[Martin Taylor 980228 22:40]

Bill Powers (980228.0308 MST)

I'm really beginning to understand Franz Kafka a lot better these last

couple of days, but now we seem to have gone through the looking glass

with him, to join Alice, as well.

Bill Powers (980227.0814 MST)--

I suppose that if I said "I was at the lake today" and you knew my wife

had been there with me, you would say "No you weren't; you were

not there alone."I don't understand how this illustration applies.

I said "p is a function of d" and you said I was wrong, because p isn't

a function _only_ of d. I merely point out how nonsensical the same

transformation is in another, equivalent, circumstance. Perhaps I should

have been more direct, and pointed out that if y is y(x1, x2, x3, ...),

y is normally said to be a function of each of its arguments. In particular

y is a function of x1. Only one of Kafka's bureaucrats would find a way to

insert "only" into the statement so that the suspect could be shown thereby

to be a criminal.

In any case, I went further, and said that when the other arguments are

held constant, y is a pure function of x1 (as opposed to being simply

a function of x1).

Anyway, as Rick points out, your formula is not correct for the present

discussion. It should be written asp = Fd(d)/(1+G) + Gr/(1+G)

That is _not_ what Rick pointed out. Rick suggested that my formula was

wrong. You show that you believe it to be right. In another message

directed to Rick, I have provided its derivation. For the "present

discussion," Rick uses "d" for the influence of the disturbance on

the CEV, as in the formula "p = o+d" which he often repeats, and which

you have not criticised.

In this Wander-land Through the Looking Glass, the validity of a mathematical

expression obviously depends on who says it.

nor is there any general basis for choosing one solution when the

input function is multiple-valued (as it most often is).Are you saying (today) that p can take many values for a given condition

of the Controlled CEV (i.e. a particular set of sensory values, along with

their histories if the PIF is a time-function such as an integrator or

differentiator)? That's what a muliple-valued function is.No, it's not.

I suggest you look in some kind of a mathematical dictionary, if you won't

take my word for the difference between a multi-valued function and a

function with multiple arguments. A multi-valued function has more than

one possible value for a particular set of values of its argument. A

square root is a trivially simple example. A function with multiple

arguments is like p(x1, x2) = x1 + x2, with its two arguments. A function

with multiple arguments may or may not be multiple-valued. A multi-valued

multi-argument function might be p(x1, x2) = sqrt(x1+x2).

Even if you won't take my word, I find it interesting to know that

(today) the CCEV is not a scalar quantity, as it was yesterday, and

probably will again be, tomorrow. And this isn't even a Leap Year.

Another simple function that has the property of multiple values is

p = x1 + x2.

OK, I'll believe you if you tell me what the multiple values are for, say,

x1 = 3 and x2 = 4.

We can have the SAME value of p for an infinity of pairs x1,x2. I think

you've got the problem reversed in your head.

I'm afraid I have. It's only through the Looking Glass that a function's

output determines its input; in the everyday world in which I usually live,

the input determines the output. In the world in which I live, any pair

of values of the two inputs results in only _one_ possible value of the

output.

It's not that the same

environment can produce two values of the controlled perception; it's that

the same value of the controlled perception can represent a multiplicity of

states of the environment.

But only one value of the CCEV, unless you are introducing some quite new

concept here, such as that the CCEV is multiple-valued, or perhaps that

the CCEV is multidimensional. Or both.

The controlled perception is the given; the

state of the environment needed to produce that given perception is what is

multiple-valued. Many different states of the environment can produce the

same perception; this is why a _single_ perception can't be used to deduce

the state of anything in the environment.

Except the state of the CCEV, perhaps?

Are you asserting that it most often is

true that perceptual functions are like that?Yes, but not in the reversed sense you're talking about. In general, most

perceptions are functions of multiple input variables, and so the state of

a given perception can't be used to deduce the actual state of the input

variables.

In the world seen from the viewpoint of the control system, there is

only one input variable for which a state is accessible, and that is

the scalar variable called the Controlled Complex Environmental Variable.

The value of the perceptual signal is the internal representation of

the state of that variable.

An external analyst may observe that the CCEV is a complicated function

of many observables (available to _his/her_ perception), together with

their histories, but that complication doesn't affect the single control

system. (It can be important in analyzing the interactions among simple

control systems).

All that matters in analyzing the operation of an Elementary Control

loop is the set of relations among several scalar variables, namely the

two inputs to the loop (the reference signal and the disturbance signal),

the two outputs from the loop (the perceptual signal and the "output"

signal) and two internal variables (the error signal and the--usually

unnamed--influence of the output signal on the CCEV). The functions that

relate the variables each transform one scalar variable into another scalar

variable, except that the comparator and the CCEV both have two arguments,

the reference signal and the perceptual signal in the case of the comparator,

and the output signal and the disturbance signal in the case of the CCEV.

perceptual signal ^ | reference signal

> V

>____________-_____error signal___

> comparator |

perceptual signal | |

> output function

input function |

> >output signal

CCEV + -----environment function-------|

> >

disturbance signal| |output signal

^ V

It's quite irrelevant to the control analysis whether the link between

one scalar variable in the loop and another goes by one wire or a million.

And if it goes by a million, the values in each micro-wire never

affect the operation of the loop independently of the value of the scalar

variable.

Apparently, the PCT world is one in which what matters is not what is

said but who said it, where functions go from output to input, where

functions of several arguments are automatically multiple-valued, and

where what is true on Tuesday is false on Thursday.

I do not care to continue to try to discuss matters using technical and

logical arguments in a world in which such arguments are invalid. And

I do not care to substitute personal for technical argument. When

you and Rick can come to an agreement on which terminology I should use,

when you can distinguish a function's arguments from its value, and

its output from its input, when you understand the difference between

"X is true" and "only X is true," then perhaps there might be merit in

continuing.

Martin