So far we have considered only questions concerning the foundations of the mathematical sciences. Indeed, the study of the foundations of a science is always particularly attractive, and the testing of these foundations will always be among the foremost problems of the investigator. Weierstrass once said, "The final object always to be kept in mind is to arrive at a correct understanding of the foundations of the science.[19] But to make any progress in the sciences the study of particular problems is, of course, indispensable." In fact, a thorough understanding of its special theories is necessary to the successful treatment of the foundations of the science. Only that architect is in the position to lay a sure foundation for a structure who knows its purpose thoroughly and in detail. So we turn now to the special problems of the separate branches of mathematics and consider first arithmetic and algebra.
[14] Cf. Bohlmann, "Ueber Versicherungsmathematik", from the collection: Klein and Kiecke, Ueber angewandte Mathematik und Physik, Leipzig, 1900.
[15] Die Mechanik in ihrer Entwickelung, Leipzig, 4th edition, 1901.
[16] Die Prinzipien der Mechanik, Leipzig, 1894.
[17] Vorlesungen über die Principe der Mechanik, Leipzig, 1897.
[18] Einführung in das Studium der theoretischen Physik, Leipzig, 1900.
[19] Math. Annalen, vol. 22, 1883.
7. IRRATIONALITY AND TRANSCENDENCE OF CERTAIN NUMBERS.
Hermite's arithmetical theorems on the exponential function and their extension by Lindemann are certain of the admiration of all generations of mathematicians. Thus the task at once presents itself to penetrate further along the path here entered, as A. Hurwitz has already done in two interesting papers,[20] "Ueber arithmetische Eigenschaften gewisser transzendenter Funktionen." I should like, therefore, to sketch a class of problems which, in my opinion, should be attacked as here next in order. That certain special transcendental functions, important in analysis, take algebraic values for certain algebraic arguments, seems to us particularly remarkable and worthy of thorough investigation. Indeed, we expect transcendental functions to assume, in general, transcendental values for even algebraic arguments; and, although it is well known that there exist integral transcendental functions which even have rational values for all algebraic arguments, we shall still consider it highly probable that the exponential function