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.

Despite the importance of Kant's claims about mathematical cognition for his philosophy as a whole and for subsequent philosophy of mathematics, there is still no consensus on his philosophy of arithmetic, and in particular the role he assigns intuition in it. This inquiry sets aside the role of intuition for the nonce to investigate Kant's conception of natural number. Although Kant himself doesn't distinguish between a cardinal and an ordinal conception of number, some of the properties Kant attributes to number can be characterized as cardinal or ordinal. This essay argues that Kant's conception of number includes both cardinal and ordinal elements; it suggests that the cardinal elements provide the basis of a conception of number in general, while the ordinal elements contribute to specifying the exact size of particular collections. In considering these elements, roles for intuition begin to emerge, setting the stage for a reevaluation of the role of intuition in Kant's arithmetic.

Reviewed by: Kant’s Construction of Nature: A Reading of the by Michael Friedman Daniel Sutherland Michael Friedman. Kant’s Construction of Nature: A Reading of the Metaphysical Foundations of Natural Science. Cambridge-New York: Cambridge University Press, 2013. Pp. xix + 624. Cloth, $111.95. Kant’s Construction of Nature (KCN) marks a major milestone in Kant scholarship. Friedman draws on over thirty years of research to make Kant’s deep engagement with Newtonian natural science accessible, and shows its relevance to Kant’s critical philosophy, especially the Critique of Pure Reason (CPR). Friedman has benefited from other excellent scholarship on Kant’s Metaphysical Foundations (MF), but no other scholar combines such detailed textual analysis and understanding of the issues with a systematic presentation of the whole and its connection to Kant’s critical philosophy. The MF is an extreme distillation of Kant’s views on a complex set of topics, but Friedman writes exceptionally clearly, and includes many cross-references that make it easy to review an issue when it resurfaces. The connection between the MF and the four sets of synthetic a priori principles receive extended treatment, and introductory and concluding chapters focus helpfully on the role of the MF in Kant’s critical philosophy. This makes KCN particularly valuable to anyone working on Kant’s critical metaphysics and epistemology. Friedman clarifies how Kant simultaneously pursues multiple themes in the MF, and how he reforms both Leibnizian metaphysics and Newton’s fundamental concepts and principles, for example, how Kant rejects Newton’s conception of matter in favor of a dynamical theory and redefines substance in terms of permanence rather than unity. In addition, Kant reconstructs Newtonian physics without presupposing absolute space and time; Friedman explains Kant’s strategy to exploit Newton’s procedure for successively determining a privileged frame of reference to reconceptualize absolute space and time as the never fully attained outcome of this procedure. Friedman also emphasizes that Kant sought to explain the possibility of Newtonian mathematical physics and hence how the properties of matter “acquire their mathematical structure” (103–4). This includes an account of how Kant’s mechanical laws of motion make it possible to measure the quantity of matter. Throughout, Friedman reveals Kant’s depth of understanding of Newtonian physics. In a particularly nice example, he explains why Kant appeals to Newton’s arguments concerning the oblate shape of the earth: in order to argue for the crucial feature of matter that every part acts on every other part. Friedman’s interpretation of the MF includes a subtle understanding of Kant’s project and how he carries it out. In Friedman’s view, Kant did not simply attempt to provide an a priori deduction of Newton’s laws of motion or his dynamical conception of matter by applying the categories and principles of the CPR to the empirical concept of matter. The MF is instead a broader exploration of the conditions for the possibility of Newtonian [End Page 173] mathematical physics that also relies on empirical data. Thus, for Friedman, the laws of motion turn out to be a priori in the sense that they are conditions for mathematizing quantity of matter. Friedman also argues that Kant does not attempt an a priori construction of the concept of matter itself; instead, Kant’s “construction” is a demonstration of how the various properties of matter acquire their mathematical structure. Friedman provides an interpretation of the relationship between the MF and the CPR: the former is not required to complete the deduction of the latter, but provides the first and most important instantiation and only full realization of the latter’s concepts and principles in the phenomenal world. While the CPR provides a perspective on the phenomenal world grounded in the transcendental unity of apperception, the MF provides a more specific perspective on the phenomenal world as mathematically realized in pure natural science. A strength of Friedman’s work is that it brings so much to bear on puzzling passages and makes sense of the whole. This sometimes leads Friedman to depart from the most natural reading of a passage in favor of one that makes more overall sense. Only a philosopher and historian of...

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 colorful and controversial life of American artist James Abbott McNeill Whistler took many twists and turns. One of the most dramatic and least explicable of the turns came in 1866, when he traveled to Valparaiso, Chile. Generations of Whistler biographers have been puzzled by the reasons for this sojourn, but a recently discovered collection of documents provides significant clues for solving the mystery. Whistler, it seems, had become involved in a scheme to provide weapons to the Chilean navy in a war against Spain. The scheme turned into a fiasco, dashing Whistler's hopes of handsome financial profits, but his weeks in South America significantly shaped his development as an artist.

There is evidence in Kant of the idea that concepts of particular numbers, such as the number 5, are derived from the representation of units, and in particular pure units, that is, units that are qualitatively indistinguishable. Frege, in contrast, rejects any attempt to derive concepts of number from the representation of units. In the Foundations of Arithmetic, he softens up his reader for his groundbreaking and unintuitive analysis of number by attacking alternative views, and he devotes the majority of this attack to the units view, with particular attention to pure units. Since Frege, the units view has been all but abandoned. Nevertheless, the idea that concepts of number are derived from the representation of units has a long history, beginning with the ancient Greeks, and was prevalent among Frege's contemporaries. I am not interested in resurrecting the units view or in righting wrongs in Frege's criticisms of his contemporaries. Rather, I am interested in the program of deriving concepts of number from pure units and its history from Kant to Frege. An examination of that history helps us understand the units view in a way that Frege's criticisms do not, and in the process uncovers important features of both Kant's and Frege's views. I will argue that, although they had deep differences, Kant and Frege share assumptions about what such a view would require and about the limits of conceptual representation. I will also argue that they would have rejected the accounts given by some of Frege's contemporaries for the same reasons. Despite these agreements, however, there is evidence that Kant thinks that space and time play a role in overcoming the limitations of conceptual representation, while Frege argues that they do not.

Executive Summary In December of 2007, the Department of Homeland Security (DHS) Privacy Office convened a two-day public workshop to examine best practices for government use of camera technology, commonly referred to as closed circuit television (CCTV). Titled CCTV: Developing Privacy Best Practices, the Workshop examined how technology, local and international communities, law enforcement, government agencies, and privacy advocates are shaping the use of CCTV and what safeguards should be in place as the use of CCTV expands. The Workshop served as a valuable resource to the Privacy Office in its joint effort with the DHS Office for Civil Rights and Civil Liberties to develop an informational guide to best practices for government use of CCTV. The best practices guide, along with sample templates for Privacy Impact Assessments (PIAs) customized for CCTV and the DHS template for Civil Liberties Impact Assessments, are included in the Appendix to this Workshop report. The DHS Privacy Office and Office for Civil Rights and Civil Liberties hope that government agencies will consider these resource materials in developing their CCTV programs and policies. These resources may be useful in helping government agencies build privacy and civil liberties protections into the design and implementation of a CCTV program. Failure to address privacy and civil liberties can undermine public support for the use of CCTV and erode confidence in government's ability to protect privacy and civil liberties while protecting the Homeland. Government agencies can avoid project delays and gather public support by ensuring that the appropriate safeguards and policies are in place before launching CCTV systems. The Workshop brought together leading academics, international government officials, researchers, law enforcement representatives, technologists, community leaders, and policy experts. These panelists identified a range of challenges facing local governments, communities, and law enforcement regarding privacy and use of CCTV. The key topics discussed at the Workshop included: • CCTV technology and its impact on privacy; • International perspectives on the use of CCTV; • Law enforcement use of CCTV; • Community perspectives on use of CCTV; • Legal and policy considerations regarding the use of CCTV; and • Best practices for the implementation and use of CCTV. The panel on Technology Perspectives opened the Workshop by providing a basic understanding of the current CCTV technologies in use. This discussion led into a more in-depth discussion on the capabilities of the technology and video analytics being used today. The panel then discussed …

Kant on Arithmetic, Algebra, and the Theory of Proportions Daniel Sutherland Kant's philosophy of mathematics has both enthralled and exercised philosophers since the appearance of the Critique of Pure Reason. Neither the Critique nor any other work provides a sustained and focused account of his mature views on mathematical cognition, forcing readers to glean what they can from disparate contexts. Despite these hurdles, Kant's views have been of great interest to philosophers of mathematics. They have also been of interest to philosophers wishing to understand Kant's philosophy more generally, since Kant maintains that mathematical judgments are synthetic a priori and that the type of synthesis underlying mathematics is the same as that underlying the perception of objects (CPR, B 202–03, A 163/B 204). 1 Our understanding of Kant's views has been greatly improved during the last four decades by work on Kant's philosophy of mathematics. 2 [End Page 533] Despite these advances, however, I think the recent work has largely neglected the importance of Kant's theory of magnitudes and the extent to which that theory is influenced by the Eudoxian theory of proportions presented in Euclid's Elements. As a consequence, an important feature of Kant's philosophy of mathematics has been overlooked: the role of intuition in the representation of magnitudes. 3 Eudoxus developed the theory of proportions to provide a mathematical treatment of continuous magnitudes, in particular spatial magnitudes, and there is strong evidence for this interpretation with respect to Kant's geometry. It is unclear, however, what bearing this account has on discrete magnitudes, and hence arithmetic. In general, Kant's claim that intuition is required for mathematical cognition has seemed less comprehensible and plausible in the case of arithmetic than in the case of geometry. This paper will attempt to make it more comprehensible and plausible by focusing on the place of arithmetic in Kant's theory of magnitudes. Kant's views on arithmetic and discrete magnitudes can be best clarified by considering two closely related issues. First, does Kant think that there is a universal mathematics common to all mathematical disciplines, and if so, how does it relate to arithmetic? Second, what is the role of intuition in our cognition of arithmetical propositions? I will argue that Kant thought of algebra as an overarching universal mathematics that includes arithmetic, that it takes magnitudes as its object of study, and that algebra expresses the Eudoxian theory of proportions. 4 Furthermore, on Kant's view, intuition plays a crucial role in arithmetic through representing discrete magnitudes. My investigation will require discussion of the Greek mathematical tradition and an examination of the development of algebra in the early modern period, since early modern conceptions of arithmetic were strongly influenced by both. In the first part of the paper, I examine Kant's arithmetic in relation to the theory of proportions. Section 1 sets the stage with a brief introduction to the Greek conception [End Page 534] of number and the Eudoxian theory of proportions. Section 2 describes Kant's project of explaining mathematical cognition by explaining the cognitions presupposed by the theory of proportions. It suggests a natural way of assimilating arithmetic and number into his account of geometrical cognition, but raises three problems with this assimilation. These problems motivate a closer examination of the relationship between arithmetic and algebra, which I investigate in the second part of the paper. Sections 3 and 4 provide needed background on the relation between arithmetic and algebra in the early modern period and in Kant. They argue that Kant thought of algebra as a universal mathematics and that algebra expresses the Eudoxian theory of proportions. With this important background in place, Section 5 argues for a different understanding of Kant's conception of arithmetic and number from that suggested in Section 2. 1. The Greek Conception of Number and the Theory of Proportions The way in which numbers were understood by Greek mathematicians...

Kant's Critique of Pure Reason makes important claims about space, time and mathematics in both the Transcendental Aesthetic and the Axioms of Intuition, claims that appear to overlap in some ways and contradict in others. Various interpretations have been offered to resolve these tensions; I argue for an interpretation that accords the Axioms of Intuition a more important role in explaining mathematical cognition than it is usually given. Appreciation for this larger role reveals that magnitudes are central to Kant's philosophy of mathematics and to the part that intuition plays in it.

Equality, similarity and congruence are essential elements of Kant's theory of geometrical cognition; nevertheless, Kant's account of them is not well understood. This paper provides historical context for treatments of these geometrical relations, presents Kant's views on their mathematical definitions, and explains Kant's theory of their cognition. It also places Kant's theory within the larger context of his understanding of the quality-quantity distinction. Most importantly, it argues that the relation of equality, in conjunction with the categories of quantity, plays a pivotal and wide-ranging role in Kant's account of mathematical cognition.

In the Critique 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.

Mathematics and Necessity contains essays by M. F. Burnyeat, Ian Hacking, and Jonathan Bennett based on lectures given to the British Academy in 1998. All concern the history of the philosophical treatment of mathematics, necessity, or both, although there is no single thread running through all three essays.