Do so would be to give up the hope of making single-valued pre-
Dictions. Fortunately, experience has long since shown what s 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 “the
System is that thing there”. This method, however, has a funda-
Mental 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 struc-
Ture, chemical impurities, some radio-activity, velocity, reflecting
Power, tensile strength, a surface film of moisture, bacterial con-
39
Fig. 3/10/1
↓
The usefulness of the phase-space (II) can now be seen, for the
Whole range of trajectories in the system can be seen at a glance, fro-
Zen, as it were, into a single display. In this way it often happens that
Some property may be displayed, or some thesis proved, with the
Greatest ease, where the algebraic form would have been obscure.
Such a representation in a plane is possible only when the vec-
Tor has two components. When it has three, a representation by a
Three- dimensional model, or a perspective drawing, is often still
Useful. When the number of components exceeds three, actual
Representation is no longer possible, but the principle remains, and
A sketch representing such a higher-dimensional structure may
Still be most useful, especially when what is significant are the
General topological, rather than the detailed, properties.
The words “phase space” are sometimes used to refer to the
Empty space before the arrows have been inserted, i.e. the space
Into which any set of arrows may be inserted, or the diagram, such
As II above, containing the set of arrows appropriate to a particular
Transformation. The context usually makes obvious which is
Intended.)
38
A N I N T R O D UC T I O N T O C Y B E R NE T I C S
TH E D ET ERM IN A TE MA C HI N E
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Tamination, an optical absorption, elasticity, shape, specific grav-
Ity, and so on and on. Any suggestion that we should study “all”
The facts is unrealistic, and actually the attempt is never made.
What is try is that we should pick out and study the facts that are
Relevant to some main interest that is already given.
The truth is that in the world around us only certain sets of facts
Are capable of yielding transformations that are closed and single.
The discovery of these sets is sometimes easy, sometimes diffi-
Cult. The history of science, and even of any single investigation,
Abounds in examples. Usually the discovery involves the other
Method for the defining of a system, that of listing the variables
That are to be taken into account. The system now means, not a
But a list of variables. This list can be varied, and the experi-
Menter’s commonest task is that of varying the list (“taking other
Variables into account”) until he finds a set of variables that he
Required singleness. Thus we first considered the pendulum as if
It consisted solely of the variable “angular deviation from the ver-
Tical”; we found that the system so defined did not give single-
Ness. If we were to go on we would next try other definitions, for
Instance the vector:
Angular deviation, mass of bob),
Which would also be found to fail. Eventually we would try the
Angular deviation, angular velocity)
And then we would find that these states, defined in this way,
Would give the desired singleness (cf. Ex. 3/6/14).
Some of these discoveries, of the missing variables, have been
Of major scientific importance, as when Newton discovered the
Importance of momentum, or when Gowland Hopkins discovered
The importance of vitamins (the behaviour of rats on diets was not
Single-valued until they were identified). Sometimes the discovery
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Is scientifically trivial, as when single-valued results are obtained
Only after an impurity has been removed from the water-supply, or
A loose screw tightened; but the singleness is always essential.
Sometimes what is wanted is that certain probabilities shall be
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