Mathematics and metaphysics in William James’s Some Problems of Philosophy

Francesca Bordogna

University of Notre-Dame

In his posthumously published Some Problems of Philosophy (James 1911/1979) William James drew a distinction between a broader and older conception of philosophy, according to which philosophy “must include the results of all the sciences,” and a more recent and narrower conception of philosophy as the discussion of specific metaphysical problems. While in the older sense philosophy could not “be contrasted with the sciences,” in the more recent and more technical sense of “metaphysics,” philosophy was “contrasted with the sciences.” James announced that the rest of the book would be devoted to philosophy in this latter sense, and that he would “let both religion and the results of the sciences alone.” Yet, in Some Problems James’s treatment of the metaphysical problem of novelty involved a detailed, critical discussion of mathematical theories and conceptions, including those of continuity and infinity. James noted that in perceptual experience change occurs through a finite number of discrete, sensible, however small, increments, and compared those “minima sensibilia” to “drops, “buds,” or “steps … coming wholly when they do come, or coming not at all.” Whether James at this stage actually embraced an atomistic conception of time and change or not, to his mind our perceptual experiences of change coming along in such “drops” avowed for the possibility of genuine novelty, and, hence, of free will. He argued that conceptual translations of change violated perceptual insight either by assuming that the “drops” have no “duration” or “extension,” or by assuming that they are not real minima, but can be divided ad infinitum. James found that, in order to make the case for his finitist position and for the possibility of genuine novelty, he needed first to challenge those assumptions. Going beyond his discussion of continuity in Pluralistic Universe (James 1909/1977), in Some Problems James opened up for scrutiny a series of concepts and theories adopted by many late 19th-century mathematicians, including the concept of the actual infinite, Cantor’s theory of transfinite numbers, and the “substitution of the arithmetization” of the continuum for the “intuition” of the geometrical line. This paper has two goals. First, I will use James’s critical discussion of mathematical concepts in order to gain a better understanding of the conception of the nature of metaphysics and of its role with the respect to the sciences, which James embraced toward the end of his life. Second, I will locate James’s discussion of the mathematical concepts of the actual infinite and the arithmetization of the continuum in the context of the early 20th-century analyses of the crisis of the foundations of mathematics and the crisis of the logicist program, including, e.g., Henri Poincaré analysis of the “Cantorian antinomies” in “Les mathématiques et la logique” (1905-1906), one of James’s main sources. I will use this contextualization in order to shed new light on some aspects of James’s late metaphysics, especially James’s commitment to what early twentieth-century philosopher W. P. Montague described as “the logic of irrationalism.”