Each possible state of the machine corresponds uniquely to
A particular element in the graph, and vice versa. The correspon-
Dence is one-one.
Each succession of states that the machine passes through
Because of its inner dynamic nature corresponds to an unbroken
Chain of arrows through the corresponding elements.
If the machine goes to a state and remains there (a state of
Equilibrium, S.5/3) the element that corresponds to the state will
Have no arrow leaving it (or a re-entrant one, S.2/17).
If the machine passes into a regularly recurring cycle of
States, the graph will show a circuit of arrows passing through the
Corresponding elements.
The stopping of a machine by the experimenter, and its
Restarting from some new, arbitrarily selected, state corresponds,
In the graph, to a movement of the representative point from one
Element to another when the movement is due to the arbitrary
Action of the mathematician and not to an arrow.
When a real machine and a transformation are so related, the
Transformation is the canonical representation of the machine,
And the machine is said to embody the transformation.
Ex. 1: A culture medium is inoculated with a thousand bacteria, their number
Doubles in each half-hour. Write down the corresponding transformation
Ex. 2: (Continued.) Find n after the 1st, 2nd, 3rd, . . ., 6th steps.
Ex. 3: (Continued.) (i) Draw the ordinary graph, with two axes, showing the cul-
Ture’s changes in number with time. (ii) Draw the kinematic graph of the sys-
Tem’s changes of state.
Ex. 4: A culture medium contains 109 bacteria and a disinfectant that, in each
Minute, kills 20 per cent of the survivors. Express the change in the number
Of survivors as a transformation.
Ex. 5: ( Continued.) (i) Find the numbers of survivors after 1, 2, 3, 4, 5 minutes.
Ii) To what limit does the number tend as time goes on indefinitely?
Ex. 6: Draw the kinematic graph of the transformation in which n' is, in a table
of four-figure logarithms, the rounded-off right-hand digit of log10 (n+70).
What would be the behaviour of a corresponding machine?
Ex. 7: (Continued, but with 70 changed to 90).
Ex. 8: (Continued, but with 70 changed to 10.) How many basins has this
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Graph?
Closure. Another reason for the importance of closure can
Now be seen. The typical machine can always be allowed to go on
In time for a little longer, simply by the experimenter doing noth-
ing! This means that no particular limit exists to the power that the
Transformation can be raised to. Only the closed transformations
Allow, in general, this raising to any power. Thus the transforma-
Tion T
T: ↓
A b c d e f g
E b m f g c f
Is not closed. T4(a) is c and T5(a) is m. But T(m) is not defined, so
T6(a) is not defined. With a as initial state, this transformation
Does not define what happens after five steps. Thus the transfor-
Mation that represents a machine must be closed. The full signif-
Icance of this fact will appear in S.10/4.
The discrete machine. At this point it may be objected that
Most machines, whether man-made or natural, are smooth-work-
Ing, while the transformations that have been discussed so far
Change by discrete jumps. These discrete transformations are,
However, the best introduction to the subject. Their great advan-
Tage IS their absolute freedom from subtlety and vagueness, for
Every one of their properties is unambiguously either present or
Absent. This simplicity makes possible a security of deduction that
Is essential if further developments are to be reliable. The subject
Was touched on in S.2/1.
In any case the discrepancy is of no real importance. The discrete
Change has only to become small enough in its jump to approximate
As closely as is desired to the continuous change. It must further be
Remembered that in natural phenomena the observations are almost
Invariably made at discrete intervals; the “continuity” ascribed to
Natural events has often been put there by the observer’s imagina-
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Tion, not by actual observation at each of an infinite number of
Points. Thus the real truth is that the natural system is observed at
Discrete points, and our transformation represents it at discrete
Points. There can, therefore, be no real incompatibility.
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