The Foundation of Kant's Philosophy of Mathematics

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

Project Summary The Foundation of Kant's Philosophy of Mathematics What sort of things are numbers? Does mathematics have a foundation, and if so, what is it? How do we know mathematical truths such as .5+7=12,. or the Pythagorean Theorem? There is still no consensus on the answers to these questions, and, despite advances in mathematics and mathematical logic during the last few centuries, Kant's deep influence on these debates is still felt. There are nevertheless important aspects of Kant's philosophy of mathematics that have not been properly understood, in no small part due to the tremendous changes in mathematics since the early modern period. Above all, Kant's philosophy of mathematics and mathematical cognition has been obscured by the centrality of his theory of magnitudes. Since the 19th century, numbers rather than magnitudes have played the central role in our understanding of mathematics. Intellectual Merit This research project will provide a new, systematic, and historically sensitive account of Kant's philosophy of mathematics based on his theory of magnitudes. It will revive a significant perspective on mathematics very different from the modern view, one that can offer insights into current debates. Both the early modern and Kant's understanding of magnitudes can be traced back to the Eudoxian theory of proportions found in Euclid's Elements. The Eudoxian theory rests on the cognition of the part-whole and equality relations of magnitudes, and Kant adopts this mereological approach to magnitudes in his own theory. His mereology of magnitudes contrasts sharply with the modern view, in which mathematics is founded on set theory. Furthermore, Kant holds that mathematical cognition requires a non-conceptual quasi-perceptual kind of mental representation he calls intuition.. The modern set-theoretic approach can be best described as appealing to conceptual representation alone. Reconstructing the early modern influences on Kant's thought and Kant's own theory of magnitudes will therefore articulate a very different way of looking at both mathematics and mathematical cognition. The first part of the project will investigate the early modern views of magnitude, number, and mathematical properties that influenced Kant.. Early modern views were in flux, however, and differed in important respects from the Greek mathematical tradition. Reconstructing the early modern views that influenced Kant will explain further aspects of Kant's own theory. The second part of the present project will further investigate Kant's own views on three important issues concerning the role of intuition in mathematical cognition. Together, the two parts will provide a complete, systematic account of Kant's philosophy of mathematics. The proposed project will build on earlier studies and will complete this work to provide a complete and systematic treatment of Kant's philosophy of mathematics. Broader Impacts The results will be widely disseminated; they will appear in various presentations, at least four articles, and will form the basis of a book. The project will more broadly impact the history of mathematics and will offer insight into contemporary debates in the philosophy of mathematics. The results of this research will be incorporated into undergraduate courses focusing on the interaction between philosophy, mathematics, and science during the scientific revolution. These courses put students in the position of a natural philosopher of the 17th and 18th centuries and ask them to reflect on magnitudes such as distance, force, and weight, beginning with levers and building up to a (simplified version) of Newton's derivation of the law of universal gravitation. A discussion of the mathematical character of magnitudes is an effective way to make mathematics and science accessible to humanities students while giving historical and philosophical understanding to science and engineering students. Further insight into the early modern view of magnitudes and their relation to mathematics will improve this successful pedagogical approach.