ONT Differential Logic -- Series B
DLOG. Note B1
| The most fundamental concept in cybernetics is that of "difference",
| either that two things are recognisably different or that one thing
| has changed with time.
| William Ross Ashby,
|'An Introduction to Cybernetics',
| Chapman & Hall, London, UK, 1956,
| Methuen & Company, London, UK, 1964.
Linear Topics. The Differential Theory of Qualitative Equations
This chapter is titled "Linear Topics" because that is the heading
under which the derivatives and the differentials of any functions
usually come up in mathematics, namely, in relation to the problem
of computing "locally linear approximations" to the more arbitrary,
unrestricted brands of functions that one finds in a given setting.
To denote lists of propositions and to detail their components,
we use notations like:
!a! = <a, b, c>, !p! = <p, q, r>, !x! = <x, y, z>,
or, in more complicated situations:
x = <x_1, x_2, x_3>, y = <y_1, y_2, y_3>, z = <z_1, z_2, z_3>.
In a universe where some region is ruled by a proposition,
it is natural to ask whether we can change the value of that
proposition by changing the features of our current state.
Given a venn diagram with a shaded region and starting from
any cell in that universe, what sequences of feature changes,
what traverses of cell walls, will take us from shaded to
unshaded areas, or the reverse?
In order to discuss questions of this type, it is useful
to define several "operators" on functions. An operator
is nothing more than a function between sets that happen
to have functions as members.
A typical operator F takes us from thinking about a given function f
to thinking about another function g. To express the fact that g can
be obtained by applying the operator F to f, we write g = Ff.
The first operator, E, associates with a function f : X -> Y
another function Ef, where Ef : X x X -> Y is defined by the
Ef(x, y) = f(x + y).
E is called a "shift operator" because it takes us from contemplating the
value of f at a place x to considering the value of f at a shift of y away.
Thus, E tells us the absolute effect on f that is obtained by changing its
argument from x by an amount that is equal to y.
Historical Note. The protean "shift operator" E was originally called
the "enlargement operator", hence the initial "E" of the usual notation.
The next operator, D, associates with a function f : X -> Y
another function Df, where Df : X x X -> Y is defined by the
Df(x, y) = Ef(x, y) - f(x),
Df(x, y) = f(x + y) - f(x).
D is called a "difference operator" because it tells us about the
relative change in the value of f along the shift from x to x + y.
In practice, one of the variables, x or y, is often
considered to be "less variable" than the other one,
being fixed in the context of a concrete discussion.
Thus, we might find any one of the following idioms:
1. Df : X x X -> Y,
Df(c, x) = f(c + x) - f(c).
Here, c is held constant and Df(c, x) is regarded
mainly as a function of the second variable x,
giving the relative change in f at various
distances x from the center c.
2. Df : X x X -> Y,
Df(x, h) = f(x + h) - f(x).
Here, h is either a constant (usually 1), in discrete contexts,
or a variably "small" amount (near to 0) over which a limit is
being taken, as in continuous contexts. Df(x, h) is regarded
mainly as a function of the first variable x, in effect, giving
the differences in the value of f between x and a neighbor that
is a distance of h away, all the while that x itself ranges over
its various possible locations.
3. Df : X x X -> Y,
Df(x, dx) = f(x + dx) - f(x).
This is yet another variant of the previous form,
with dx denoting small changes contemplated in x.
That's the basic idea. The next order of business is to develop
the logical side of the analogy a bit more fully, and to take up
the elaboration of some moderately simple applications of these
ideas to a selection of relatively concrete examples.