In 1932, Professor Cassius Jackson Keyser called for “the disciplining of men, women, and children in the art of ‘postulate detection.’” An essential goal of education, he believed, was to teach Americans how to find the hidden assumptions that determine every system of thought, from politics to religion to philosophy. As Adrian Professor of Mathematics at Columbia University, Keyser had spent part of his career studying the foundational assumptions, or axioms, at the heart of mathematical reasoning. While the axioms of mathematics had traditionally been regarded as “self-evident” truths, mathematicians around the turn of the twentieth century became wary of such claims and began to distinguish between “axioms” and “postulates.” Unrelated to self-evidence or truth, postulates were simply agreed-upon statements used to construct a system of reasoning. For Keyser and other American mathematicians, postulate sets became their own field of mathematical interest tied to contemporary considerations of rigor. In this talk, I discuss how the study of postulates and their properties—like consistency (no postulate contradicts another) and independence (no postulate can be derived from another)— was used to criticize the abstract, out-of-touch practices of modern mathematical research as well as to celebrate its artistry and virtues. Overall, I situate what would later be referred to as “American Postulate Theory” within the landscape of modern mathematics as well as the early- twentieth-century growth of the American mathematics community.

In her Modern Introduction to Logic (1930), often considered the first textbook of analytic philosophy, British philosopher L. Susan Stebbing (1885–1943) presented a coherent long durée vision of the science of logic. Contrary to a caricature (popular then and now) that presents mathematical logic as an irruption of genius redeeming a heretofore worthless discipline, Stebbing construed the advent of mathematical notation and its attendant methods in logic as a cumulative triumph. She positioned the domains of logic, mathematics, and scientific method in relation to each other and within a reorganized disciplinary matrix she indicated new possibilities for the person of the philosopher. Over the course of her career Stebbing exhibited a vision of the philosopher as a public figure at a time when it remained rare for a woman to be recognized as a philosopher at all. She asserted the need for the rigors of logic in public discourse and likewise asserted herself as an authoritative representative of that science. Striving to render the latest mathematical logic accessible to as wide an audience as possible, she used its methods to analyze found examples of misleading political discourse and stressed the importance of argumentative clarity amidst the turmoil of the 1930s. By asserting the specifically democratic value of mathematized rigor, she posited a particular social role for the philosopher as an intermediary between modern science and everyday experience—a role she held to be urgently needed in the face of pervasive unscrupulous rhetoric in the age of fascism.

In the early 1960s, a semiautonomous Division of Computer Science existed within Stanford University’s Department of Mathematics. While the division had initially grown out of interest in numerical analysis within the mathematics department, members of the computer science division became increasingly frustrated with the limits their relationship with mathematics placed on their growth and their ability to direct the future directions of computer science at Stanford. The computer science faculty was interested in branching out from its early emphasis on numerical analysis, but the division’s position within the mathematics department complicated efforts to include contested or less “mathematical” subfields such as artificial intelligence. I will argue that the mathematicians and computer scientists used different notions of what was mathematical and what made a good mathematician to pursue certain desired relationships between mathematics and computer science at Stanford. While the historical scholarship addressing the relationship between mathematics and computing or computer science has largely focused on specific subdisciplines or on public conversations and widespread discourses (Dick, Ensmenger, MacKenzie, Mahoney), this paper expands on this literature by examining how the institutionalized relationship between mathematics and computer science within the university context mattered to the early development of computer science as an autonomous discipline. In doing so, it contributes to the history of mathematics, the history of computing, and the history of the disciplines.