*MATHEMATICAL INSTRUCTION*

By Professor G. A. MILLER

LELAND STANFORD JUNIOR UNIVERSITY

SEVERAL prominent writers have suggested that 'pure mathematics' should be called 'free mathematics' in view of the great latitude of freedom both in subject matter and in methods of work.^{[1]} This view is diametrically opposed to the one commonly held. The student of elementary text-books on mathematics can not fail to be impressed by the close similarity in subject matter and in methods. In our arithmetics and algebras we find problems which are quite similar to those in the work written by Ahmes, an Egyptian priest living seventeen hundred years before Christ. Our geometries bear such a close resemblance to Euclid's, written three hundred years before Christ, that the terms Euclid and elementary geometry are still practically synonymous in the minds of many teachers.

Moreover, there are numerous *a priori* reasons to regard mathematics as a slave rather than as a free, living, pulsating being, exhibiting the many exhilarating changes characteristic of youthful development. The reasonableness of the main postulates of mathematics has never been questioned. In most cases contrary hypotheses would appear perfectly absurd to those who have not been trained to mistrust their intuitions. The main function of mathematics has been, and probably always will be, to draw necessary conclusions from such postulates. Whether the postulates have been explicitly stated or not is a secondary matter. Hence mathematics has become a vast structure which is perfectly invulnerable except possibly at its foundation, and here the attack seems merely a matter of words. The most that has been done in an effectual manner is to erect structures on other sets of postulates. It should be emphasized that the main duty of the mathematician is to build upon given postulates. In these operations he will always be free from attack, since he can not arrive at any conclusion without proving that it is the only possible one.^{[2]}

- ↑ Cf. Liebmann,
*Jahresbericht der Deutschen Mathematiker-Vereinigung*, Vol. 14 (1905), p. 231. - ↑ No attempt is here made to define the term mathematics. The views expressed are equally true whether mathematics is defined as a method, or whether the objects which it considers should enter into the definition. From

vol. lxviii.—11.