gination 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.

We reach certainty in matters of fact by direct perception, or by
inferences from other facts, when they transcend the testimony of our
senses and memory. These arguments from experience are of an entirely
different sort from the rational demonstrations of mathematics; as the
contrary of a fact is always thinkable (the proposition that the sun will
not rise to-morrow implies no logical contradiction), they yield, strictly
speaking, probability only, no matter how strong our conviction of their
accur