Kant's Mathematical World: Mathematics, Cognition, and Experience by Daniel Sutherland (review)

Reviewed by: Kant's Mathematical World: Mathematics, Cognition, and Experience by Daniel Sutherland David Hyder Daniel Sutherland. Kant's Mathematical World: Mathematics, Cognition, and Experience. Cambridge: Cambridge University Press, 2021. Pp. 300. Hardcover, $99.99. In this lengthy book, Daniel Sutherland proposes to rectify our long neglect of Kant's theory of mathematics by means of both historical and systematic analyses. This is a worthy undertaking, since the scope and significance of that theory were lost from view during the twentieth century. In fact, the theory of mathematics spans the first several hundred pages of the first Critique. In the Transcendental Aesthetic, the pure unquantified homogeneous multiplicities of space and time are posited as structures of human perception. The Analytic begins by defining 'number' set-theoretically, as a property of sets of elements that can be linearly ordered, while explicating the concept of a set (Menge) in terms of pure logic, augmented by the abstract concept of a multiplicity. In the Axioms of Intuition, the concept of space "as it is required in geometry" is defined as the concept of a continuous extended geometrical magnitude, that is to say, a homogeneous manifold whose elements can be described by a coordinate system. In the Antinomies of the Dialectic, Kant addresses problems of completeness and incompleteness, which emerge when we try to extend this concept ad infinitum, and so follows through on Leibniz's distinction between finitary and infinitary proofs. Finally, in the Metaphysical Foundations of Natural Science (1786), Kant turns to the applied mathematical foundations of Eulerian mechanics, producing a relativistic derivation of the sine law for the composition of velocities. By attaching his theory to the structures that guided physicists and mathematicians over the next century, Kant ensured his own work would be carried on the wave. The topics just mentioned were taken up, criticized, and modified by Helmholtz, Klein, Cournot, Hamilton, Frege, Cantor, Russell, Hilbert, Poincaré, Einstein, Wittgenstein, and Weyl. This theory was, in other words, the backbone of the nineteenth-century tradition that became what we today call "philosophy of science" and "philosophy of mathematics." Unfortunately, Sutherland's book says nothing about that theory, nor about the traditions that preceded it or followed in its wake. In fact, if Sutherland's interpretation is correct, Kant's project cannot succeed. For Sutherland, it is essential to our understanding of Kant, and of eighteenth-century science and mathematics more generally, that we recognize that period as essentially Hellenistic. It is fundamentally different from what he calls "our modern" point of view, according to which mathematics is a science of number, which only emerged "over the course of the nineteenth century" (4–6). Therefore, it is to the Greeks that we must turn to understand Kant, not to the mathematicians whose works he owned and with whom he actively corresponded (163–93). Because arithmetization and algebra are not in question for Sutherland, the central concepts under investigation—magnitude and homogeneous—are also to be explicated with reference to Greek sources, above all the geometry of Euclid. This approach means that Sutherland fails to link the Principle of the Axioms of Intuition to the definition of number in the Schematism, and this, in my view, largely destroys Kant's project. Thus, and despite their centrality to his thesis, the first term, 'magnitude,' is never adequately defined by Sutherland, not even in the section entitled "Kant's Definition of the Concept of Magnitude" (66–75), where we find a long discussion of Vaihinger. The second, 'homogeneous,' is explicated with reference to genus/species relations "as its [End Page 713] etymology reflects" (199). Kant never actually uses this Greek term in the Critique, except in one passage where, indeed, genus/species relations, and not the theory of magnitudes, are in question (B 686). For, as indicated, Kant's primary references for this entire theory are works emanating from the Basel school (the Bernoullis, Euler, etc.), and the German gleichartig translates their homogène, which by this time was used, for instance, to describe "homogeneous functions" in the exact sense that contemporary mathematicians use the term today. Unfortunately, this type of error characterizes this book just as well in the large as in the small...