Reply to Bill C's note

[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.