THEORIES AND MODELS 237 



ing what they mean. Certainly the symbol F in the calculation of a 

 lunar orbit has only the most tenuous association with our muscle 

 sensations and experience of springs. Yet we do conceive analogy 

 here, and we grasp F in terms of the comparatively irrelevant sensa- 

 tions and experience. 



In the microcosm Newtonian mechanics functions as in the macro- 

 cosm. The primitive concepts may now refer to entities totally un- 

 observable. In the kinetic theory of gases we do not observe molecu- 

 lar velocities, but infer them, perhaps from measurements of density 

 and pressure. We do not measure the mass of a molecule, but infer 

 it, perhaps by way of observations of the Brownian motion of much 

 larger particles. Again the effective semantic rules attach to deriva- 

 tive concepts, such as pressure. Again Newton's explicit definitions 

 do not supply those rules, but again they lead us to the conception 

 of simple models from which, as before, we find it easy to draw the 

 rules. And we find entirely parallel situations in many other theories 

 involving microcosmic unobservables. Mendel's conception of ge- 

 netic factors and Kekule's conceptions of valence bonds stipulate 

 unobservables we can rather easily grasp in terms of fairly straight- 

 forward models. But we must now look at other microcosmic theories 

 in which use of such models is no longer possible. 



Third case: Indirect denotations and hierarchic models. Dirac 

 prefaces his exposition of his theory with the following explanatory 

 statement. 



We introduce certain symbols which we say denote physical things 

 such as states of a system or dynamical variables. These symbols we 

 shall use in algebraic analysis in accordance with certain axioms which 

 will be laid down. To complete the theory we require laws [i.e., 

 identifications, or semantic rules] by which any physical conditions 

 may be expressed by equations between the symbols and by which, 

 conversely, physical results may be inferred from equations between 

 the symbols. A typical calculation in quantum mechanics will now 

 run as follows: One is given that a system is in a certain state in which 

 certain dynamical variables have certain values. This infonnation is 

 expressed by equations involving the symbols that denote the state 

 and the dynamical variables. From these equations other equations 

 are then deduced in accordance with the axioms governing the sym- 

 bols and from the new equations physical conclusions are drawn. 

 One does not anywhere specify the exact nature of the symbols em- 

 ployed, nor is such specification at all necessary. They are used all 



