*THE POPULAR SCIENCE MONTHLY*

On the main idea of what mathematical proof should be, the mathematician can hardly agree with Schopenhauer. Schopenhauer considers a succession of separate logical conclusions, which are contained in a rigorous mathematical proof, as insufficient and unendurable; he wants to be convinced of the truth of a theorem instantaneously, by an act of *intuition.* He advances the theory that, besides the severely logical deductions there is another method of proving mathematical truths, that of direct perception and intuition. We agree with Schopenhauer that intuition should play an important part, especially in preliminary courses, before children enter upon courses in demonstrative geometry; but eventually the logical proof must be made to follow before we are prepared to accept a proposition as established. Schopenhauer directs his criticisms particularly against Euclid's proof of the Pythagorean Theorem and then offers his own proof, which is practically the same as the Hindu proof and can be given by drawing the figure and then explaining, as did the Hindus, "Behold." But Schopenhauer's is not a *general* proof; it holds only for a special case, namely, for the isosceles right triangle.

Eeally, Euclid's proof of the Pythagorean Theorem consists of a number of steps, each of which is quite evident to the eye. Thus a square is represented as the sum of two rectangles, which is an intuitive relation. Then each rectangle is shown to be equal to double a triangle of the same base and altitude. This again the child accepts the more readily as more or less intuitively evident. And so on. Every step appears quite reasonable to one depending on intuition alone. It does seem as if Schopenhauer could have made a better selection from Euclid for his point of attack.

From what we have said it appears that Schopenhauer's attack bears only indirectly upon the question relating to the mind-training value of mathematics; his criticism is focused directly upon questions of logic, of mode of argumentation and of sufficiency of proof.

I pass now to a third attack upon mathematics, made in 1869 by the naturalist, Thomas H. Huxley. So far as I know, Huxley was not influenced either by Hamilton or Schopenhauer, though the words he used remind us of a sentence in Hamilton. Hamilton had said: "Of Observation, Experiment, Induction, Analogy, the mathematician knows nothing."^{[1]} Huxley, in the June number of the *Fortnightly Review,* 1869, said: Mathematics is that study "which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation."^{[2]} Huxley and Hamilton both name observation, experiment, induction, but they differ in the fourth process. Hamilton says "analogy"; Huxley says "causation."

In the same year there appeared in print an after-dinner speech de-