Pose, for instance, that the machine were A:
a b c d eA: ↓ e b a b e
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This is the machine as seen by the first observer (call him One).
Suppose now that another observer (call him Two) was unable to
Distinguish states a and d, and also unable to distinguish b and e.
Let us give the states new names for clarity:
A d c b e
K L M
The second observer, seeing states K, L or M would find the
Machine’s behaviour determinate. Thus when at K (really a or d)
It would always go to M (either b or e), and so on. He would say
That it behaved according to the closed transformation
K L M ↓ M K M
And that this was single-valued, and thus determinate.
The new system has been formed simply by grouping together
Certain states that were previously distinct, but it does not follow
That any arbitrary grouping will give a homomorphism. Thus sup-
Pose yet another observer Three could distinguish only two states:
A b c d e
PQ
He would find that P changed sometimes to Q (when P was really
At a) and sometimes to P (when P was really at b or c). The change
From P is thus not single-valued, and Three would say that the
Machine (with states P and Q) was not determinate. He would be
Dissatisfied with the measurements that led to the distinction
Between P and Q and would try to become more discriminating,
So as to remove the unpredictability.
A machine can thus be simplified to a new form when its states
Are compounded suitably. Scientific treatment of a complex sys-
Tem does not demand that every possible distinction be made.
Ex. 1: What homomorphism combines Odd and Even by the operation of addi-
Tion ?
Ex. 2: Find all possible simplifications of the four-state system
a b c d ↓ b b d c
Which leaves the result still a determinate machine.
Ex. 3: What simplification is possible in
x' = – y
2 y' = x + y,
If the result is still to be a determinate machine ?
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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 BL AC K B O X
The deliberate refusal to attempt all possible distinctions,
And the deliberate restriction of the study of a dynamic system to
Some homomorphism of the whole, become justified, and in fact
Almost unavoidable, when the experimenter is confronted with the
System of biological origin.
We usually assumed, in the earlier chapters, that the observer
Knew, at each moment, just what state the system was in. It was
Assumed, in other words, that at every moment his information
About the system was complete. There comes a stage, however, as
The system becomes larger and larger, when the reception of all
The information is impossible by reason of its sheer bulk. Either
The recording channels cannot carry all the information, or the
Observer, presented with it all, is overwhelmed. When this occurs,
What is he to do? The answer is clear: he must give up any ambi-
Tion to know the whole system. His aim must be to achieve a par-
Tial knowledge that, though partial over the whole, is none the less
Complete within itself, and is sufficient for his ultimate practical
Purpose.
These facts emphasise an important matter of principle in the
Study of the very large system. Faced with such a system, the
Observer must be cautious in referring to “the system”, for the
Term will probably be ambiguous, perhaps highly so. “The sys-
Tem” may refer to the whole system quite apart from any observer
To study it— the thing as it is in itself; or it may refer to the set of
Variables (or states) with which some given observer is con-
Cerned. Though the former sounds more imposing philosophi-
Cally, the practical worker inevitably finds the second more
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Important. Then the second meaning can itself be ambiguous if
The particular observer is not specified, for the system may be any
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