244 THEORIES AND MODELS 



"system" and "state" can be used in a statement amenable to an em- 

 pirical test. 



The transformation of energy from one form into another can be 

 understood in terms of this intei-pretation. Mechanical, electric, mag- 

 netic, chemical, and radiant energy produce heat and make the en- 

 gine run; and all these various kinds of energy can be precisely meas- 

 ured through the mechanical equivalent of heat. The concept of en- 

 ergy becomes more comprehensive but the model shows how to use 

 it, since it can be interpreted ultimately in terms of mechanics. 



Second: Formal thermodynamics is a subject found difficult even 

 by very able students. In the absence of suitable overt models, they 

 cannot "see" their way through a thermodynamic deri\^ation as they 

 do through, say, a geometric derivation intrinsically no less difficult. 

 Not only students have trouble with thermodynamics: their instruc- 

 tors do, too! We find textbooks on thermodynamics almost invariably 

 blemished by one or several egregious errors— errors far less frequent 

 in expositions of theories that, though perhaps involving more diffi- 

 cult formalisms, do offer o\^ert models. 



Third: A theory conspicuously difficult to master, thermodynamics 

 is also a theory far less fruitful of discoveries than the corpuscular 

 hypotheses so scorned by Duhem, Mach, and Comte. Precisely he- 

 cause the highly formal theory recommended by Duhem lacks all 

 superfluity, it must be a comparatively dull heuristic tool. For con- 

 sider the observation made, in another context, by Hesse. 



. . . one of the main functions of an analogy or model is to suggest 

 extensions of the theory by considering extensions of the analogy, 

 since more is known about the analogy than is known about the sub- 

 ject matter of the theory itself. ... A collection of observable con- 

 cepts in a purely formal hypothesis suggesting no analogy with any- 

 thing would consequently not suggest either any directions for its own 

 development. 



We do not make thermodynamics a more effective instrument of dis- 

 covery by giving it a still more starkly mathematical formulation. 

 I may find a real sense of beauty in Caratheodory's formulation of 

 the second law of thermodynamics, which apparently removes all 

 vestige of reference to physical models. But I do not know of a single 

 major scientific discovery consequent to Caratheodory's work. On the 

 other hand, I know a great many such discoveries originating in 

 Boltzmann's conception of the second law, which at last explains it 



