This paper examines the role of Kant’s theory of mathematical cognition in his phoronomy, his pure doctrine of motion. I argue that Kant’s account of how we can construct the composition of motion rests on the construction of extended intervals of space and time, and the representation of the identity of the part–whole relations the construction of these intervals allow. Furthermore, the construction of instantaneous velocities and their composition also rests on the representation of extended intervals of space and time, reflecting the general approach to instantaneous velocity in the eighteenth century.

The aim of this paper is to show that attention to Kant’s philosophy of mathematics sheds light on the doctrine that there are two stems of the cognitive capacity, which are distinct, but equally necessary for cognition. Specifically, I argue for the following four claims: (i) The distinctive structure of outer sensible intuitions must be understood in terms of the concept of magnitude. (ii) The act of sensibly representing a magnitude involves a special act of spontaneity Kant ascribes to a capacity he calls the productive imagination. (iii) Contrary to what is assumed by many commentators, it is not the case that the Two Stems Doctrine implies that a representation is either sensible or spontaneity-dependent, but not both. (iv) Outer sensible intuitions are both sensible and spontaneity-dependent – they are sensible because they exhibit the kind of structure Kant takes to be distinctive of outer sensible intuitions, and they depend on spontaneity because they are cases of sensibly representing a magnitude.

I advance a novel interpretation of Kant’s argument that our original representation of space must be intuitive, according to which the intuitive status of spatial representation is secured by its infinitary structure. I defend a conception of intuitive representation as what must be given to the mind in order to be thought at all. Discursive representation, as modelled on the specific division of a highest genus into species, cannot account for infinite complexity. Because we represent space as infinitely complex, the spatial manifold cannot be generated discursively and must therefore be given to the mind, i.e. represented in intuition.

It is often maintained that one insight of Kant’s Critical philosophy is its recognition of the need to distinguish accounts of knowledge acquisition from knowledge justification. In particular, it is claimed that Kant held that the detailing of a concept’s acquisition conditions is insufficient to determine its legitimacy. I argue that this is not the case at least with regard to geometrical concepts. Considered in the light of his pre-Critical writings on the mathematical method, construction in the Critique can be seen to be a form of concept acquisition, one that is related to the modal phenomenology of geometrical judgement.

This paper gives a contextualized reading of Kant’s theory of real definitions in geometry. Though Leibniz, Wolff, Lambert and Kant all believe that definitions in geometry must be ‘real’, they disagree about what a real definition is. These disagreements are made vivid by looking at two of Euclid’s definitions. I argue that Kant accepted Euclid’s definition of circle and rejected his definition of parallel lines because his conception of mathematics placed uniquely stringent requirements on real definitions in geometry. Leibniz, Wolff and Lambert thus accept definitions that Kant rejects because they assign weaker roles to real definitions.

The consensus view in the literature is that, according to Kant, definitions in philosophy are impossible. While this is true prior to the advent of transcendental philosophy, I argue that with Kant’s Copernican Turn definitions of some philosophical concepts, the categories become possible. Along the way I discuss issues like why Kant introduces the ‘Analytic of Concepts’ as an analysis of the understanding, how this faculty, as the faculty for judging, provides the principle for the complete exhibition of the categories, how the pure categories relate to the schematized categories, and how the latter can be used on empirical objects.

In his 1763 Prize Essay, Kant is thought to endorse a version of formalism on which mathematical concepts need not apply to extramental objects. Against this reading, I argue that the Prize Essay has sufficient resources to explain how the objective reference of mathematical concepts is secured. This account of mathematical concepts’ objective reference employs material from Wolffian philosophy. On my reading, Kant’s 1763 view still falls short of his Critical view in that it does not explain the universal, unconditional applicability of mathematical concepts.

This paper shows that Kant’s investigation into mathematical purposiveness was central to the development of his understanding of synthetic a priori knowledge. Specifically, it provides a clear historical explanation as to why Kant points to mathematics as an exemplary case of the synthetic a priori, argues that his early analysis of mathematical purposiveness provides a clue to the metaphysical context and motives from which his understanding of synthetic a-priori knowledge emerged, and provides an analysis of the underlying structure of mathematical purposiveness itself, which can be described as unintentional, but also as objective and unlimited.

This paper tries to make sense of Kant’s scattered remarks about conic sections to see what light they shed on his philosophy of mathematics. It proceeds by confronting his remarks with the source that seems to have informed his thinking about conic sections: the Conica of Apollonius. The paper raises questions about Kant’s attitude towards mathematics and the way he understood the cognitive resources available to us to do mathematics.