530 



SCIENCE. 



N. S. Vol. XI. No. 275. 



more accessible to English readers by the 

 treatises of Forsyth and of Harkness and 

 Morley. 



The critical spirit of our age is, in a large 

 measure, due to the study of the theory 

 of analytic functions. "Newton assumes 

 without hesitation the existence, in every 

 case, of a velocity in a moving point, with- 

 out troubling himself with the inquiry 

 whether there might not be continuous 

 functions having no derivative."* When 

 it was discovered that such functions exist 

 and that the works of some of the great- 

 est mathematicans of the preceding cen- 

 turies have to be modified in some instances, 

 mathematicians naturally became much 

 more exacting in regard to rigor, and thus 

 ushered in an age which may be compared 

 with the times of Euclid with respect to its 

 demands for rigor. Whether our critical 

 age will produce a work which, like Euclid, 

 will serve as a model for millenniums can- 

 not be foretold, but it seems certain that 

 works which can stand the critical tests of 

 this age will stand the tests of all ages. 



The critical spirit of our times is the 

 foundation of what has been styled the 

 arithmetization of mathematics. This move- 

 ment which the late Weierstrass knew so 

 well to lead is pervading more and more 

 the whole mathematical world. We are 

 rapidly banishing from our treatises the 

 term quantity and replacing it by the word 

 number. Our geometric intuitions are 

 forced into the background and logical de- 

 ductions from definitions are taking their 

 places. Who can conceive of curves which 

 have no tangent at any of their rational 

 points in a given interval ? Nevertheless it 

 is well known that such curves exist. An 

 account of such functions was first pub- 

 lished by Hankel in ISTO.f 



Mathematicians find themselves in a 



* Klein, Evansion Colloquium, 1894, p. 41. 

 fCf. Pierpont, Bulletin of the American Mathematical 

 Society, vol. 5, p. 398. 



great dilemma at this point. Geometric 

 intuition has been such a strong instrument 

 of research and has given so much life and 

 beauty to mathematical investigation that 

 mathematicians cling to it, as their own 

 lives. It is an enormous price when rigor 

 can be purchased only with geometric in- 

 tuition. Yet, in the present stage of mathe- 

 matical thought, this seems to be the only 

 thing that will be accepted, and mathema- 

 ticians stand helpless before this decree. 



A few examples may throw some light on 

 this subject. What do we understand by 

 the length of a continuous curve ? The in- 

 tuitionalist says, if we connect different 

 points of the continuous curve by straight 

 lines and find the sum of the lengths of 

 these straight lines, then the length of the 

 curve will be the limit of this sum as the 

 number of the points is indefinitely in- 

 creased. Jordan was the first to call atten- 

 tion to the fact that this sum need not have 

 a limit. Hence there are continuous curves 

 which do not have any length according to 

 the ordinary definition of length. In fact 

 a number of area-filling curves have re- 

 cently been studied, and Cantor has shown 

 that a multiplicity of any number of dimen- 

 sions can be put in a one to one correspond- 

 ence with a multiplicity of one dimension. 



These are some of the facts which have 

 compelled mathematicians to construct their 

 own worlds — the number worlds. Conclu- 

 sions drawn in one number world do not 

 necessarily apply to another. When a 

 problem is under consideration the number 

 world is so chosen as to meet the demands 

 of the problem. For instance, the construc- 

 tions and demonstrations of Euclid's geom- 

 etry seem to require only a space composed 

 of quadratic numbers.* Hence it appears 

 that we do not need to assume that space is 

 continuous in order to demonstrate the 

 theorems of elementary geometry. Simi- 



*Cf. Strong, Bulletin of the American Mathematical 

 Socie/y, vol. 4, 1898, p. 443. 



