June 1, 1917] 



SCIENCE 



551 



ing, mathematics clearly dealt with matters 

 of secondary importance. Is not the finger 

 more important than the number attached 

 to it in view of its position in relation to 

 the other fingers? Why should the primi- 

 tive races then have turned their thoughts 

 away from the most important to matters 

 of secondary importance? As I under- 

 stand it, this turning away from un- 

 fathomable but most enticing difficulties to 

 the fathomable but less enticing ones is the 

 great keynote of science, and mathematics 

 was perhaps the first scientific subject to 

 sound this keynote with decided clearness. 

 It was, however, not always sounded with 

 clearness by mathematicians. The Pytha- 

 goreans, for instance, endeavored to make 

 it appear that numbers were endowed with 

 noble properties which are entirely foreign 

 to them. They were not yet sufficiently 

 modest to study mathematics most efi'ee- 

 tively, and their spirit has its representa- 

 tives even in our day. 



The picture representing the attitude of 

 mind which contributed most powerfully 

 to mathematical investigation would, how- 

 ever, not be complete without uniting with 

 modesty discretion and a love for mental 

 travel and exploration. The traveler and 

 the explorer are usually first attracted to 

 regions which are easily accessible and 

 whence it is easy to retrace one's steps. 

 The attitude of mind which is exhibited by 

 the common expression "safety first" has 

 been largely responsible for the trend of de- 

 velopments in mathematics. With the pass- 

 ing of years these safe permanent thought 

 roads have acquired historic interest and 

 they have naturally been used as models by 

 those who aim to open up scientific regions 

 where mire and quicksand impede progress, 

 and sometimes engulf roads which had been 

 supposed to be secure. 



The chief function of mathematics in 

 scientific research is, however, not the ciilti- 



vation of a feeling of modesty or of a desire 

 for mental travel and exploration. The 

 problem of the mathematical investigators 

 is a much more difficult one, since it relates 

 to the discovery and development of unify- 

 ing processes which are sufficiently compre- 

 hensive to avoid bewilderment as a result 

 of a maze of details, and yet sufficiently 

 close to the concrete to become useful in the 

 widely separated fields of scientific en- 

 deavor. With the growth of scientific 

 knowledge in various fields, the problems of 

 the mathematical investigators becomes 

 more complex, and the world has never been 

 in greater need for more mathematics than 

 at the present time, since civilization never 

 before called so loudly for perspicuity in 

 science. 



It is not an easy matter to characterize 

 briefly and yet clearly the function of 

 mathematics in the broad field of scientific 

 endeavor. A prominent feature of mathe- 

 matical work is that it changes postulates 

 or assumptions into different forms, which 

 are sometimes more readily accessible for 

 experimentation than the original postu- 

 lates were or are more directly useful in 

 the solution of other scientific difficulties. 

 The transformation of postulates, or ac- 

 cepted conclusions, is not peculiar to mathe- 

 matics, but is common to all sciences. In 

 mathematics these transformations, or the 

 results derived therefrom, serve as a means 

 to obtain new transformations, while in the 

 other sciences they serve as a means to 

 wrest from nature a new truth. Mathe- 

 matics is commonly guided by current hy- 

 potheses about nature in the selection of 

 her postulates, but after these are once se- 

 lected, she is interested in building up a 

 world for herself by constructions which, 

 are necessary consequences of these postu- 

 lates. If these constructions represent only 

 a small fraction of the interesting questions 

 which appear to become clarified by logical 



