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. As to the historical relation between metaphysics and mathematics, the subject is so vast that we shall only attempt a very rapid generalisation of its results on the growth of the conception of the Finite and the Infinite. (Of course, there are many other sides of the relation which might be studied.) The general result of the inquiry has been, as far as we can judge, that metaphysics has exercised an inspiring force on mathematics, and mathematics has defined and strengthened the conceptions of metaphysics at every critical stage in the history of philosophy. But where metaphysics has been treated as the proof of science, where it has been laid down as the foundation for exact knowledge, the results have not corresponded with the truth of experience, and the quality of thought has become degenerate. Progress depends on the right perception of the relations between the sciences and parts of philosophy.