I agree with the qualifications you make. In particular there will be
cases where chaos is just in the model and not in the system being
Maybe the weather, or perception/cognition, are not chaotic after all. (03)
Let's take another perspective on that. Would it be a good thing if
our cognition were not chaotic? (Predictable weather, I grant you,
might be a good thing :-) (04)
I think it is worth looking at chaos from a positive point of view for
a change. Instead of fearing our cognition is chaotic let's hope it is
To me it is somewhat evocative of Kolmogorov complexity. Normally we
think of a random string as being empty, without information. But as
Kolmogorov/Chaitin pointed out, a random string will actually contain
more information than an ordered string (because the ordered string
can be predicted.) (06)
I don't know how Kolmogorov complexity and chaos theory are related
formally, but it makes sense to me that just as by Kolmogorov random
strings are the most compact, chaotic systems might be considered to
contain more information than systems which are not chaotic. (Crudely,
it takes more information to specify their behaviour.) (07)
Actually this looks like the other side of Randall's "rounding error"
issue. To copy a chaotic system you need to record more information,
ergo the system contains more information (than a classical system
recorded to the same number of decimal places.) (08)
So chaos need not be all bad. It would make sense if we stored our
information this way. Otherwise we would just be wasting bits. (09)
On Feb 6, 2008 2:18 AM, Ed Barkmeyer <edbark@xxxxxxxx> wrote:
> Rob Freeman wrote:
> > On Feb 5, 2008 3:54 AM, Ed Barkmeyer <edbark@xxxxxxxx> wrote:
> >> Chaotic response is a behavior of a model under certain conditions. It
> >> is proper to say that the model is probably not "valid" or "good enough"
> >> under those conditions to predict anything about the system being
> >> modeled.
> > Are you saying, not only "meaning" or the weather, but all chaotic
> > behaviour is really only a manifestation of models which are not "good
> > enough"?
> No. I should have been more careful about what I wrote.
> We have observed in many cases that mathematical models or discrete
> simulation models of certain systems demonstrate chaotic responses to
> certain stimuli, when the systems in question simply respond by moving
> to a different, although less predictable, understood dynamic state. In
> those cases, the problem is clearly that the model isn't faithful to the
> behaviors of the system.
> There are systems that actually have chaotic response to very minor
> perturbations in certain regions, or in the self-symmetric case, in some
> parts of every region.
> My point was that one cannot conclude from chaotic behavior in a model
> that that behavior is reflected in the behavior of the modeled system.
> It may just be that the model breaks down in that region. We must come
> to recognize true chaotic behavior by experimental observation.
> And trying to validate that kind of stimulus-response performance
> experimentally is quite tricky. You have to ensure that the values of
> the two experimental stimuli are different but close enough to
> demonstrate the chaotic response, and that usually requires very fine
> control and measurement of the stimulus. Further, you have to be
> convinced that there is no other uncontrolled variable operating to
> produce the difference in effect. And most of the simulation model
> failures occur precisely because they don't take account of some
> unexpected influential variable. (The mathematical models tend to fail
> because the model uses an understood function to approximate an unknown
> one, and the behaviors of those functions diverge in some region. Once
> you make reliable experimental observations in the region, it is easy to
> see the functional divergence. That was the point of Mary Payne's
> observation -- the tangent of the closest machine value to pi/2 is still
> representable as a machine value, and the tangent of the next value
> beyond that has a different sign.)
> -Ed (011)
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