THEORIES AND MODELS 217 



formalisms we arrive at colligative relations that fail the test of 

 experience. But a properly constructed Riemannian geometry eventu- 

 ates in colligative relations that, in the range of terrestrial magnitudes 

 and to the sensitivity of terrestrial measurements, yield the same pre- 

 dictions we draw from the Euclidean relations. Thus, in terrestrial 

 applications Euclidean geometry is not forced on us but, rather, pre- 

 ferred by us for its superior "simplicity." Euclidean geometry is then 

 again typically a scientific theory, judged for sufficiency like any 

 other scientific theory. 



Euclidean geometry highlights an important point characteristic 

 of scientific theories generally, but not elsewhere always as evident 

 as here. Alicays a scientific theory is the aggregate of a formalism and 

 a model. The formalism constitutes the deductive machinery re- 

 quired for the theory's function as a correlative device; the model 

 gives rise to the multitude of semantic rules and, as we will see, 

 much more besides. In view of this conclusion, my account of scien- 

 tific theories will proceed by way of separate treatment of formalisms 

 on the one hand and models on the other. But we must keep well in 

 mind that important interactions between model and formalism must 

 be allowed for when, as we will, we come to theories superficially 

 very difiFerent from Euclidean geometry. 



Logic and Mathematics 



With reference to the basic notion of congruence, Poincare comments 

 that "if there were no solid bodies in nature, there woidd be no 

 geometry." Thus greatly conditioned by experience, formal geometry 

 is however not at all concerned with experience: it tells us nothing 

 whatever about solid bodies, or anything else, to be found in the 

 world. Affiliating semantic rules with the formalism, we enter the 

 realm of the physical scientist who, however indirectly or approxi- 

 mately, must always somehow render account of the world of ex- 

 perience. But precisely as we make that affiliation we part company 

 with the geometer, and quit the realm of strictly formal disciplines. 

 Profoundly involved in all scientific advance, and substantially in- 

 fluenced by such advance, pure logic and pure mathematics are ob- 

 viously not themselves sciences. 



Formal logic imposes certain syntactic rules for the combination 

 and transformation of sets of symbols that never represent anything 



