I would love to share those sources. The Sante Fe Institute (which has a WWW
home page but I don't know the address) is using nonlinear dynamics. Some of
what they have done is good, some bad. Stuart Kauffman is one of the leading
advocates in the field. Dawkins has some morph forms but they really aren't
nonlinear dynamics. I will list the bibliography at the end of this.
I use the term "phase space' in the sense that Yockey uses it. If you
have a 2-nucleotide unit long DNA, the phase space is basically a 2
dimensional surface which has a point for each possiblity. There are 4
nucleotides so for this 2-nucleotide there are 16 different possibilities
outlined below.
a . . . .
t . . . .
c . . . .
g . . . .
g c t a
If you add a third nucleotide to this sequence you have to add another axis
to the phase space. Now you have 4 x 4 x 4 =256 different possibilities in
this 3 dimensional phase space. Add a fourth, there are 1024 possibilities
in a 4 dimensional space. And so forth.
Each point in the phase space represents the entire DNA. A mutational path
through the multi-billion dimensional DNA phase space is a line connecting
mutation 1 to mutation 2 to mutation 3 etc. In the programs I have been
using I have only a 4 dimensional phase space In the Selection program, the
successively chosen mutations form a line through that phase space.
Gordon wrote:
"By the way, I have run all three of your compiled programs. They are good
fun, and do seem pretty interesting. But I can sympathize with those who
view them as something of a black box. I view them as *illustrative* of
what *might* be occurring with mutational processes. Skeptics like Jim
Bell might easily find them totally unconvincing. On the other hand,
those who find evolution easier to accept are likely to find your programs
more satisfying. Ideally, one would like to be able to close up the gap by
making the link between the real and modeled mutational processes as
clearly stated as possible. "
Well, if you remember, I used to post the code for these programs but no body
would program them. I decided to put the code in an executable so that
people could run them. This would appear to be a "darned if you do; darned
if you don't" situation. I have had more success with people running them by
putting the code in executable form. I agree that some people might view
these as a black box. I will try to explain it below.
Here is how I veiw the way the situation works. All nonlinear systems
involve some type of equation like
x-new = function of x-old.
When x-new is calculated, it then becomes x-old and the process is repeated.
A simple well behaved function is
x-new = x-old * x-old.
Starting at x-old =1 you iterate this equation and you get a series:
1,1,1,1,1,1.........
Start at x-old = 2 you get the series
2,4,16,256,65536......
If you use a function like x(n+1):=Rx(n)[1-x(n)] things get quite
interesting. This is the logistics equation and models what happens in
populations in the wild. The n in the above equation is generation number.
If R<1 then no matter what x(0) you start with, the population will go
extinct. After a certain number of generations x(n)=0. If R>1 and < 3 the
population is stable and does not rise or fall. If R > 3 then the population
becomes cyclic. The population values fall around particular values and not
anywhere near other values. The closer R gets to 3.57, the more values the
population will try to satisfy. At R=3.57 and above, the population number
is totally chaotic and is as likely to go extinct as to survive. There is no
predictability.
Gordon wrote:
"Jim does have a point. Mathematics does allow a great deal of flexibility
for model building. While not all things are possible, there are lot of
possibilities that incorrectly describe reality. A good illustration can
be given in physics: What happens when Newton's inverse square law is
replaced by an "inverse cube law" in the gravitational equation? Nothing
that resembles reality. But it is no more than a mathematical model - and
one can easily illustrate its effects on the computer. But it is just the
wrong model. Nothing more.
A "proof" of a good model is that it makes good predictions. The inverse
square law certainly meets this test. At present, does any mathematical
model of evolutionary processes make good predictions? "
Yes, mathematics does provide a lot of flexibility and lots of models are
wrong. That is why I freely posted my code in the past. I think it is
important that everyone get a chance to examine the code. I hate the
scientific articles which discuss the results of a computer model that the
authors wrote, but you can't ever see the code. There are so many ways that
a model can go wrong that it seems to me that examining the code is also
important. I will give the code to anyone who wants it.
But I was getting the impression that Jim was arguing that no model would
be satisfactory. If I am wrong, then I apologize, Jim. But if my impression
was correct, then I would strongly disagree that models have no bearing upon
reality.
A correct model is one which captures all the essential features of the
phenomena and then carries forth the calculations in a proper manner. My
computer program captures all the essential features.
1. the computer code, by itself is incapable of doing anything
DNA without a cell is incapable of doing anything
2. Random mutations to a computer code generate far more harmful mutations
that beneficial ones.
Random mutations to DNA does the same.
3. Certain locations in my program can be mutated at will and the output
alters tremendously but the program still runs on the computer.
Certain locations in DNA can be mutated at will and the output alters but
the code still runs on the cellular machinery.
4. The code produces (with the help of the computer) a mutated copy of
itself and produces a new generation with an altered morphology
DNA produces (with the help of the cell) a mutated copy of itself and
produces the individual of the next generation with altered morphology.
I can't off hand think of something that makes this system non-analogous to
the DNA-cell system.
One final important point, when the model is run, one MUST compare the output
to the behavior of reality. This is what Russ Maatman did with his baseball
model. His model matched reality. (Russ, I bet you could sell that program
to bookies for a lot of money. :-) ) If the model does not match reality,
then the model must be altered. In the case of my CAMBEXPL and EVOLVE
programs, the things I wanted to model was punctuated equilibrium and
disparity before diversity. These are two features of the fossil record that
Christian apologists say disprove evolution. Both those programs use the
same algorithm and they matched the behavior of the fossil record on both
counts! Thus, my model was successful (what else can you expect of a model?)
Since I have a reproductive algorithm which can match reality, it seems to
me that Christian apologists should be careful in their statements that there
is no mechanism for punctuated equilibrium and disparity before diversity.
In answer to your question about the inverse cubed law, I think the curve is
called a Cotes spiral. With an inverse cubed law the object spirals into the
gravitating body, if it is an inverse <2 law, the object spirals outwards I
believe.
(I haven't checked this).
Bibliography
Bill Hamilton put out an excellent bibliography on nonlinear dynamics. I got
most of my info from articles.
Quantifying Chaos- Computers in Physics, Nov/dec 1989. p. 86-89
This article discusses the logistic difference equation which models
population dynamics and explains why populations rise and fall in
numbers chaotically.
It also discusses the Henon attractor, the Lorenz equations (which
model convection cells), a nonlinear oscillator (which models real
physical
pendulums
The Arithmetic of Chaos - Nature Oct 22, 1987, p. 670-671
Minimal chaos and stochastic webs Nature April 9,1987 p. 559-563
Fast dynamo action in a steady chaotic flow Nature apr. 11, 1991, p. 483-485
The Classic paper which started it all is
Edwin Lorenze "Maximum Simplification of the Dynamic Equations, Tellus
August 1960, pp243-254
Deterministic Nonperiodic Flow, Edward N. Lorenz, Journal of the Atmospheric
Sciences, march 1963, p. 130-141
Spatial structure and chaos in insect population dynamics Nature Sept. 19,
1991, pp 255-258
Self-organized criticality in the game of life Nature Dec 14, 1989 p. 780-781
Dimension of weather and climate attractors, E. N. Lorenz Nature sept. 19,
1991, p. 241-244
Strange kinetics - Nature May 6, 1993, p. 31-37.
Two books are worthy of mention other than those Bill put out.
F. Morrison, _The Art of Modeling Dynamic Systems_, Wiley, 1991,
Thompson and Stewart, Nonlinear Dynamics and Chaos, Wiley, 1986
There is a program which models the motion of a star in a galaxy in
Steven E. Koonin, _Computational Physics_ Benjamin Cummings , 1986,
pp235-246. Under certain energy levels, the orbit of such a star is very
chaotic. I used to have the original article on this from the Astrophysical
Journal, but I have mis-filed it so I can't give you a reference.
glenn