THE MACHINE WITH INPUT 4/1 



the special case in which the several transformations act on the 

 same set of operands. Thus, if the four common operands are 

 a, b, c, and d, there might be three transformations, R^, R2, and R^: 



.abed .abed .abed 



e d d b bade d c d b 



These can be written more compactly as 



which we shall use as the standard form. (In this chapter we shall 

 continue to discuss only transformations that are closed and single- 

 valued.) 



A transformation corresponds to a machine with a characteristic 

 way of behaving (S.3/1); so the set of three — Ri, R2, and ^3 — if 

 embodied in the same physical body, would have to correspond to a 

 machine with three ways of behaving. Can a machine have three 

 ways of behaving? 



It can, for the conditions under which it works can be altered. 

 Many a machine has a switch or lever on it that can be set at any 

 one of three positions, and the setting determines which of three 

 ways of behaving will occur. Thus, if^ a, etc., specify the machine's 

 states, and Ri corresponds to the switch being in position 1, and 

 R2 corresponds to the switch being in position 2, then the change of 

 R's subscript from 1 to 2 corresponds precisely with the change of the 

 switch from position 1 to position 2; and it corresponds to the 

 machine's change from one way of behaving to another. 



It will be seen that the word "change" if applied to such a machine 

 can refer to two very different things. There is the change from 

 state to state, from a to b say, which is the machine's behaviour, and 

 which occurs under its own internal drive, and there is the change 

 from transformation to transformation, from 7?i to R2 say, which is a 

 e/iange of its way of behaving, and which occurs at the whim of the 

 experimenter or some other outside factor. The distinction is 

 fundamental and must on no account be slighted. 



R's subscript, or any similar symbol whose value determines 

 which transformation shall be applied to the basic states will be 

 called a parameter. If numerical, it must be carefully distinguished 

 from any numbers that may be used to specify the operands as 

 vectors. 



43 



