DESIGN FOR A BRAIN 12/4 



This operational test provides the ' atom ' of independence. 

 Two transitions are needed: the concept of 'independence' is 

 meaningless with less. 



12/4. In general, what happens when the test is applied to one 

 pair of initial states docs not restrict what may happen if it is 

 applied to other pairs. The possibility cannot be excluded that 

 the test may give results varying arbitrarily over the possible 

 pairs. Often, however, it happens that, for some given value of 

 all other variables or parameters, Z, W f . . . , Y is independent 

 of X for all pairs of initial states that differ only in the value of X. 

 In this case, for that particular field and for that particular value 

 of the other variables and parameters, Y is independent of X in a 

 more extended sense. Provided the field and the initial values 

 of Y, Z, W, etc., do not change, Y's transition is unaffected by 

 X's initial value. In this case, Y is independent of X over a 

 region (in the phase space) represented by a line parallel to the 

 .Y-axis, ' independent ' in the sense that whenever the representa- 

 tive point, moving on a line of behaviour, leaves this region, Y 

 will undergo the same transition. 



Sometimes it may happen that Y is independent of X not only 

 for all values of X but also for all values of the other variables 

 and parameters — Z, W, etc. In the previous paragraph a change 

 of Z's value might have changed the field or region so that Y 

 was no longer independent of X. In the present case, F's transi- 

 tion (from a uniform Y- value) is the same regardless of the initial 

 values of X, Z, W, etc. Y is then independent of X unconditionally. 



It will be seen that two variables may be ' independent ' to 

 varying degrees : at two points, over a line, over a region, over the 

 whole phase-space, over a set of fields. The word is thus capable 

 of many degrees of application. The definitions given above are 

 not intended to answer the question (of doubtful validity) ' what 

 is independence really ? ' but simply to show how this word must 

 be used if a speaker is to convey an unambiguous message to his 

 audience. Clearly, the word often needs supplementary specifica- 

 tion (e.g. does i Y independent of X ' mean l over this field ' or 

 1 over all fields ' ?); the supplementary specification must then be 

 given, either by the context or explicitly. 



The word ' independent ' is thus similar to the word ' stable ' : 

 both words are often useful in that they can convey information 



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