This paper argues that kant's general epistemology incorporates a theory of algebra which entails a less constricted view of kant's philosophy of mathematics than is sometimes given. To extract a plausible theory of algebra from the "critique of pure reason", It is necessary to link kant's doctrine of mathematical construction to the idea of the "schematism". Mathematical construction can be understood to accommodate algebraic symbolism as well as the more familiar spatial configurations of geometry.

In theCritique of Pure Reason,Kant argues for two principles that concern magnitudes. The first is the principle that ‘All intuitions are extensive magnitudes,’ which appears in the Axioms of Intuition (B202); the second is the principle that ‘In all appearances the real, which is an object of sensation, has an intensive magnitude, that is, a degree,’ which appears in the Anticipations of Perception (B207). A circle drawn in geometry and the space occupied by an object such as a book are (...) paradigm examples of extensive magnitudes, while the intensity of a light is a paradigm example of an intensive magnitude. These principles justify and explain the possibility of applying mathematics to objects of experience. The Axioms principle also explains the possibility of any mathematical cognition at all. (shrink)

Prolegomena §38 is intended to elucidate the claim that the understanding legislates a priori laws to nature. Kant cites various laws of geometry as examples and discusses a derivation of the inverse-square law from such laws. I address 4 key interpretive questions about this cryptic text that have not yet received satisfying answers: How exactly are Kant's examples of laws supposed to elucidate the Legislation Thesis? What is Kant's view of the epistemic status of the inverse-square law and, relatedly, of (...) the legitimacy of the geometric derivation of that law? Whose account of laws, the understanding, and space is Kant critiquing in the passage? What positive account of the relationship between laws, the understanding, and space is Kant offering in the passage? My answer to depends crucially on my answers to –. As I interpret Kant, he holds that a wide range of a priori laws—including geometric laws, the inverse-square law, and the universal laws discussed in the Analytic of Principles—are ‘grounded’ in categorial syntheses rather than the intrinsic nature of the space given to us in pure intuition. (shrink)

In his Metaphysische Anfangsgründe der Naturwissenschaft, Kant claims that chemistry is a science, but not a proper science (like physics), because it does not adequately allow for the application of mathematics to its objects. This paper argues that the application of mathematics to a proper science is best thought of as depending upon a coordination between mathematically constructible concepts and those of the science. In physics, the proper science that exhausts the a priori knowledge of objects of the outer sense, (...) only motions and concepts reducible to motions can be legitimately coordinated with mathematical constructions. Since chemistry can neither achieve its own a priori principles of coordination nor be reduced to the coordinated doctrine of motion, it is a merely improper science. (shrink)

Consider the laws of nature—the laws of physics, for example. One familiar philosophical question about laws is this: what is it to be a law of nature? More specifically, is a law of nature a regularity, or a generalization stating a regularity? Or is it something else? Another philosophical question is: how, and to what extent, can we have knowledge of the laws of nature? I am interested here in Kant's answers to these questions, and their place within his broader (...) theoretical philosophy during the period spanning from the first to the third Critique. (shrink)

Kant's distinction between analytic and synthetic method is connected to the so-called Aristotelian model of science and has to be interpreted in a (broad) directional sense. With the distinction between analytic and synthetic judgments the critical Kant did introduced a new way of using the terms 'analytic'-'synthetic', but one that still lies in line with their directional sense. A careful comparison of the conceptions of the critical Kant with ideas of the precritical Kant as expressed in _Ãœber die Deutlichkeit, leads (...) to the interpretation defended here of Kant's distinction between analytic and synthetic judgments within his philosophy of mathematics. (shrink)

It's well-known that Kant believed that intuition was central to an account of mathematical knowledge. What that role is and how Kant argues for it are, however, still open to debate. There are, broadly speaking, two tendencies in interpreting Kant's account of intuition in mathematics, each emphasizing different aspects of Kant's general doctrine of intuition. On one view, most recently put forward by Michael Friedman, this central role for intuition is a direct result of the limitations of the syllogistic logic (...) available to Kant. On this view, Kant's reasons for introducing intuition are taken to be logical or mathematical, rather than philosophical. The other tendency, which I shall try to develop here, emphasizes an epistemological or phenomenological role for intuition in mathematics arising out of what may loosely be called Kant's ‘antiformalism.’This paper, which focuses specifically on the case of geometry, falls into two parts. First, I consider Kant's discussion of intuition in the Metaphysical Exposition of the concept of space. (shrink)

The interpretation of Kant's Critical philosophy as a version of traditional idealism has a long history. In spite of Kant's and his commentators’ various attempts to distinguish between traditional and transcendental idealism, his philosophy continues to be construed as committed to various features usually associated with the traditional idealist project. As a result, most often, the accusation is that his Critical philosophy makes too strong metaphysical and epistemological claims.In his The Revolutionary Kant, Graham Bird engages in a systematic and thorough (...) evaluation of the traditionalist interpretation, as part of perhaps the most comprehensive and compelling defence of a revolutionary reading of Kant's thought. In the third part of this special issue, the exchanges between, on the one hand, Graham Bird and, on the other, Gary Banham, Gordon Brittan, Manfred Kuehn, Adrian Moore and Kenneth Westphal focus on specific aspects of Bird's interpretation of Kant's first Critique. More exactly, the emphasis is on specific aspects of Bird's interpretation of the Introduction, Analytic of Principles and Transcendental Dialectic of Kant's first Critique.The second part of the special issue is devoted to discussions of particular topics in Bird's construal of the remaining significant parts of the first Critique, namely, of the Transcendental Aesthetic and the Analytic of Concepts. Written by Sorin Baiasu and Michelle Grier, these articles examine specific issues in these two remaining parts of the Critique, from the perspective of the debate between the traditionalist and revolutionary interpretation. The special issue begins with an Introduction by the guest co-editors. This provides a summary of the exchanges between Bird and his critics, with a particular focus on the debates stemming from the differences between traditional and revolutionary interpretations of Kant. (shrink)

This paper gives an interpretation of Kant's argument for transcendental idealism in the Transcendental Aesthetic. I argue against a common way of reading this argument, which sees Kant as arguing that substantive a priori claims about mind-independent reality would be unintelligible because we cannot explain the source of their justification. I argue that Kant's concern with how synthetic a priori propositions are possible is not a concern with the source of their justification, but with how they can have objects. I (...) argue that Kant's notion of intuition needs to be understood as a kind of representation which involves the presence to consciousness of the object it represents, and that this means that a priori intuition cannot present us with a mind-independent feature of reality. (shrink)