6/13 AN INTRODUCTION TO CYBERNETICS 



does not observe all the many detailed processes going on in the 

 individual members. The biologist thus usually studies only a 

 small fraction of the system that faces him. Any statement he 

 makes is only a half-truth, a simplification. To what extent can 

 systems justifiably be simplified? Can a scientist work properly 

 with half-truths? 



The practical man, of course, has never doubted it. Let us see 

 whether we can make the position clear and exact. 



Knowledge can certainly be partial and yet complete in itself. 

 Perhaps the most clear-cut example occurs in connexion with 

 ordinary multiplication. The complete truth about multiplication 

 is, of course, very extensive, for it includes the facts about all 

 possible pairs, including such items as that 



14792 X 4,183584 = 61883,574528. 



There is, however, a much smaller portion of the whole which 

 consists simply in the facts that 



Even X Even = Even 

 Even X Odd = Even 

 Odd X Even = Even 

 Odd X Odd = Odd 



What is important here is that though this knowledge is only an 

 infinitesimal fraction of the whole it is complete within itself. (It 

 was, in fact, the first homomorphism considered in mathematics.) 

 Contrast this completeness, in respect of Even and Odd, with the 

 incompleteness shown by 



2x2 = 4 



2x4-8 



4x2 = 8 



4x4=16 



which leaves unmentioned what is 4 x 8, etc. Thus it is perfectly 

 possible for some knowledge, though partial in respect of some 

 larger system, to be complete within itself, complete so far as it 

 goes. 



Homomorphisms may, as we have seen, exist between two different 

 machines. They may also exist within one machine: between the 

 various possible simplifications of it that still retain the character- 

 istic property of being machine-like (S.3/1). Suppose, for instance, 

 that the machine were A: 



, . a b c d e 

 A: i 



e b a b e 



104 



