368 THE POPULAR SCIENCE MONTHLY 



On the main idea of what mathematical proof should be, the mathe- 

 matician can hardly agree with Schopenhauer. Schopenhauer consid- 

 ers 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 Schopen- 

 hauer that intuition should play an important part, especially in pre- 

 liminary courses, before children enter upon courses in demonstrative 

 geometry; but eventually the logical proof must be made to follow be- 

 fore we are prepared to accept a proposition as established. Schopen- 

 hauer directs his criticisms particularly against Euclid's proof of the 

 Pythagorean Theorem and then offers his own proof, which is practi- 

 cally the same as the Hindu proof and can be given by drawing the fig- 

 ure and then explaining, as did the Hindus, " Behold." But Schopen- 

 hauer'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 in- 

 fluenced 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."^® Huxley, in the June number of the Fortnightly Review, 

 1869, said: Mathematics is that study "which knows nothing of ob- 

 servation, nothing of experiment, nothing of induction, nothing of 

 causation."^'' Huxley and Hamilton both name observation, experi- 

 ment, induction, but they differ in the fourth process. Hamilton say& 

 " analogy " ; Huxley says " causation." 



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



^^ Edinburgh Bevieiv, Vol. 22, p. 433. 



=" Fortnightly Beviexo, London, Vol. 5, 1869, p. 667. 



