THEORIES AND MODELS 229 



tactic rules of the formalism seems always aflFected by our awareness 

 of the semantic rules we draw from the model. No scientific theory 

 meaningfully considered as such has yet been completely formalized, 

 and it is much to be doubted that any such theory ever will be. 

 Quite generally then we find, in making our derivations, that nothing 

 in the formalism supplies, or even implies, what we need to pass 

 from one step to the next. We make that passage all the same, using 

 assumptions borrowed, on the strength of the model, from other 

 branches of science or from common sense. When the need for such 

 assumptions is recognized, they are made explicit. Probably more 

 often the assumptions are only implicit, wholly unmentioned be- 

 cause the principles we take for granted make them so completely 

 "self-evident." Thus, for example, the modern professional geometer 

 easily shows that Euclid's proofs require assumptions underivable 

 from the postulates he states. As nonprofessional geometers we make 

 all the requisite assumptions implicitly, even as Euclid did. Given the 

 model that makes the Euclidean theory, the assumptions become so 

 "obvious" we easily see our way to and through them. But, precisely 

 because they are so "obvious," we fail entirely to see them. 



A conceptual device of immense power, the physical model is in- 

 separably associated with a grave hazard. The very thematic proc- 

 ess which so strongly sustains us may also, on occasion, betray us. 

 Thus Bruner notes that when we reason in concrete rather than ab- 

 stract terms we are predisposed toward certain conclusions that have 

 for us an air of verisimilitude "in spite of the fact that they are both 

 incorrect logically and readily detectable as such when they appear 

 in a neutral form." We proceed to those expected conclusions by way 

 of implicit assumptions wholly uninspected because their presence 

 is wholly unsuspected. A substantial check to progress may thus de- 

 velop: long years may pass before anyone discovers the implicit as- 

 sumption that, once recognized as such, simply will not bear inspec- 

 tion. Some have thought to bypass this hazard, by giving scientific 

 theories a completely explicit formal development— reaching its op- 

 timal expression in the "purely mathematical" scientific theory that 

 involves no physical model. Is this a program to be taken seriously? 

 A multitude of historical precedents suggests the invariable presence 

 of implicit assumptions in scientific theories. Moreover, as we might 

 by now suspect, the whole idea of a purely mathematical scientific 

 theory is itself delusory. 



