Everyone will acknowledge the considerable difference between a sensation actually present (of heat, for instance) and the mere idea of one previously experienced, or shortly to come.  This consists in the greater force, liveliness, and vividness of the former.  Although these two classes of states (the idea of a landscape described by a poet and the perception of a real one, anger and the thought of anger) are only quantitatively distinct, they are scarcely ever in danger of being confused—­the most lively idea is always less so than the weakest perception.  The actual, outer or inner, sensations may be termed impressions; the weaker images of memory or imagination, which they leave behind them, ideas.  Since nothing can gain entrance to the soul except through the two portals of outer and inner experience, there is no idea which has not arisen from an impression or several such; every idea is the image and copy of an impression.  But as the understanding and imagination variously combine, separate, and transpose the elements furnished by the senses and lingering in memory, the possibility of error arises.  A hidden, and, therefore more dangerous source of error consists in the reference of an idea to a different impression than the one of which it is the copy.  The concepts substance and causality are examples of such false reference.

The combination of ideas takes place without freedom, in a purely mechanical, way according to fixed rules, which in the last analysis reduce to three fundamental laws of association:  Ideas are associated (1) according to their resemblance and contrast; (2) according to their contiguity in space and time; (3) according to their causal connection.  Mathematics is based on the operation of the first of these laws, on the immediate or mediate knowledge of the resemblance, contrariety, and quantitative relations of ideas; the descriptive and experimental part of the sciences of nature and of man on the second; religion, metaphysics, and that part of physical and moral science which goes beyond mere observation on the third.  The theory of knowledge has to determine the boundaries of human understanding and the degree of credibility to which these sciences are entitled.

The objects of human thought and inquiry are either relations of ideas or matters of fact.  To the former class belong the objects of mathematics, the truths of which, since they are analytic (i. e., merely explicate in the predicate the characteristics already contained in the subject, and add nothing new to this), and since they concern possible relations only, not reality, possess intuitive or demonstrative certainty.  It is only propositions concerning quantity and number that are discoverable a priori by the mere operation of thought, without dependence on real existence, and that can be proved from the impossibility of their opposites—­mathematics is the only demonstrative science.