Master of Arts
Theory and Criticism
In the Critique of Pure Reason, Immanuel Kant takes the ostensive constructions characteristic of Euclidean-style demonstrations to be the paradigm of both mathematical proofs and synthetic a priori cognition in general. However, the development of calculus included a number of techniques for representing infinite series of sums or differences, which could not be represented with the direct geometrical demonstrations of the past. Salomon Maimon’s Essay on Transcendental Philosophy addresses precisely this disparity. Maimon, owing much to G. W. Leibniz, proposes that differentials of sensation achieve what Kantian constructions could not. More importantly, Maimon develops a kind of symbolic cognition that is not delimited by the constraint of the pure forms of intuition. The mind does not construct its objects, but constructs itself through inquiry into the real objects of thought.
Summary for Lay Audience
With the Critique of Pure Reason Immanuel Kant reorients metaphysics away from things considered independently of the mind, towards the invariant structures of cognition and how the mind must suppose these structures in every experience of things. While this might bring order to the flurry of affections that impose themselves on the mind, a consequence of the critical turn is that the non-empirical objects of mathematics must also find their ultimate source and validation in the experience of empirical objects. Kant develops his philosophy of mathematics according to the use, tradition, and rigor of constructions in geometry. By the early eighteenth century, however, calculus had challenged the role of perception and sensation in the mathematical sciences. And not only this, the prominence of symbolic notation over geometrical construction allowed mathematicians and philosophers to think real objects that could never be given in the domain of experience. This line of thought is inaugurated by G.W. Leibniz and further developed by Salomon Maimon in the eighteenth century. In the first chapter, I develop Kant’s notion of ‘construction’ and its place within his philosophy of mathematics, and critical philosophy more generally. The second chapter explicates Kant’s relationship with calculus, specifically with Isaac Newton’s Method of Fluxions, and shows how the indefinite iteration of constructions cannot adequately represent the relevant properties of infinite series. Chapter 3 goes on to develop Maimon’s response to Kant - the 'differentials of sensation' - together with Leibniz’s analytic method of infinitesimals. The fourth and final chapter illustrates the use of differentials in cognition with an example from Leibniz’s De quadratura and goes on to explicate some consequences for Kant’s critical philosophy. It concludes by indicating a passage from representation to reality, where cognition determines the thinking subject as much as it determines the object of experience.
Birmingham, Nicholas A. J., "Cognition without Construction: Kant, Maimon, and the Transcendental Philosophy of Mathematics" (2020). Electronic Thesis and Dissertation Repository. 6932.
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