The Role of Intuition in Kant's Philosophy of Mathematics and Theory of Magnitudes

Dissertation, University of California, Los Angeles (1998)
Abstract
The way in which mathematics relates to experience has deeply engaged philosophers from the scientific revolution to the present. It has strongly influenced their views on epistemology, mathematics, science, and the nature of reality. Kant's views on the nature of mathematics and its relation to experience both influence and are influenced by his epistemology, and in particular the distinction Kant draws between concepts and intuitions. My dissertation contributes to clarifying the role of intuition in Kant's theory of mathematical cognition. It focuses in particular on Kant's theory of magnitudes. This theory explains the applicability of mathematics to experience while bridging his philosophy of mathematics and his philosophy of science; it is therefore central. Kant's views on magnitudes, however, have been neglected. ;The dissertation divides into four parts. It begins by analyzing Kant's concept of magnitude and the closely related concepts of quanta, quantitas, quantity and number. These clarifications lay the groundwork for reconstructing and evaluating Kant's views. ;Understanding Kant's theory of mathematical cognition requires understanding how the quantitative forms of judgment and the categories of quantity are employed in mathematics. I argue for an interpretation that reflects the influence of 18$\sp{\rm th}$ Century metaphysics on Kant's thought, and reveals Kant's attempt to find a place for a mathematical conception of the world within that metaphysics. ;I also examine Kant's arguments for two fundamental properties of space--infinite divisibility and unlimited extent. I argue that intuition plays a perceptual role that can only be understood within the context of Kant's theory of imagination. ;Finally, I argue that according to Kant, a homogeneous manifold in intuition is required to represent composition and a mathematical version of the part/whole relation. The dissertation uses a modern understanding of the properties of relations to analyze the part/whole relations Kant refers to in his logic, and contrast them with the part/whole relations in his theory of magnitudes. Modern measurement theory aids in identifying Kant's insights into the conditions for the applicability of mathematics to experience. Kant's insights constitute an important part of his views on the role of intuition in mathematics.
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