228 ANNUAL EEPORT SMITHSONIAN INSTITUTION, 19 30 



do not consider what the atom is made of nor what its structure is, 

 but usually feel justified in assuming properties which are as simple 

 as possible. For example, in the elementary kinetic theory it is 

 assumed that the molecules are hard, elastic spheres, not because 

 anyone really believes that molecules have these properties, but 

 merely because these are the simplest properties we can think of 

 which are consistent with the known facts. 



What we really do, therefore, is to replace in our minds the actual 

 gases which we observe and which have many properties which we 

 do not fully understand by a simplified model, a human abstraction, 

 which is so designed by us that it has some of the properties of the 

 thing we wish to displace. 



There is thus a difference of degree rather than of kind between 

 the adoption of a mechanical model and the development of a math- 

 ematical theor}?- such as Euclidian geometry. When the mathe- 

 matical physicist develops an abstract theory of actual phenomena — 

 for example, Hamilton's equations to summarize the laws of me- 

 chanics — he is in reality constructing a mathematical model. Math- 

 ematical equations have certain definite properties or rather they 

 express certain relationships between the symbols which enter them. 

 In a mathematical theory of physical phenomena the equations are 

 so chosen that the relation between the symbols corresponds in some 

 simple way to that which is observed between measurable physical 

 quantities which are the bases of our concepts of physics. 



Within recent years, especially in the development of the rela- 

 tivity and quantum theories, physicists have been making increasing 

 use of mathematical forms of expression, and have been giving less 

 attention to the development of mechanical models. The older gen- 

 eration of physicists and chemists and those among the younger men 

 who are less skilled in the use of mathematics are inclined to believe 

 that this is only a temporary stage and that ultimately we must be 

 able to form a concrete picture or model of the atom, that is, to get 

 a picture of what the atom is really like. It seems to be felt that a 

 mechanical model whose functioning can be understood without the 

 aid of mathematics, even if it only gives the qualitative representa- 

 tion of the phenomena in question, can represent the truth in some 

 higher sense than a mathematical theory whose symbols perhaps can 

 be understood only by a mathematician. 



There is, I believe, no adequate justification for this attitude. 

 Mechanical models are necessarily very much restricted in scope. 

 The relationships of their parts are limited to those that are already 

 known in mechanics (or in electricity or magnetism). Mathematical 

 relationships are far more flexible ; practically any conceivable quan- 

 titative or qualitative relationship can be expressed if desired in 



