. We investigate Kant’s constructive continuum and we compare it to the Aristotelian continuum, emphasising their similarities, in particular with respect to the notion of contact between regions. We then compare Kant’s perspective to Poincaré’s philosophy of space, suggesting some possible interactions between the two perspectives. We propose a parallel between the work of the figurative synthesis in Kant and the gradual process of abstraction that leads, in Poincaré’s philosophy, to the formation of a mathematical continuum from a physical continuum. Finally, we produce a formal model for Kant’s spatial continuum, adopting a mereological approach. Starting from a set of finite structures (Boolean algebras and their dual Stone spaces), which represent the spatial extent of possible experiences, we build a direct system and an inverse system. The limit of the inverse system, together with a relation of proximity, is the formal correlate to space as the formal intuition. The proximity relation (dual to a contact relation on Boolean algebras), is the key to obtain a continuum of points that are the emerging boundaries of regions.