On the theory of the infinite in modern thought

ON THE THEORY OF THE INFINITE IN MODERN THOUGHT
TWO INTRODUCTORY STUDIES
DOCTOR OF THE UNIVERSITY OF PARIS VICE-PRINCIPAL, ST. HUGH’S HALL, OXFORD
LONGMANS, GREEN AND CO. 39 PATERNOSTER ROW, LONDON NEW YORK, BOMBAY, AND CALCUTTA 1911
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Of the two papers here reproduced, the first was given in 1905 to a meeting of the women science students in Oxford; the second, in 1908, to the Philosophical Society of this College. They are printed by request, with the author’s apologies for their incompleteness. The lecture form has been retained. I am indebted to my brother, Mr. P. Jourdain, for help in preparing the first lecture, and for his revision of the text.
E. F. JOURDAIN.
St. Hugh’s Hall, Oxford, January, 1911 .

The influence of mathematics on philosophy and vice versâ can be inferred from the historical progress of both studies, though it has not been possible till about within the last fifteen years to give a logical explanation for the relations between them. As long as it was believed, according to the Kantian view, that the science of mathematics was based on intuitions of time and space, the alliance between philosophy and mathematics could not be proved to be closer than that between philosophy and experimental science, although the historical fact remained that philosophy and mathematics exercised a mutual stimulus, and developed at the same periods of history.
The use of the word philosophic in this connection suggests a necessity for further definition. Philosophy is held to include at least two great branches—​Metaphysics and Ethics. The influence of mathematics is most evident on the metaphysical side of philosophy; in fact, the grouping of mathematics and metaphysics as allied sciences tends to bring out the essential distinction between metaphysics and ethics, and—​though not by any means to imply a break in their real relation—​to show where this has been misunderstood. No philosophy has been equally strong on both sides; they represent different forms of activity of the human mind; but it is still true, and from the conditions always must be, that an ethical system grows out of metaphysics as practice follows precept and conduct implies belief. The new definition of mathematics does not touch these consequences; it merely marks the limits within which philosophy on the metaphysical side can submit to, or rest upon, the conclusions of mathematics.