220 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1933 



When a subject is developed from concepts, the concepts play 

 the part of the terms occurring in the axioms of geometry. It may 

 be convenient to have some sort of idea of what the concepts stand 

 for, to give crude illustrations of them, as for example when we 

 make to ourselves a crude picture of a geometrical point or line. 

 But actually no use is made of these crude illustrations. The con- 

 cepts are undefined save as being governed by propositions of which 

 they are subjects. To say of energy that it is that which is con- 

 served in processes of certain types takes us no further in under- 

 standing what energy means, nor has the proposition any content 

 unless a measurement process is specified by which conservation can 

 be tested. As Mr. Bertrand Russell remarks in an oft-quoted sen- 

 tence, the upshot of this is that — in geometry for example — mathe- 

 maticians do not know what they are talking about and do not care ; 

 they do not care because to care would be irrelevant. 



In the other method of procedure, a synthetic process is followed. 

 Things experienced or observed come first, and then combinations of 

 these are constructed which have such aspects of generality that they 

 give insight into the relations between the things observed. The 

 things observed lead to generalizations which involve terms with 

 an observational meaning, and these generalizations sum up many 

 possible observations. 



Physical science in its theoretical development tends, not to oscil- 

 late between the two methods, but to replace the first method by the 

 second. The method of concepts often requires the greater imagina- 

 tion and the deeper insight to isolate the concepts which are to prove 

 useful. But it is more primitive. In a way, it saves trouble. The 

 method of defining each term in terms of things already observed 

 requires much painstaking analysis. But it has the ultimate merit of 

 avoiding the unnecessary. The method of concepts affords no test 

 as to whether the concepts are essential. Indeed, the method of con- 

 cepts is a matter of mental economy, but not of logical economy. It 

 is a pioneer method, without which progress would be often slow or 

 impossible. Concepts wisely chosen lead to discoveries, to phenom- 

 ena previously unsuspected. Often the conceptual character disap- 

 pears of itself, when things introduced as concej^ts become objects of 

 observation. At other times unnecessary concepts are only bundled 

 out by a great revolution in thought. The atomic theory of matter 

 and the dynamical theory of gases both started with concepts, 

 the concept of atomic entities of different species and the concept of 

 these atomic or molecular entities in motion interacting dynamically 

 with one another. The atomic theory of matter was not implied by 

 the rules of chemical combination — the conservation of mass and the 



