COLLIGATIVE RELATIONS AND SCIENTIFIC LAWS 127 



A relation of Euclidean geometry. "The sum of the angles of a 

 triangle is 180°." With straight lines defined by the paths of light 

 rays, Gauss carefully carried through a large scale triangulation 

 that showed this relation sound within the (narrow) limits of what 

 could legitimately be regarded as experimental error. Kant notwith- 

 standing, Gauss apparently considered that a failure of the relation 

 would be recognizable. Even at the most fundamental level of our 

 conception of space, a colligative relation would not then be so con- 

 ventionalized as to be forever made "true." Indeed, once we agree on 

 the denotations attaching to the conceptual entities of geometry, it 

 becomes possible to prefer one geometry to another. In macrocosmic 

 applications most physicists today prefer to Euclid's a Riemannian 

 geometry in which the relation stated above is no more than an 

 approximation. 



THE "PROOF" AND ENDURANCE OF COLLIGATIVE RELATIONS 



What is a legitimate generalization is not given us a priori, but al- 

 ways we seek maximum deployment of each relation. Not content 

 simply to interpolate, we also extrapolate; e.g., we extend a graphed 

 relation into regions where we do not have, and perhaps cannot get, 

 any experimental points whatever. Such extrapolations we recognize 

 as potentially hazardous; for whenever experience takes us into new 

 and unfamiliar realms, as Bridgman observes, 



. . . we must be prepared to find, and as a matter of fact we have 

 often found, that we encounter phenomena of an entirely novel char- 

 acter for which previous experience has given us no preparation. 



Yet, so profound is our confidence in the principle of continuity, we 

 still remain bold enough to essay the most extreme extrapolations. At 

 the very worst, we think, the relation will not blow up abruptly but 

 fail only gradually— becoming a progressively poorer approximation. 

 And, such is our faith in continuity, ^ven today we are sometimes 

 lamentably slow to recognize the failure, even as approximations, of 

 apparently plausible extrapolations. 



A theory may be harmful if it encourages us to forget that such 

 extrapolations may prove fallible. But it is harmful, too, if it too 

 strongly discourages generalization and extrapolation. Provided that 

 we permit no relation to become permanently conventionalized, 



