The Euclidean Tradition and Kant’s Thoughts on Geometry
Authors: Duncan Howard 
Published: June 2016

While not paramount among Kant scholars, issues in the philosophy of mathematics have maintained a position of importance in writings about Kant’s philosophy, and recent years have witnessed a rejuvenation of interest and real progress in interpreting his views on the nature of mathematics. My hope here is to contribute to this recent progress by expanding upon the general tacks taken by Jaakko Hintikka concerning Kant’s writings on geometry.Let me begin by making a vile suggestion: Kant did not have a philosophy of mathematics. When Kant was writing about mathematics, essentially he was reporting the views of others. The texts provide sufficient evidence to make this suggestion plausible. Generally, when Kant writes about mathematics in his mature works, he does so in order to illustrate or argue for a philosophical point. There are important references to mathematical method in the preface to the 1787 edition of Critique of Pure Reason; however, Kant’s purpose is to describe those basic features of a method that he intended to incorporate in his theory of philosophical method: ‘our new method of thought, namely, that we can know a priori of things only what we ourselves put into them.’ Indeed, Kant makes it clear in this preface that he thought there to be no extant problems to be solved in mathematical methodology; such was the state of the science, he thought. It was for this reason that Kant felt some confidence in borrowing from this method to improve the state of metaphysics; it is also for this reason that one should not expect to find Kant engaging in basic research in mathematical methodology. Similarly, the material on syntheticity added to the second edition Introduction to Critique of Pure Reason occurs in the context of a discussion of the syntheticity of metaphysical principles; that the propositions of both disciplines are synthetic a priori lends credence to the extrapolation of some features from the mathematical method for use in developing a metaphysics. Many writers find a philosophy of mathematics in the ‘Transcendental Aesthetic’; it is clear, however, that in this section his concern is to support his theory of the nature of space, time, and sensation. What is said about geometry, for example, is restricted to those of its features relevant to the subjectivity of space. The other major discussion of mathematics and its method is found in the section, ‘Doctrine of Method.’ Here we find Kant’s fullest account of the mathematical method and of constructions. It must be borne in mind, though, that his purpose is to argue against views that the proper methods of mathematics and metaphysics (philosophy generally) are identical, that the disciplines differ in subject matter alone. The result of the discussion is not a theory of mathematical method, but an account of the method proper to the philosopher.2 Kant simply is mentioning certain features of mathematical method sufficient to support the claim that the philosopher cannot incorporate it lock, stock, and barrel.’ In short, we do not find a systematic theory of mathematics or its method described by Kant in the first Critique, nor do we find discussions of mathematics other than in contexts where philosophical positions are being developed. This holds for Kant’s other works, too.