THEORIES AND MODELS 233 



we know it to represent the spreading of a wave through space. It is 

 the interpretation which is attached to the equation due to previous 

 apphcation that we need for describing our experiments. 



This reference to "previous application" explains why to the unin- 

 itiated quantum mechanics appears a purely mathematical system: 

 unaware of any previous application of like formalisms, they fail to 

 detect the model implicit in the formalism. The same reference ex- 

 plains also how some of those professionally concerned with quan- 

 tum mechanics can still misconceive it purely mathematical: taking 

 the previous applications and the implicit interpretations entirely for 

 granted, they do not realize how much is thus conveyed to them. 



Formalism implicit in model. Until quite recently the vast majority 

 of all important scientific theories have been founded on the con- 

 ception of some physical model, and make no explicit reference to 

 any formalism. Often syntactic rules may well pass unmentioned be- 

 cause the theory requires, and simply assumes, no more than the 

 simplest kinds of everyday reasoning. But even when more elaborate 

 formal operations are involved they may be given little or no overt 

 notice. How then do we acquire the formalism needed to complete 

 the theory? The answer is foreshadowed by our earlier discovery 

 that the apparently purely formal scientific theory always conveys 

 an implicit model. We now encounter the complementary case: the 

 scientific theory that apparently proposes nothing but a model actu- 

 ally conveys an implicit formalism. 



Consider, for example, the Mendelian theory which postulates 

 "genes" as carriers of hereditary traits. The appearance, or nonap- 

 pearance, of each trait in a given organism is then supposed to arise 

 from the random combinations and recombinations, in its ancestors, 

 of allelic forms of some particular gene. The genes are completely 

 hypothetical invisible entities, conceived on the model of "discrete 

 objects." From that model the Mendelian theory at once acquires a 

 formalism— precisely that earlier found applicable to statistical de- 

 scription of random combinations in certain systems of discrete ob- 

 jects. Given the Mendelian model, we are thus led to set up certain 

 axiomatic relations we expect to apply to the independent assortment 

 of genes and, in reasoning from these axioms, we follow the (syn- 

 tactic ) procedures earlier found appropriate. 



This example is noteworthy only in its comparative simplicity, for 

 even a very primitive model may sufiice to suggest a very highly 



