time or plurality in space. Magnitude exists in a variety of forms or systems relative to one another, and the nature of the system in which we live is an empirical problem to be solved by experience alone. Moreover, the 'theory of transformation', by which we pass from one system of magnitudes to another, reveals the fact that the origin of the notion of magnitude exerts no determining influence on either the development or certainty of mathematics. Mathematics does not rest on a groundwork of certain judgments of a priori origin. The question whether mathematics has a 'certain' basis is unmeaning, for mathematics has no basis at all. The sole principle taken absolutely for granted is that of all reasoning or discourse, the principle of contradiction. Two judgments, contradictory if admitted simultaneously, may be admitted successively without contradiction, but any principle contradictory in its successive consequences must be rejected. The certainty of mathematics is due to the nature of its subject matter and of its method of procedure. Mathematics is a deduction of attributes from elements by judgments exclusively analytic. It affirms that if such elements have such attributes, there exists between them such relations. Hence, its conclusions are relative to premises and have a certainty proportional to that of the premises. Yet the conclusions are necessary, because no opposite judgments can be proposed to render them false, or doubtful, for the premises have been stated in all possible forms. Furthermore, the validity of the premises cannot be called in question, because they are not to be regarded from an absolute point of view. Though true when stated, they yield immediately to inverse judgments, when the latter are stated. Mathematical principles have only the validity we wish to attribute to them with reference to the system of magnitudes under consideration; but this validity we can make as great as we please. Mathematical principles are neither a priori syntheses nor empirical facts; they are conventions or labels for cataloguing various systems of magnitudes, some applicable to all, some to only one system. But experience alone can enable us to choose from all possible conventions those applicable to our own world.
This article is introductory to a series of articles which will advocate a kind of an analysis of the simplest and most pervading aspect of experience, namely acquaintance. It is a preliminary survey of the data of experience. Faint and peripheral sensations are included in experience. Attention is not a prerequisite for experience, for attention is a selection among objects that are 'before the mind' and, therefore, presupposes a larger field. Some facts—those which we see for ourselves—are included in our present experience. We now experience past things which we remember, for the remembering refers to something known to be in the past, a definite experience in the past, and not to the image which we call up now. We know that the group of things now experienced is not all-embracing, partly because of our belief in
