236 THEORIES AND MODELS 



ory which it is hoped may be expanded in the future to explain more 

 generahzations than it was originally designed to explain must allow 

 more freedom to its theoretical terms than would be given them were 

 they to be logical constructions out of observable entities. 



Newton sought for his system applications far beyond the familiar 

 realm of molar teiTestrial phenomena. And so of course he does not 

 shackle his concepts with explicit denotations applicable only in that 

 realm. One cannot expect to determine the mass of an atom with the 

 equal-arm balance, or to apply a spring balance to measure the force 

 of attraction between earth and moon. In its macrocosmic and micro- 

 cosmic applications the Newtonian system takes on a quite different 

 aspect. 



Second case: Indirect denotations and simple models. The "given" 

 in celestial dynamics is represented by certain directional co-ordi- 

 nates at which a certain planet, say, has been seen at certain times. 

 From these we calculate, by the methods of the formalism, various 

 distances, velocities, accelerations, and the like; and from these, 

 finally, we may calculate the masses of the planets and the forces 

 acting upon them. The Newtonian system is here linked with ex- 

 perience only by way of its derivative concepts: the semantic rules 

 attach to these and not to the primitive concepts. Indeed the primi- 

 tive concepts are now in some sense unobservables: we no longer 

 measure force and mass with any directness but instead infer their 

 values from other quite different values we do measure. 



In its macrocosmic applications Newtonian dynamics thus takes 

 on something of the appearance of quantum mechanical theories. 

 For in these also the primitive concepts are unobservables only very 

 indirectly evaluable, through the medium of the theory and the im- 

 plicit definitions constituted by its axioms. Whatever the similarity, 

 however, there is also a striking difference: in celestial dynamics we 

 find the primitive concepts readily conceivable. Newton's explicit 

 definitions of his primitive concepts are here just as inactive as in 

 the domain of molar terrestrial phenomena: they do not themselves 

 supply the working denotations. But they are also just as active here 

 as there: they lead us to conceive the analogies and models from 

 which we draw the working denotations— which here we attach to 

 the derivative terms in the formalism. And, in terms of these models 

 and analogies, w^e find we can represent even the primitive concepts 

 in concrete models and diagrams, and have always the sense of know- 



