Mathematical Space and Time

Abstract

Physical Space and Time are thus one with mental space and time, or, more strictly, portions of the one Space and Time may be enjoyed and are identical with parts of the one contemplated Space and Time. Space and Time as we have regarded them are empirical or experienced extension or duration, though as continua of moments or points they have been described by help of conceptual terms. Are Space and Time so regarded the Space and Time of the mathematicians, and if so, what is the difference between the metaphysics and the mathematics of them? Our answer will be, that directly or indirectly mathematics is concerned with empirical Space and Time, and that, however remote from them mathematics may seem and be, they are never in mathematics torn away from their original. But a difficulty meets us at the outset because of the different conceptions of mathematics entertained by mathematicians themselves. According to some, Space and Time are the absolute or total Space and Time consisting of entities called points or instants. According to others, and they are the more philosophical mathematicians, Space and Time are not extension or duration, not in any sense stuff or substance,¹ as Descartes, to all intents and purposes, conceived particular spaces to be, but relations between material things which move in them.