MATHEMATICS, PHILOSOPHY OF

the study of the nature of mathematics, including underlying assumptions of the discipline and its scope. A chief point of interest that has emerged from modern attempts to characterize philosophy is the importance of distinguishing dialectical or analytical inquiries about meaning from empirical inquiries about fact. A primary, traditional task of the philosopher has been to present things in such a light that human feelings may be reasonably grounded. The need for this is especially obvious in the case of the moral philosopher or the aesthetician, whose work treats explicitly of subjective concerns. But the need remains the same in all philosophical inquiries, including even discussions of the foundations of mathematics. For it is obvious to a careful observer that persons who put forward theses about the nature of mathematics are involved not just intellectually but also emotionally in their pursuits; while it must be supposed that in some cases this involvement stems from or inevitably leads to intellectual confusion, it must also be allowed that a certain emotional commitment may perhaps be a necessary condition for the making of discoveries. Thus it can scarcely be an accident that no great mathematician has ever accepted the conventionalist view according to which mathematical truths are man-made. The inquiry into the nature, underlying assumptions, and scope of mathematics has emerged in the 20th century as a subdiscipline of mathematics itself, known as the study of foundations. For a full historical treatment of this field, see the article mathematics, foundations of. The material below, edited from an article originally written by Alfred North Whitehead for the 11th edition of the Encyclopdia Britannica, treats mathematics itself as an object of philosophical investigation. It has been usual to define mathematics as the science of discrete and continuous magnitude. Even Leibniz, who initiated a more modern point of view, follows the tradition in thus confining the scope of mathematics properly so called, while apparently conceiving it as a department of a yet wider science of reasoning. A short consideration of some leading topics of the science will exemplify both the plausibility and inadequacy of the above definition. Arithmetic, algebra, and the infinitesimal calculus, are sciences directly concerned with integral numbers, rational (or fractional) numbers, and real numbers generally, which include incommensurable numbers. It would seem that the general theory of discrete and continuous quantity is the exact description of the topics of these sciences. Furthermore, can we not complete the circle of the mathematical sciences by adding geometry? Now geometry deals with points, lines, planes, and cubic contents. Of these all except points are quantities. Also, as the Cartesian geometry shows, all the relations between points are expressible in terms of geometric quantities. Accordingly, at first sight it seems reasonable to define geometry in some such way as the science of dimensional quantity. Thus every subdivision of mathematical science would appear to deal with quantity, and the definition of mathematics as the science of quantity would appear to be justified. We have now to see why the definition is inadequate.