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Benoit Mandelbrot (RIP) and the quest for a theory of really everything

The views expressed are those of the author and are not necessarily those of Scientific American.


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Benoit MandelbrotThe passing of the mathematician Benoit Mandelbrot has triggered in me a wave of nostalgia for the 1980s, when Mandelbrot and other researchers seemed to be creating a scientific revolution. They hoped that sophisticated new mathematical techniques, plus increasingly powerful computers, could help them fathom a wide range of complex, nonlinear phenomena—from brains and immune systems to economies and climate—that had resisted analysis by the reductionist methods of the past.

The journalist James Gleick brilliantly described this research in his 1987 bestseller Chaos: Making a New Science. Mandelbrot, an applied mathematician who dabbled in a wide variety of fields, was a hero of Gleick’s book. Beginning in the 1960s Mandelbrot realized that many real-world phenomena—clouds, snowflakes, coastlines, stock market fluctuations, brain tissue—have similar properties. They display "self-similarity," patterns that recur at smaller and smaller scales; and they have fuzzy boundaries.

Mandelbrot found that he could model these phenomena with mathematical objects that he called fractals. The name Mandelbrot setrefers to a property called fractional dimensionality: fractals are fuzzier than a line but never quite fill a plane. The most famous fractal is the Mandelbrot set, which is generated by repeatedly solving a simple mathematical function and plugging the answer back into it.

When plotted by a computer, the Mandelbrot set produces a very odd-looking object, resembling a warty snowman toppled on its side. As you look at the object with higher and higher resolution, you see that the snowman’s borders are as vague as the borders of a flame; that is what fractional dimensionality looks like. Certain patterns, such as the warty snowman, keep recurring at smaller scales with subtle variations.

Other mathematicians had explored similar phenomena since at least the early 20th century. Moreover, in 1990 several mathematicians claimed to have explored the Mandelbrot set before Mandelbrot did, a claim that I reported on in an article for Scientific American, "Who discovered the Mandelbrot set?" After I called him for a comment, Mandelbrot contacted my editor to protest the story’s publication, but I don’t blame him for that. He clearly deserved credit for drawing the attention of other researchers to fractals—including the Mandelbrot set—and pointing out their peculiar similarities to a host of natural phenomena.

What resulted, finally, from attempts to model the physical world with fractals? Or all the other trendy mathematical and computational methods described in Chaos and a host of sequels by other science writers? Let me remind you of some of the other buzzwords from that heady era: cellular automata, artificial life, genetic algorithms, self-organized criticality. All purported to be methods for modeling, and hence understanding, complex phenomena. All these approaches found applications, but none turned out to be as powerful as adherents had hoped.

Indeed, chaos theory and its successor, complexity (which was really just chaos in a glossy new wrapping), followed the same boom–bust cycle as two previous scientific movements: cybernetics and catastrophe theory. Cybernetics (a neologism coined from the Greek term kubernetes, or steersman) was conceived by the mathematician Norbert Wiener. In his 1948 book Cybernetics: Control and Communication in the Animal and the Machine he proclaimed that cybernetics could in principle model the operation of not only machines but also all biological phenomena, from single-celled organisms up through the economies of nation–states.

Cybernetics became extremely popular, especially in Russia, but by the 1960s it was already losing its luster. In 1961 the electrical engineer John Pierce sneered that the term cybernetics "has been used most extensively in the press and in popular and semiliterary, if not semiliterate, magazines." (Cybernetics lives on, of course, in the pop-culture term "cyberspace," coined by sci-fi writer William Gibson.)

The next big idea was catastrophe theory, a set of equations that the French mathematician Rene Thom claimed could model phenomena exhibiting abrupt, "catastrophic" discontinuities. Thom and his followers suggested that catastrophe theory could help to explain not only events such as earthquakes but also biological and social phenomena, such as the emergence of life, the metamorphosis of a caterpillar into a butterfly and the collapse of civilizations.

One reviewer of Thom’s 1972 book Structural Stability and Morphogenesis compared it with Newton’s Principia, arguably the most important scientific treatise of all time. By the late 1970s, critics were complaining that Thom’s work "provides no new information about anything" and is "exaggerated, not wholly honest." An essay in Nature called catastrophe theory "one of many attempts to deduce the world by thought alone," which is "a dream that cannot come true." (Scientific American is part of Nature Publishing Group.)

Chaos theory followed the same pattern. In 1991, just four years after Chaos was published, David Ruelle, who like Mandelbrot was a pioneer in mathematical modeling of chaotic systems, complained that chaos, "in spite of frequent triumphant announcements of ‘novel’ breakthroughs, has had a declining output of interesting discoveries."

Every now and then, some ambitious soul proclaims once again that he has created an all-powerful mathematical theory. One high-profile example was the physicist and mathematical-software mogul Stephen Wolfram, who declared that his 2002 self-published book A New Kind of Science would engender, well, a new kind of science. But Wolfram’s "new" approach to solving all scientific puzzles was just cellular automata, the computational modeling system invented by John Von Neumann in the 1950s.

Perhaps it is time to acknowledge that no single mathematical or computational system can model all of reality. Mandelbrot, who was not known for modesty, suggested as much. "Most emphatically, I do not consider the fractal point of view as a panacea," he wrote in his strange and wonderful 1977 book The Fractal Geometry of Nature. The similarity of a fractal to a natural phenomenon, he noted, did not necessarily yield deep insights into its underlying physical mechanisms.

But fractals have proved useful for financial modeling, image-compression and other applications. And Mandelbrot was absolutely right that computer-generated fractal images revealed "a world of pure plastic beauty unsuspected until now." I can still remember the thrill I got when I first saw the colored images of the Mandelbrot set, which shimmered with intimations of infinity. I will always be grateful to Mandelbrot for that.

Image credits: Wikicommons

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  1. 1. jtdwyer 7:29 am 10/18/2010

    It should go without saying that all my statements are expressions of my own opinions based on personal experience and do not necessarily represent any scientifically established system of knowledge.

    Arbitrary analytical mathematical modeling methods, no matter how fascinating, are useful to describe the behavior of any system only to the extent that the system produces arbitrary mathematical results.

    There is no singular mathematical formula that represents or simulates all aspects of the physical processing characteristics of all systems. While there are some that represent fundamental processes that are repeated at varying scales, such as the point radiation of energy, not all objects are spherically symmetrical distributions of mass and cannot be successfully be treated as such.

    Producing mathematical models containing arbitrary equations that happen to accurately predict the results of processes to which it is applied does not indicate that all aspects of the process are understood. That the standard model of particle physics, for example, accurately predicts the results of interactions among material quanta to which it is applied does not prove that all aspects of those interactions are correctly represented by arbitrary equations chosen solely on the basis of their apparent effectiveness.

    Complete understanding of physical processes can only be proven through the accurate demonstration of all aspects of their processing results.

    These are only personal opinions, thank you.

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  2. 2. candide 8:56 am 10/18/2010

    "There is no singular mathematical formula that represents or simulates all aspects of the physical processing characteristics of all systems."

    I agree with that and have said as much myself. But it begs the question – why do people so much WANT everything to boil down to s single theory/equation? For me a set of a few basic theories/equations would be just fine.

    However, that said, I personally do not think we will ever understand/explain everything. The unexplainable "mystery" may become smaller and more obscure, but (IMO) it will always be there.

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  3. 3. Laertes 9:21 am 10/18/2010

    "generated by repeatedly solving a simple mathematical function and plugging the answer back into it." I’m sure you meant "simple" in a figurative sense. A complex mathematical function is required to produce a fractal. I don’t think it works without the imaginary part of the number.

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  4. 4. yrral86 11:32 am 10/18/2010

    The function is simple. The operand is complex. f(n) = n^2 + c, where n and c are complex, and c is a constant. The Mandelbrot Set is the set of complex numbers C such that for every c in C, repeated application of f(n) = n^2 +c remains bounded. That is, it doesn’t grow without limit.

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  5. 5. Steve D 3:39 pm 10/18/2010

    Very few things ever struck me as being as silly and vacuous as Mandelbrot’s "fractal Pangaea" and his claim that the real Pangaea could be accounted for simply if it actually had a the fractal dimension he envisioned. As brilliant and original as his work was, he just had no clue that some things simply aren’t fractals, period. He obviously had no idea what defines the edges of continents.

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  6. 6. ttreker 4:50 pm 10/18/2010

    I hope SciAm gets somebody with a better understanding of math and physics to evaluate the value of Mandlebrot’s work, and it’s context in the field of non-linear dynamics, in the monthly publication. I don’t know much about this column, however, if I were to judge it by this post I would place it squarely in the realm of the mediocre.

    Hogan seems to have misunderstood Mandlebrot’s work. Popularized as though Chaos Theory was in the popular press, it was never touted as a theory of everything (I don’t understand this post’s title so maybe I am missing something). Fractals, and fractal geometry are more than just a bunch of "plastic" looking pictures generated by computers. From Mandlebrot on forward, trees, mountains, circulatory systems, clouds, and on and on will be understood as fractals. They are everywhere all around us. Before Mandlebrot, nothing tied these things we encounter everyday of our lives together mathematically. (We should be teaching fractals to our children in grammar school as they are as fundamental as circles.) This alone would be a pretty remarkable insight, but in hardly begins to scratch the surface of the value of this new world view.

    The mathematical view of non-linear systems changed just as fundamentally. Chaotic dynamics represents a remarkable change in world view brought on by the era of digital computing.

    I am not going to spend the time to articulate the value here. SciAm editors should understand it and get somebody that knows what they are talking about to put Mandlebrot’s contribution into perspective. This post is a terrible tribute.

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  7. 7. ttreker 6:00 pm 10/18/2010

    Let me make this clearer to SciAm: It was your publication that highlighted the Mandlebrot Set with a cover photo. Why on earth would you have such a terrible tribute to this brilliant work posted on your website at his death.

    Go look at the New Scientist’s tribute. Pull it together here, SciAm.

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  8. 8. natlawson 11:10 pm 10/18/2010

    There are those whose insights open the door to our survival. Ignorance is vast and consuming. Imagine a person whose great stride was to explain fundamental certainty.
    Fractals and topology will survive in our lexicon as long as we survive. Mandelbrot achieved his humanity and wore it like the shining light his sacrifice, his love, is to the rest of our kind.

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  9. 9. rloldershaw 11:30 pm 10/18/2010

    I fully agree with ttrekker.

    To treat the fundamental and universal concepts of chaos and fractals as under-achievers like "catastrophe theory" is remarkably ignorant of the important roles these physical concepts play in many scientific and social fields, and grossly underestimates the even more important role they will probably play in the future.

    If Dr Horgan and/or others want to see an evidence-based fractal paradigm for modeling nature that can make 38 fundamental retrodictions and more than 10 definitive predictions, I suggest that they take a look at www3.amherst.edu/~rloldershaw .

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  10. 10. rloldershaw 11:37 pm 10/18/2010

    I fully agree with ttrekker.

    To treat the fundamental and universal concepts of chaos and fractals as under-achievers like "catastrophe theory" is remarkably ignorant of the important roles these physical concepts play in many scientific and social fields, and grossly underestimates the even more important role they will probably play in the future.

    If Dr Horgan and/or others want to see an evidence-based fractal paradigm for modeling nature that can make 38 fundamental retrodictions and more than 10 definitive predictions, I suggest that they take a look at www3.amherst.edu/~rloldershaw . This research is based on scores of papers published in peer-reviewed scientific journals.

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  11. 11. jtdwyer 6:20 am 10/19/2010

    Thanks – I think you raise a very telling question that I’ve tried to answer in the past by explaining the apparent strong connection between gravitational and quantum particle force theories: both are obviously missing a mechanical or analytical explanation for the effect of mass.

    Of course this shortcoming was the focus of Einsteins’ consuming preoccupation with producing a ‘Grand Unified Theory’ that could explain the four fundamental forces that had been identified by theorists (strong, weak, electromagnetic and gravitation).

    Te GUT effort became redefined by later physicists and/or the popular press, being misidentified as a ‘Theory of Everything’ which seemingly must explain, well, everything anyone might want explained! This, of course, is fantastic folly!

    In my opinion, fractal equations are useful for visually illustrating the effect of a common process that can affects edge geometries at all scales to which is applied. That this scalar geometric similarity has some application in describing the patterns of the natural environment has little to do with fractals’ ability to describe physical forces, except that the process defined is iterative. Otherwise it has little if any use in describing the effects of any fundamental force(s).

    In my opinion, string theory can be better described as ‘grasping at straws’ – an attempt to produce a mechanical process that physically represents the effects produced by a candidate ‘Grand Unified Force’.

    I think that the actual grand unified force is all around us, staring us in the face: it is the fundamental force that imparts motion to matter. In the special conditions of the early universe it alone directed the development of the universe and the production of the four derivative processes developed from it that were earlier identified as the four fundamental forces.

    This, of course, is only my personal opinion regarding unification theories of anything: I think there is great value in pursuing the unification of particle force theory and gravitation; beyond that lie the dreams of fools.

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  12. 12. rloldershaw 10:46 pm 10/19/2010

    Hello David,

    The basic message at the Fractal Cosmology website is as follows.

    (1) Nature is organized hierarchically (probably without upper or lower bounds on size/mass scales).

    (2) The hierarchy is divided into discrete "Scales", like the …, Atomic, Stellar, Galactic, … Scales.

    (3) The cosmological Scales are either exactly or almost exactly self-similar, such that for any system of Scale N there is a specific self-similar analogue with highly analogous physics on all other Scales N+/- x.

    In this paradigm, nature has a symmetry property called discrete scale invariance. If the self-similarity is exact then you have full relativity of scale, to go along with relativity of position, time, orientation, state of motion. "Big" and "small" only have meaning relative to an arbitrarily chosen Scale.

    In the exact self-similarity of Discrete Scale Relativity, the only difference between subatomic nuclei, neutron stars and galaxies/quasars/AGN, is their spatial, temporal and mass scales. Analogues on different Scales can appear somewhat different if they are in different energy states, i.e., a highly rotationally excited nucleus has different properties from a ground state nucleus. Ditto for excited vs n=1 atoms.

    Anyone else ready for a new paradigm that may identify the path to Einstein’s dream of a unified physics?

    Rob O

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  13. 13. jtdwyer 11:32 pm 10/19/2010

    Because of ‘exact self-similarity of Discrete Scale Relativity’ then, shouldn’t the percentage of total system mass located in the orbital planes of spiral galaxies and atoms be identical? What about the Solar system – does it qualify? Thanks in advance for your explanations.

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  14. 14. rloldershaw 11:59 pm 10/19/2010

    Hi JT,

    Firstly in case people wonder, when you include their dark matter components, the shapes of galaxies are spheroidal, elliptical and prolate. Those are the same shapes as subatomic nuclei. Lesson: what you SEE is sometimes less than all of what is there.

    Regarding your question, galaxies and atoms are definitely not proper analogues, since the discrete self-similar scaling rules would predict that Galactic Scale atoms are vastly larger (by ~10^5 times) than a typical galaxy. The correct Atomic Scale analogue for a galaxy is a subatomic particle or nucleus. This can be checked by their radii, spin periods and oscillation periods (done on Fractal Cosmology website). According to Discrete Scale Relativity both classes of objects are best described currently as Kerr-Newman ultracompacts (black holes) which are solutions of the Einstein-Maxwell equations.

    < http://www3.amherst.edu/~rloldershaw > offers a great deal information on the scaling rules of DSR, tests of those rules, evidence for the self-similarity of proposed analogues, predictions, etc.

    Best, Rob

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  15. 15. jtdwyer 2:05 am 10/20/2010

    Unfortunately, I cannot accept that dark matter exists, since its requirement became established by the invalid initial presumption that Kepler’s orbital characteristics of the central mass Solar system should apply to vastly distributed mass spiral galaxies. Without that initial expectation there is not galaxy rotation problem and no requirement for dark matter, simply an extremely complex distribution of enormous mass that is very difficult model gravitationally.

    I can’t agree with your presumptions about unseen characteristics of galaxies. I think that the rotational characteristics of spiral galaxies is solely a product of the self gravitating mass of the galactic disk in conjunction with the largely independent spheroidal orbital system of the central bulge. I don’t think there can be any inherent proportionality between large scale structures a quantum particles, unless one constructs the missing proportional mass from imaginary matter.

    Sorry for being obstinate – best wishes, anyway. You do have an impressive body of cohesive work.

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  16. 16. sgeiger2009 3:11 pm 11/7/2010

    Define what "complex" is! That is one of the beautiful things that fractals help explain: What is complex may be simple at the same time. Complex mathematics are needed to produce a fractal? Hardly. All you need is feedback and a side effect. The Mandelbrot Set is produce with z=z^2+c. The "complex plane" on which it is graphed is not "complex" in the sense of complexity but complex in the sense of composed of more than one part. There’s nothing inherently complex in imaginary numbers.

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  17. 17. sridattadev 2:54 pm 11/12/2010

    Who am I? I am in this universe as much as it is in I?
    What is I? I is sphere full of love. I is the singularity.
    Imagination is more important than knowledge, for all that we know is just an imagination.
    Theory of everything is there is absolutely nothing.

    http://sridattadev-theoryofeverything.blogspot.com/2010_01_01_archive.html

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