One of the many sub-machines provided by homomorphism. Why
All these meanings should be distinguished is because different
Sub-machines can have different properties; so that although both
Sub-machines may be abstracted from the same real “thing”, a
Statement that is true of one may be false of another.
It follows that there can be no such thing as the (unique) behav-
Iour of a very large system, apart from a given observer. For there
Can legitimately be as many sub-machines as observers, and
Therefore as many behaviours, which may actually be so different
As to be incompatible if they occurred in one system. Thus the
State system with kinematic graph
h ← k →
m → l ← j →
106
Sub-machine (with states r and s) given by the transformation
H b j k l
Rs
has graph s → r, with one basin and no cycle. Both statements are
Equally true, and are compatible because they refer to different
Systems (as defined in S.3/11).
The point of view taken here is that science (as represented by
The observer’s discoveries) is not immediately concerned with
Discovering what the system “really” is, but with co-ordinating
The various observers’ discoveries, each of which is only a por-
Tion, or an aspect, of the whole truth.
Were the engineer to treat bridgebuilding by a consideration of
Every atom he would find the task impossible by its very size. He
Therefore ignores the fact that his girders and blocks are really
Composite, made of atoms, and treats them as his units. As it hap-
Pens, the nature of girders permits this simplification, and the
Engineer’s work becomes a practical possibility. It will be seen
Therefore that the method of studying very large systems by stud-
Ying only carefully selected aspects of them is simply what is
Always done in practice. Here we intend to follow the process
More rigorously and consciously.
The lattice. The various simplifications of a machine have
Exact relations to one another Thus, the six forms of the system of
Ex. 6/13/2 are:
A, b, c, d
(2) a + b, c, d
(3) a, b, c + d
|
|
|
(4) a + b, c + d
(5) a, b + c + d
(6) a + b + c + d
where, e.g. “a + b” means that a and b are no longer distinguished.
Now (4) can be obtained from (3) by a merging of a and b. But (5)
Cannot be obtained from (4) by a simple merging; for (5) uses a
Distinction between a and b that has been lost in (4). Thus it is
Soon verified that simplification can give:
From (1): all the other five,
And (6),
And (6),
,, (4): (6),
,, (5): (6),
None.
107
↓
Has two basins, and always ends in a cycle. The homomorphic
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 various simplifications are thus related as in the diagram, in
Which a descending line connects the simpler form (below) with
The form from which it can be directly obtained (above):
1
2
3
Tem with a vast number of interacting parts, going through a trade
Cycle, to the simple form of two states:
Boom
Slump
Thus, the various simplifications of a dynamic system can
Ordered and related.
Models. We can now see much more clearly what is meant
By a “model”. The subject was touched on in S.6/8, where three
Systems were found to be isomorphic and therefore capable of
Being used as representations of each other. The subject is some
Of importance to those who work with biological systems, for in
Many cases the use of a model is helpful, either to help the worker
Think about the subject or to act as a form of analogue computer.
The model will seldom be isomorphic with the biological sys-
Tem: usually it will be a homomorphism of it. But the model is
Itself seldom regarded in all its practical detail: usually it is only
|
|
|
Some aspect of the model that is related to the biological system;
Thus the tin mouse may be a satisfactory model of a living
Mouse— provided one ignores the tinniness of the one and the
Дата добавления: 2019-11-16; просмотров: 213; Мы поможем в написании вашей работы! |
Мы поможем в написании ваших работ!