[From Bill Powers (2011.11.24.0634 MDT)]

(Gavin Ritz

2011.11.24.17.38NZT)

Im listening to music and Im singing in my head, no vocal sound. Lying

on the bed very still in a room, no movement.

Is the sound a disturbance or a controlled

variable?

The sound is a controlled variable. A control system does not perceive

disturbances (as causes) – they are known to the control system only as

unintended changes in controlled variables. Of course another system

inside you may perceive the cause of the disturbance, but that’s a

different control system. It doesn’t have to exist for the first control

system to work. And “you,” the conscious observer, may be aware

of both the controlled variable and the cause of the disturbance, and

many other perceptions going on at the same time. But “you” are

only a spectator.

What is the output variable? (is it the singing in my

head)

The output is a reference signal which normally enters the comparator of

at least one lower-order control system, telling it what amount of its

perceptual variable it is to produce. In the imagination mode, however,

that output is switched to enter the input function of the higher control

system instead of the lower comparator, and that signal takes the place

of the perceptual signal that would normally be generated by the lower

system. So you get the impression of hearing singing even though no

singing is coming into your ears. This follows from the postulate that

perceptions exist only as afferent signals generated by perceptual input

functions.

Most of this is discussed in B:CP.

As to the “function” problem,

those definitions you were citing are from some very abstract branches of

mathematics, not the sort of math that a mere physicist/engineer like me

uses. Unless you have mastered those higher realms of mathematics, I

recommend sticking to the simple practical ideas that are used in

engineering. If you have mastered them, then you will have a

communication problem here until you learn to translate from the abstract

to the particular kinds of math at my level of understanding.

At my level, part of the problem may simply be language. We engineers

often say that a variable symbolized as x has a numerical value, say 2.5,

in some units of measure. But we don’t say it that way: we just say

“x is 2.5.” Of course x is NOT 2.5, it’s an alphabetic symbol

standing for a number which might be any real (i.e., on an infinitely

fine scale) number that just happens to be, at the present moment,

2.500… .

So when we say a function is a variable, what we mean is that by

evaluating the mathematical expression defining the function, we can

compute a magnitude

that we call the value of the function, and represent that value by a

symbol for another variable. Because the arguments of the function are

variables, the value computed from them is also variable, so the value of

the function is a variable, too. Another variable can be defined and set

equal to the value of the function, and our shorthand for that is to say

that the variable is a function of the arguments. This does not mean that

a function f(v) “is” a variable, any more than we would mean

that x “is” 2.5. We mean that the value of the function is

variable and can be symbolized as such.

p = f(v) is read out loud as “p is the function f of the variable

v”. Of course this does not mean that the FORM of the function is

symbolized as p. It means that the VALUE of the function, obtained by

some set of operations on the value of v, is symbolized as the value of

p.

Again in engineering terms, mathematical functions are used as

descriptions of the operation of various components of a system. A

capacitor, for example, can be treated as a function that converts the

flow of a certain number of electrons into a voltage. Using E for

“electromotive force” which just means voltage, Q for quantity

of charge, and C for capacitance (itself a function of the area and

separation of the plates of the capacitor), we have in general

E = f(Q,C) or specifically

E = Q/C

The first form simply says that the voltage E depends on Q and C in some

unspecified way indicated by the symbol f. The second form spells out

exactly how E depends on those other variables, so we don’t need

the functional notation. Of course we can assign the symbol f to the

expression Q/C and use other letters for other forms of functions. Then

when we refer to f(Q,C) we mean only the expression Q/C and not any other

form.

The functional notation is used when we want to state a dependency

without giving the details of its form (perhaps because we don’t know it

yet). There are all sorts of theorems in advanced algebra that deal with

functions without ever saying what their forms are. But in engineering we

want to know what the forms are, eventually. In PCT we speak of input

functions and output functions, without knowing what forms they may take

in a specific application. Different functions will involve different

mathematical expressions, so we give them different names.

So you can see that you’re right in thinking of a function as a

transformation or mapping of the arguments into the value of the

function. But the shorthand engineer-talk doesn’t disagree with that. It

just leaves out some details.

Best,

Bill P.