James and Math

Abstract In the last thirty years of his life, William James reflected periodically on two questions concerning infinite totalities: is the notion of an infinite totality logically contradictory? And, if not, are infinite totalities endowed with “extra-logical,” “extra-mathematical” existence? As he addressed these questions, James drew a distinction between two kinds of infinities: “growing infinities,” such as the infinite series of the natural numbers 1, 2, …, n, …, and “standing infinities,” as would be the stars under the assumption that infinitely many stars exist. He rejected as self-contradictory the notion that a growing infinite could constitute or be viewed as a totality. James never called into question, either logically or factually, the possibility of the existence of infinitely many things (e.g., the infinitely many stars), provided those things did not constitute (or are not viewed or experienced by some knower as) a totality. Because of his methodological commitment to human experience, which for James was finite, he banned (standing) infinite totalities from his pluralistic universe. In addition to explaining how James reached these conclusions, this chapter also argues that James’s engagement with questions concerning infinite totalities played a major role in shaping his understanding of his colleague Josiah Royce’s conception of the Absolute, which Royce had mathematized as an actually infinite totality, and, as a result, helped him develop in new ways his own pluralistic metaphysics.