If an absolute demonstration of the existence of an immaterial principle in us be insisted upon, we must reply, that philosophy, like geometry, has its axioms, that is to say, its self-evident truths, which need not, or, if we choose to say so, which cannot be mathematically demonstrated. The existence of the soul is one of those axioms of philosophy. Diogenes answered a rhetorician who denied movement by walking in his presence. By expressing any thought, by saying "yes," or "no," we may prove the existence of the immortal soul to the sophists who would attempt to contest it.
We have just said that geometry has its axioms. Let us remember that an entire school of geometricians amused themselves by disputing the axioms, under the pretext that it was impossible to demonstrate them. We were present, in December, 1866, at a curious sitting of the Institute, during which M. Lionville, a celebrated mathematician, and professor at the Sorbonne, explained this strange polemic with great skill.
In attempting to demonstrate the propositions of geometry, certain axioms, _i.e._, self-evident truths, must be admitted in the first place. Otherwise, the primary reasoning will have no basis. But, among the numerous propositions of this kind which present themselves to the mind, and which result from the admission of one of their number, which is the most evident? That depends on the nature of the mind of each of us, and therefore it is that there is not, and that there never will be, an argument on this question.
There is a school of geometry which pretends to demonstrate everything. There is another, the true and good school, which, recognizing that the human mind has limits, and that everything is not accessible by our thoughts, lays down, under the name of axioms, certain truths which do not require proof, or, which is often the same thing, are incapable of proof.
Among the number of self-evident truths, or truths difficult of demonstration, we find the question of parallel lines. What are two parallels? Two lines which never meet each other. But how can we prove this property of two lines by reasoning? That is not, exactly speaking, possible, since the notion of the infinite is not admitted, or not understood by everybody, and cannot, therefore, serve as the basis of an absolutely rigorous argument.