Mat 18, 1917] 



SCIENCE 



469 



sciously while many of them are apparently 

 unperceived. In other words, while we 

 are applying mathematics we are at the 

 same time making use of the customary 

 large bundle of prejudices and preconcep- 

 tions which we have not yet f!ound a way 

 to avoid whenever we have to treat the 

 phenomena of nature. 



In order to illustrate the way in which 

 applied mathematics is shot through with 

 assumptions, let us take a single case. 

 When we begin to apply numbers in the 

 measurement of physical quantities we as- 

 sume at once that the true measure may as 

 well be an irrational number as a rational 

 number. For this assumption we have no 

 a priori grounds or experimental reasons. 

 We simply find it the most convenient as- 

 sumption to make and we make it, having 

 no other justification than convenience for 

 our procedure. It may very well be true 

 that the universe is so constructed that 

 every measurement in it yields essentially 

 a rational number. This would be true if 

 all material things, all force, all energy, 

 etc., were granular in structure and the 

 mutual ratios of all granules were com- 

 mensurable. For instance, if it should turn 

 out that mass is to be estimated by count- 

 ing the number of granules (all alike) in a 

 body, then the mass of the body would be 

 expressed essentially by a rational num- 

 ber. If electricity consists altogether of 

 electrons of equal charge, then the measure 

 of any charge of electricity is essentially a 

 rational quantity. But we treat the phe- 

 nomena mathematically as if essentially 

 irrational quantities occur generally in na- 

 ture. Even the continuity of space appar- 

 ently rests upon just such sheer assump- 

 tions. 



Starting with this fundamental hypoth- 

 esis of applied mathematics, we might fol- 

 low the subject from these remotely ab- 

 stract regions into the things of more com- 



mon thought, and in doing so we should 

 find that such fundamental assumptions 

 obtrude themselves at every point. If we 

 looked persistently only at this side of the 

 matter we should probably lose all con- 

 fidence in our theoretical interpretations; 

 but fortunately we are able always to test 

 our results approximately with experi- 

 mental data. It is not that we are testing 

 out our processes of reasoning (we have 

 another method for doing that) ; but rather 

 that we are examining as to whether we 

 have found a construction and interpre- 

 tation which fit in with phenomena to a 

 satisfying degree. 



After these remarks I will hardly need 

 to urge the necessity of testing in the labo- 

 ratory all results obtained from given 

 hypotheses by logical processes. For we 

 can never know the truth of any hypoth- 

 esis, or even understand its import, until 

 we know the consequences which flow from 

 it. Whether the conclusions reached in this 

 be determined only by an appeal directed 

 to experience. 



Among the fields of applied mathematics 

 that of rational mechanics has been most 

 completely transfused by the mathematical 

 spirit and it is here that the latter has ex- 

 hibited some of its most characteristic con- 

 quests. It has here shown how high mathe- 

 matical skill on the part of some investi- 

 gators is necessary to the greatest progress 

 of science, illustrating the way in which 

 the mathematical spirit and method fur- 

 nish a bond of union to the separate divi- 

 sions of physical science. 



So far we have considered the character 

 of the provisions made by mathematics for 

 the needs of science. It remains to give 

 some specific details as to the past and some 

 indication of apparently probable lines of 

 development in the future. Clearly, a 

 catalogue of specific provisions would be 

 impossible in this address; such a thing 



