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SCIENCE 



[N. S. Vol. XLI. No. 1056 



the history of thought — such a conception 

 as will at least enable them as educated 

 men to mention the subject without a feel- 

 ing of sham or to hear it mentioned without 

 a feeling of shame. A few well-considered 

 lectures should suffice. At all events it 

 would not require many to show the his- 

 torical background of the calculus, to ex- 

 plain the nascence and nature of the scien- 

 tific exigencies that gave it birth, to make 

 clear the concepts of derivative and integral 

 as the two central notions of its two great 

 branches, and to present a few simple ap- 

 plications of these notions to intelligible 

 problems of typical significance. Even the 

 idea of a differential equation could be 

 quickly reached, the nature of a solution 

 explained, and simple examples given of 

 physical and geometric interpretations. As 

 to the range and power of the calculus, a 

 sense and insight can be given, in some 

 measure of course by a reference to its 

 literature, but much more effectively by a 

 few problems carefully selected from vari- 

 ous fields of science and skillfully explained 

 with a view to showing wherein the meth- 

 ods of the calculus are demanded and how 

 they serve. Is not all this elementary and 

 undergraduate 1 In point of nomenclature, 

 yes. It is not necessary, however, to let 

 words deceive us. We teach whole numbers 

 to young children, but even Weierstrass was 

 not aware of the logico-mathematical deeps 

 that underlie cardinal arithmetic. 



The calculus, however, is hardly the topic 

 with which the course would naturally 

 begin. A principal aim of the course 

 should be to show what mathematics, in' its 

 inner nature, is — to lay bare its distinctive 

 character. Its distinctive character, its 

 structural nature, is that of a "hypothetico- 

 deductive ' ' system. Probably, therefore, it 

 would be well to begin with an exposition 

 of the nature and function of postulate 

 systems and of the great role such systems 



have always played in the science, espe- 

 cially in the illustrious period of Greek 

 mathematics and even more consciously and 

 elaborately in our own time. It is plain 

 that such an exposition can be made to 

 yield fundamental insight into many 

 matters of interest and importance not only 

 in mathematics, but in logic, in psychology, 

 in philosophy, and in the methodology of 

 natural science and general thought. The 

 material is almost superabundant, so numer- 

 ous are the postulate systems that have been 

 devised as foundations for many different 

 branches of geometry, algebra, analysis, 

 Mengenlehre and logic. A general survey 

 of these, were it desirable to pass them all 

 in review, would not be sufficient. It wiU 

 be necessary to select a few systems of 

 typical importance for minute examination 

 with reference to such capital points as 

 convenience, simplicity, adequacy, inde- 

 pendence, compatibility and categoricalness. 

 The necessity and presence of undefined 

 terms in any and all systems will afford a 

 suitable opportunity to deal with the highly 

 important, much neglected and little under- 

 stood subject of definition, its nature, vari- 

 eties and function, in light of the recent . 

 literature, especially the suggestive hand- 

 ling of the matter by Enriques in his 

 "Problems of Science." A given system 

 once thus examined, the easy deduction 

 of a few theorems will suffice to show the 

 possibility and the process of erecting upon 

 it a perfectly determinate and often impos- 

 ing superstructure. And so will arise 

 clearly the just conception of a mathe- 

 matical doctrine as a body of thought com- 

 posed of a few undefined together with 

 many defined ideas and a few primitive or 

 postulated propositions with many demon- 

 strated ones, all concatenated and welded 

 into a form independent of will and tem- 

 poral vicissitudes. Revelation of the charm 

 of the science will have been begun. A 



