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Kant defines the schema of reality as the continuous and uniform generation of a quantity ascribed to sensation through which it can more or less fill time. In this paper, I ask why Kant has to attribute uniformity to the schema of reality. Through an interpretation that takes the uniformity thesis as a crucial element in Kant’s formulation, I contend that, in contrast to prevailing scholarly literature, Kant’s schema of reality must be comprehended mathematically without relying on regulative principles of understanding like that of causality.

Kant’s Prize Essay of 1764 emphasizes the importance for mathematical cognition of manipulating signs according to rules, which has led some recent commentators to ask whether Kant’s position there is a species of mathematical formalism. While most have hesitated to find formalism in the Prize Essay, this hesitation derives from misconceptions about what formalists actually believe. I therefore examine some nineteenth century formalists who were in dialogue with Kant, using their views as a model against which to compare the Prize Essay. I argue that Kant’s view in the Prize Essay is continuous with their formalism, since the Prize Essay shares the essential aspects of their views about the sign-signified relation in mathematics and the role of signs in mathematical reasoning.

This paper revisits Kant’s 1768 incongruent counterpart argument that space is absolute. Most commentators today dismiss Kant’s argument as begging the question against the relationalist. I argue that this dismissal is too quick, and that we have something to learn by considering what might have led him to argue as he does. My focus is on the role of geometrical intuitions and the extent to which they can provide defeasible warrant for claims about space. By “geometrical intuitions” I mean both the geometrical intuitions of spatial figures in imagination as well as the claims that seem to directly follow from them. My focus includes uncovering the metaphysical assumptions that are implicated in those geometrical intuitions. Whether or not Kant’s argument ultimately succeeds, the role of geometrical intuitions is worth exploring. Moreover, Kant appeals to geometrical intuitions, and any adequate interpretation of Kant will have to take them into account.

For Kant, inquiry into nature requires seeking to explain all material wholes merely mechanically, in terms of their parts. There is no consensus on how he justifies this Principle of Mechanism. I argue that Kant seeks to derive this claim about part and wholes neither from his laws of mechanics, nor from the mere discursivity of our understanding (two standard options in the literature), but instead from a priori principles laid out in the first Critique , which govern parts, wholes, and magnitudes. These principles are also fundamental to Kant’s account of mathematics. Therefore, Kant’s Principle of Mechanism and his philosophy of mathematics have common foundations.

Kant repeatedly uses the biangle as an example of an impossible figure. In this paper, I offer an account of these passages and their significance for the possibility of geometry as a science. According to Kant, the constructibility of the biangle would signal the failure of geometry. Whereas Wolff derives the no-biangle proposition from the axiom that between two points there can be only one straight line, Kant gives it axiomatic status as a synthetic a priori principle possessing immediate certainty. Because we are unable to generate a schema for the biangle, the failure of the attempt to construct it is intuitively clear. The parallel between mathematical and empirical concepts is instructive because both involve the synthesis of disparate intuitions into a unity. We do not, strictly speaking, even possess a well-formed concept of the biangle, because its representation cannot fulfill certain basic requirements of concept formation.