THEORIES AND MODELS 227 



duction. Ordinarily quite simple and familiar machinery serves our 

 needs. Only the theories of modern physics, chemistry, and astron- 

 omy require elaborate techniques of modern mathematics that are 

 neither less, nor more, rational by virtue of their sophistication and 

 unfamiliarity. 



Throughout the discussion of induction I took deduction for 

 granted. Indeed, we all do seem to share the same reasonably clear 

 idea of what is meant by deduction. Moreover, unlike induction, de- 

 duction does seem reducible to methodical operations in accordance 

 with definable (syntactic) rules. This appearance is not wholly de- 

 ceptive. But Polanyi's analysis of an opinion of Poincare's rather 

 strongly suggests that even completely formal deduction is not free 

 from all complications. 



To look at a mathematical proof by merely verifying each con- 

 secutive step— says Poincare— is like watching a game of chess, noting 

 only that each step obeys the rules of chess. The least that is re- 

 quired is a grasp of the logical sequence as a purposeful procedure: 

 what Poincare describes as "the something which constitutes the unity 

 of the demonstration." It is this "something"— perhaps in the form of 

 an outline embodying the main steps in the proof— for which the stu- 

 dent will grope, if baffled by a sequence of operations which convey 

 no sense to him, and it is again this outline, embodying the general 

 principle or general structure of the mathematical proof, which will 

 be remembered when the details of the proof are forgotten. 



The "something" no doubt involves an element of familiarity. But, in 

 view of Jeffreys' comment, it seems to me that the competent mathe- 

 matician must bring that something even to formal operations unlike 

 any he has before attempted. 



Deduction cannot be completely fomaalized, as was shown by Lewis 

 Carroll in "what the Tortoise said to Achilles". . . . 



. . . even deductive logic needs a qotion of appreciation that can- 

 not be formally expressed within the system, but depends on looking 

 at it from outside; . . . 



The "notion of appreciation" that escapes expression even in com- 

 pletely formal systems looms still larger in our use of scientific the- 

 ories. There, however, the "notion" may perhaps be more narrowly 

 definable. Let us now abandon the useful but arbitrary absolute 

 dichotomy hitherto maintained between the formalism and the 



