To pass from the logical world to that of reality, all that is required is a fact to serve as a bridge. If this fact be offered to the understanding, the two banks are joined, and science commences. I feel, I think, I exist: these are facts of consciousness; combine any one of them with the principle of contradiction, and what before were sterile intuitions become prolific ratiocinations, embracing at once the world of ideas, and that of reality.
140. Even in the purely ideal order, the principle of contradiction is sterile unless joined with particular truths of the same order. In geometry, for example, it is often argued thus: such a quantity is either greater or less than another, or equal to it; for otherwise it would be both greater and less, equal and unequal, at the same time, which is absurd. Here the principle of contradiction is effectively applied, not alone, but together with a particular truth which makes such an application available. Thus, in the above argument, no use could be made of the principle of contradiction, to prove equality or inequality were not the existence or non-existence of one of the two previously proved or supposed: since this neither does nor can result from the principle of contradiction which includes, not a particular idea, but the most general ideas presented to the human mind.
141. General truths, of themselves, even in the purely ideal order, lead to nothing, because of the indeterminateness of the ideas which they contain; and, on the other hand, particular truths of themselves produce no result, because they are limited to what they are, making reasoning, which cannot take one step without the aid of general ideas and propositions, impossible. Light results from the union of one with the other; separated they afford only an abstract and vague intuition, or the contemplation of a particular truth, which, limited to a contracted sphere, can give no knowledge of beings considered under a scientific aspect.
142. We shall see when we come to treat of ideas, that our mind has two very distinct classes of them; the one supposes space, and cannot abstract it, such are all geometrical ideas: the other does not relate to space, and includes all non-geometrical ideas. These two orders of ideas are separated by an impassable abyss, if the two orders are not approximated by a simultaneous use of both. The ideal order is not complete without this approximation; and the real order of the universe is turned into a chaos, or rather disappears, if real and ideal truths are not combined, in both the geometrical and non-geometrical orders. From all geometrical ideas imaginable, considered in all their ideal purity, nothing would result for the ideal non-geometrical order, for the world of material, much less of immaterial realities; and, on the other hand, from non-geometrical ideas alone we could not get so much as the idea of a right line. This observation shows that there is for us in the ideal order no one truth, the origin of all other truths; for if we take the geometrical order, we are limited to those combinations which do not go out of it; if the non-geometrical order, we lack the idea of space, without which we lose even the possibility of conceiving the corporeal world.(14)