Unlocking the Second Antinomy: Kant and Wolff

But how in this business can metaphysics be reconciled with geometry, when it seems easier to mate griffins with horses than to unite transcendental philosophy with geometry?¹

Kant, 1756

The second antinomy, treating the proof and refutation of bodies as composed of simple substances, is one of the more puzzling sections of the Critique of Pure Reason. The thesis argument especially baffles commentators. Edward Caird in 1889 said: “Kant’s statement of this argument is very obscure.” Among commentators of our era, T. E. Wilkerson complains: “This argument is very muddy.” Jonathan Bennett remarks: “The text, however, is not clear enough for us to be sure exactly what Kant is relying on in his argument for the Thesis.”² Nor have commentators spared the antithesis argument. No one, to my knowledge, has interpreted the text so that the thesis and antithesis arguments come out valid and constitute a genuine antinomy. Yet Kant stated on more than one occasion that all the antinomy arguments are valid. The proofs “are not deceptions, but are well-founded, under the supposition that appearances or a sensible world which comprehends them all are things in themselves” (B535).³ In the Prolegomena he writes: “The thesis, as well as the antithesis, can be shown by equally clear, evident, and irresistible proofs—for I pledge myself as to the correctness of all these proofs.”⁴

It would be pleasant to report that Kant, in the Second Antinomy, has uncovered a universal and timeless flaw in human understanding, involving general concepts of simple and composite things. After two centuries of Kant interpretation, I believe that we can safely conclude that, whatever the Second Antinomy proves, it does not demonstrate such a flaw. Hence I will take a different tack in this paper. The Second Antinomy arguments, in my view, maintain whatever cogency they have in the context of eighteenth-century metaphysics and mathematics.⁵ The arguments look much more respectable in that light. Further, the antinomy is not merely two separate arguments; there is no conflict unless both sets of premises—those of the thesis and those of the antithesis—are affirmed together. The setting where that happens is Kant’s own system, with the addition of the false presupposition that appearances are things in themselves.

My aims in this paper are (a) to rehabilitate the arguments of Kant’s Second Antinomy through eighteenth-century substance philosophy, especially the philosophy of Christian Wolff, and the mathematics of the period; (b) to investigate how Kant’s system can support both the thesis and the antithesis; and (c) to consider how the antinomy supports transcendental idealism.

1. the substance philosophy framework

Within the philosophy of the Early Modern period, there was only one highly developed metaphysics—that of substance and accident. Substances were the basic ontological units on which the existence of everything else depended. Every substance metaphysician accepted the fundamental principle that properties and relations cannot exist without being properties and relations of something.

During this period there was no other metaphysical system able to compete with substance theories for supremacy in philosophy. Instead, some critics adopted a skeptical stance: they removed some or all of the substances from systems such as Cartesian metaphysics and tried to make do with the remaining components, principally ideas or representations in the mind.

In the case of corporeal substances, the kind of interest in the Second Antinomy, the fixed characteristics (essence) assigned to the substances were supposed to provide a foundation for scientific explanations of the behavior of bodies. As physical science evolved, so did the essences postulated by the philosophers. The bodies that we perceive have two sorts of characteristics which engage physical scientists. They take up space and they participate in causal processes. The Second Antinomy treats the problem of explaining the extension of bodies within a substance framework. The antinomy abstracts from questions involving forces and interactions of substances, as we will see.

A detailed exposition of the various substance philosophies in the Early Modern Period and a thoroughgoing account of how substance fits into Kant’s transcendental idealism are beyond the scope...