Mathematics, Logic, and Method in Kant's Transcendental Philosophy

Friday - Saturday, 13-14 April 2012
Center for Philosophy of Science
817 Cathedral of Learning
University of Pittsburgh
Pittsburgh, PA USA

Emily Carson, McGill University
Number, Quantity and Non-Conceptual Content in Kant

Abstract: In this paper, I consider recent readings of Kant by Robert Hanna and Lucy Allais which take Kant to be a proponent of nonconceptual content. I will argue that careful consideration of Kant’s views on the central role of the concept of number—and with it the category of quantity—in the first Critique speak against such readings.  I also (might) offer tentative suggestions about how recognising a special status for the category of quantity might do justice to some of the motivation for the nonconceptualist reading.

Jeremy Heis, University of California, Irvine
Should Kant Have Thought That Logic Was Complete Since Aristotle?

Abstract: Russell argued (in Principles of Mathematics) that the development of
modern, relational logic provides the definitive refutation of Kant's
philosophy, whose doctrine of intuition depends on the weakness of
traditional logic. Ernst Cassier, in his 1907 review of Russell's
book, argued provocatively that modern, relational logic is a more
natural fit for Kantian philosophy than traditional syllogistic. For
him, the characteristic activity of the understanding is synthesis,
and the structures of relational logic are "nothing other than a sum
total of such synthetic forms." In this paper I will argue that
Cassirer's claim, though presented by Cassirer himself in a confused
way, is correct.

My main argument depends on distinguishing the "logical structures,"
from Kant's point of view, of intuitions, schemata, and concepts. My
contention is that, though Kant strongly emphasized that the structure
of intuitions is quite different from the structure of concepts, he
was also committed to there being a unique structure for schemata. And
inasmuch as schemata serve various roles in his philosophy as
intermediaries between concepts and intuitions, Kant's account of
schematic structure has implications for his account of conceptual
structure -- the structures studied by pure general logic. After
discussing the historical background to Cassirer's claim about
Russell, I substantiate my argument by considering three topics in
Kant's philosophy: first, Kant's account of real definitions in
mathematics; second, Kant's sketchy remarks on "intuitive marks"; and
third, Kant's curious attitude toward concept formation by
abstraction.

Anja Jauernig, University of Pittsburgh
Transcendental Logic, the Applicability of Mathematics, and the Schematism of the Understanding

Abstract: The question of how a priori concepts can relate to really possible objects—which is part and parcel of the famous question “how are synthetic judgments a priori possible?”—lies at the heart of Kant’s theoretical philosophy in general and of his philosophy of mathematics in particular. The core of Kant’s answer to this question is contained in his account of the so-called ‘schematism’ of the understanding: a priori concepts can relate to really possible objects if their schemata constitute rules that determine how our imagination generates these objects in the first place. After briefly defending the foregoing claims, I will further explore how Kant conceives of the constructive activity of our imagination. The outcome of this exploration will be a better understanding of how, on Kant’s view, transcendental philosophy and the various exact sciences, including arithmetic, algebra, geometry, and the pure theory of motion, are related and represent facets of one unified project, namely, the making explicit of the rules according to which our imagination operates.

Lisa Shabel, Ohio State University
On the Possibility of Mathematics: Revisiting Kant's "Argument From Geometry"

Abstract: In a previous paper ("Kant's 'Argument from Geometry'", 2004) I offered an interpretation of the Transcendental Aesthetic according to which Kant there grounds the possibility of pure mathematics on the forms of intuition via a distinctively "synthetic" argument. In the present paper I investigate Kant's explanation of the possibility of pure mathematics as he presents it in §§6-11 of his Prolegomena to any Future Metaphysics, where he seems to make a genuine "argument from geometry." An understanding of the relation between the Critique's synthetic (or progressive) argument and the Prolegomena's analytic (or regressive) argument can make sense of Kant's claim that the Prolegomena functions as "preparatory exercises" for the Critique, while also illuminating the broad, methodological sense of Kant's analytic/synthetic distinction.

Daniel Sutherland, University of Illinois, Chicago
Kant’s Conception of Number

Abstract:Almost all of Kant’s texts concerning arithmetical cognition refer to counting or enumeration, but they do not make Kant’s views clear.  Most attempts to understand Kant’s philosophy of arithmetic have looked to what Kant says about space and time to help determine the role of intuition in his account of arithmetic, to clarify his conception of number, and to explain his philosophy of arithmetic more generally.   This paper will take a different tack.  It will focus first on determining the extent to which Kant holds either a cardinal or an ordinal conception of number.  The hope is that progress on this question will provide new insight into Kant’s philosophy of arithmetic and the role of intuition in particular.