[From Bill Powers (961019.1730 MDT)]
Bruce Gregory (961019.1610 EDT) --
I agree with Marken I and Martin.
I disagree with Martin II and Bill. An attractor
discribes the trajectories through phase space
generated by a set of dynamical equations.
...
The danger which I was concerned about was
addressed by Marken I. Marken II and Bill have
apparently fallen victim to the reification monster.
Uh-uh. An attractor is not the trajectory actually followed, but an
imaginary trajectory toward which the actual trajectory in phase-space
continually returns after perturbations, as if attracted to it. A
point-attractor is an imaginary point in phase-space toward which the actual
points tend to move, again as if -- AS IF -- attracted to that point. In
some cases there is a literal attractor (a mass distorting space or
alternatively exerting a pull); in others, for example a control system,
there is only a signal specifying a goal-state toward which actions will
_push_ the controlled variable.
The reification of which you speak comes in when you say that just because
the same mathematical equation approximates the behavior of two different
processes, it represents some natural force or principle that "governs" the
processes in some trans-physical way. I am not a devotee of the faith that
preaches that the universe is mathematical. I see mathematics as a
particularly disciplined mode of description, nothing more. The map is not
the territory. That, of course, make me a heretic, but I'm assuming we
heretics don't get burned at the stake any more.
Suppose that the image which Poincare' and others had originally come up
with had been different. Suppose that someone influential enough had
suggested that we think in terms of "constrictors." A constrictor has the
effect of confining the behavior of a point's trajectory within some volume
of phase-space, such as a toroid or a curved tubular region with a
particular closed shape. A point-constrictor would be like a vector field
with arrows all pointing inward, forcing the trajectory toward smaller and
smaller regions as time goes on. Of course the equations describing
constrictors would be exactly the same as those describing attractors. The
only difference would be that we would be describing all these processes,
verbally, in terms of pushes instead of pulls, which would be incorrect for
the OTHER set of processes, those we model as involving pulls.
Or suppose someone had proposed "steering functions." A steering function
acts on a trajectory to bend it in some systematic way, sometimes in closed
paths and at other times converging to particular places (in phase space, of
course). The various kinds of steering functions would be described, I
imagine, by equations that are like the first time-space derivatives of the
equations describing attractors. Now the image would be right for certain
control systems, but not for marbles in a bowl (which are pushed toward the
center) or for planets (which are pulled toward the central region).
However, the equations would be the same, or exactly equivalent, in all
cases, except for the deviations from ideal that typify the various kinds of
real systems.
Mathematical descriptions of processes are nice because we don't use words
in them; we use arbitrary symbols which in themselves have no meanings
except the ones explicitly given to them for purposes of the development.
But as I have found to my sorrow, a description given in words, no matter
how carefully and repeatedly they are defined, leads not to one logical
development but to a large number, as word-associations come into play,
unbidden and bring in meanings that are not intended. Why deliberately
encourage that sort of thing?
Best,
Bill P.