MATHEMATICS. The science of mathe matics— what shall it be said to be? A ques tion much discussed by philosophers and mathe maticians in the course of more than 2,000 years, and especially with deepened interest and insight in our own times. Many have been the answers, but none has approved itself as final. All of them, by nature belonging to the ((liter ature of knowledge? fall under its law and tend to "perish by supersession." Naturally enough conception of the science has had to grow with the growth of the science itself. For it must not be inferred that mathematics, because it is so old, is dead. Old it is indeed, classic already in Euclid's. day, being surpassed in point of antiquity by but one of the fine arts and by none of the ((natural° sciences; but it is not only the oldest science, it is also as not as any, living and flourishing to-day as never before, advancing in a -thousand directions by leaps and bounds. It is not merely as a giant tree throwing out and aloft myriad branching arms in the upper regions of clearer light and plunging deeper and deeper root in the darker soil beneath. Rather is it an immense forest of such oaks, which, however, literally grow into each other, so that, by the junction and intercrescence of limb with limb and root with root and trunk with trunk, the manifold wood becomes a single living organic growing whole. A vast complex of interpenetrating theories such the science now actually is, but it is more wondrous still potentially, component theories continuing more and more to grow and multiply beyond all imagination and beyond the com pass of any single genius, however gifted. What is this thing so marvelously vital? What does it undertake? What is its motive? How is it related to other modes and interests of the human spirit? One of the oldest, at the same time the most familiar, of the definitions conceived mathe matics to be the science of magnitude, where magnitude, including multitude as a special kind, signified whatever was "capable of in crease and decrease and measurement? Capa bility of measurement was the essential thing. That was a most natural definition of the science, for magnitude is a singularly funda mental notion, not only inviting but demand ing consideration at every stage and turn of life. The necessity of finding out how many
and how much was the mother of counting and measurement and mathematics, first from ne cessity and then from joy, so busied itself with these things that they came to seem its whole employment. But now the ordinary notion of measurement as the repeated application of a constant finite unit has been so refined and gen eralized, on the one hand through the creation of the so-called irrational and imaginary num bers (see AL tints; COMPLEX VARIABLE) , and on the other by use of a scale, as in non Euclidean geometry (see N ON- EUCLIDEAN GEOMETRY; ANALYTICAL METRICS), where the unit appears to suffer lawful change from step to step of its application, that to retain the old words and call mathematics the science of measurement seems quite inept as no longer telling either what the science has actually be come or what its spirit is bent upon.
Moreover, the most striking measurements, as of the volume of a planet, the valency of atoms, the velocity of light or the distance of star from star, are not done by direct repeated application of a unit. They are all accom plished by indirection. Perception of this fact it was which led to the famous definition by the philosopher and mathematician, Auguste Comte, that mathematics is the "science of indirect measurement? Here doubtless we are in presence of a finer insight and a larger view, but the thought is not yet either wide enough or deep enough.
For it is obvious that there is much admit tedly mathematical activity that is not in the least concerned with measurement whether direct or indirect. In projective geometry (which see), for example, it was observed that metric considerations were either absent subordinate. The fact, to take a simplest example, that the two points determine a line uniquely, or that the intersection of a sphere and a plane is a circle, is not a metric fact: it is not a fact about size or quantity or magni tude. In this field it was position rather than size that to some seemed the centre of interest, and so it was proposed to call mathematics the science of magnitude or measurement and posi tion.