Kant, Infinite Space, and Geometry

New research by Aaron Wells

My research includes an ongoing project on the philosophy of mathematics and the metaphysics of magnitude among early modern thinkers such as Leibniz, Wolff, and Du Châtelet. One goal of the project is to better understand the historical context of Kant’s much-discussed philosophy of mathematics, which also requires detailed engagement with Kant’s ideas.

One example is the following problem. Kant claims we intuit infinite space. However, Kant also thinks full awareness of infinite space requires synthesis—the act of putting representations together and comprehending them as one. But our ability to synthesize is finite. Tobias Rosefeldt has argued in a recent paper that Kant’s notion of decomposing synthesis offers a solution, so long as determinate space in Kant is understood as only potentially infinite. An example of decomposing synthesis is becoming determinately aware of a circle as distinct from its background, which involves dual awareness of the circle and of its background as unbounded.

In an upcoming talk at the American Philosophical Association (Eastern Division), also to be presented at the 2024 Kant-Congress, I criticize this approach and suggests an alternative.

An initial conceptual problem is that decomposing synthesis on its own only shows that we do not in fact intuit a boundary of a given space, not that the space really is unlimited or potentially infinite. Two further textual problems stem from Kant’s definitions of ‘infinity’ and his account of construction in geometry. First, Kant’s relevant definitions of ‘infinity’ invoke concepts, but decomposing synthesis is supposed to be non-conceptual. Second, Rosefeldt’s account appeals to geometrical construction, but Kant takes such construction to involve conceptual activity, so it cannot be understood in terms of decomposing synthesis alone.

Instead, I argue, ‘composing’ synthesis that runs from parts to wholes is sufficient for potentially infinite space. Consider, for example, Euclid’s third postulate for constructing a circle of arbitrary size. Given concepts of circle and of straight line (for the radius), a determinate circle can be made as large as we wish, where this involves constructing a circle part-by-part via composing synthesis. Parallel procedures can be used to construct an arbitrarily large sphere. Decomposing synthesis, then, plays no essential role in the representation of potentially infinite space. In other work, Wells has argued that Kant’s potentialism is in many ways similar to a position laid out earlier by Emilie Du Châtelet.

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