Bilities), show that if the number of states is large, the variety will fall at the
First step, in the ratio of 1 to 1 - 1/e, i.e. to about two-thirds. (Hint: The prob-
Lem is equivalent (for a single step) to the following: n hunters come sud-
Denly on a herd of n deer. Each fires one shot at a deer chosen at random.
Every bullet hits. How many deer will, on the average, be hit? And to what
Does the average tend as n tends to infinity?)
And apply it to some set of the operands, e.g.
B B A C C C A A B A
The result is C C B C C C B B C B
What is important is that the variety in the set has fallen from 3
To 2. A further transformation by Z leads to all C’s, with a variety
Of 1.
The reader can easily satisfy himself that such a set, operated on
By a single-valued transformation, can never increase in variety,
And usually falls. The reason for the fall can readily be identified.
In the graph, a confluence of arrows
Can occur, but a diver-
Genceis impossible. Whenever the transformation makes two
States change to one, variety is lost; and there is no contrary proc-
Ess to replace the loss.
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Set and machine. We must now be clear about how a set of
States can be associated with a machine, for no real machine can,
At one time, be in more than one state. A set of states can be con-
Sidered for several reasons.
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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
Q UA N TI TY O F V AR IE TY
We may, in fact, not really be considering one machine, however
Much we speak in the singular (S.7/3), but may really be consider-
Ing a set of replicates, as one might speak of “the Model T Ford”, or
“the anterior horn cell”, or “the white rat”. When this is so we can
Consider all the replicates together, one in one state and one in
Another; thus we get a set of states for one transformation to act on.
A set of states can also arise even if the machine is unique. For
We may wish to consider not only what it may do at one time from
One state but also what it may do at another time from another
State. So its various possible behaviours at a set of times are natu-
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Rally related to a set of states as operands.
Finally, a set may be created by the fiat of a theoretician who,
Not knowing which state a particular machine is at, wants to trace
Out the consequences of all the possibilities. The set now is not the
Set of what does exist, but the set of what may exist (so far as the
Theoretician is concerned). This method is typically cybernetic,
For it considers the actual in relation to the wider set of the possi-
Ble or the conceivable (S.1/3).
Decay of variety. Having, for one of these reasons, a set of
States and one single-valued transformation, we can now, using
The result of S.7/22, predict that as time progresses the variety in
The set cannot increase and will usually diminish.
This fact may be seen from several points of view.
In the first place it gives precision to the often made remark that
Any system, left to itself, runs to some equilibrium. Usually the
Remark is based on a vague appeal to experience, but this is unsat-
Isfactory, for the conditions are not properly defined. Sometimes
The second law of thermodynamics is appealed to, but this is often
Irrelevant to the systems discussed here (S.1/2). The new formu-
Lation shows just what is essential.
In the second place it shows that if an observer has an absolute
System, whose transformation he knows but whose states cannot,
For any reason, be observed, then as time goes on his uncertainty
About its state can only diminish. For initially it might be at any
One of all its states, and as time goes on so does the number of its
Possible states diminish. Thus, in the extreme case in which it has
Only one basin and a state of equilibrium, he can, if initially uncer-
Tain, ultimately say with certainty, without making any further
Observation, at which state it is.
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The diminution can be seen from yet another point of view. If
The variety in the possible states is associated with information, so
That the machine’s being at some particular state conveys some
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Particular message, then as time goes on the amount of informa-
Tion it stores can only diminish. Thus one of three messages might
Be carried to a prisoner by a cup of coffee, the message depending
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