[From Gabriel 921211 11:12CST]

A wondrous bright light Bill, and nobody shoving an M16 in your back

either. Let me respond mathematically, but without formality - I don't

get to do that often which is why you shone such a great light on the

subject. Usually it's not possible to set out the idea without the

formalism. And I really enjoy talking about my real research to

somebody other than myself.

The reason why my 04:30 note is appealing but not proven is that it's

an imagined neuroanatomical implementation of the mathematical

abstractions of search spaces, ordinary differential equations,

optimal decision theory, and Hamiltonian system dynamics.

There are lots of other possible implementations, and one can only

tell between them by experiment outside the realm of the common

mathematical abstraction. That is to say, the mathematics has degrees

of freedom which are lost once you go a physical system of any kind.

I can implement the procedure in neuroanatomical wetware (perhaps),

organisational wetware (certainly but I have serious constraints

about who I choose to put in the organisation if I want it to be

cost effective) silicon (at great capital expense, but practically

zero cost of replication) software (same as silicon except easier

to change when a find a mistake).

BUT since the mathematical abstraction has lots of properties we

can observe to be approximately true outside the neuroanatomical

black box, and actually observe in operations like Desert Capture,

it's useful independent of the details of wetware or software or

silicon or organisations. Its usefulness lies in the observed

fact that it's a detailed abstract implementation of the phenomena

we observe, or a "model." It's predictive, it elucidates strong

recognition reflexes from those who know the physical implementations,

and so on. That is to say, it has the properties of PCT. The

basis for PCT in neuroanatomy is not bad, but certainly not

conclusive for anything much bigger than the Moths and the Bats.

The real justification for PCT is that we can build the Little Man

and Little Arm models in software, and they have lots of the

properties of their analogues in wetware, but they are built from

the piece parts of the PCT premise.

Now, there are some other universal abstractions that have been very

useful, and it's worth giving them a passing glance because they

are meta-meta.....meta theories. Hard to use, but very powerful.

And they have to do with constricting degrees of freedom which

is one of the things that interest us.

If you look at planetary dynamics for instance, you find there are

some transformations of coordinates that don't change the equations

of motion. For example those that arise because gravitation is a

central force, and so, although orbits are not circular, one ellipse

is as good an orbit as another of the same size but rotated in 3 space,

and two ellipses with the same T**2/A**3 are both equally good. And

that (r**2)*(d theta/dt) is the same at all points in the orbit.

These facts, observed by Kepler, can be made to yield some astonishing

results. That acceleration is directed towards the sun, that it is

inversely proportional to r**2, and that the constant of proportionality

is the same for all the planets.

Now we make the great gedanken experiment. Introduce an imaginary

new planet, and assume the same things are true. We can at once

conclude the usual statements of Newton's Laws and Gravitation are

true for this imaginary new planet, and so on .... BUT the conclusion

depends on two things, the Bayesian priors, and the assumption that

they are true for the new planet - which was OK until Einstein, and

still good enough for Govt. work.

Now, when we do the same kind of gedanken experiment for the next

level up in the hierarchy, we arrive at the idea of symmetry

operators for the system, and the theorem that initial conditions

and the symmetry operators alone determine an orbit.

This has a counterpart in psychology, it's Gestalt theory, and

invariants, and although I still don't really know what reorganisation

is, I suspect it has to do with throwing away some invariants, as

distinct from simply changing initial conditions.

Now back to software, which can mimic any physical system, so it's

potentially a useful abstraction too. If we have a program

y = f(x)

if it's going to be useful it had better yield reproducible reproducible

results, so that for each input x, there is only one possible output y.

That is to say, f is a many:1 mapping - several x may each give the same

answer, but a particular x better not give different answers Mon Tue Wed,

from Thur Fri Sat - on the seventh day f takes a rest.

This divides the set of all possible x into subsets, such that for every

x in a subset, y=f(x) is the same.

You can see there are all kinds of symmetry lollygagging around -

the "brotherhood" of all the x giving the same y is just a restriction

of necessary varieties from the set of all x to the set of all y, i.e.

a Ross Ashby necessary variety. If you leave a few brotherhoods out

of your consideration of possible inputs, you have left out some

y values, i.e. possible outcomes of running the dynamical system

(campaign, TACWAR model....) represented by f().

Better get off my soapbox before it breaks. Merry Christmas to All.

This gets mathematical at about the same rate as Bill's example

of mining the Straits of Tsushima, damaged the Japanese prosecution of

the war in the Pacific. But the idea of symmetries, brotherhoods of

scenarios all leading to (nearly enough) the same outcome, and Gestalt,

and human perception of same, and search amongst them is very much our

business.

John

PS I think I just did Ross Ashby wrong. The set of all brotherhoods

is the necessary variety, and we need just one representative from

each to get all the possible outcomes. But, if a brotherhood ain't

really a brotherhood, i.e. we've put two different phenomena in the

same class (f(x) can be a classifier) we may get an unpleasant

surprise if the actual scenario we face is not the one belonging

with the representative we chose for the "brotherhood". For the

mathematician, there is less error in the abstraction than in

the implementation - that's how I did Ross A. wrong, and why

there are bugs in programs.