[Hans Blom, 970127]

(From Bill Powers (970124.0500 MST))

I suppose I will go to my grave still trying to figure out this

problem ... Somehow I can't bring myself to believe either the

mathematician or the Frenchman. Why is that?

Thanks for a nice and thoughtful post.

As Hans Blom has mentioned once or twice in passing, this basic

method depends on the existence of an inverse of the function f over

at least some limited region. Nothing is said about how the inverse

is calculated. While we can perhaps imagine an actual simulation of

the environment existing in the forward form, it is hard to imagine

how the inverse of that same form could also be calculated. Since,

after adaptation, is it only the existence of the inverse that is

critical, we have to ask how the correct inverse could be obtained.

Yes, that's the riddle. My thoughts, experiments, calculations and

simulations show (prove may be too strong a word) _that_ an inverse

function (of the environment) is needed in a (any) controller if it

is to "calculate" how to act. The riddle is _how_ such an inverse or

its approximation can be "calculated" by the actual nervous system.

As an engineer, I know a variety of approaches from the technical

literature. A well-known one -- and the one that appears in PCT

models -- "calculates" the inverse in the feedback loop of a high

gain amplifier, much like an op-amp circuit can "transform" a

capacitance into an inductance. Another approach does not invert but

calculates the inverse directly. Other approaches again do invert,

but in a manner seemingly far removed from what math says about

inverting a function or a matrix; Martin's wasp-waisted neural

network comes to mind.

So how does the nervous system do it? Our models so far have been

mostly _functional_ models, which take the presupposed possibilities

and constraints of the nervous system more or less lightly. You do

the same when you use "nerve currents" rather than sequences of

action potentials. That is fine; every theory needs to be built up

from a satisfying, simple set of primitives. That eliminates the

infinite search for "deeper" meaning and provides the opportunity for

explanation in terms of "intuitively obvious" notions.

I guess much of your post stresses the difference between the

territory and its map. Some of our maps are pretty good -- they allow

us to be in control and to make fine predictions. But we are often

greatly puzzled about the _type_ of the mapping from "reality" to

those maps of ours. What is real and what is "just" a model?

We'll never know. But the occasional awareness that we can only talk

in terms of maps and mappings helps, maybe.

Greetings,

Hans