On the theory of the infinite in modern thought — Context and Discussion

Mathematics as a Science of Relations

Jourdain opens by asserting that mathematics, as now understood, is “based, like formal logic, on the prerequisites of thought, not on the notions of space and time.” This marks a departure from the Kantian view that mathematics rests on intuitions of space and time. She argues that the new mathematics is a “science of relations rather than of mere number,” and that it can exist “out of conditions of time and space as we know them.” This redefinition allows mathematics to serve as a “valuable aid and illustration to philosophy,” while philosophy can in turn “imagine lines for the exercise of the constructive power involved in mathematics.” The lecture thus sets up a reciprocal relationship between the two disciplines, grounded in the logical structure of thought itself.

The Pragmatist Challenge and the Absolute

Jourdain engages critically with pragmatist thinkers, particularly William James, whom she quotes as saying, “I personally give up the Absolute. I find it entangles me in metaphysical paradoxes that are inacceptable.” She also examines the arguments of Bax, who posits a “moving synthesis” as the goal of thought, and contends that such a view inadvertently proves the existence of an Absolute: “the appearance of the goal of thought as a moving synthesis would presuppose an Absolute as a ground reality.” Jourdain’s rebuttal relies on the mathematician’s concept of the series of all finite and transfinite ordinal numbers, which she sees as leading to the proof of an “Absolute Infinite.” The structure of her argument moves from specific philosophical positions to a broader mathematical foundation.

Fact, Law, and the Logic of Denial

In the second lecture, Jourdain turns to the pragmatist appeal to facts over scientific law and logic. She notes the inconsistency: “If he consistently denied logic, his position would be unassailable by logic, but he uses the method he denies, and is thus open to attack.” She acknowledges the pragmatist point that there is “no actual continuity between a fact and a law,” but counters that laws are concepts—the result of mental activities—and that the denial of logic itself requires logical reasoning. This section is notable for its tight, almost syllogistic structure, as Jourdain systematically exposes the self-referential problems in the pragmatist position. The lecture form allows her to build her case step by step, addressing objections as they arise.

The Recurring Image of Background and Ground

Throughout both lectures, Jourdain returns to the image of a background or ground against which change is perceived. She cites Bax’s argument that everything we perceive is seen “against a background which itself moves,” and that this implies no fixed Absolute. Jourdain inverts this: only the permanent can produce the phenomena of change. She also references Ormond’s “Foundations of Knowledge” and Illingworth’s “Doctrine of the Trinity” to support the idea that “the evolutionary aspect of the world itself” must be connected to “a ground reality that is stable.” This recurring visual metaphor—of shifting figures against a stable ground—gives the abstract argument a concrete, almost spatial dimension, and ties together the mathematical and philosophical threads of the work.

Jourdain’s lectures are best approached as a pair of linked arguments rather than a comprehensive treatise. The first lecture establishes the mathematical foundation; the second applies it to contemporary philosophical debates. Readers unfamiliar with the work of Cantor or the pragmatist school may find it helpful to consult a primer on transfinite numbers or William James’s pragmatism. Jourdain’s prose is clear and direct, but the arguments are dense and repay careful reading. The lectures offer a window into early twentieth-century attempts to reconcile new mathematical discoveries with enduring metaphysical questions.

There’s something strangely comforting in how those 1905 lectures treated the infinite not as a mystery to fear, but as a quiet structure we could think with. It reminds me of an old volume on commerce I once found, Morals in Trade and Commerce — Themes and Context, where fairness felt similarly abstract, yet reachable. Both left me with a gentler sense of order in things.