238 THEORIES AND MODELS 



the time in an abstract way, the algebraic axioms that they satisfy 

 and the connection [semantic rules] between equations involving 

 them and physical conditions being all that is required. The axioms, 

 together with this connection, contain a number of physical laws, 

 which cannot conveniently be analyzed or even stated in any other 

 way. 



The symbols are never defined, yet they have for us some physical 

 signification. How do they acquire it? Concluding a careful analysis 

 of Dirac's theory, Hesse asks : 



What then is the significance of the concepts in Dirac's theory? The 

 answer is clearly to be found in terms of the classical analogy. It is 

 this that gives meaning to the purely formal statements of the hy- 

 pothesis and hence gives rules for manipulation of the concepts. 

 Dirac's discussions about measurement and observability become 

 meaningful if we realize that he has in his mind, not practically pos- 

 sible experimental measurements, but a highly idealized system of 

 particles like those considered in classical dynamics, only with the 

 difference that complete information about the positions and mo- 

 menta of the particles cannot be obtained . . . 



No simple particle model can possibly represent what Dirac's theory 

 seeks to convey. But let me now try to show how a hierarchic model 

 can, and does. 



From the extreme abstraction of the formalism that offers neither 

 overt model nor direct denotations for its primitive concepts, we 

 Avork our way back through a sequence of progressively less abstract 

 theories. W'ithout even thinking much about it, we loosely relate the 

 concept of stationary state, in an advanced quantum mechanics, to 

 the far more readily intelligible concept of electronic orbit in the 

 older quantum theory. Thence, presumably, we work still further 

 back, to classical mechanics— and perhaps at last to the simplest 

 common-sense concepts. No one model or analogy here suffices. Only 

 through an entire series of incomplete but overlapping models and 

 analogies do we contrive ultimately to grasp the nature of the primi- 

 ti\'e concepts. As Hutten correctly emphasizes, the very same chain 

 of successive partial interpretations— terminating in familiar concepts 

 liaving clear denotations— is also precisely what we need to link up 

 the derivative concepts in the abstract theory with what we actually 

 ■do and see in our experiments. 



The hierarchic model is a somewhat diffuse composite or super- 



