December 15, 1916] 



SCIENCE 



839 



fields of the natural sciences sees the result 

 — the conclusion — before him and tries to 

 work back to underlying causes. Nature 

 has laid a foundation and reared thereon a 

 mighty superstructure, through which the 

 natural scientist wanders amid a maze of 

 halls and chambers, scratching the surface 

 a little here and a little there trying to 

 find what sort of a foundation can support 

 all this that he sees. The natural scientist 

 accepts as his foundation that theory which 

 best explains the results. The theory may 

 be wrong, but it serves all the purposes of 

 a scientific theory if it explains to a fair 

 degree of satisfaction observed phenomena. 

 I seem to remember to have read the state- 

 ment of a physicist that we should prob- 

 ably explain some phenomena of light on 

 the wave theory and other phenomena on 

 the atomic theory. Whenever a theory is 

 contradicted by experiments, the natural 

 scientist seeks another. This may seem a 

 rather "hit or miss" way of scientific re- 

 search, but it is the best that man can hope 

 for with his human intellect trying to find 

 first causes underlying the workings of a 

 universe. 



The mathematician is not thus restricted. 

 He lays his own foundation. Some natural 

 science may furnish the material for this 

 foundation, but the mathematical mason 

 handles each stone and sets it in proper 

 relationship in the mortar of consistency. 



By being an exact science, mathematics 

 serves the natural sciences in two ways. In 

 the first place, the methods of mathematical 

 deduction offer a convenient means of test- 

 ing the consistency of a theory. Mathe- 

 matics will take the essential elements of a 

 theory as postulates and deduce the neces- 

 sary conclusions. If this leads to a con- 

 tradiction of experiment, the incorrectness 

 of the theory is shown. It might even be 

 possible in certain cases to locate exactly 

 what part of the theory is at fault. If the 



deduced results agree with experiment our 

 faith in our theory is strengthened. An ex- 

 ample of this sort of thing is to be found in 

 Carmiehael 's ' ' Theory of Relativity. ' ' The 

 author, a mathematician, has taken as his 

 postulates statements whose truth is ac- 

 cepted by a number of physicists. He has 

 arrived, by purely mathematical means, at 

 results whose truth or falsity are susceptible 

 of experimental proof. The results of such 

 an experiment as he suggests would dis- 

 prove or increase our faith in the truth 

 of his postulates. 



A second way in which this exact science 

 can serve the natural sciences, and which 

 does not differ much from the way already 

 mentioned, is in the matter of discovery. 

 If the postulates of a natural theory are 

 true, then all its consequences are true. 

 Mathematics offers a tool for finding out 

 these consequences. A classic example of 

 discovery in this manner is Maxwell's pre- 

 diction of the pressure due to heat or light 

 radiation, which was not experimentally 

 demonstrated for several years after Max- 

 well's death. Sir W. R. Hamilton's predic- 

 tion of conical refraction is another such 

 example. This prediction was experimen- 

 tally verified by his colleague Lloyd within 

 a short time after it was announced. This 

 mathematical working out of the conse- 

 quences of a theory has, in my judgment, 

 not received its due at the hands of the 

 natural scientist. 



A more universal service rendered by 

 mathematics has been the furnishing of a 

 system of shorthand that is as exact and 

 much more workable than the completely 

 written out statement. If you do not be- 

 lieve in the value of a well-chosen symbol- 

 ism, try to calculate the value of 22-J dozen 

 eggs at 39|- cents per dozen by using Roman 

 notation. A well-known example of the 

 value of symbolism is furnished by mathe- 

 matics itself in the development of the 



