248 THEORIES AND MODELS 



quantum mechanics does not derive primarily from its formal struc- 

 ture. 



I am far from arguing that no important discoveries have been 

 obtained from highly formal theories. Consider, for example, the 

 enormous suggestiveness of the purely formal analogy, between phys- 

 ical optics and classical dynamics, to which Hamilton called atten- 

 tion. Moreover, just as a physical model leads us to construe one field 

 in the terms of another, the transfer of a mathematical formalism may 

 perform much the same function. Thus Maxwell remarks (not very 

 charitably) that Mossotti obtained the theory of electrostatic induc- 

 tion, from Poisson's mathematical theory of magnetic induction, sim- 

 ply by translating the magnetic language into electric and the French 

 into Italian. Even highly formal theories that dispense with the "su- 

 perfluity" of explanation, in terms of overt physical models or analo- 

 gies, can then function as heuristic devices. I maintain only that they 

 do not function as well as theories that accept some such superfluity. 

 Bridgman, who shares Duhem's point of view, argues : 



. . . the mathematical model is just as good as the physical model if 

 it only enables us to answer any question that we may propose about 

 the behavior of the physical system, nevertheless we have an uncom- 

 fortable feeling that we have lost something. 



I think that we discover on analysis that it is the explanation which 

 we feel we have lost. . . . the mathematical model gives up the 

 possibility of explanation in the usual sense. 



The mathematical model cannot be "just as good": to the extent that 

 we give up "explanation in the usual sense" our theory must become 

 a less productive source of questions "we may propose about the 

 behavior of the physical system." 



Heuristic power from "superfluous explanation.'^ Tlie physical 

 model or analogy that makes explanation makes also an instrument 

 of discovery. For it is precisely "by considering extensions of the 

 analogy" that we arrive at the pregnant questions which, Hesse says, 

 "suggest extensions of the theory." Toulmin cites, as a good exam- 

 ple of this possibility, the conception of light as a "something that 

 travels in straight lines." Beyond explaining the manifold phenomena 

 of shadow-casting, this conception suggests new and important ques- 

 tions (e.g., if light is a something that travels, how fast does it 

 travel?). The model gives rise also to questions ultimately recognized 



