ABSTRACT
Mathematics is the first of the sciences. The fact that mathematics is thus both demonstrative and constantly applied to nature, leads to the following seemingly paradoxical triad: the propositions of mathematics deal with the material world, material propositions are not necessary truths and the propositions of mathematics are necessary truths. The distinction between pure and applied mathematics has been brought to light by recent developments not only in mathematics but also in physics and logic. To sum up, geometry as a statement of the properties which any entity must logically have if it has certain other properties is pure mathematics; but geometry as a systematic description of the nature of space is applied mathematics, or applied logic, or physics. A modified form of the Kantian attempt to explain the fruitfulness of mathematical reasoning on an intuitive rather than logical basis is that of Poincare.
