232 THEORIES AND MODELS 



Although the theory of observ^ables appears to be a theory about 

 measurements, it is still far removed from measurements as they are 

 actually carried out, and presupposes a particle analogy which is not 

 directly given in experimental data. 



The analogy invests the formalism with experiential relevance, and 

 so makes of it a scientific theory; the analogy allows us to hold the 

 physical ideas in mind, and so makes it possible for us to use that 

 theory. Dirac's theory, classic exemplar of the scientific theory "purely 

 mathematical," is nothing of the sort. 



Model implicit in formalism. \Mien we look at Dirac's quantum 

 mechanics we nowhere find explicit reference to a model. How then 

 does a model or analogy come to be associated with a theory that 

 o£Fers as overt premises only a set of equations? It is entirely a matter 

 of our previous experience with equations of related types. Given 

 that experience, we at once associate with the formalism a model 

 or analogy founded on the system(s) to which we have earlier ap- 

 plied such equations. A model or analogy is then implicit in the 

 formalism. 



Like Dirac's system, Schrodinger's wave mechanics appears at 

 first sight a purely mathematical consti'uction. By now we know that 

 must be an illusion, and in this case the analogy is utterly inseparable 

 from the wave equation just because, as Hutten says, 



. . . whenever we see a certain differential equation of the second 

 order (in the space and time co-ordinates), we think of waves, . . . 



Is what we are thus led to think of any importance? Schrodinger 

 stresses the centrality of the e.xtra-mathematical character of his 

 theory, in terms very similar to those we found Dirac applying to his. 



... I do not refer to the mathematical difficulties, which eventually 

 are always trivial, but rather to the conceptual difficulties. 



The "probability waves" of an electron are quite unlike any familiar 

 wa\'es, but the analogy posed by the implicit wave model is still 

 sufficient to permit us to master those "conceptual difficulties." 



A "purely mathematical" system can, then, function as a genuine 

 physical theory, if its equations imply to us something not purely 

 mathematical. Hutten's summary seems entirely just. 



There are ... no mathematical models in physics: the equation by 

 itself is not the model. The wave equation is a model only because 



