216 THEORIES AND MODELS 



geometry. Precisely as we thus establish the set of denotations (what 

 logicians call semantic rules), we cease to deal with a formal system 

 that is neither true nor false but only logically consistent. We come 

 instead to deal with a physical theory that may, or may not, prove 

 sufficient to impose a rational order on elements of our experience. 

 I mark this transition by calling the abstract system studied by math- 

 ematicians the "Euclidean formalism," while calling the "Euclidean 

 theory" that aggregate of formalism with semantic rules the non- 

 mathematician takes for granted when he speaks of "Euclidean 

 geometry." 



In die Euclidean theory the derived theorems are rendered col- 

 ligative relations by virtue of the denotations attaching to their con- 

 ceptual terms. These denotations, though designated "semantic 

 rules," cannot of course be reduced to any hard set of explicit rules. 

 On the contrary, like the denotations of scientific concepts generally, 

 they reflect the characteristic compromise forever struck among re- 

 liability, generality, and simplicity. If we describe a "point" as a 

 0.5 mm pencil spot on paper we give a semantic rule adequate for 

 the high-school student of geometry. But for the machinist this is 

 much too broad a "point"; and for the engineer concerned with the 

 guidance of long-range missiles this is much too narrow a "point" 

 ( for him a "point" may be several miles in diameter) . 



As in so many other cases, our grasp of meaning is secured by our 

 sense of purpose. We carry in our mind's eye some image of the 

 "ideal" entities of a Euclidean theory in which the theorems are 

 simply truths by definition. We thus frame conceptual models against 

 which we measure elements of our experience which afford us only 

 approximate cognates of points with no parts, lines with no breadth, 

 solids with no deformability, etc. For the purpose in hand we may 

 or may not judge the approximations "close enough" to proceed by 

 the methods of Euclidean geometry. And, if we do so proceed, our 

 predictions may quite well fail if we have erred in judging what 

 are adequate approximations to the ideal entities of the Euclidean 

 theory. 



By and large, predictions drawn from the colligative relations ac- 

 commodated in the Euclidean theory are excellently borne out. The 

 Euclidean theory thus proves sufficient to the test of experience, but 

 this rather mild adjective is the strongest justifiable. To be sure, if 

 we attach the same set of semantic rules to most of the non-Euclidean 



