Chapter 4 



THE MACHINE WITH INPUT 



4/1. In the previous chapter we studied the relation between 

 transformation and machine, regarding the latter simply as a unit. 

 We now proceed to find, in the world of transformations, what 

 corresponds to the fact that every ordinary machine can be acted 

 on by various conditions, and thereby made to change its behaviour, 

 as a crane can be controlled by a driver or a muscle controlled by a 

 nerve. For this study to be made, a proper understanding must be 

 had of what is meant by a "parameter". 



So far, each transformation has been considered by itself; we 

 must now extend our view so as to consider the relation between one 

 transformation and another. Experience has shown that just the 

 same methods (as S.2/3) applied again will suffice; for the change 

 from transformation A to transformation B is nothing but the 

 transition A-> B. (In S.2/3 it was implied that the elements of a 

 transformation may be anything that can be clearly defined: there 

 is therefore no reason why the elements should not themselves be 

 transformations.) Thus, if Tj, T2, and Ti, are three transformations, 

 there is no reason why we should not define the transformation U: 



U: i ^' ^^ ^' 

 ^ T2 T2 Ti 



All that is necessary for the avoidance of confusion is that the 

 changes induced by the transformation T^ should not be allowed to 

 become confused with those induced by U\ by whatever method is 

 appropriate in the particular case the two sets of changes must 

 be kept conceptually distinct. 



An actual example of a transformation such as U occurs when a 

 boy has a toy-machine T^, built of interchangeable parts, and then 

 dismantles it to form a new toy-machine T2. (In this case the 

 changes that occur when Ti goes from one of its states to the next 

 (i.e. when Ti "works") are clearly distinguishable from the change 

 that occurs when Tj changes to T2.) 



Changes from transformation to transformation may, in general, 

 be wholly arbitrary. We shall, however, be more concerned with 



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