[Martin Taylor 920824 15:40]

(Bill Powers 920824.1100)

I think that when you shift types as you go up levels, the degrees-of-

freedom problem takes on some new aspects. You already showed how you

could get 18 degrees of freedom out of 9, just by adding first

derivatives. Couldn't we get 9 more by adding second derivatives, and

so on? The question now is, what ELSE can be added that takes us out

of the domain defined by our original concept of intensities ->

sensations -> configurations?

The answer to the last question there is "Nothing." At least in respect of

the number of available degrees of freedom. Certainly adding a second

derivative gives you 9 more, but you need another independent time sample

to get it. You can get as many degrees of freedom as you want by taking

sufficiently many independent time samples.

Bandwidth really isn't the problem here -- we could keep all

variations well within the available bandwidth and still have a

problem to solve.

Well, in a sense it is THE problem, but when you get non-linear you can't

interchange time and frequency freely, so you need some kind of surrogate

for bandwidth. But for heuristic discussion, bandwidth will do.

All nine reference levels can thus vary independently with

respect to oscillation frequency.In addition, the amplitude of oscillation can be varied independently

for all nine systems. There's an outer envelope set by bandwidth; the

bandwidth and momentary frequency sets the maximum rate of variation

in amplitude, but within that envelope, there is complete freedom. Now

we have nine more dimensions.

No you don't, if I understand your layout. You only have the selection of

values within the original nine dimensions. You do change the information

rate if you change the power level (SNR) within a dimension, but you don't

add any degrees of freedom.

So the same nine systems

can be used to construct an infinity of different patterns of

variation at the event level. We now have not only nine derivatives

(at least) at an instant, but a time-spanning characteristic called

amplitude, and a time-spanning pattern characteristic that can extend

indefinitely through time.

Sure. There are an infinity of possible settings of any single variable, too.

But that variable would have only one degree of freedom instantaneously, and

2*W degrees of freedom per second, where W is the equivalent rectangular

bandwidth of the variation of that variable (in a linear system).

[Discussion of making melodic phrases, which can be compared with ones

in memory]

We've now created a conceptual space in which various

components of a perception can change while holding other components

the same. We can define axes arbitrarily in such spaces, can't we?

Sure, but no more informationally independent ones than the original set

of notes allows- 2NWT where N is the number of simultaneously hearable notes

and T is the duration of the phrase. (I assume chords are allowed in your

phrases). You can define an infinite number of different axes, but measurements

on one won't be independent of measurements on another.

(Parenthetically: I think exactly this does happen in any "real" control

hierarchy, and it is what allows the neural architecture to deviate from

strictly reciprocal connections in which reference links parallel perceptual

links between higher and lower ECSs.)

When we

think of levels of perception in which temporal patterns are the

variable, the idea of simultaneity no longer applies. We can solve a

control problem that requires the nine systems to achieve their

reference states ONE AT A TIME or IN A CERTAIN PATTERN or IN CERTAIN

SPACE-TIME RELATIONSHIPS and so on. These are actually solutions to

conflicts that would arise if we demanded that all nine systems reach

zero error at once. These solutions may explain why higher levels

exist.

Yep, that solution to the problem is directly implicit in the degrees of

freedom argument. If there were no ouput bottleneck in degrees of muscular

freedom, the patterns WOULD be simultaneously achievable and there would be

no conflict. It is the limitation of available df for control that leads

to all conflict situations (or most, anyway; I'm not prepared to stake my

reputation on "all").

Even if we think of only nine letters on a keyboard with one finger

for each letter, typing the word "keyboards" is impossible to do with

a linear combination of finger-presses. We can clearly control each

finger simultaneously and independently with nine control systems, but

when we try to do so, the keyboard won't respond and we won't see the

word "keyboards" on the screen. We'll see whichever letter was hit

ahead of the others by a millisecond.

So here, the df bottleneck is in the external world. The keyboard has one

instantaneous df, and as many df per second as it will accept keystrokes.

And with only the nine fingers plus two more degrees of freedom (x and

y), we can type everything that it is possible to type. How many

degrees of freedom have I used in saying all this so far?

I find it easier with ten. But I don't know how many df you have used. A

rough estimate would be to count the letters, but that could be either an

understimate or an overstimate. It could be an underestimate because you

may have done a lot of erasing and retyping, which does not show in the post,

and it could be an overestimate because of the sequential redundancies among

the letters, which remove from you the choice of which key to strike on

some occasions if you want to make sense. But the number of characters is

a reasonable order-of-magnitude estimate.

Martin