Ands. Consider the set of vectors, each of three components,
A, T(a), T2(a)),
With a given all its possible values in turn. How many degrees of freedom
Has the set ?
Ex. 7: In what way does the ordinary graph, of y on x, show constraint ?
Ex. 8: How many degrees of freedom has an ordinary body— a chair say— in
Three dimensional space?
Science looks for laws; it is therefore much concerned with
Looking for constraints. (Here the larger set is composed of what
Might happen if the behaviour were free and chaotic, and the
Smaller set is composed of what does actually happen.)
This point of view, it will be noticed, conforms to what was said
In S.1/5. Cybernetics looks at the totality, in all its possible rich-
Ness, and then asks why the actualities should be restricted to
Some portion of the total possibilities.
Ex. 1: How is the chemist’s Law of Simple Proportions a constraint?
Ex. 2: How is the Law of Conservation of Energy a constraint?
I M P OR TA N CE OF C O N ST RA IN T
Constraints are of high importance in cybernetics, and will
Be given prominence through the remainder of this book, because
When a constraint exists advantage can usually be taken of it.
Shannon’s work, discussed chiefly in Chapter 9, displays this
Thesis clearly. Most of it is directed to estimating the variety that
Would exist if full independence occurred, showing that con-
Straints (there called “redundancy”) exist, and showing how their
Existence makes possible a more efficient use of the channel.
The next few sections will also show something of the wide
Applicability and great importance of the concept.
Laws of Nature. First we can notice that the existence of any
Invariant over a set of phenomena implies a constraint, for its exist-
Ence implies that the full range of variety does not occur. The gen-
Eral theory of invariants is thus a part of the theory of constraints.
Further, as every law of nature implies the existence of an
Invariant, it follows that every law of nature is a constraint. Thus,
The Newtonian law says that, of the vectors of planetary positions
And velocities which might occur, e.g. written on paper (the larger
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Set), only a smaller set will actually occur in the heavens; and the
Law specifies what values the elements will have. From our point
Of view, what is important is that the law excludes many positions
And velocities, predicting that they will never be found to occur.
130
Object as constraint. Constraints are exceedingly common
In the world around us, and many of our basic concepts make use
Of it in an essential way. Consider as example the basic concept of
A “thing” or “object”, as something handled in daily life. A chair
Is a thing because it has coherence, because we can put it on this
Side of a table or that, because we can carry it around or sit on it.
The chair is also a collection of parts.
Now any free object in our three dimensional world has six
Degrees of freedom for movement. Were the parts of the chair
Unconnected each would have its own six degrees of freedom; and
This is in fact the amount of mobility available to the parts in the
Workshop before they are assembled. Thus the four legs, when
Separate, have 24 degrees of freedom. After they are joined, how-
Ever, they have only the six degrees of freedom of the single
Object. That there is a constraint is obvious when one realises that
If the positions of three legs of an assembled chair are known, then
That of the fourth follows necessarily— it has no freedom.
Thus the change from four separate and free legs to one chair
Corresponds precisely to the change from the set’s having 24
Degrees of freedom to its having only 6. Thus the essence of the
Chair’s being a “thing”, a unity, rather than a collection of inde-
Pendent parts corresponds to the presence of the constraint.
Seen from this point of view, the world around us is
Extremely rich in constraints. We are so familiar with them that
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We take most of them for granted, and are often not even aware
That they exist. To see what the world would be like without its
Usual constraints we have to turn to fairy tales or to a “crazy” film,
And even these remove only a fraction of all the constraints.
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