454 



SCIENCE 



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



est; and that impx'ession is just. At the 

 same time another impression will be 

 gained, namely, that the various branches 

 rest, each of them, upon a foundation of 

 its own. This impression will have to be 

 corrected. It will have to be shown that 

 the branch-foundations are not really fun- 

 damental in the science but are, literally 

 and genuinely, component parts of the 

 superstructure. It will have to be shown 

 that mathematics as a whole, as a single 

 unitary body of doctrine, rests upon a basis 

 of primitive ideas and primitive proposi- 

 tions that lie far below the so-called branch- 

 foundations and, in supporting the whole, 

 support these as parts. The course will, 

 therefore, turn to the task of acquainting 

 its students with those strictly fundamental 

 researches which we associate with such 

 names as C. S. Peirce, Schroeder, Peano, 

 Frege, Russell, Whitehead and others, and 

 which have resulted in building underneath 

 the traditional science a logico-mathematical 

 sub-structure that is, philosophically, the 

 most important of modern mathematical 

 developments. 



It must not be supposed, however, that 

 the instruction must needs be, nor that it 

 should preferably be, confined to questions 

 of postulate and foundation, and I will 

 devote the remainder of the time at my 

 disposal to indicating briefly how, as it 

 seems to me, a large or even a major part 

 of the course may concern itself with mat- 

 ters more traditional and more concrete. 



Any one can see that there is an abun- 

 dance of available material. There is, for 

 example, the history and significance of the 

 great concept of function, a concept which 

 mathematics has but slowly extracted and 

 gradually refined from out the common con- 

 tent and experience of all minds and which 

 on that account can be not only defined pre- 

 cisely and intelligibly to such laymen as 

 are here concerned, but can also be clarified 



in many of its forms by means of manifold 

 examples drawn from elementary mathe- 

 matics, from the elements of other sciences, 

 and from the most familiar phenomena of 

 the work-a-day world. 



Another available topic is the nature and 

 role of the sovereign notion of limit. This, 

 too, as every mathematician knows, admits 

 of countless illustration and application 

 within the radius of mathematical knowl- 

 edge here presupposed. In this connection 

 the structure and importance of what 

 Sylvester called "the Grand Continuum," 

 which so many scientific and other folk 

 talk about unintelligently, will offer itself 

 for explanation. And if the class fortu- 

 nately contain students of philosophic 

 mind, they will be edified and a little 

 astonished perhaps when they are led to see 

 that the method and the concept of limits 

 are but mathematicized forms of a process 

 and notion familiar in all domains of spir- 

 itual activity and known as idealization. 

 Not improbably some of the students will 

 be sufficiently enterprising to trace the 

 mentioned similitude in some of its mani- 

 festations in natural science, in psychology, 

 in philosophy, in jurisprudence, in litera- 

 ture and in art. 



I have not mentioned the modern doe- 

 trine variously known as Mengenlehre, 

 Mannigfaltigkeitslehre, the theory of point- 

 sets, assemblages, manifolds or aggregates: 

 a live and growing doctrine in which ex- 

 pert and layman are about equally inter- 

 ested and which, like a subtle and illu- 

 minating ether, is more and more pervading 

 mathematics in all its branches. For the 

 avocational instruction of lay students of 

 "maturity and power" how rich a body 

 of material is here, with all its fascinating 

 distinctions of discrete and continuous, 

 finite and infinite, denumerable and non- 

 denumerable, orderless, ordered, and well- 

 ordered, and with its teeming host of near- 



