Friday, December 1, 2017

Kant’s Philosophy of Mathematics

Immanuel Kant’s philosophy of mathematics is a part of his broader philosophical project. His view that all mathematical cognition is synthetic and a priori is in line with his theory of pure sensibility, doctrine of transcendental idealism, and his view on appropriate and successful methods of reasoning.

In The Cambridge Companion to Kant and Modern Philosophy (Edited by Paul Guyer), Lisa Shabel has an insightful essay on Kant’s view of mathematics (Chapter 3, “Kant’s Philosophy of Mathematics”). Here’s an excerpt:
Kant, a long-time teacher and student of mathematics, developed his theory of mathematics in the context of the actual mathematical practices of his predecessors and contemporaries, and he produced thereby a coherent and compelling account of early modern mathematics. As is well known, however, mathematical practice underwent a significant revolution in the nineteenth century, when developments in analysis, non-Euclidean geometry, and logical rigor forced mathematicians and philosophers to reassess the theories that Kant and the moderns used to account for mathematical cognition. Nevertheless, the basic theses of Kant’s view played an important role in subsequent discussions of the philosophy of mathematics. Frege defended Kant’s philosophy of geometry, which he took to be consistent with logicism about arithmetic; Brouwer and the Intuitionists embraced Kant’s idea that mathematical cognition is constructive and based on mental intuition; and Husserl’s attempt to provide a psychological foundation for arithmetic owes a debt to Kant’s characterization of mathematics as providing knowledge of the formal features of the empirical world. 

In the later twentieth century, by contrast, most philosophers accepted some version of Bertrand Russell’s withering criticism of Kant’s account, which he based on his own logicist program for mathematics. But now it is clearly time to reassess the relevance of Kant’s philosophy of mathematics to our own philosophical debates. For just a few examples, contemporary work in diagrammatic reasoning and mereotopology raise issues that engage with Kant’s philosophy of mathematics; Lakatos-style antiformalism is arguably a descendant of Kant’s constructivism; and our contemporary understanding of the relation between pure and applied mathematics, especially in the case of geometry, is illuminated by Kant’s conception of the sources of mathematical knowledge. More generally, because we persist in considering mathematics to be a sort of epistemic paradigm, our current investigations into the possibility of substantive a priori knowledge would surely benefit from reflection on Kant’s own subtle and insightful account of mathematics. 

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