3/2 AN INTRODUCTION TO CYBERNETICS 



Ex. : Name two states that are related as operand and transform, with 

 time as the operator, taking the dynamic system from : 



(a) Cooking ; (b) Lighting a fire ; (c) The petrol engine ; (d) Embryo- 

 logical development ; (e) Meteorology ; (/) Endocrinology ; (g) Econ- 

 omics ; (h) Animal behaviour ; (/) Cosmology. (Meticulous accuracy is 

 not required.) 



3/2. Closure. Another reason for the importance of closure can 

 now be seen. The typical machine can always be allowed to go on 

 in lime for a little longer, simply by the experimenter doing nothing! 

 This means that no particular limit exists to the power that the 

 transformation can be raised to. Only the closed transformations 

 allow, in general, this raising to any power. Thus the transforma- 

 tion T 



a b c d e f g 



e b m f g c f 



is not closed. T\a) is c and T\a) is /;/. But T{m) is not defined, 

 so T\a) is not defined. With a as initial state, this transformation 

 does not define what happens after five steps. Thus the transforma- 

 tion that represents a machine must be closed. The full significance 

 of this fact will appear in S.10/4. 



3/3. The discrete machine. At this point it may be objected that 

 most machines, whether man-made or natural, are smooth-working, 

 while the transformations that have been discussed so far change by 

 discrete jumps. These discrete transformations are, however, the 

 best introduction to the subject. Their great advantage is their 

 absolute freedom from subtlety and vagueness, for every one of their 

 properties is unambiguously either present or absent. This sim- 

 plicity makes possible a security of deduction that is essential if 

 further developments are to be reliable. The subject was touched 

 on in S.2/1. 



In any case the discrepancy is of no real importance. The discrete 

 change has only to become small enough in its jump to approximate 

 as closely as is desired to the continuous change. It must further 

 be remembered that in natural phenomena the observations are 

 almost invariably made at discrete intervals; the "continuity" 

 ascribed to natural events has often been put there by the observer's 

 imagination, not by actual observation at each of an infinite number 

 of points. Thus the real truth is that the natural system is observed 

 at discrete points, and our transformation represents it at discrete 

 points. There can, therefore, be no real incompatibility. 



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