 Cassirer’s first “systematic” work, Substance and Function [Cassirer 1910], takes an essential philosophical step beyond Cohen by explicitly engaging with the late nineteenth-century developments in the foundations of mathematics and mathematical logic that exerted a profound influence on twentieth-century philosophy of mathematics and natural science. Cassirer begins by discussing the problem of concept formation, and by criticizing, in particular, the “abstractionist” theory characteristic of philosophical empiricism, according to which general concepts are arrived at by ascending inductively from sensory particulars. This theory, for Cassirer, is an artifact of traditional Aristotelian logic; and his main idea, accordingly, is that developments in modern formal logic (the mathematical theory of relations) allows us definitively to reject such abstractionism (and thus philosophical empiricism) on behalf of the genetic conception of knowledge. In particular, the modern axiomatic conception of mathematics, as exemplified especially in Richard Dedekind’s work on the foundations of arithmetic and David Hilbert’s work on the foundations of geometry, has shown that mathematics itself has a purely formal and ideal, and thus entirely non-sensible meaning. Pure mathematics describes abstract “systems of order” – what we would now call relational structures – whose concepts can in no way be accommodated within abstractionist or inductivist philosophical empiricism. Cassirer then employs this “formalist” conception of mathematics characteristic of the late nineteenth century to craft a new, and more abstract, version of the genetic conception of knowledge. We conceive the developmental process in question as a series or sequence of abstract formal structures (“systems of order”), which is itself ordered by the abstract mathematical relation of approximate backwards-directed inclusion (as, for example, the new non-Euclidean geometries contain the older geometry of Euclid as a continuously approximated limiting case). In this way, we can conceive all the structures in our sequence as continuously converging, as it were, on a final or limit structure, such that all previous structures in the sequence are approximate special or limiting cases of this final structure. The idea of such an endpoint of the sequence is only a regulative ideal in the Kantian sense – it is only progressively approximated but never in fact actually realized. Nevertheless, it still constitutes the a priori “general serial form” of our properly empirical mathematical theorizing, and, at the same time, it bestows on this theorizing its characteristic form of objectivity.