Of M7.— if cut into quadrants?)
Transform M by the many-one transformation T:
i
j
a b c d e i j k lT: ↓ h h h g g β β α α
(which is single-valued but not one-one as in S.6/9) and we get
↓
Which is isomorphic with N.
Examination of M shows now where the resemblance to N lies.
Within M the transitions occur in blocks; thus a, b and c always
Go to some one of a, b or c. And the blocks in M undergo transi-
Tions in the same way as the states in N. N is thus equivalent to a
Simplified version of M.
The relation can be displayed in another way. Suppose first the
Two machines are viewed by some one who can distinguish all the
Five states of M; he will report simply that M is different from N
(i.e. not isomorphic) and more complex. Suppose next that they
Are viewed by some observer with less power of discrimination,
One who cannot discriminate between a, b, and c, but lumps them
All together as, say, A; and who also lumps d and e together as B,
i and j as I', and k and l as d. This new observer, seeing this sim-
Plified version of M, will report that it is isomorphic with N. Thus
Two machines are homomorphic when they become alike if one is
Merely simplified, i.e. observed with less than full discrimination.
Formally, if two machines are so related that a many-one trans-
Formation can be found that, applied to one of the machines, gives
A machine that is isomorphic with the other, then the other (the
Simpler of the two) is a homomorphism of the first.
Ex.: Is isomorphism simply an extreme case of homomorphism?
Problem: What other types of homomorphism are there between machine
And machine?
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
g
h
h
g
g
g
h
h
g
g
β
β
α
α
If the methods of this book are to be applied to biological
Systems, not only must the methods become sufficiently complex
To match the systems but the systems must be considerably sim-
Plified if their study is ever to be practical. No biological system
Has yet been studied in its full complexity, nor is likely to be for a
Very long time. In practice the biologist always imposes a tremen-
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Dous simplification before he starts work: if he watches a bird
Building its nest he does not see all the intricate pattern of detailed
Neuronic activities in the bird’s brain; if he studies how a lizard
Escapes from its enemies he does not observe the particular
Molecular and ionic changes in its muscles; if he studies a tribe at
103
102
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
Its council meeting he does not observe all the many detailed
Processes going on in the individual members. The biologist thus
Usually studies only a small fraction of the system that faces him.
Any statement he makes is only a half-truth, a simplification. To
What extent can systems justifiably be simplified? Can a scientist
Work properly with half-truths?
The practical man, of course, has never doubted it. Let us see
Whether we can make the position clear and exact.
Knowledge can certainly be partial and yet complete in itself.
Perhaps the most clear-cut example occurs in connexion with
Ordinary multiplication. The complete truth about multiplication
Is, of course, very extensive, for it includes the facts about all pos-
Sible pairs, including such items as that
14792 × 4,183584 = 61883,574528.
There is, however, a much smaller portion of the whole which
Consists simply in the facts that
Even × Even = Even
Even × 0dd = Even
0dd × Even = Even
0dd × 0dd = 0dd
What is important here is that though this knowledge is only an
Infinitesimal fraction of the whole it is complete within itself. (It
Was, in fact, the first homomorphism considered in mathematics.)
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Contrast this completeness, in respect of Even and Odd, with the
Incompleteness shown by
2 × 2 = 4
2 × 4 = 8
4 × 2 = 8
4 × 4 = 16
which leaves unmentioned what is 4 × 8, etc. Thus it is perfectly
Possible for some knowledge, though partial in respect of some
Larger system, to be complete within itself, complete so far as it
Goes.
Homomorphisms may, as we have seen, exist between two dif-
Ferent machines. They may also exist within one machine:
Between the various possible simplifications of it that still retain
The characteristic property of being machine-like (S.3/1). Sup-
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