[Footnote 1:  Thus it results that knowledge, although dependent on experience for all its materials, extends beyond experience.  The understanding is completely bound in the reception of simple ideas; less so in the combination of these into complex ideas; absolutely free in the act of comparison, which it can omit at will; finally, again, completely bound in its recognition of the relation in which the ideas it has chosen to compare stand to one another.  There is room for choice only in the intermediate stage of the cognitive process; at the beginning (in the reception of the simple ideas of perception, a, b, c, d), and at the end (in judging how the concepts a b c and a b d stand related to each other), the understanding is completely determined.]

More certain than our sensitive knowledge of the existence of external objects, are our immediate or intuitive knowledge of our own existence and our mediate or demonstrative knowledge of the existence of God.  Every idea that we have, every pain, every thought assures us of our own existence.  The existence of God, however, as the infinite cause of all reality, endowed with intelligence, will, and supreme power, is inferred from the existence and constitution of the world and of ourselves.  Reality exists; the real world is composed of matter in motion and thinking beings, and is harmoniously ordered.  Since it is impossible for any real being to be produced by nothing, and since we obtain no satisfactory answer to the question of origin until we rise to something existent from all eternity, we must assume as the cause of that which exists an Eternal Being, which possesses in a higher degree all the perfections which it has bestowed upon the creatures.  As the cause of matter and motion, and as the source of all power, this Being must be omnipotent; as the cause of beauty and order in the world, and, above all, as the creator of thinking beings, it must be omniscient.  But these perfections are those which we combine in the idea of God.

Intuitive knowledge is the highest of the three degrees of knowledge.  It is gained when the mind perceives the agreement or disagreement of two ideas at first sight, without hesitation, and without the intervention of any third idea.  This immediate knowledge is self-evident, irresistible, and exposed to no doubt.  Knowledge is demonstrative when the mind perceives the agreement (or disagreement) of two ideas, not by placing them side by side and comparing them, but through the aid of other ideas.  The intermediate links are called proofs; their discovery is the work of the reason, and quickness in finding them out is termed sagacity.  The greater the number of the intermediate steps, the more the clearness and distinctness of the knowledge decreases, and the more the possibility of error increases.  In order for an argument (e. g., that a = d) to be conclusive, every particular step in it (a = b, b = c, c = d) must possess intuitive certainty.  Mathematics is not the only example of demonstrative knowledge, but the most perfect one, since in mathematics, by the aid of visible symbols, the full equality and the least differences among ideas may be exactly measured and sharply determined.