SUMMING UP.
149. The examination of the idea of the infinite is of the greatest importance, because it is inseparably united with the idea of God.
150. We have the idea of the infinite; but the disputes concerning its nature, and even its existence, denote its obscurity.
151. The finite is that which has limits.
152. The infinite is not the same as the indefinite. The infinite is that which has no limits—thenot-finite; the indefinite is that to which no limits are assigned—thenot-defined.
153. The difference between the infinite and the finite is founded on the principle of contradiction: the finite affirmslimits; the infinite denies them: there is no medium between yes and no.
154. Limit is the negation of a being, or of something real, applied to a being: the limit of a line is the point which terminates it; the limit of a force is the point beyond which it does not extend.
155. The idea of the infinite, denying limit, denies a negation; therefore it is an affirmative idea: the idea of the finite is negative, because it affirms a negation.
156. The idea of the infinite is applied to many orders of beings, and presents strange anomalies, which seem contradictions. A line produced to infinity in only one direction appears infinite, since it is greater than all finite lines; and it is not infinite, because it has a limit in the point where it starts. The same thing is verified in surfaces and solids. To explain these anomalies we must attend to the following observations.
157. The idea of the infinite is not intuitive. We have no intuition of an object either absolutely or relatively infinite.
158. The idea of the infinite is an indeterminate conception formed by the union of the two indeterminate ideas of being in general, and the negation of limit in general.
159. The indeterminate conception of the infinite gives us no knowledge of any thing infinite.
160. The anomalies and apparent contradictions, which we find in the application of the idea of the infinite, vanish when we reflect that the difference of the results depends on the different conditions under which we apply the idea of the infinite. Things which would be infinite under one condition cease to be so when considered under other conditions: the apparent contradiction is caused by one not remarking the change of conditions.
161. We have the conception of infinite number, for wecan unite in our mind the two indeterminate conceptions of number and the negation of limit.
162. We have the conception of infinite extension, for we can unite the two indeterminate ideas of extension and the negation of limit.
163. The possibility or non-contradiction of conceptions in the purely ideal order does not prove their possibility in the real order. When the conceptions are realized, their reality is not in an abstract extension or an abstract number, but in individual extended beings, or individual numbers: the determinateness implied by the reality may involve contradiction to the true infinity, although it be impossible for us to discover any contradiction in the indeterminate conception, which abstracts the conditions of their realization.
164. Although we have the conception of infinite extension, it is impossible for us to imagine it.
165. No extrinsic or intrinsic repugnance can be discovered in the existence of infinite extension.
166. We cannot know by purely philosophical means whether the extension of the universe is infinite or finite.
167. Although an absolutely infinite number may be indeterminately conceived, it is not susceptible of any arithmetical or geometrical expression: no series of what mathematicians call infinite expresses an absolutely infinite number.
168. The intrinsic impossibility of anactualinfinite number may be demonstrated from the intrinsic repugnance of theco-existenceof certain things which may benumbered.
169. The idea of the absolutely infinite real being cannot be indeterminate: it necessarily involves positive and formal perfections.
170. All that does not imply a contradiction must beaffirmed of the infinite being. That which is absurd is not a perfection.
171. Analyzing indeterminate and intuitive ideas, we find that all the reality contained in them is affirmed of God.
172. The absolutely infinite being must be intelligent.
173. Intelligence is a perfection which does not imply contradiction.
174. Will and liberty must also be found in the absolutely infinite being.
175. The indeterminate idea of the infinite is favored by the combination of the ideas of being and not-being.
176. The idea of an absolutely infinite being consists in the idea of a union of all being that involves no contradiction.
177. The indeterminate idea of a real infinite being, or of God, is formed from the idea of an absolutely infinite being, combined with the intuitive ideas of intelligence, will, liberty, causality, and all others that can be conceived without imperfection, in any infinite degree.