An output at y that copies that at v. Cover the central parts of the
Mechanism and the two machines are indistinguishable through-
Out an infinite number of tests applied. Machines can thus show
The profoundest similarities in behaviour while being, from other
Points of view, utterly dissimilar.
Nor is this all. Well known to mathematicians are equations of
The type
Dzd z
--a ------- + b ---- + cz = w
Dt
Dt
By which, if a graph is given showing how w varied with time (t),
The changes induced in z can be found. Thus w can be regarded as
An “input” to the equation and z an “output”. If now a, b, and c are
Given values suitably related to L, R, S, etc., the relation between
W and z becomes identical with those between u and v, and
Between x and y. All three systems are isomorphic.
The great practical value of isomorphisms is now becoming
Apparent. Suppose the problem has arisen how the mechanical
System will behave under certain conditions. Given the input u,
The behaviour v is required. The real mechanical system may be
Awkward for direct testing: it may be too massive, or not readily
accessible, or even not yet made! If, however, a mathematician is
Available, the answer can be found quickly and easily by finding
The output z of the differential equation under input w. It would be
Said, in the usual terms, that a problem in mathematical physics
Had been solved. What should be noticed, however, is that the
Process is essentially that of using a map— of using a convenient
Isomorphic representation rather than the inconvenient reality.
It may happen that no mathematician is available but that an
Electrician is. In that case, the same principle can be used again.
The electrical system is assembled, the input given to x, and the
Answer read off at y. This is more commonly described as “build-
Ing an electrical model”.
Clearly no one of the three systems has priority; any can substi-
Tute for the others. Thus if an engineer wants to solve the differ-
Ential equation, he may find the answer more quickly by building
The electrical system and reading the solutions at y. He is then usu-
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Ally said to have “built an analogue computer”. The mechanical
System might, in other circumstances, be found a more convenient
Form for the computer. The big general-purpose digital computer
Is remarkable precisely because it can be programmed to become
Isomorphic with any dynamic system whatever.
The use of isomorphic systems is thus common and important.
96
2
It is important because most systems have both difficult and easy
Patches in their properties. When an experimenter comes to a dif-
Ficult patch in the particular system he is investigating he may if
An isomorphic form exists, find that the corresponding patch in the
Other form is much easier to understand or control or investigate.
And experience has shown that the ability to change to an isomor-
Phic form, though it does not give absolutely trustworthy evi-
Dence (for an isomorphism may hold only over a certain range), is
Nevertheless a most useful and practical help to the experimenter.
In science it is used ubiquitously.
It must now be shown that this concept of isomorphism, vast
Though its range of applicability, is capable of exact and objective
Fig. 6/9/1
Definition. The most fundamental definition has been given by
Bourbaki; here we need only the form suitable for dynamic sys-
Tems It applies quite straightforwardly once two machines have
Been reduced to their canonical representations.
Consider, for instance, the two simple machines M and N, with
Canonical representations
↓
a
b
c
d
↓
g
h
j
k
a c d c δ k j h g
Ν: ε b a d ck h g g
They show no obvious relation. If, however, their kinematic
Graphs are drawn, they are found to be as in Fig. 6/9/1. Inspection
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Shows that there is a deep resemblance. In fact, by merely rear-
Ranging the points in N without disrupting any arrow (S.2/17) we
Can get the form shown in Fig. 6/9/2.
These graphs are identical with M’s graphs, apart from the label-
Ling.
More precisely: the canonical representations of two machines
Are isomorphic if a one-one transformation of the states (input and
97
α
Μ: β
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
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