THE DETERMINATE MACHINE 3/11 



Ex.: Sketch the phase-spaces, with detail merely sufficient to show the main 

 features, of some of the systems in S.3/4 and 6. 



3/11. What is a ''system'"? In S.3/1 it was stated that every real 

 determinate machine or dynamic system corresponds to a closed, 

 single-valued transformation; and the intervening sections have 

 illustrated the thesis with many examples. It does not, however, 

 follow that the correspondence is always obvious; on the contrary, 

 any attempt to apply the thesis generally will soon encounter certain 

 difficulties, which must now be considered. 



Suppose we have before us a particular real dynamic system — a 

 swinging pendulum, or a growing culture of bacteria, or an auto- 

 matic pilot, or a native village, or a heart-lung preparation — and 

 we want to discover the corresponding transformation, starting 

 from the beginning and working from first principles. Suppose it is 

 actually a simple pendulum, 40 cm long. We provide a suitable 

 recorder, draw the pendulum through 30° to one side, let it go, and 

 record its position every quarter-second. We find the successive 

 deviations to be 30° (initially), 10°, and —24° (on the other side). 

 So our first estimate of the transformation, under the given condi- 

 tions, is 



I 30° 10° 

 Y 10° -24° 



Next, as good scientists, we check that transition from 10°: we draw 

 the pendulum aside to 10°, let it go, and find that, a quarter-second 

 later, it is at +3°! Evidently the change from 10° is not single- 

 valued — the system is contradicting itself. What are we to do now ? 



Our difficulty is typical in scientific investigation and is funda- 

 mental: we want the transformation to be single-valued but it will 

 not come so. We cannot give up the demand for singleness, for 

 to do so would be to give up the hope of making single-valued 

 predictions. Fortunately, experience has long since shown what 

 is to be done: the system must be re-defined. 



At this point we must be clear about how a "system" is to be 

 defined. Our first impulse is to point at the pendulum and to 

 say "the system is that thing there". This method, however, has a 

 fundamental disadvantage: every material object contains no less 

 than an infinity of variables and therefore of possible systems. The 

 real pendulum, for instance, has not only length and position; it 

 has also mass, temperature, electric conductivity, crystalline structure, 

 chemical impurities, some radio-activity, velocity, reflecting power, 

 tensile strength, a surface film of moisture, bacterial contamination, 



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