Linear Topics : The Differential Theory of Qualitative Equations

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, Cybernetics

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:

\[\mathbf{a} = (a, b, c),\ \mathbf{p} = (p, q, r),\ \mathbf{x} = (x, y, z),\!\]

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 \(\operatorname{F}\) 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 \(\operatorname{F}\) to \(f\!\), we write \(g = \operatorname{F}f.\)

The first operator, \(\operatorname{E}\), associates with a function \(f : X \to Y\) another function \(\operatorname{E}f\), where \(\operatorname{E}f : X \times X \to Y\) is defined by the following equation:

\[\operatorname{E}f(x, y) = f(x + y).\]

\(\operatorname{E}\) 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, \(\operatorname{E}\) 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 "shift operator" \(\operatorname{E}\) was originally called the "enlargement operator", hence the initial "E" of the usual notation.

The next operator, \(\operatorname{D}\), associates with a function \(f : X \to Y\) another function \(\operatorname{D}f\), where \(\operatorname{D}f : X \times X \to Y\) is defined by the following equation:

\[\operatorname{D}f(x, y) = \operatorname{E}f(x, y) - f(x),\]

or, equivalently,

\[\operatorname{D}f(x, y) = f(x + y) - f(x).\]

\(\operatorname{D}\) 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:

\[\operatorname{D}f : X \times X \to Y,\]

\[\operatorname{D}f(c, x) = f(c + x) - f(c).\]

Here, \(c\!\) is held constant and \(\operatorname{D}f(c, x)\) is regarded mainly as a function of the second variable \(x\!\), giving the relative change in \(f\!\) at various distances \(x\!\) from the center \(c\!\).

\[\operatorname{D}f : X \times X \to Y,\]

\[\operatorname{D}f(x, h) = f(x + h) - f(x).\]

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. \(\operatorname{D}f(x, h)\) 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.

\[\operatorname{D}f : X \times X \to Y,\]

\[\operatorname{D}f(x, \operatorname{d}x) = f(x + \operatorname{d}x) - f(x).\]

This is yet another variant of the previous form, with \(\operatorname{d}x\) 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.