St. Hugh’s Hall, Oxford,
January, 1911.
CONTENTS
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.
Such a science might exist out of conditions of time and space as we know them. It is a science of relations rather than of mere number. Founded, then, on the laws of symbolic logic, it is a valuable aid and illustration to philosophy; philosophy, on the other hand, can imagine lines for the exercise of the constructive power involved in mathematics. It is the object of this paper to show that the close though apparently accidental union of philosophy and mathematics throughout the history of thought can now be explained, and that the problems with which pure mathematics is now concerned are those which lie at the core of philosophic thought and speculation. (Symbolic logic has developed to meet the new demands made upon it. It does not now reduce itself to the syllogism, as Aristotle thought it did; the prerequisites of thought are shown to be manifold instead of single.[3])
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.