



CHAPTER 19 



The State- determined System 



19/1. The mathematics necessary for the study of adaptation 

 does not consist simply of the solution of a particular mathe- 

 matical problem. The problem, to the bio-mathematician, ranges 

 from the identification of the basic logic necessary for the repre- 

 sentation of the basic concept of mechanism, through its develop- 

 ment into various branches (such as from the discrete to the 

 continuous and from the non -metric to the metric), to the eventual 

 use of specialised techniques for special particular problems. 



Since the problems that interest the biologist usually come 

 from systems of very great complexity, in which treatment of all 

 the facts is not possible, special importance must be given 

 to methods, such as that of topology, that allow simple answers 

 to be given to simple questions, even though the basic facts are 

 complex. The mathematical basis should therefore be sufficiently 

 general to allow specialisation into the methods of topology. 

 Here we have been greatly aided by the magnificent work of the 

 French school that writes, collectively, under the pseudonym of 

 N. Bourbaki. In their great Elements de, Mathematiques this 

 school has shown how the theory of sets, in a simple basic form, 

 can be gradually extended and developed, without the least loss 

 of precision or the least change in the fundamental concepts, into 

 the realms of topology, algebra, geometry, theory of functions, 

 differential equations, and all the various branches of mathematics. 



How the theory of sets, essentially in the form used by Bourbaki, 

 gives a secure basis for the logic of mechanism, has already been 

 displayed in Part I of J. to C. (That book does not use Bourbaki's 

 symbols explicitly, but his concepts are used throughout and in 

 exactly his form; so the reader who wishes to correlate /. to C. 

 with Bourbaki's work will find that the correlation is in most 

 places obvious.) 



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