466 



SCIENCE 



[N. S. Vol. XLV. No. 1168 



be understood, and that without mathe- 

 matics no science can develop to maturity. 

 One delights in observation and the record 

 of facts, believing that he has understood 

 a class of phenomena when he has given a 

 general description of their relations, order 

 a;nd connections; the other considers the 

 mathematical formula as "the point 

 through which all the light gained by sci- 

 ence must pass in order to be of use in 

 practise. ' ' 



But it is true, I believe, that mathe- 

 matics is generally recognized as essential 

 at least to scientific progress; and what 

 I intend to do this evening is to discuss 

 the nature of the provision made by it for 

 the needs of science. But I can not consent 

 to enter upon a treatment of this topic 

 without pausing a moment, first of all, to 

 combat a certain dangerous and apparently 

 rather widespread error among scientists, 

 namely, that the primary and perhaps the 

 sole function of mathematics is to assist in 

 the solution of the problems of science. If 

 this were the essential purpose of mathe- 

 matics, you would not find it cultivated as 

 it is to-day. Instead of this the men who 

 now give all their energy to its develop- 

 ment would be allied with the particular 

 natural sciences and would study mathe- 

 matics only as auxiliary to their central 

 concern. One could find no incentive to 

 labor otherwise. 



Fundamentally mathematics is a free 

 science. The range of its possible topics 

 appears to be unlimited; and the choice 

 from these of those actually to be studied 

 depends solely on considerations of inter- 

 est and beauty. It is true that interest has 

 often been, and is to-day as much as ever, 

 prompted in a large measure by the prob- 

 lems actually arising in natural science, and 

 to the latter mathematics owes a debt only 

 to be paid by essential contributions to the 



all, the fundamental motive to its activity 

 is in itself and must remain there if its 

 progress is to continue. 



Some mathematicians are glad when 

 their fields of thought touch other sciences 

 (or even practical matters) ; others concede 

 the fact unwillingly or without interest; 

 others still would perhaps consider them- 

 selves unorthodox in their feeling if they 

 allowed any matter of connections with 

 other things to affect at all their interest 

 in their own fields. It was perhaps an ex- 

 treme ease of this feeling which prompted 

 Sylvester to the pious or impetuous hope 

 that no use would ever be found for the 

 theory of invariants which he was devel- 

 oping with so much delight. 



But it has turned out, as the mathe- 

 matician is now well aware, that it is these 

 same invariants which afford us an ex- 

 pression for the laws of nature. An in- 

 variant is simply a thing or a relation 

 which remains unaltered when the elements 

 with which it is connected undergo a cer- 

 tain class or group of transformations. 

 When we know the transformations to 

 which a class of phenomena are subject 

 the matter of finding out the laws con- 

 necting these phenomena is a problem of 

 invariants. It may be that a particular 

 physical law was discovered long before the 

 idea of invariants arose; but it is never- 

 theless true that a useful connection among 

 such laws is afforded by this notion. On 

 the other hand, we may have an equation 

 expressing a fundamental relation among 

 a large class of phenomena and find that 

 this equation is invariant when the ele- 

 ments in it are subjected to a certain group 

 of transformations. We may be sure that 

 this group of transformations has some- 

 thing essential to do with the phenomena 

 in consideration, and that its invariants ex- 

 press (partially or completely) the laws 



i 



interpretation of phenomena. But after governing these phenomena. Our problem 



