24 INTRODUCTION TO PHILOSOPHY OF SCIENCE 



questionable predictions, doubtful methods. Thus when the 

 scientist looks at his product with a view to the presup- 

 positions which were employed in its construction, with a 

 view to the inherent weaknesses in its internal structure, 

 with a view to the possible inadequacies which may later 

 come to light — when a scientist does this he ceases to be a 

 scientist and becomes a philosopher. 



Two examples of such crises in science, both drawn from 

 the last century, may be offered for consideration. 



The first is taken from mathematics, and, more specifically, 

 from the field of geometry. Probably no one of the sciences 

 has had a more complacent history, a more continuous de- 

 velopment, and fewer regressive movements. Euclidean 

 geometry was a monument to scientific ingenuity, exhibiting 

 internally a high degree of consistency and affording ex- 

 ternally a highly accurate and extremely convenient method 

 for measuring land, constructing buildings, and meeting 

 innumerable problems of practical life. Then came two 

 individuals, about the middle of the last century, Riemann 

 and Lobachewski, who showed in a conclusive manner that 

 there could be alternative systems of geometry, i.e., that the 

 Euclidean system of geometry was not the geometry but 

 merely one among a large number of such systems. Each 

 such system was shown to be internally perfectly consistent; 

 from a group of axioms and postulates accepted as true the 

 total body of propositions could be deduced by strictly 

 logical processes. Rut between systems there was an essen- 

 tial contradiction. For example, in the Euclidean system, 

 the sum of the angles of a triangle is equal to 180° but in the 

 Lobachewskian system the sum of the angles of a triangle 

 is always less than 180°; and in the Riemannian system the 

 sum is always greater than 180°. Immediately the question 

 arises, which system is true? Clearly they cannot all be 

 true for they are incompatible; yet one of them must be true. 



To answer this question would take one into the very 

 heart of recent physical science. What should be pointed 

 out here is the change in character which geometry experi- 



