Cassirer on the Concept of Number: A Neo-Kantian Perspective on Dedekindian Abstraction

The concept of number occupies a special place in Cassirer’s epistemology. Cassirer identified it as the first and paradigmatic example of a concept determined by a serial principle alone in his seminal article “Kant and modern mathematics” (1907), and he placed it at the basis of the logical determinability of the objects of knowledge in his first major epistemological work Substance and Function (1910). Cassirer articulated this view further in the third volume of the Philosophy of Symbolic Forms (1929) by taking into consideration the phenomenological basis of mathematical concept formation. From the broader perspective of Cassirer’s mature philosophy of culture, the concept of number tended to become paradigmatic of a purely symbolic configuration of reality, while retaining its fundamental role in Cassirer’s later accounts of scientific objectivity (see, e.g., the fourth volume of The Problem of Knowledge, from 1940). The unifying tendency of Cassirer’s account, however, is sometimes seen to be in some tension with his reliance on specific notions, such as an ordinal conception of number. This paper aims to shed light on the peculiar status of the concept of number in Cassirer’s view, by reconsidering the neo-Kantian background of the notion of abstraction that emerges from his account of the structural procedures at work in seminal mathematical theories, from real analysis to projective geometry and foundational inquiries. Whereas the universal concepts of traditional Aristotelian logic are abstracted from a certain number of objects by disregarding their distinguishing characteristics, the logic underlying modern mathematics in Cassirer’s account shows a constructive side of abstraction as leading to relational concepts with infinitely many possible instantiations. Cassirer maintained that this notion of abstraction was foreshadowed in developments of logic and related philosophical disciplines in the Nineteenth century and found its clearest expression in Dedekind’s way to define numbers as sui generis objects that are independent of spatio-temporal notions and uniquely determined in and through their mutual relations. The first part of the paper will discuss how Cassirer’s view led him to take a different stance from his Marburg teachers concerning the arithmetization of mathematics. Whereas Cohen grounded the notion of number on that of infinitesimal quantity, Cassirer followed the view – shared by most nineteenth-century mathematicians – that a rigorous foundation of analysis ought to avoid all notions of quantity and be carried out by arithmetical means. It will be pointed out that, on the other hand, Cassirer relied on his interpretation of abstraction to provide a new account of the conceptual construction of the objects of experience in the wake of Marburg neo-Kantianism. The second part will discuss the rediscovery of Cassirer’s interpretation in some recent attempts to give a logical reading of Dedekindian abstraction in the context of number theory, and make a suggestion for a possible extension of the structuralist methodology that emerges from Cassirer’s interpretation to other mathematical domains.