# What Role Plays Intuition in Mathematics and Science? On the Borders Between Several Conceptions of What It Means to Intuit

## Abstract

In discussions about Euclidean and non-Euclidean geometries, to take one central example, the notion of intuition plays a decisive role. According to Immanuel Kant, no geometry is possible without being based in a priori or pure intuition of space. The arrival of non-Euclidean geometries has generally been understood to falsify this claim, in particular according to Henri Poincaré’s introduction of conventionalism in the philosophy of mathematics. However, even on the surface, qualifications are introduced into the discussion, such as “the intuition of representative space” and “rational intuition”, indicating that the term “intuition” covers up serious ambiguities. Furthermore, the models of so-called non-Euclidean geometries, allegedly based in conventionally chosen sets of formal axioms, are directly or indirectly the very same objects claimed to be given to intuition in Euclidean space. The paper explores the nuances in the use of the term “intuition” in Kant and Poincaré, thereby discovering deep paralogisms in the discussion about geometry and hence resolving some of the problems pestering modern epistemology, by focusing on the border zones of conceptions covered by the same or similar terms.

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