228 THEORIES AND MODELS 



model present in each scientific theory. For, beyond semantic rules, 

 we draw also from the model the notion of appreciation that gives 

 us full command of the formalism. Consider for example that, though 

 presumably we might have been, in secondary school we were not 

 taught the abstract formalism of the modern geometer. Instead we 

 were taught Euclidean geometry with the aid of diagrams and real or 

 imaginary constructions. The geometric formalism was thus supported 

 by models, e.g., when the concept of congruence was represented to 

 us as the physical superposability of two rigid figures. We could 

 then easily "see" our way through relatively complex proofs because 

 we had in view, sometimes with the physical eye and always with 

 the mind's eye, a relevant physical situation. From physical concep- 

 tions entirely foreign to the geometric formalism, we thus acquired 

 the notion of appreciation we used in producing workmanlike Eu- 

 clidean proofs from Euclidean axioms. And this is but a familiar spe- 

 cial case of an important general situation. The work of Bruner and 

 his associates provides convincing evidence that humans reason bet- 

 ter when they work with concrete materials rather than with ab- 

 stract logical symbols presented in propositions of identical form. 



... in the concrete realm verisimilitude provides a way of checking 

 upon validity. . . . 



. . . much of human reasoning is supported by a kind of thematic 

 process rather than by an abstract logic. 



It is the model that supplies verisimilitude and evokes the thematic 

 process that supports our operations with the formalism of a scien- 

 tific theory. 



Given the importance of this thematic process, the absolute di- 

 chotomy of syntactic and semantic rules becomes quite as indefensi- 

 ble as the related dichotomy of formalism and model just now aban- 

 doned. And, beyond any vague "thematic process," the model offers 

 the formalism several more specific elements of support. Thus, for 

 example, in making derivations from theoretic postulates, often we 

 simply drop out certain terms, or groups of terms, we hold negligible. 

 What suggests and justifies these approximations? Nothing in the 

 formalism as such but, rather, considerations of relative magnitude 

 that can arise only as we impute "meaning" to the abstract terms of 

 the formalism. 



Even when we need make no approximations, our use of the syn- 



