By a state of a system is meant any well-defined condition or
Property that can be recognised if it occurs again. Every system
Will naturally have many possible states.
When the beads are released, their positions (P) undergo a
Series of changes, P0, P1, P2 ...; this point of view at once relates
The system to a transformation
↓
P0 P1 P2 P3 …
P1 P2 P3 P4 …
Clearly, the operands of the transformation correspond to the
States of the system.
The series of positions taken by the system in time clearly cor-
Responds to the series of elements generated by the successive
Powers of the transformation (S.2/14). Such a sequence of states
Defines a trajectory or line of behaviour.
Next, the fact that a determinate machine, from one state, can-
Not proceed to both of two different states corresponds, in the
Transformation, to the restriction that each transform is sin-
Gle-valued.
Let us now, merely to get started, take some further examples,
Taking the complications as they come.
A bacteriological culture that has just been inoculated will
Increase in “number of organisms present” from hour to hour. If
At first the numbers double in each hour, the number in the culture
Will change in the same way hour by hour as n is changed in suc-
cessive powers of the transformation n' = 2n.
If the organism is somewhat capricious in its growth, the sys-
Tem’s behaviour, i.e. what state will follow a given state, becomes
Somewhat indeterminate So “determinateness” in the real system
Evidently corresponds’ in the transformation, to the transform of
A given operand being single-valued.
Next consider a clock, in good order and wound, whose hands,
Pointing now to a certain place on the dial, will point to some
Determinate place after the lapse of a given time. The positions of
Its hands correspond to the transformation’s elements. A single
Transformation corresponds to the progress over a unit interval of
time; it will obviously be of the form n' = n + k.
In this case, the “operator” at work is essentially undefinable for
It has no clear or natural bounds. It includes everything that makes
The clock go: the mainspring (or gravity), the stiffness of the brass
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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
In the wheels, the oil on the pivots, the properties of steel, the inter-
Actions between atoms of iron, and so on with no definite limit. As
We said in S.2/3, the “operator” is often poorly defined and some-
What arbitrary— a concept of little scientific use. The transforma-
Tion, however, is perfectly well defined, for it refers only to the facts
Of the changes, not to more or less hypothetical reasons for them.
A series of changes as regular as those of the clock are not
Readily found in the biological world, but the regular courses of
Some diseases show something of the same features. Thus in the
Days before the sulphonamides, the lung in lobar pneumonia
passed typically through the series of states: Infection → consol-
idation → red hepatisation → grey hepatisation → resolution →
Health. Such a series of states corresponds to a transformation that
Is well defined, though not numerical.
Next consider an iron casting that has been heated so that its
Various parts are at various but determinate temperatures. If its
Circumstances are fixed, these temperatures will change in a
Determinate way with time. The casting’s state at any one moment
Will be a set of temperatures (a vector, S.3/5), and the passage
from state to state, S0 → S1 → S2 → …, will correspond to the
Operation of a transformation, converting operand S0 successively
to T(S0), T2(S0), T3(S0),…, etc.
A more complex example, emphasising that transformations do
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