KiASS. y,^7 CHAPTER VIII 



Theories and Models 



'oNCEWiNG scientific theories as 

 postulational systems, we begin again with that famihar paradigm, 

 EucHdean geometry. Viewed as it is by mathematicians, of what does 

 EucHdean geometry consist? We find (1) certain posits: "point," 

 "Hne," etc.; and (2) a set of stipulated axioms. Following these come 

 specifications (very incompletely indicated by Euclid himself) of 

 (3) certain syntactic rules, the formal manipulations we are author- 

 ized to use in drawing deductions from the axioms. We then readily 

 produce (4) the manifold of theorems of Euclidean geometry. 



Euclidean geometry so conceived is a purely formal system. The 

 axioms, expressing relations among the posits that figure in them, 

 supply some implicit definitions of "point," "line," and so forth. But 

 nowhere are denotations associated with these terms. All the other 

 entities of Euclidean geometry "constructed" from the primary posits 

 are then left equally devoid of experiential relevance. Through this 

 novel conception of a wholly inapplicable abstract geometry (fully 

 achieved only in the 19th century ) we come to grasp the possibility of 

 non-Euclidean geometries turning on somewhat different sets of 

 axioms. These geometries are less familiar than Euclid's but— if, like 

 his, each is a consistent development of a set of noncontradictory 

 axioms— they are equally valid, because equally insusceptible to con- 

 firmation or falsification in the world of experience with which they 

 have no link whatever. 



Now all this seems absurd: we "know" that Euclidean geometry 

 applies to the spatial relations of our terrestrial world. Quite so, but 



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