ATTACKS UPON THE STUDY OF MATHEMATICS 367 



Often a reductio ad absurdum shuts all the doors one after another, until only one 

 is left through which we are therefore compelled to enter. Often, as in the 

 proposition of Pythagoras, lines are drawn, we don't know why, and it after- 

 wards appears that they were traps which close unexpectedly and take prisoner 

 the assent of the astonished learner . . . (page 94). Euclid's logical method 

 of treating mathematics is a useless precaution, a crutch for sound legs . . . 

 (page 95). The proposition of Pythagoras teaches us a qualitas occulta of the 

 right-angled triangle; the stilted and indeed 

 fallacious demonstration of Euclid forsakes 

 us at the tohy, and a simple figure, which we 

 already know, and which is present to us, 

 gives at a glance far more insight into the 

 matter, and firm inner conviction of that 

 necessity, and of the dependence of that quality 

 upon the right triangle: 



In the case of unequal catheti also, and indeed generally in the case of every 

 possible geometrical truth, it is quite possible to obtain such a conviction based 

 on perception . . . (page 96). It is the analytical method in general that I wish 

 for the exposition of mathematics, instead of the synthetical method which 

 Euclid made use of. 



In the above we have Schopenhauer's famous characterization of 

 mathematical reasoning as "mouse-trap proofs" (Mausefallenbeweise). 

 These quotations and other passages which space does not permit us to 

 quote indicate that his objections are directed almost entirely against 

 Euclid. Schopenhauer discloses no acquaintance with such modern 

 mathematical concepts as that of a function, of a variable, of coordinate 

 representation, and the use of gi-aphic methods. With him Euclid and 

 mathematics are largely synonymous. Because of this one-sided and 

 limited vision we can hardly look upon Schopenhauer as a competent 

 judge of the educational value of modern mathematics. 



If Schopenhauer's criticism of Euclid is taken as the expression of 

 the feelings, not of an ? Ivanced mathematician, but of a person first 

 entering upon the study of geometry and using Euclid's " Elements," 

 then we are willing to admit the validity of Schopenhauer's criticisms, 

 in part. Euclid did not write his geometry for children. It is a his- 

 torical puzzle, difficult to explain, how Euclid ever came to be regarded 

 as a text suitable for the first introduction into geometry. Euclid is 

 written for trained minds, not for immature children. Of interest is 

 Schopenhauer's reference to the method of proof, called the reductio 

 ad absurdum. The experience of teachers with this method has been 

 much the same in all countries. Some French critics called it a method 

 which "convinces but does not satisfy the mind." De Morgan says: 

 " The most serious embarrassment in the purely reasoning part is the 

 reductio ad absurdum, or indirect demonstration. This form of argu- 

 ment is generally the last to be clearly understood, though it occurs 

 almost on the threshold of the ' Elements.' We may find the key to the 

 difficulty in the confined ideas which prevail on the modes of speech 

 there employed." 



