ON THE THEORY OF THE INFINITE IN MODERN THOUGHT ON THE THEORY OF THE INFINITE IN MODERN THOUGHT TWO INTRODUCTORY STUDIES BY E. F. JOURDAIN 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 All rights reserved 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_. CONTENTS I PAGE THE PROBLEM OF THE FINITE AND THE INFINITE 1 II PRAGMATISM AND A THEORY OF KNOWLEDGE 31 I THE PROBLEM OF THE FINITE AND THE INFINITE 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. But mathematics, as now defined, is independent of intuitions of space and time, and also of axioms and hypotheses.[1] Mathematics, as now understood, is based, like formal logic, on the prerequisites of thought, not on the notions of space and time. Here there is no definition of number or space, but the conception of number and space,[2] which is more complicated, can be derived from them. All other complicated mind processes can, in the same way, be reduced to the simple elements of the prerequisites of thought.