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

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Authors: Jourdain, Eleanor F. (Eleanor Frances), 1863-1924, Jourdain, Philip E. B. (Philip Edward Bertrand), 1879-1919 [Contributor] Digital Source: Project Gutenberg Gutenberg Release Year: 2024 First Publication Year: Not confirmed

Key Word: Knowledge, Theory of; Pragmatism; Infinite Target Reader: Readers of public-domain and historical texts

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Two lectures from 1905 and 1908 examine the relationship between mathematics and philosophy, arguing that modern mathematics, based on symbolic logic rather than space-time intuitions, provides a framework for understanding the infinite and refuting pragmatist objections to an Absolute.

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should lead from the known to the unknown, simple to complex, and had defined the first as that which could be known without the help of the second. This logical order of reasoning has been attributed to mathematics, but has been considered to be inapplicable to philosophy. Mathematics, in its recent development, by the argument from the Finite to the Infinite and back again, starts from two propositions, neither of which can be said to be axiomatic, because each in turn can be proved from the other, but in the course of argument from either mathematics makes use of the logical process. The real axiom, as has been shown, is that of _existence_ or _being_. A metaphysical argument has the same root--that of existence--but a metaphysical problem deals with paradoxes, with questions which are sometimes defined as having two answers, each equally correct, and sometimes as yielding no answer at all. The method of thesis, antithesis, and synthesis is in the Hegelian logic applied to their solution.

A mathematical and a metaphysical problem are not, then, problems of the same kind to be solved by the same method; nor is the conception of the mathematical Absolute reached in the same way as that of the metaphysical Absolute. We are even unable to say how far they correspond except in respect of their absoluteness.[14] But the contention of the mathematician to-day and of the epistemologist school of philosophy is not the identity of methods and results in the two sciences. It is the axiom of existence on which they both depend: the law of thought by which all methods are developed, and, above all, the _correlative value of each science to the other_, which allows us, in developing our knowledge from the standpoint of the two sciences, to recognise something of the greatness of the Absolute principle to which they both reach up, and in which their being consists.

[1] Of course, if we comprehend in our view only elementary geometrical and algebraical science, it is easy to show that they _do_ demand both axioms and intuitions. Take, _e.g._ Euclid I. I., where in the construction it is necessary to employ intuition for the assertion that the arcs really cut one another. There is no logical certainty that they do; in fact, in some other conditions, _e.g._ in those of other space dimensions, they might not.

[2] This is, of course, not the space of experience. Logic and mathematics deal with implications of thought. See B. Russell (_Hibbert Journal_, 1904, pp. 809-12), who has shown that in all pure mathematics it is only the implications that are asserted, not the premiss or the consequence, as mathematicians used formerly to assume.

[3] De Morgan, Peirce, Schröder, and B. Russell have worked out the logic of relations as well as the syllogism.

[4] See Taylor, “Elements of Metaphysics,” p. 13.

[5] See Dr. Caird, “Evolution of Theology in the Greek Philosophers.”

[6] So Galileo, Newton, Huygens were philosophers in science. Descartes, Pascal, Leibniz were mathematicians as well as philosophers.

[7] See S. Augustine, _De Civitate Dei_, Book XII. ch. xix.: “Ita vero suis quisque numerus proprietatibus terminatur, ut nullus eorum par esse cuicumque alteri possit. Ergo et dispares inter se atque diversi sunt, et singuli quique finiti sunt, et omnes infiniti sunt.”

[8] See R. Dedekind, _Was sind und was sollen die Zahlen?_ 1893.

[9] Two transfinite aggregates can have an ordinal correspondence with one another.

[10] See G. Cantor, _Zur Lehre vom Transfiniten_. 1890.

[11] _e.g._ Mr. P. Jourdain, _Philosophical Magazine_. 1904.

[12] The same result is hinted at by Mr. Taylor. Taylor, “Elements of Metaphysics,” p. 22.

[13] Linear order, 1, 2, 3, &c. Circular. A CD B, A CD B.… The latter, it is true, involves the idea of separation. But this idea can be developed from those of inclusion and exclusion, which belong to the fundamental laws of thought.

[14] The Absolute, according to a recent metaphysical thinker, is “a conscious life which embraces the totality of existence, all at once, and in a perfect systematic unity, as the content of its experience.”--Taylor, “Elements of Metaphysics,” p. 60.

PRAGMATISM AND A THEORY OF KNOWLEDGE

The question before us is the relation of Pragmatism to a body of knowledge.

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The two lectures that make up this volume were delivered five years apart—the first to women science students at Oxford in 1905, the second to the Philosophical Society of St. Hugh’s Hall in 1908. Their author, Eleanor F. Jourdain, was Vice-Principal of St. Hugh’s and held a doctorate from the University of Paris. The lectures are printed here with minimal revision, retaining the direct address of the spoken form. Jourdain’s central claim is that the redefinition of mathematics as a science of relations, independent of space and time intuitions, allows a new alliance with philosophy. She traces this shift through the work of thinkers such as Cantor, Bradley, and James, and uses it to argue against pragmatist denials of an Absolute.

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.