THEORIES AND MODELS 221 



which, as Weizsacker observes, we learn something about the world 

 of which logic and mathematics as such say nothing. 



A real path of light is for a living, observing man a road of wonder; a 

 possible path of light is a geometrical curve and nothing more. But 

 the "possible things" are on the other hand necessary instiTiments of 

 thought. For since we do not know in advance the full truth, we can 

 think of the real only by separating it out from the plenitude of the 

 possible. 



Abstract formalism and scientific law. My primary concern is the 

 role of formalisms in co-ordinating scientific laws in theories. I ap- 

 proach this matter by way of very brief consideration of the second 

 major scientific application of logic and mathematics, in the discov- 

 ery and expression of such laws. Weyl comments that: 



. . . Kepler's astronomical discovery would have been impossible 

 without the Greek geometer's preceding discovery of the ellipses as a 

 mathematically simple class of curves. 



In modern times the idea of quanta first occurs to Planck only after 

 he had expressed the data on blackbody radiation in a formula he 

 could not have obtained had he known nothing of exponential func- 

 tions. Beyond thus furnishing (1) essential expressions for our laws, 

 logic and mathematics provide ( 2 ) techniques for the analysis of our 

 data. Whether the analyses are as sophisticated as those required of 

 Kepler and Planck, or as simple as that required to discover Boyle's 

 law in Boyle's J-tube data, such analyses are clearly indispensable. 



Closely related to the last function of abstract analysis, observe 

 that it gives us (3) capacity to calculate empirically Inaccessible 

 "data" that may manifest some significant regularity. I explicitly dis- 

 tinguish "data" from data: "data" are inferences drawn, from data 

 observed, with the aid of previously established colligative relations 

 and implicit invocation of the principle of continuity. As an example 

 consider that, like Cannizzaro, I suspect and seek some regularity in 

 the relative densities of various gases under the same conditions of 

 temperature and pressure. For practical reasons ( involatility, insta- 

 bility, etc. ) all the densities cannot be measured under the same con- 

 ditions. I then make my measurements at difiFerent temperatures and 

 pressures and, applying Boyle's and Charles' laws, mathematically 

 reduce my data to "data" allegedly representative of some one stand- 



