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Cybernetics in the 3rd Millennium (C3M) -- Volume 2 Number 1, Jan. 2003
Alan B. Scrivener --- www.well.com/~abs --- mailto:abs@well.com
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Why I Think Wolfram Is Right
www.amazon.com/exec/obidos/ASIN/1579550088/hip-20 )
(If you missed Part One see the C3M e-Zine archives -- the link is at
the end of this article.)
Psychedelic guru Timothy Leary's last book, "Chaos & Cyber Culture"
(1994), was a collection of his recent short writings.
( www.amazon.com/exec/obidos/ASIN/0914171771/hip-20 )
In one essay he talked about Hermann Hesse, whose "Siddhartha" (1922),
and "Steppenwolf" (1927) were counter-culture favorites in the 1960s:
Poor Hesse, he seems out of place up here in the high-tech,
cybercool, Sharp catalog, M.B.A., upwardly mobile 1990s.
But our patronizing pity for the washed-up Swiss sage may be
premature. In the avant-garde frontiers of the computer culture,
around Massachusetts Avenue in Cambridge, around Palo Alto, in
the Carnegie-Mellon A.I. labs, in the back rooms of the
computer-graphics labs in Southern California, a Hesse comeback
seems to be happening. This revival, however, is not connected
with Hermann's mystical, eastern writings. It's based on his
last, and least-understood work, "Magister Ludi, or The Glass
Bead Game."
( www.amazon.com/exec/obidos/ASIN/022461844X/hip-20 )
This book... is positioned a few centuries in the future, when
human intelligence is enhanced and human culture elevated by
a device for thought-processing called the glass bead game.
Up here in the Electronic Nineties we can appreciate what Hesse
did at the very pinnacle (1931-42) of the smoke-stack mechanical
age. He forecast with astonishing accuracy a certain
postindustrial device for converting thoughts to digital
elements and processing them. No doubt about it, the sage of
the hippies was anticipating an electronic mind-appliance that
would not appear on the consumer market until 1976.
I refer, of course, to that Fruit from the Tree of Knowledge
called the Apple Computer.
Hesse described the glass bead game as "a serial arrangement, an
ordering, grouping, and interfacing of concentrated concepts of
many fields of thought and aesthetics. ...the Game of games had
developed into a kind of universal language through which the
players could express values and set these in relation to one
another."
This sounds to me remarkably like Wolfram's vision of the
significance of Cellular Automata.
I don't think I've made it a secret that I think Wolfram is onto
something big here. In this issue I will address why I think he's
right, and why everybody doesn't get it.
WHY I THINK HE'S RIGHT
In increasing order of significance, I think Wolfram is right because:
1) many of his discoveries and assertions match my own intuitions
2) he's gotten some great results so far using his new methodology
3) his new methodology prescribes much future work (it gives the
graduate students something to do)
3) his new methodology may actually address the problem of
unsolvability which has plagued science since Newton
Intuition
Many of Wolfram's discoveries and assertions match my own intuitions.
For example:
* I've long believed that braids, rope, knots, and weaving are
somehow important to the understanding of cybernetics and
systems theory. (See Wolfram p. 874.)
* It has been clear to me for some time that even simple
substitution systems, such as the early text editor "ed"
and its variants dating back to minicomputers, can yield
complicated results. (See Wolfram p. 82 and p. 889.)
* Similarly, I have long believed that simple programs can
yield complex results, if only by playing with some of the
winners of "Creative Computing" Magazine's "two line basic
program" contest on the Apple II in the early 1980s.
* Because computers allow us to go way beyond our current theories
of mathematics and logic, computational experiments are essential
to further progress.
My mentor, Gregory Bateson, taught me at length that "logic is a
poor model of cause and effect." He elaborated this later in
"Mind and Nature: A Necessary Unity" (1979).
( www.amazon.com/exec/obidos/ASIN/1572734345/hip-20 )
For example, both classical and symbolic (Boolean) logic hold that
it is a "contradiction" and therefore illegal to assert "A equals
Not A." But if you take computer logic chips and wire them up so
that a "Not" gate's output goes back into its input, it doesn't
blow up like on "Star Trek"; it simply oscillates. Furthermore,
this kind of feedback of output to input is essential to creating
the most fundamental of computer circuits, the "flip flop," which
is a one-bit memory. See the description of the "Basic RS NAND
Latch" at:
www.play-hookey.com/digital/rs_nand_latch.html
This follows from the fact that logic has no concept of time.
In the C programming language this problem is resolved by
having two different types of equals:
A == !A
is a test, which always resolves as zero ("false"),
while:
A = !A
is a command, which sets A to its logical opposite. If
its initial value is 1 ("true) it becomes 0 ("false") and
vice-versa. Clearly, if even these simple constructs are
beyond the theories of mathematical logic, but trivial on
a computer, there are many places a computer can take you
which are "off the map" of theory.
Before I'd ever owned a computer, back in 1977, I visited my
friend Bruce Webster who had just borrowed $800 to buy a
programmable calculator. On a lark, we investigated what
happened if we took the cosine of the log of the absolute
value of a number, over and over. We found that, given a
starting value of 0.3, that the result converged quickly to
zero. What this meant we couldn't say. But we figured that
it was probably the case that no one had done the experiment
before, and we were quite sure there was no robust theory to
explain what we were doing.
What we were messing around with is what is called
"iterated maps," and it was a very young field in 1977.
In 1976 Robert May had published some of the first clues
to the existence of chaos using a similar methodology
to explore the "logistics equation" -- albeit more
systematically than we'd been. See:
astronomy.swin.edu.au/~pbourke/fractals/logistic/
Later, Benoit Mandelbrot used similar computational experiments
(in 2D) to discover the Mandelbrot set; see:
aleph0.clarku.edu/~djoyce/julia/julia.html
And earlier, Edward Lorenz used computational experiments
to find the first "strange attractor" in 3D; see:
astronomy.swin.edu.au/~pbourke/fractals/lorenz/
All of this is chronicled in James Gleick's wonderful book
for the lay reader, "Chaos: Making a New Science" (1987).
www.amazon.com/exec/obidos/ASIN/0140092501/hip-20
Wolfram represents the first attempt I have seen to
systematize the field of computational experiments and
to draw broad conclusions from the effort.
* In 1975 I read an amazing book by Gregory Bateson's daughter
(with Margaret Meade), Mary Catherine Bateson: "Our Own Metaphor;
A Personal Account of a Conference On the Effects of Conscious
Purpose On Human Adaptation" in which she described a conference
organized by her father.
( www.amazon.com/exec/obidos/ASIN/0394474872/hip-20 )
One session had cyberneticist Anatol "Tolly" Holt presenting
a notation for representing systems that change over time which
he called Petri Diagrams. From this notation he derived the
result that simultaneity is only meaningful when parts of a
system are in communication. It made perfect sense to me. See:
www.well.com/~abs/Cyb/4.669211660910299067185320382047/OOM1.jpg
for a portion of that argument. Wolfram asserts the same result
(p. 517) and again I find his intuition agrees with mine.
Bateson used to say quite often that the antidote for scientific
arrogance is lots of data running through your brain from "natural
history," by which he meant the way things are and have been in the
natural world. (He used the story of Job in the Bible as an example
of this. Job's sin was "piety," a kind of religious arrogance, and
God's response -- the voice out of the whirlwind, was "Knowest thou
the time when the wild goats of the rock bring forth? or canst thou
mark when the hinds do calve?") I say that Wolfram has made it his
business to pump an awful lot of the "natural history" of computation
through his brain, and it has cured him of much of the scientific
arrogance of our time, and sharpened his intuition dramatically.
Results
As if to show that he didn't just spend ten years in a dark room
staring at cellular automata and vegging out, Wolfram produces
some remarkable results. For example, a new, simpler Turing Machine.
In one of the few mostly positive reviews of Wolfram's book,
Ray Kurzweil wrote:
What is perhaps the most impressive analysis in his book,
Wolfram shows how a Turing Machine with only two states
and five possible colors can be a Universal Turing Machine.
For forty years, we've thought that a Universal Turing
Machine had to be more complex than this. [Wolfram p. 707]
( www.kurzweilai.net/articles/art0464.html?printable=1 )
Wolfram also reveals that the "rule 30" cellular automaton, with a
single black cell as initial condition, produces very high quality
random bits down its center column. He uses well-accepted tests
for randomness to show that this source is far superior to any
other in commercial use, and reveals that this approach has been
the source of random numbers in Mathematica for a long time, with
no complaints. (p. 1084)
Wolfram also tackles the paradox of the 2nd Law of Thermodynamics.
Newton's equations of motion teach us that all the little
collisions between molecules in a fluid are reversible, so if you
do the classic experiment of removing the barrier in a tank of
half water and half fruit punch, wait for the red and clear fluids
to mix through diffusion, and then reverse the path of every molecule,
it should all go back to the partitioned state. But the 2nd Law says
entropy will always increase, so it CAN'T go back to the partitioned
state. This contradiction is said to lave lead to Bolzmann's suicide
in 1906. Nobel prize-winning chemist Ilya Prigogine addressed this
quandary in his 1984 book, "Order Out of Chaos: Man's New Dialogue
With Nature."
( www.amazon.com/exec/obidos/ASIN/0553343637/hip-20 )
Wolfram's approach is fundamentally similar, but more concise
(17 pages including diagrams) and, in my opinion, more rigorous
and yet easier to understand. (pp. 441-457)
But what I find to be his most remarkable result is his proof
that Cellular Automaton number 110 is Turing Complete; in other
words, it can be made to do any computation that is possible with
any digital computer (with unlimited time and memory). Like
Conway's "Life" before it, this simple system astonishes us by
matching the complexity of any algorithmic machine.
In addition, just to ground his results in reality, Wolfram shows
how a Cellular Automaton can almost exactly reproduce some of the
patterns on seashells.
Future Work for Grad Students
I've been told that in the "Source Citations Index," which maps all
citations in scientific literature, the most-cited work is the book
"The Structure of Scientific Revolutions" by Thomas S. Kuhn.
( www.amazon.com/exec/obidos/ASIN/0226458075/hip-20 )
Perhaps this is because Kuhn describes how new theories are resisted
by the scientific establishment, and every crackpot with a new idea
likes to point to the resistance he or she faces as evidence of their
genius. But, of course, hidden among the crackpots are the brilliant
new theories of tomorrow.
Kuhn is perhaps most famous for popularizing the phrase "paradigm
shift," and he describes how it usually takes the retirement of the
"old guard" for a new paradigm to be accepted. He also points out
that the real test of a new theory isn't if it is "true." All
scientific theories are eventually proved false; it's only matter
of time. No, the real test is if it provides new directions of
research for the graduate students. Of course, initially the
graduate students are not allowed to pursue the research. My own
taxonomy of the stages of acceptance of a new scientific theory
goes like this:
- hoots and catcalls
- boos and hisses
- forbidding the grad students to work on the theory
- associate professors sneaking peeks at the work covertly
- long official silence
- cheery admissions that everyone has known for some time
that the theory is right
In my lifetime I have seen these phases in the acceptance of
continental drift theory in geology, chaos theory in applied physics,
Bucky Fuller's geodesic geometry in chemistry, and the law of
increasing returns (the "Fax Effect") in economics. Currently it
seems to be going on with the Atkins Diet in medicine, and in the
commercial world with the adoption of the Linux operating system.
This is where I think Wolfram's contribution is really outstanding.
Some of you may recall the "Grand Challenges" in the late 1980s.
( www.cs.clemson.edu/~steve/Parlib/faq/gccommentary )
From "A Research and Development Strategy for High Performance
Computing," Executive Office of the President, Office of Science
and Technology Policy, November 20, 1987:
A "grand challenge" is a fundamental problem in science or
engineering, with broad applications, whose solution would be
enabled by the application of high performance computing
resources that could become available in the near future.
Examples of grand challenges are:
(1) Computational fluid dynamics for the design of hypersonic
aircraft, efficient automobile bodies, and extremely quiet
submarines, for weather forecasting for short and long
term effects, efficient recovery of oil, and for many
other applications;
(2) Electronic structure calculations for the design of new
materials such as chemical catalysts, immunological
agents, and superconductors;
(3) Plasma dynamics for fusion energy technology and for safe
and efficient military technology;
(4) Calculations to understand the fundamental nature of
matter, including quantum chromodynamics and condensed
matter theory;
(5) Symbolic computations including speech recognition,
computer vision, natural language understanding,
automated reasoning, and tools for design, manufacturing,
and simulation of complex systems.
I don't know about you, but I don't find all of these challenges
that grand. (Though number 5 has promise.) But Wolfram has
definitely thrown down the grandest challenge of all: to map out
the state space of all possible computations.
One thing I admire about Wolfram is that he is very careful to
delineate when he is asserting something is true -- he usually proves
it on the spot -- and when he is making a conjecture based on his
observations and intuition. Two of his conjectures strike me as
providing fertile ground for future research: his classification
of the four types of behaviors of computations, and his Principle
of Computational Equivalence.
Unfortunately, Wolfram doesn't give explicit names to his four classes
of behavior (p. 231). I would call them:
1) constant
2) oscillating
3) mild chaos
4) going ape
A promising research project would be to do a larger search through
the state-space of Cellular Automata (and related systems) with more
varied initial conditions, bigger memories and longer runs, to see
if any other behavioral classes can be identified.
The Principle of Computational Equivalence asserts that systems fall
basically into two categories: trivial, and Turing-complete. Once
a system is Turing-complete it can be arbitrarily complex, or --
depending on what you believe about the fundamental limits of
human and other natural intelligence -- as complex as anything
will ever be.
This raises two questions: Is there an intermediate level of
complexity beyond the trivial but short of the Turing machine?
Is there a higher level of complexity beyond the Turing machine?
Ultimately the answer to this last question will be found
in determining whether human intelligence occupies another level
beyond the algorithmic. Roger Penrose has tackled this question
(unconvincingly) in "The Emperor's New Mind: Concerning Computers,
Minds, and the Laws of Physics" (1989),
( www.amazon.com/exec/obidos/ASIN/0192861980/hip-20 )
and it remains an open question, at least to me. A good starting
point is: why can a human determine that the Godel String -- which
asserts "this statement cannot be proven" -- is true, even though
it can't be proved (or determined to be true) by a Turing machine?
(Penrose's intuition is that it has to do with quantum gravity, but
he offers little more than hand-waving to support this.)
Get to work, grad students!
Unsolvability
I have found that approximately 100% of non-scientists, and a
majority of scientists, seem unaware of the "dirty little secret"
of quantitative science: that all the really important problems
involve nonlinear Ordinary Differential Equations (ODEs) -- or worse
-- and that in principle these equations are mostly unsolvable.
The way this manifests for most students is that they are taught
Newtonian mechanics -- typically in 9th grade -- but told "we will
ignore friction for now." In the lab, great efforts are made to
keep friction out: dry ice pucks are used, even though we never
encounter them "in the wild." But if you follow a physics curriculum
through graduate school you eventually find that the friction question
remains unsolved in the general case no matter how far you go. Sure,
for the computation of the "terminal velocity" of a falling body you
learn the solution for a perfect sphere, but to this day aerospace
companies like Lockheed-Martin, Northrop-Grumman, TRW and Boeing use
the biggest supercomputers they can afford to compute the terminal
velocities of re-entering spacecraft and missile shapes on a
case-by-case basis. The scandalous thing is that most high school
physics teachers don't know this, and glibly promise students that
they will be taught how to solve problems with friction "later."
How did we get here? Let's briefly review the history of quantative
science. An excellent summary is found in Dr. Alan Garfinkel's
groundbreaking paper, "A Mathematics for Physiology" (1983),
American Journal of Physiology 245: Regulatory, Integrative and
Comparative Physiology 14: R455-66. He describes Newton's success
at inventing the Ordinary Differential Equation form, defining
the two-body gravitational system using eight ODEs, and solving
the equations to prove, for example, that Kepler's three laws of
planetary motion follow from the inverse square law of gravitational
force (Newton's Law of Gravity) and the definition Force equals mass
times acceleration (F=ma). Garfinkel goes on:
...the beauty of Newton's solution to the two-body problem did
not seem to be extendable. In the typical cases, even in
systems only slightly more complex than the two-body problem,
one could write equations based on first principles, but then
it was completely impossible to say what motions would ensue,
because the equations could not be solved. A classic case was
the three-body problem. This is a more realistic model of the
solar system, because it can take into account the nonnegligible
gravitational effects of Jupiter. A great deal of attention
was focused on this problem, because it expressed the stability
of the solar system, a question that had profound metaphysical,
even religious, consequences. Mathematicians attempted to pose
and answer this, some spurred on by a prize offered by King
Oscar of Sweden. Several false proofs were given (and
exposed), but no real progress was made for 150 years. The
situation took a revolutionary turn with the work of Poincare
and Bruns in 1890, which showed that the equations of the
three-body problem have no analytic solution.
The story continues with Poincare inventing topology to answer
qualitative questions (like "is this orbit stable?") about systems
with no analytical solution. (And I always thought topology was
invented to determine whether or not you could morph a coffee cup
into a donut.)
This problem has persisted, and dogged attempts to extend quantitative
science into other realms. In "General System Theory: Foundations,
Development, Applications" (1968) by Ludwig Von Bertalanffy,
( www.amazon.com/exec/obidos/ASIN/0807604534/hip-20 )
A. Szent-Gyorgyi is quoted as relating in 1964:
[When I joined the Institute for Advanced Study in Princeton]
I did this in the hope that by rubbing elbows with those great
atomic physicists and mathematicians I would learn something
about living matters. But as soon as I revealed that in any
living system there are more than two electrons, the physicists
would not speak to me. With all their computers they could
not say what the third electron might do. The remarkable
thing is that it knows exactly what to do. So that little
electron knows something that all the wise men of Princeton
don't, and this can only be something very simple.
Bertalanffy also reproduces this table of the difficulty of
solving equations, which is pretty grim:
www.well.com/~abs/equations.html
In his seminal book "Cybernetics, or Control and Communication in
the Animal and the Machine" (1948),
( www.amazon.com/exec/obidos/ASIN/026273009X/hip-20 )
Norbert Wiener tells the story (in the preface to the second,
1961 edition):
When I came to M.I.T. around 1920, the general mode of putting
the questions concerning non-linear apparatus was to look for a
direct extension of the notion of impedance which would cover
linear as well as non-linear systems. The result was that the
study of non-linear electrical engineering was getting into a
state comparable with that of the last stages of the Ptolemaic
system of astronomy, in which epicycle was piled on epicycle,
correction upon correction, until a vast patchwork structure
ultimately broke down under its own weight.
Interestingly, Wiener also reveals that he made early suggestions
for the construction of a digital computer when analog computers
-- especially integrators, electronic analogs to Lord Kelvin's
disk-globe-and-cylinder -- were failing him. His original
justification for the construction of a general purpose calculating
device was to simulate unsolvable ODEs. (And I always though the
computer was invented to crack German codes and later to calculate
ballistics tables for big Naval guns.)
Another take on this history is found in a textbook on control
system design, "Linear Control Systems" (1969), by James Melsa
and Donald Schultz. (It is long out of print and I couldn't find a
reference to it on Amazon.com, but a more recent revised edition
with the same title, published in 1992, by Charles E. Rohrs, James
Melsa, and Donald Schultz -- also out of print, alas -- is listed:
www.amazon.com/exec/obidos/ASIN/0070415250/hip-20 )
From the 1969 edition:
In [1750] Meikle invented a device for automatically steering
windmills into the wind, and this was followed in 1788 by Watt's
invention of the flywheel governor for regulation of the steam
engine.
However, these isolated inventions cannot be construed as
reflecting the application of any automatic control THEORY.
There simply was no theory although, at roughly the same time
as Watt was inventing the flywheel governor, both Laplace and
Fourier were developing the two transform methods that are now
so important in electrical engineering and in control in
particular. The final mathematical background was laid by
Chaucy (1789-1857), with his theory of the complex variable...
Although... it was not until about 75 years after his death
that an actual control theory began to evolve. Important
early papers were "Regeneration Theory," by Nyquist, 1932,
and "Theory of Servomechanisms," by Hazen, 1934. World War
II produced an ever-increasing need for working automatic
control systems and this did much to stimulate the development
of a cohesive control theory. Following the war a large number
of linear-control-theory books began to appear, although the
theory was not yet complete. As recently as 1958 the author
of a widely-used control text stated in his preface that
"Feedback control systems are designed by trial and error."
With the advent of new or modern control theory about 1960,
advances have been rapid and of far-reaching consequence.
The basis of much of this modern theory is highly mathematical
in nature and almost completely oriented to the time domain.
A key idea is the use of state-variable-system representation
and feedback, with matrix methods used extensively to shorten
the notation.
So what we see is that while virtually no progress was made on
solving non-linear ODEs, more and more sophisticated analytic
techniques were being developed for the linear cases. This is
great if you have the luxury of choosing which problems you will
solve. (Theoreticians it seems have long looked down their noses
at the poor wretches doing APPLIED mathematics, who have to take
their problems as they come.) The danger in this was that a
whole generation of scientists and engineers were building up
intuitions based only on linear systems.
An extremely notable event occurred with the investigation of the
so-called "Fermi-Pasta-Ulam" problem at Los Alamos around 1955.
A copy of their landmark paper can be found at:
www.osti.gov/accomplishments/pdf/A80037041/A80037041.pdf
More recently it has been called the Fermi-Ulam-Pasta problem;
I don't know if this is because -- though Fermi, the "grand old
man" of the group, passed away in 1954 -- Ulam went on to do some
related work for years, including early investigations into iterated
maps and Cellular Automata. Or maybe it's so the problem can be
called "FUP" for short.
After the first electronic computers were built there was a
long-running debate between those who wanted to do things that
could always be done, only now they were faster, and those who
wanted to do things that were previously impossible. The
former group, including the Admirals who wanted those ballistics
tables, usually had the best funding. But finally, probably as a
prize for good work in inventing A- and H-bombs, the folks at Los
Alamos were permitted to use their "MANIAC I" computer to do some
numerical experiments on the theory of non-linear systems, starting
in 1954. Fermi and company wanted to study a variety of non-linear
systems, but for starters they picked a set of masses connected by
springs, like so:
+----+ +----+ +----+ +----+
| |_/\/\_| |_/\/\_| |_/\/\_| |
| | | | | | | |
+----+ +----+ +----+ +----+
Here I show 4 masses connected by 3 springs. The FUP group first
modeled 16 masses, and later 64 masses. They were going to displace
a mass vertically and then observe the vibrations of the system.
If the springs were linear, so that the force was exactly
proportional to the spring's displacement:
F = -kx
by some constant factor "k," the system was fully understood.
It had vibrational modes which could be studied analytically
using Fourier and Laplace's transforms. The physicists had grown
up with their intuitions shaped by these techniques. They explained:
The corresponding Partial Differential Equation (PDE)
obtained by letting the number of particles become
infinite is the usual wave equation plus non-linear terms
of a complicated nature.
They knew that the linear case had vibratory "modes" which
would each exhibit a well-defined portion of the system's energy,
independent of each other. They also knew that non-linear systems
tended to be "dissipative," in that the energy was expected to
"relax" through the modes, until it was fairly equally shared
between them. This was called "thermalization" or "mixing."
Of course, computer time was horrendously expensive at this point,
and more importantly the paradigm was still analytic, so the goal
was to do just a few experiments, use the results to "tweak" the
linear case somehow to generalize it to non-linear, and continue
as they were accustomed. But when they plugged only slightly
non-linear force equations into their computer, they were astonished
at the results. In fact, they first suspected hardware or software
errors. But ultimately they were forced to report:
Let us say here that the results of our computations show
features which were, from the beginning, surprising to us.
what they saw instead was the energy being passed around by the
first three modes only, dominating one and then another in turn,
until 99% of it ended back in a mode where it had been before;
this pattern repeated nearly periodically for a great number
of iterations. Nothing in their intuition had prepared them for
this.
The title of the paper was:
STUDIES OF NONLINEAR PROBLEMS, I
and the group stated that they had a whole series of numerical
experiments planned:
This report is intended to be the first one of a series
dealing with the behavior of certain nonlinear physical
systems where the nonlinearity is introduced as a perturbation
to a primarily linear problem....
Several problems will be considered in order of increasing
complexity. This paper is devoted to the first one only.
The approach they were taking was intended to enhance mainstream
"Ergodic Theory." The web site:
www2.potsdam.edu/MATH/madorebf/ergodic.htm
offers the best definition I have found: "In simple terms Ergodic
theory is the study of long term averages of dynamical systems."
This is a little odd when you consider that the "average" position
of the Earth in its annual orbit is close to being at the center
of the Sun.
One of the things that the FUP group realized in retrospect was
that they were rediscovering "solitons" which are non-dissipative
non-linear waves. According to:
physics.usc.edu/~vongehr/solitons_html/solitons.html
solitons have this history:
A solitary wave was firstly discussed in 1845 by J. Scott Russell
in the "Report of the British Association for the Advancement of
Science". He observed a solitary wave traveling along a water
channel. The existence and importance was disputed until D.J.
Korteweg and G. de Vries gave a complete account of solutions to
the non-linear hydrodynamical [partial differential] equation in
1895.
Apparently this mathematical basis for solitons was mostly
forgotten until the FUP group found the phenomenon in their
simulations and revived interest. (Solitons also explain tsunamis
or so-called "tidal waves.") The web site:
www.ma.hw.ac.uk/solitons/
has further information about solitons, including the story of how
the canal was recently renamed in honor of Russell, how in 1995
the wave he observed was recreated on the canal, how solitons are
now widely used in fiber optics because they don't dissipate, and
how a fiber optic channel now runs along the path of that very canal.
(Let me say here parenthetically that I just love being right.
I concluded that the FUP experiment was important as a precursor to
Wolfram's work BEFORE reading his Notes, in which he makes the same
observation., on page 879. There is also discussion on Wolfram's
site at:
scienceworld.wolfram.com/physics/Fermi-Pasta-UlamExperiment.html
And a recent book, "The Genesis of Simulation in Dynamics: Pursuing
the Fermi-Pasta-Ulam Problem" by Thomas P. Weissert
www.amazon.com/exec/obidos/ASIN/0387982361/hip-20
delves deeply into the problem from a modern perspective, though
I haven't read it.)
So what exactly does FUP have to do with "A New Kind of Science"
anyway? As Sir Winston Churchill once said, "Men occasionally
stumble over the truth, but most of them pick themselves up and
hurry off as if nothing ever happened." The curious thing is that
there was never a "STUDIES OF NONLINEAR PROBLEMS, II." I really
don't know why. Was their result too "weird" to get any more
funding? Did it scare them off? Did Fermi's death derail them?
Can we credit Ulam with carrying on the work in his own way?
But one thing is clear, their work suggested that a methodical
examination of ODEs and PDEs was in order to bring clarity to the
extremely muddled field of nonlinear analysis. I sort of assumed
that sooner or later this happened. But Wolfram tells us (p. 162):
Considering the amount of mathematical work that has been
done on partial differential equations, one might have thought
that a vast range of different equations would by now have been
studied. But in fact almost all of the work -- at least in one
dimension -- has been concentrated on just ... three specific
equations ... together with a few others that are essentially
equivalent to them.
(For some associated illustrations see:
www.wolframscience.com/preview/nks_pages/?NKS0165.gif )
Well! The lesson of FUP has still not been absorbed!
Let me try and explain this another way. Prior to Darwin
the argument for an omniscient and omnipotent creator went
something like this: If I am able to learn to walk, let's say,
I must have an innate ability-to-learn-to-walk, which must
have been designed in by a creator who knew how to create
such an ability. Alternately, I might have learned how to
learn how to walk. But then, I must have had an innate
ability-to-learn-how-to-learn-how-to-walk, which must have
been designed in by an even wiser creator who knew how to
design such an ability to learn how to learn. And so on.
Darwin suggested that I might instead just be lucky, or at
least descended from lucky ancestors. And all of my potential
ancestors didn't have to have this luck -- only the survivors.
Experiments in genetic algorithms have already shown that computer
programs can be "bred" to do things they don't intrinsically
"know how to do." One of the reasons this works is that there
are only so many things that a computer program can do in the
first place. If one of those things can solve a problem
presented to the environment of "breeding" genetic algorithms,
one of them is bound to find it sooner or later.
Recall what A. Szent-Gyorgyi said: "So that little electron
knows something that all the wise men of Princeton don't, and
this can only be something very simple." The problems we can't
solve analytically we may be able to solve with luck, especially
if our computers allow us to create billions, trillions, or more
opportunities for the luck to appear.
WHY EVERYBODY DOESN'T GET IT
It has been hard for me to find reviews of Wolfram's book
that praise him. A typical review appeared in "The New York
Review of Books" on October 24, 2002: "Is the Universe a Computer?"
by Steven Weinberg:
I am an unreconstructed believer in the importance of the word,
or its mathematical analogue, the equation. After looking at
hundreds of Wolfram's pictures, I felt like the coal miner in
one of the comic sketches in "Beyond the Fringe," who finds the
conversation down in the mines unsatisfying: "It's always just
'Hallo, 'ere's a lump of coal.'"
(For the whole thing see:
www.nybooks.com/articles/15762
For a thorough list of reviews available on-line, see:
www.math.usf.edu/~eclark/ANKOS_reviews.html )
The most thoughtful review I have found is "A Mathematician
Looks at Wolfram's New Kind of Science," by Lawrence Gray. He
is careful, thorough, and not too unkind, but I think he gets it
wrong.
Why does this book inspire so much opposition? A pat answer would
be that Wolfram is wrong side of a paradigm shift, and this is just
par for the course. The two most common objections to a new
breakthrough are "this is crazy" and "there's nothing new here."
Consider that when Einstein published his Theory of Relativity (the
Special variety) the complaint was made (besides that it was crazy)
that he was just collating a bunch of results from others. For
example, it was already known that the Lorenz contractions --
shrinking in the direction of motion near the speed of light --
could be derived from Maxwell's electromagnetics equations, already
half a century old. But by systematizing and integrating a bunch
of material Einstein paved the way for new results, new experiments,
and new theories.
Certainly Wolfram faces the problem that most of his contemporaries
have not worked as he has so extensively with computational
experiments, and so lack his intuition on the subject. I think
I share some of his intuitions because I've been performing my own
experiments for my whole adult life. I've simulated solitaire
games on a mainframe with punch cards, modeled Ross Ashby's
"homeostat" learning machine on a minicomputer time-sharing system,
observed the bizarre behavior of self-modifying code on an early
personal computer, explored the Mandelbrot set on a supercomputer
and the Lorenz attractor on a powerful graphical workstation, and
wandered through problems in number theory on a Windows system.
I've see some of the same chimera he has, and I trust his hunches
nearly as much as my own.
But there's something more here in the opposition to Wolfram.
Early in the Notes (p. 849), he says:
Clarity and modesty.
There is a common style of understated scientific writing to
which I was once a devoted subscriber. But at some point
I discovered that more significant results are usually
incomprehensible if presented in this style. For unless one
has a realistic understanding of how important something is,
it is very difficult to place or absorb it. And so in writing
this book I have chosen to explain straightforwardly the
importance I believe my various results have. Perhaps I might
avoid some criticism by a greater display of modesty, but the
result would be a drastic reduction in clarity.
Recall how John Lennon was crucified by the press when he said
of the Beatles, "Now we're more famous than Jesus." Even Jesus
had John the Baptist to declare His divinity instead of proclaiming
it personally. I think Wolfram would have benefited by having some
sort of "shill" to trumpet the profundity of his work, while he stood
to the side and said "aw shucks." But it's too late for that now.
It has been said that coincidence is a researcher's best ally.
I coincidentally happened to be reading the science fiction novel
"Contact" by Carl Sagan while working on this essay.
( www.amazon.com/exec/obidos/ASIN/0671004107/hip-20 )
I found a few things that are relevant. A religious leader is
talking to a scientist about the implications of a message from
the intelligent aliens near the star Vega which has been detected
on earth:
"You scientists are so shy," Rankin was saying. "You love to
hide your light under a bushel basket. You'd never guess what's
in these articles from the titles. Einstein's first work on the
Theory of Relativity was called 'The Electrodynamics of Moving
Bodies.' No E=mc squared up front. No sir. 'The Electrodynamics
of Moving Bodies.' I suppose if God appeared to a whole gaggle
of scientists, maybe at one of those big Association meetings,
they'd write something all about it, and call it, maybe, 'On
Spontaneous Dendritoform Combustion in Air.' They'd have lots
of equations, they'd talk about 'economy of hypothesis'; but
they'd never say a word about God."
Later when the message is decoded, it contains instructions for
building an elaborate machine of unknown purpose:
For the construction of one component, a particularly intricate
set of organic chemical reactions was specified and the resulting
product and the resulting product was introduced into a swimming
pool-sized mixture... The mass grew, differentiated, specialized,
and then just sat there -- exquisitely more complex than anything
humans knew how to build. It had an intricately branched network
of fine hollow tubes, through which perhaps some fluid was to
circulate. It was colloidal, pulpy, dark red. It did not make
copies of itself, but it was sufficiently biological to scare a
great many people. They repeated the procedure and produced
something apparently identical. How the end product could be
significantly more complicated than the instructions that went
into building it was a mystery.
Here Sagan, himself a scientist, puts his finger on two of Wolfram's
problems: he isn't shy enough, and he challenges the widely held
belief that "the end product could [not] be significantly more
complicated than the instructions that went into building it."
Recently I interviewed my friend and associate Art Olson at Scripps
Research Institute for a future C3M column. He was very interested
in my reaction to Wolfram. I told him how I thought he provided
directions for research by the grad students. I said I thought
biology -- especially genomics -- could get a lot of out of his
approach. He challenged me, "How would I advise a grad student to
continue this research? What exactly would I tell him to do?"
I suggested that, since Scripps has always seemed to have the
latest and greatest "big iron" in supercomputers, he have a grad
student continue the search for other categories of Cellular
Automata behaviors, as I described above. "And how would this
benefit biology?" hew asked.
I've given this question a lot of thought. I came up with
this metaphor. America used to be a nation of farmers connected
by gravel roads. It was not too tough a job to convince farmers
to improve the roads that lead directly from their farms to market.
But the U.S. highway system was much harder to sell. Why would
one rural region want to be connected to another with paved roads,
let alone all the way to Chicago? Maybe you might say, "So you
can order from the Sears Catalog."
The farmer might reply, "But I can already order from the Sears
Catalog."
"Well, the stuff would get here sooner, and shipping would cost less."
"Well, I can wait, and besides I don't really order that much from
the catalog anyway."
"But you would if it were cheaper and faster!"
And so on. Obviously, in retrospect, the U.S. highway system has
been an enormous boon to American farmers. But is always hard to
pitch the benefits of general infrastructure improvements. What
Wolfram proposes has the potential of great benefits to the general
scientific and mathematical infrastructure, but it will be hard for
a while to find specific champions.
Perhaps this metaphor is a little unfair. Art would certainly agree
that general improvements in -- say -- processor speed would benefit
biology. A better metaphor might be this: a regional park near my
house has a paved path through the oak valley, since that's where
it's easiest to put a trail, and besides, it's a nice, shady place
to walk. Along the path there are signs to identify the native plant
species. If no one had ever been to the mountain top nearby, and
someone were to propose cutting a path through the chaparral that
lead up to the peak, critics might say "Why bother? Stick to valley
because the brush is impenetrable. And besides, there's probably
nothing new up there. Stay down here where we have all the species
labeled." Well, maybe there isn't anything new up there, but there's
only one way to find out.
The late Alfred North Whitehead is said to have once introduced a
lecture by Bertrand Russell on quantum mechanics, and afterwards to
have thanked Russell for his lecture, "and especially for leaving
the vast darkness of the subject unobscured."
I would like to thank Stephen Wolfram leaving the vast darkness
of his subject unobscured.
At the risk of mixing my metaphors, I say, let's mount a few
expeditions into that darkness.
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