DESIGN FOR A BRAIN 19/2 



The logic of mechanism 



19/2. Our starting-point is the idea, much more than a century 

 old, that a ' machine ' is that which, whenever it is in given 

 conditions and at a given internal state, goes always to a parti- 

 cular state (i.e. not to different states on different occasions). 

 This definition at once shows its formal correspondence with 

 Bourbaki's ' algebraic law of external composition '. For if the 

 external conditions can be at any one of a set Q, and the internal 

 states of the machine at any one of a set E, then the machine 

 defines, by its behaviour, a mapping (Bourbaki's ' application') 

 of Q x E into E. The concept of ' machine ' thus corresponds 

 exactly to one of the most basic concepts in mathematics. 



After this basic identification many others follow at once. 

 The mapping of E into E given by holding the value of Q constant 

 corresponds to the machine when isolated. An element of E 

 that is invariant in the algebra (for some value of Q) corresponds 

 to a state of equilibrium of the machine when the input (or 

 surrounding conditions) is held constant (i.e. for a given field). 

 The compatibility (or not) of an equivalence relation with an 

 external law of composition corresponds to whether or not a 

 proposed simplification of a state-determined system leaves the 

 new system still state-determined. If it does, then the algebraic 

 quotient-law corresponds to the new, simplified, canonical repre- 

 sentation. And so on, in a manner that deserves extensive 

 treatment. 



It is not my intention here to develop the subject ab initio 

 and extensively. As this book is concerned primarily with the 

 brain and with systems in which continuity is common, we need 

 only notice that Bourbaki has shown how the basic concepts, 

 stated in discrete form, can be specialised to the continuous forms 

 and to those with a metric. In this Appendix we will deal only 

 with such forms as are continuous and provided with a metric. 



(N.B. Throughout this chapter the emphasis is on the system 

 that is isolated and left alone to show what it will do, apart from 

 occasional interferences from the experimenter. The statements 

 made should be interpreted accordingly. Chapter 21 deals 

 explicitly with the system that is being subjected to changes in 

 its conditions, or at its input.) 



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