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Abstract: Ernst Cassirer’s book Substanzbegriff und Funktionsbegriff (1910) – despite the recognition it has recently received from philosophers of mathematics and science – is a difficult book for contemporary readers to understand. Its topic, the theory of concept formation, engages with debates and authors that are largely unknown today. And its “historical” style violates the philosophical standards of clarity that characterize analytic philosophy. Cassirer, for instance, never says explicitly what he means by “substance-concept” and “function-concept.”

In this paper, I answer three questions: Why did Cassirer choose to focus on the topic of concept formation? What did Cassirer mean in contrasting “substance-concepts” and “function-concepts”? and How does Cassirer’s polemic against traditional theories of concept formation lead to the distinctive theories in the philosophies of mathematics and science that he defends later in the book? I argue that Cassirer’s contrast between substance-concepts and function-concepts includes four kinds of interrelated contrasts – contrasts that touch on issues in logic, metaphysics, epistemology, and the theory of objectivity. I argue that the book presents a progressively unfolding and intricate argument that begins with epistemological problems in the theory of concept formation and culminates in a defense of mathematical structuralism, a new theory of the a priori, and a “structural” account of scientific theory change.