218 THEORIES AXD MODELS 



more than particular marks on paper. Whether premises and con- 

 clusions are "true," in the sense of corresponding to something in 

 experience, is quite emphatically not the concern of pure logic— the 

 sole concern of which is to ensure that a conclusion is a conclusion, 

 formally necessitated by the premises. The classical syllogism, for 

 example, is a form devoid of content and, more generally, the formal- 

 isms of logic involve only vacant relations. Indeed, the necessity of 

 the conclusions drawn from such formalisms entails that vacancy. 

 Logicians then quite wisely decline the scientist's endeavor to de- 

 velop useful applications of their formalisms. For to secure such ap- 

 plications the scientist must identify the symbols of the formalism 

 with concepts having experiential denotations. Just as with geometry, 

 the logically necessary conclusions are then rendered statements 

 about experience that experience may or may not confirm. 



Some very simple illustrations can easily be given in the case of 

 mathematics where, Einstein suggests, an entirely similar situation 

 prevails. 



Mathematics deals exclusively with the relations of concepts to 

 each other without consideration of their relation to experience. 



Consider this elementary statement of arithmetic: 



1 + 1=2 



One stone plus another stone gives two stones; one apple plus one 

 orange gives two pieces of fruit; and so on. Is pure madiematics then 

 teaching us something about experience? Certainly not! We have 

 identified the abstract symbols with "things" we already knew con- 

 formable with the stated relation. Lacking such prior knowledge, we 

 are not always so fortunate. One barely subcritical mass of plutonium 

 plus another such mass do not yield two units of mass, but a nuclear 

 explosion with products having less than two units of rest mass; one 

 quart of alcohol plus one quart of water do not make two quarts of 

 fluid but something less; one rabbit plus another rabbit. . . . Plainly, 

 in this second group of instances, predictions founded on the arith- 

 metic statement may fail of confirmation. 



Assuming the formalism of simple algebra, we readily agree that 

 (60) (1) = (1) (60). But it does not follow that if 1 man can dig a 

 post-hole in 60 seconds then 60 men can dig that hole in 1 second. 

 Or consider a less facetious case: 



