Online research in philosophy

This paper considers Kant's understanding of conceptual representation in light of his view of geometry.

Newton characterizes the reasoning of Principia Mathematica as geometrical. He emulates classical geometry by displaying, in diagrams, the objects of his reasoning and comparisons between them. Examination of Newton’s unpublished texts (and the views of his mentor, Isaac Barrow) shows that Newton conceives geometry as the science of measurement. On this view, all measurement ultimately involves the literal juxtaposition—the putting-together in space—of the item to be measured with a measure, whose dimensions serve as the standard of reference, so that all (...) quantity (which is what measurement makes known) is ultimately related to spatial extension. I use this conception of Newton’s project to explain the organization and proofs of the first theorems of mechanics to appear in the Principia (beginning in Sect. 2 of Book I). The placementof Kepler’s rule of areas as the first proposition, and the manner in which Newton proves it, appear natural on the supposition that Newton seeks a measure, in the sense of a moveable spatial quantity, of time. I argue that Newton proceeds in this way so that his reasoning can have the ostensive certainty of geometry. (shrink)

In a crucial passage of the second-edition Transcendental Deduction, Kant claims that the concept of motion is central to our understanding of change and temporal order. I show that this seemingly idle claim is really integral to the Deduction, understood as a replacement for Locke’s “physiological” epistemology (cf. A86-7/B119). Béatrice Longuenesse has shown that Kant’s notion of distinctively inner receptivity derives from Locke. To explain the a priori application of concepts such as succession to this mode of sensibility, Kant construes (...) the mind as receptive to its own activity. As Longuenesse understands Kant’s response to Locke, “motion” becomes little more than a metaphor for the action of understanding on inner sense. For Michael Friedman, in contrast, this passage evidences Kant’s deep concern with the foundations of Newtonian science. He reads it as a reference to inertial motion, the standard by which temporal intervals are measured. I show that Longuenesse’s and Friedman’s interpretations are in fact complementary. Both Locke and Kant are deeply concerned with the quantification of time. So Longuenesse is right that §24 is meant to supplant Locke’s account, and Friedman correctly takes it as the foundation of Kant’s account of the application of quantitative concepts to time. Kant aims, specifically, to explain cognition of time as continuous magnitude. However, Longuenesse leaves Kant without an answer to Locke’s challenge that continuous magnitude cannot be understood on the basis of discrete magnitude. And on Friedman’s view Kant’s account of the understanding’s activity presupposes, and thus cannot explain, the achievements of the mathematical sciences (such as the representation of temporal continuity). On my reading, Kant’s key contention is that we must regard the segregation of temporally ordered representation into units as the work of the understanding, rather than (as Locke views it) of sensibility. By giving the understanding this role, Kant succeeds where he takes Locke to fail. I show how the power to unify temporal representation can be ascribed to the understanding on the basis, not of assumed mathematical or scientific knowledge, but of its characterization as the power of judgment. (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)