======================================================================= Cybernetics in the 3rd Millennium (C3M) -- Volume 2 Number 1, Jan. 2003 Alan B. Scrivener --- www.well.com/~abs --- mailto:abs@well.com =======================================================================

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.


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


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.


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!


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.


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. ====================================================================== newsletter archives: www.well.com/~abs/Cyb/4.669211660910299067185320382047/ ====================================================================== Privacy Promise: Your email address will never be sold or given to others. You will receive only the e-Zine C3M unless you opt-in to receive occasional commercial offers directly from me, Alan Scrivener, by sending email to abs@well.com with the subject line "opt in" -- you can always opt out again with the subject line "opt out" -- by default you are opted out. To cancel the e-Zine entirely send the subject line "unsubscribe" to me. I receive a commission on everything you purchase during your session with Amazon.com after following one of my links, which helps to support my research. ====================================================================== Copyright 2003 by Alan B. Scrivener