2/15 DYNAMIC SYSTEMS 



The Natural System 



2/15. In S. 2/5 a system was defined as any arbitrarily selected 

 set of variables. The right to arbitrary selection cannot be 

 waived, but the time has now come to recognise that both Science 

 and common sense insist that if a system is to be studied with 

 profit its variables must have some naturalness of association. 

 But what is ' natural ' ? The problem has inevitably arisen after 

 the restriction of S. 2/9, where we repudiated all borrowed 

 knowledge. If we restrict our attention to the variables, we find 

 that as every real 4 machine ' provides an infinity of variables, 

 and as from them we can form another infinity of combinations, 

 we need some test to distinguish the natural system from the 

 arbitrary. 



One criterion will occur to the practical experimenter at once. 

 He knows that if an active and relevant variable is left unobserved 

 or uncontrolled the system's behaviour will become capricious, 

 not capable of being reproduced at will. This concept may 

 readily be made more precise. We simply state formally the 

 century-old idea that a ' machine ' is something that, if its internal 

 state is known, and its surrounding conditions, then its behaviour 

 follows necessarily. That is to say, a particular surrounding 

 condition (or input, i.e. those variables that affect it) and a 

 particular state determine uniquely what transition will occur. 



So the formal definition goes as follows. Take some particular 

 set of external conditions (or input-value) C and some particular 

 state S ; observe the transition that is induced by its own internal 

 drive and laws ; suppose it goes to state S { . Notice whether, 

 whenever C and S occur again, the transition is also always to 

 S t ; if so, record that the transitions that follow C and S are 

 invariant. Next, vary C (or S, or both) to get another pair — 

 C x and S 1 say ; see similarly whether the transitions that follow 

 C ± and S 1 are also invariant. Proceed similarly till all possible 

 pairs have been tested. If the outcome at every pair was 

 4 invariant ' then the system is, by definition, a machine with 

 input. (This definition accords with that given in /. to C.) 



In the world of biology, the concept of the machine with input 

 often occurs in the specially simple case in which all the events 

 (in one field) occur in only one set of conditions (i.e. C has the 

 same value for all the lines of behaviour). The field then comes 



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