[Hans Blom, 960206c]

(Bill Powers (960203.0500 MST))

If you want a mathematical definition, please consult a mathema-

tician.

Since you seem to know what a rigorous mathematical definition would

be, I'll consult you.

What makes you think I'm a mathematician? Anyway, hubris and all, let

me pose as one for a moment.

Let Y be a physical variable.

Mathematician: "What is that? Please define what you mean by a

_physical_ variable."

By accepted concepts of physical causality,

Mathematician: "If you want to talk about causality, please consult a

philosopher. They have some theories about efficient, ultimate and

other sorts of causes, although I don't think much of them. But I've

never heard of physical causality. Where do I find a theory about

that?"

Y is a function of some set of other physical variables, x1..xn.

Mathematician: "Ah, except for the word physical, this is something

that I recognize. You're talking about a mapping from an n-dimension-

al space to an, I assume, one-dimensional space, where Y is the

mapping function."

In general all the variables are time-dependent, so x, for example,

means x(t).

Mathematician: "Oh, I misunderstood, there is an extra dimension, t.

So you're talking about a mapping from an n+1-dimensional space to

an, I assume, one-dimensional space."

Y(t) = f(x1(t)..xn(t)), or for short, Y = f(x1..xn).

Mathematician: "Oh, sorry, I misunderstood again. You're talking

about a mapping from an n+1-dimensional space to a TWO-dimensional

space, where one dimension remains the same after mapping. But now

the mapping function is called f, where you said is was Y. Please

attempt to be consistent; you confuse me. Y is called the mapping

(noun) or projection, whereas f is the function that specifies how

the mapping (verb) is done."

This is the environment equation. Is that mathematically clear?

Mathematician: "Not quite. We seem to lack a few basic notions. Let

me propose some and see whether you agree.

1) An n-dimensional space or n-space can be given a metric by

introducing n mutually orthogonal axes that meet in the special point

O, called the origin of the space. I propose that we use a Euclidean

space, where the axes are straight lines. Or did you have some other

geometry in mind?

2) Every point in n-space can be described by its n projections on

the n mutually orthogonal axes that we introduced above.

3) I anticipate that we are going to need a definition of a distance.

Will the standard Euclidean distance do?

OK so far?"

Let a system S exist, with an input-output function g.

Mathematician: "OK, you propose another mapping function, g. From

what type of space to what other type of space?"

If Y is the input to S and x1 is its output,

Mathematician: "Oh, I see, from a 2-space to a 2-space."

we can write x1 = g(Y - Y0)

Mathematician: "Now you introduce the minus-operator. What is its

meaning? Are we to see Y and Y0 as two-dimensional vectors and the

difference as a normal vector-difference?"

where Y0 is the value of Y at which x1 is zero.

Mathematician: "This becomes unclear. We had a mapping function f

which maps from n+1-space to 2-space. Now do you propose that there

is a mapping from a subspace of the n+1-space (an n-space, resulting

from the elimination or fixation of x1) to the 2-space of Y? Do I

also understand correctly that x1, which you said was a function of

t, is to be zero for all values of t? I'm afraid that the mapping has

become unclear to me."

Again, all variables are time-dependent (Y0 is a constant).

Mathematician: "This is really confusing, variables that can be

constants. I assume that Y0 is still a function of t?"

This is the system equation.

Mathematician: "So that's what you call it."

Well, that's enough, I think. Where a mathematician could help is in

expressing the problem clearly, where it is fuzzy now. Where he could

also help is in pointing out that by going from an n+1-space to a

2-space something is irredeemably lost, which cannot be regained

later where manipulation can take place only in 2-space; this is

equivalent to stating that you cannot control your perceptions but at

most a lower-dimensional projection of them.

I'm not sure all of this helps...

Greetings,

Hans