I examine here if Kant can explain our knowledge of duration by showing that time has metric structure. To do so, I spell out two possible solutions: time’s metric could be intrinsic or extrinsic. I argue that Kant’s resources are too weak to secure an intrinsic, transcendentally-based temporal metrics; but he can supply an extrinsic metric, based in a metaphysical fact about matter. I conclude that Transcendental Idealism is incomplete: it cannot account for the durative aspects of experience—or it can (...) do so only with help from a non-trivial metaphysics of material substance. (shrink)

Recent work on Kant’s conception of space has largely put to rest the view that Kant is hostile to actual infinity. Far from limiting our cognition to quantities that are finite or merely potentially infinite, Kant characterizes the ground of all spatial representation as an actually infinite magnitude. I advance this reevaluation a step further by arguing that Kant judges some actual infinities to be greater than others: he claims, for instance, that an infinity of miles is strictly smaller than (...) an infinity of earth-diameters. This inequality follows from Kant’s mereological conception of magnitudes (quanta): the part is (analytically) less than the whole, and an infinity of miles is equal to only a part of an infinity of earth-diameters. This inequality does not, however, imply that Kant’s infinities have transfinite and unequal sizes (quantitates). Because Kant’s conception of size (quantitas) is based on the Eudoxian theory of proportions, infinite magnitudes (quanta) cannot be assigned exact sizes. Infinite magnitudes are immeasurable, but some are greater than others. (shrink)

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. (shrink)

In his argument for the possibility of knowledge of spatial objects, in the Transcendental Deduction of the B-version of the Critique of Pure Reason, Kant makes a crucial distinction between space as “form of intuition” and space as “formal intuition.” The traditional interpretation regards the distinction between the two notions as reflecting a distinction between indeterminate space and determinations of space by the understanding, respectively. By contrast, a recent influential reading has argued that the two notions can be fused into (...) one and that space as such is first generated by the understanding through an act of synthesis of the imagination. Against this reading, this article argues that a key characteristic of space as a form of intuition is its nonconceptual unity, which defines the properties of space and is as such necessarily independent of determination by the understanding through the transcendental synthesis of the imagination. The conceptual unity that the understanding prescribes to the manifold in intuition, by means of the categories, defines the formal intuition. Furthermore, this article argues that it is the sui generis, nonconceptual unity of space, when takenas a unity for the understanding by means of conceptual determination, that first enables geometric knowledge and knowledge of spatially located particulars. (shrink)

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: The distinctive structure of outer sensible intuitions must be understood in terms of the concept of magnitude. The act of sensibly representing a magnitude involves a special act of spontaneity Kant ascribes to a capacity he (...) calls the productive imagination. 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. 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. (shrink)

The problem of ‘collective unity’ in the transcendental philosophies of Kant and Husserl is investigated on the basis of number’s exemplary ‘collective unity’. To this end, the investigation reconstructs the historical context of the conceptuality of the mathematics that informs Kant’s and Husserl’s accounts of manifold, intuition, and synthesis. On the basis of this reconstruction, the argument is advanced that the unity of number – not the unity of the ‘concept’ of number – is presupposed by each transcendental philosopher in (...) their accounts of the transcendental foundation of manifold, intuition, and synthesis. This presupposition is ultimately traced to Kant’s and Husserl’s responses to Hume’s philosophy of human understanding and the critical limits of what Kant calls the ‘qualitative’ unity of transcendental consciousness. These critical limits are exposed in both philosophers’ attempts to account for that ‘qualitative’ unity on the basis of the ‘quantitative’ unity of number. (shrink)

Kant's first antinomy uses a notion of infinity that is tied to the concept of (finitary) successive synthesis. It is commonly objected that (i) this notion is inadequate by modern mathematical standards, and that (ii) it is unable to establish the stark ontological assumption required for the thesis that an infinite series cannot exist. In this paper, I argue that Kant's notion of infinity is adequate for the set-up and the purpose of the antinomy. Regarding (i), I show that contrary (...) to appearance, the Critique can even accommodate a modern notion of infinity without consequences for the antinomy; and regarding (ii), that the ‘ontological’ consequence indeed follows once its covertly epistemic character is adequately understood. (shrink)

Kant’s theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant’s theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant’s theory of arithmetic can (...) be picked up, improved by reconstruction and defended under a contemporary perspective: the theory of numbers as products of rule following construction presupposing successive synthesis in time and the theory of arithmetic equations, sentences or “formulas”—as Kant says—as synthetic a priori. In order to do so, two calculi in terms of modern mathematics are introduced which formalise Kant’s theory of addition as a form of synthetic operation. (shrink)

Kant's theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant's theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant's theory of arithmetic can (...) be picked up, improved by reconstruction and defended under a contemporary perspective: the theory of numbers as products of rule following construction presupposing successive synthesis in time and the theory of arithmetic equations, sentences or "formulas"—as Kant says—as synthetic a priori. In order to do so, two calculi in terms of modern mathematics are introduced which formalise Kant's theory of addition as a form of synthetic operation. (shrink)

J. H. Lambert proved important results of what we now think of as non-Euclidean geometries, and gave examples of surfaces satisfying their theorems. I use his philosophical views to explain why he did not think the certainty of Euclidean geometry was threatened by the development of what we regard as alternatives to it. Lambert holds that theories other than Euclid's fall prey to skeptical doubt. So despite their satisfiability, for him these theories are not equal to Euclid's in justification. Contrary (...) to recent interpretations, then, Lambert does not conceive of mathematical justification as semantic. According to Lambert, Euclid overcomes doubt by means of postulates. Euclid's theory thus owes its justification not to the existence of the surfaces that satisfy it, but to the postulates according to which these "models" are constructed. To understand Lambert's view of postulates and the doubt they answer, I examine his criticism of Christian Wolff's views. I argue that Lambert's view reflects insight into traditional mathematical practice and has value as a foil for contemporary, model-theoretic, views of justification. (shrink)

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. (shrink)

Kant’s investigations into so‐called a priori judgments of pure mathematics in the Critique of Pure Reason (KrV) are mainly confined to geometry and arithmetic both of which are grounded on our pure forms of intuition, space, and time. Nevertheless, as regards notions such as irrational numbers and continuous magnitudes, such a restricted account is crucially problematic. I argue that algebra can play a transcendental role with respect to the two pure intuitive sciences, arithmetic and geometry, as the condition of their (...) possibility. It follows that Kant’s schematism of the concept of magnitude ought to be quantitatively represented in algebraic formulas in general, and also undergo several modifications in order to suit continuities. (shrink)

Extended critical discussion of Stefanie Grüne's *Blinde Anschauung*.