Friday


Stephan Banach and Alfred Tarski

Further to my recent posts on fractals and the Banach-Tarski Paradox (A Question for Philosophically Inclined Mathematicians, Fractals and the Banach-Tarski Paradox, A visceral feeling for epsilon zero, and Adventures in Geometrical Intuition), I realized how the permutations of formal methodology can be schematically delineated in regard to the finitude or infinitude of the number of iterations and the methods of iteration.

Many three dimensional fractals have been investigated, but I don't know of any attempts to show an infinite fractal such that each step of the interation involves an infinite process. One reason for this as no such fractal could be generated by a computer even in its first iteration. Such a fractal can only be seen in the mind's eye. Among the factors that led to the popularity of fractals were the beautifully detailed and colored illustrations generated by computers. Mechanized assistance to intuition has its limits.

The Banach-Tarski Paradox involves a finite number of steps, but for the Banach-Tarski paradox to work the sphere in question must be infinitely divisible, and in fact we must treat the sphere like a set of points with the cardinal of the continuum. Each step in Banach-Tarski extraction is infinitely complex because it must account for an infinite set of points, but the number of steps required to complete the extraction are finite. This is schematically the antithesis of a fractal, which latter involves an infinite number of steps, but each step of the construction of the fractal is finite. Thus we can see for ourselves the first few iterations of a fractal, and we can use computers to run fractals through very large (though still finite) numbers of iterations. A fractal only becomes infinitely complex and infinitely precise when it is infinitely iterated; before it reaches its limit, it is finite in every respect. This is one reason fractals have such a strong hold on mathematical intuition.

A sphere decomposed according to the Banach-Tarski method is assumed to be mathematically decomposable into an infinitude of points, and therefore it is infinitely precise at the beginning of the extraction. The Banach-Tarski Paradox begins with the presumption of classical continuity and infinite mathematical precision, as instantiated in the real number system, since the sphere decomposed and reassembled is essentially equivalent to the real number system. There is a sense, then, in which the Banach-Tarski extraction is platonistic and non-constructive, while fractals are constructivistic. This is interesting, but we will not pursue this any further except to note once again that computing is essentially constructivistic, and no computer can function non-constructively, which implies that fractals are exactly what Benoît Mandelbrot said that they were not: an artifact of computing. However, the mathematical purity of fractals can be restored to its honor by an extrapolation of fractals into non-constructive territory, and this is exactly what an infinite fractal is, i.e., a fractal each step of the iteration of which is infinite.

Are fractals a mere artifact of computing technology? Certainly we can say that computers have been crucial to the development of fractals, but fractals need not be limited by the finite parameters of computing.

Once we see the schematic distinction between the finite operation and infinite iteration of fractals in contradistinction to the infinite operation and finite iteration of the Banach-Tarski extraction, two other possibilities defined by the same schematism appear: finite operation with finite iteration, and infinite operation with infinite iteration. The former — finite operation with finite iteration — is all of finite mathematics: finite operations that never proceed beyond finite iterations. All of the mathematics you learned in primary school is like this. Contemporary mathematicians sometimes call this primitive recursive arithmetic (PRA). The latter — infinite operation with infinite iteration — is what I recently suggested in A visceral feeling for epsilon zero: if we extract an infinite number of spheres by the Banach-Tarski method an infinite number of times, we essentially have an infinite fractal in which each step is infinite and the iteration is infinite.

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Fractals and Geometrical Intuition

1. Benoît Mandelbrot, R.I.P.

2. A Question for Philosophically Inclined Mathematicians

3. Fractals and the Banach-Tarski Paradox

4. A visceral feeling for epsilon zero

5. Adventures in Geometrical Intuition

6. A Note on Fractals and Banach-Tarski Extraction

7. Geometrical Intuition and Epistemic Space

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Friday


In true Cartesian fashion I woke up slowly this morning, and while I tossed and turned in bed I thought more about the Banach-Tarkski paradox, having just written about it last night. In yesterday’s A Question for Philosophically Inclined Mathematicians, I asked, “Can we pursue this extraction of volume in something like a process of transfinite recursion, arriving at some geometrical equivalent of ε0?” The extraction in question is that of taking one mathematical sphere out of another mathematical sphere, and both being equal to the original — the paradox that was proved by Banach and Tarski. I see no reason why this process cannot be iterated, and if it can be iterated it can be iterated to infinity, and if iterated to infinity we should finish with an infinite number of mathematical spheres that would fill an infinite quantity of mathematical space.

All of this is as odd and as counter-intuitive as many of the theorems of set theory when we first learn them, but one gets accustomed to the strangeness after a time, and if one spends enough time engaged with these ideas one probably develops new intuitions, set theoretical intuitions, that stand one in better stead in regard to the strange world of the transfinite than the intuitions that one had to abandon.

In any case, it occurred to be this morning that, since decompositions of a sphere in order to reassemble two spheres from one original does not consist of discrete “parts” as we usually understand them, but of sets of points, and these sets of points would constitute something that did not fully fill the space that they inhabit, and for this reason we could speak of them as possessing fractal dimension. On fractal dimension, the Wikipedia entry says this of the Koch curve:

“…the length of the curve between any two points on the Koch Snowflake is infinite. No small piece of it is line-like, but neither is it like a piece of the plane or any other. It could be said that it is too big to be thought of as a one-dimensional object, but too thin to be a two-dimensional object, leading to the speculation that its dimension might best be described in a sense by a number between one and two. This is just one simple way of imagining the idea of fractal dimension.”

The first space filling-curve discovered by Giuseppe Peano (the same Peano that formulated influential axioms of arithmetic, though the axioms seem to ultimately derive from Dedekind) already demonstrated a way in which a line, ordinarily considered one dimensional, can be two dimensional — or, if you prefer to take the opposite perspective, that a plane, ordinarily considered to be two dimensional, can be decomposed into a one dimensional line. A fractal like the Koch curve fills two dimensional space to a certain extent, but not completely like Peano’s space-filling curve, and its fractal dimension is calculated as 1.26.

Hilbert's version of a space filling curve.

The Koch curve is a line that is more than a line, and it can only be constructed in two dimensions. It is easy to dream up similar fractals based on two dimensional surfaces. For example, we could take a cube and construct a cube on each side, and construct a cube on each side of these cubes, and so on. We could do the same thing with bumps raised on the surface of a sphere. Right now, we are only thinking of in terms of surfaces. The six planes of a cube enclose a volume, so we can think of it either as a two dimensional surface or as a three dimensional body. In so far as we think of the cube only as a surface, it is a two dimensional surface that can only be constructed in three dimensions. (And the cube or sphere constructions can go terribly wrong also, as if we make the iterations too large they will run into each other. Still, the appropriate construction will yield a fractal.)

This process suggests that we might construct a fractal from three dimensional bodies, but to do so we would have to do this in four dimensions. In this case, the fractal dimension of a three dimensional fractal constructed in four dimensional space would be 3.n, depending upon how much four dimensional space was filled by this fractal “body.” (And I hope you will understand why I put “body” in scare quotes.)

I certainly can’t visualize a four dimensional fractal. In fact, “visualize” is probably the wrong term, because our visualization capacity locates objects in three dimensional space. It would be better to say that I cannot conceive of a four dimensional fractal, except that I can entertain the idea, and this is a form of conception. What I mean, of course, is a form of concrete conception not tied to three dimensional visualization. I suspect that those who have spent a lifetime working with such things may approach an adequate conception of four dimensional objects, but this is the rare exception among human minds.

Just as we must overcome the counter-intuitive feeling of the ideas of set theory in order to get to the point where we are conceptually comfortable with it, so too we would need to transcend our geometrical intuitions in order to adequately conceptualize four dimensional objects (which mathematicians call 4-manifolds). I do not say that it is impossible, but it is probably very unusual. This represents an order of thinking against the grain that will stand as a permanent aspiration for those of us who will never fully attain it. Intellectual intuition, like dimensionality, consists of levels, and even if we do not fully attain to a given level of intuition, if we glimpse it after a fashion we might express our grasp as a decimal fraction of the whole.

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A seasonally-appropriate illustration of the Banach-Tarski paradox.

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Fractals and Geometrical Intuition

1. Benoît Mandelbrot, R.I.P.

2. A Question for Philosophically Inclined Mathematicians

3. Fractals and the Banach-Tarski Paradox

4. A visceral feeling for epsilon zero

5. Adventures in Geometrical Intuition

6. A Note on Fractals and Banach-Tarski Extraction

7. Geometrical Intuition and Epistemic Space

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Benoît Mandelbrot, R.I.P.

17 October 2010

Sunday


Benoît B. Mandelbrot, 20 November 1924 to 14 October 2010

Famed French mathematician Benoît Mandelbrot has passed away a few days ago at the age of 85. When someone dies who has lived such a productive life, it would sound a little odd to say that we have “lost” him. We haven’t lost Benoît Mandelbrot. His contribution is permanent. The Mandelbrot set will take its place in history beside Euclidean geometry and Cantorian transfinite numbers. And when I say “history” you might think that I am consigning it to the dead past, but what I mean by “history” is an ongoing tradition of which we are a part, and which spills out into the future, informing lives yet to be lived in unpredictable and unprecedented ways.

The Mandelbrot set is a mathematical set of points in the complex plane, the boundary of which forms a fractal. The Mandelbrot set is the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2 + c remains bounded. (from Wikipedia)

Mandelbrot will be best remembered for having invented (discovered? formulated?) fractals and fractal geometry. By an odd coincidence, I have been thinking quite a bit about fractals lately. A few days ago I wrote The Fractal Structure of Exponential Growth, and I had recently obtained from the library the NOVA documentary Fractals: Hunting the Hidden Dimension. As with many NOVA documentaries, I have watched this through repeatedly to try to get all that I can out of it, much as I typically listen through recorded books multiple times.

There are a couple of intuitive definitions of fractals to which I often refer when I think about them. You can say that a fractal is an object that retains its properties under magnification, or you can say that a fractal is an object that possesses self-similarity across orders of magnitude. But while these definitions informally capture some important properties of fractals, the really intuitive aspect of fractals is their astonishing appearance. Mandelbrot was unapologetically interested in the appearance of fractals. In the NOVA documentary he says in an interview, “I don’t play with formulas, I play with pictures. And that is what I’ve been doing all my life.”

The equation for generating the Mandelbrot set, the later (and more interesting) iterations of which were only made possible by the sheer calculating power of computers.

It is often said that mathematicians are platonists during the week and formalists on the weekend. In other words, while actively working with mathematics they feel themselves to be engaged with objects as real as themselves, but when engaged in philosophical banter over the weekend, the mathematician defends the idea of mathematics as a purely formal activity. This formulation reduced mathematical formalism to a mere rhetorical device. But if formalism is the mathematician’s rhetoric while platonism is the philosophy that he lives by in the day-to-day practice of his work, the rhetorical flourish of formalism has proved to be decisive in the direction that mathematics has taken. Perhaps we could say that mathematicians are strategic formalists and tactical platonists. In this case, we can see how the mathematician’s grand strategy of formalism has shaped the discipline.

Julia sets, predecessors of the Mandelbrot set, found within the Mandelbrot set.

It was in the name of such formalism that “geometrical intuition” began to be seriously questioned at the turn of the nineteenth to the twentieth century, and several generations of mathematicians pursued the rigorization of analysis by way of taking arithmetization as a research program. (Gödel’s limitative theorems were an outgrowth of this arithmetization of analysis; Gödel produced his paradox by an arithmetization of mathematical syntax.) It is to this tacit research program that Mandelbrot implicitly referred when we said, “The eye had been banished out of science. The eye had been excommunicated.” (in an interview in the same documentary mentioned above) What Mandelbrot did was to rehabilitate the eye, to recall the eye from its scientific banishment, and for many this was liberating. To feel free to once again trust one’s geometrical intuition, to “run with it,” as it were, or — to take a platonic figure of thought — to follow the argument where it leads, set a new generation of mathematicians free to explore the visceral feeling of mathematical ideas that had gone underground for a hundred years.

The self-similarity of fractals means that one can find smaller interations of the Mandelbrot set within itself, i.e., the Mandelbrot set microcosm within the Mandelbrot set macrocosm.

It is the astonishing appearance of fractals that made Mandelbrot famous. Since one could probably count the number of famous mathematicians in any one generation on one hand, this is an accomplishment of the first order. Mandelbrot more-or-less singlehandedly created a new branch of mathematics, working against institutionalized resistance, and this new branch of mathematics has led to a rethinking of physics as well as to concrete technological applications (such as cell phone antennas) in widespread use within a few years of fractals coming to the attention to scientists. Beyond this, fractals became a pop culture phenomenon, to the degree that a representation of the Mandelbrot set even appeared as a crop circle.

While the crop circle representation of the Mandelbrot set would seem to have brought us to the point of mere silliness, so much so that we seem to have passed from the sublime to the ridiculous, it does not take much thought to bring us back around to understanding the intellectual, mathematical, scientific, and technological significance of fractal geometry. This is Mandelbrot’s contribution to history, to civilization, to humankind.

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Fractals and Geometrical Intuition

1. Benoît Mandelbrot, R.I.P.

2. A Question for Philosophically Inclined Mathematicians

3. Fractals and the Banach-Tarski Paradox

4. A visceral feeling for epsilon zero

5. Adventures in Geometrical Intuition

6. A Note on Fractals and Banach-Tarski Extraction

7. Geometrical Intuition and Epistemic Space

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Sunday


In his Notes on the Dynamics of Human Civilization: The Growth Revolution, Part I, T. Greer of Scholar’s Stage proposed what he called a growth revolution in conscious contrast to earlier historiographical attempts to identify periods of revolutionary change in history. For example, in reference to the Industrial Revolution, Mr. Greer says that it has been, “grossly mischaracterized,” and furthermore, “The industrialization of the world economy was the result, not the cause of modernization. The nature of this radical transformation is captured better by a different title: The Growth Revolution.”

I was thinking about this today and I realized that there are several periods of exponential growth in history of which our current world can be considered a consequence, that these occur at different orders of magnitude of history, and as such we can identify a self-similarity across different orders of magnitude of history that gives to this history a fractal structure.

At the level of history where geological time meets biological time — that is to say, the longest horizon of biological time — there is what is called the Cambrian Explosion. Most of the multi-billion year history of life on earth is little more than pond scum. For billions of years the earth was essentially covered in blue green algae and stromatolites, and the development of more complex forms of life was painfully slow. Then the Cambrian explosion occurred and suddenly there were seas teaming with an astonishing variety of life. Since that time, the earth has seen increasingly complex forms of life emerge, and, with the exception of periodic mass extinctions, growing numbers of species and biodiversity. It would seem that, once having passed a certain threshold of complexity, life’s capacity of grow exponentially was actualized.

Now we move in closer to history, thinking not in terms of millions of years of tens of millions of years, but thinking of terms tens of thousands and hundreds of thousands of years. Here we find the first exponential growth spike in specifically human history, and this is the agricultural or neolithic revolution. Colin Renfrew in his Prehistory: The Making of the Human Mind, emphasizes that this cannot be connected to the evolution of the genotype or the emergence of anatomically modern human beings. It would seem that our speciation event occurred somewhere on the horizon of 150,000 years ago, more or less (give or take some tens of thousands of years), but for most of this time modern human beings lived as hunter-gatherers with no larger social structure than the tribe or the clan. Then the agricultural revolution occurs, cities emerge, social differentiation and hierarchy emerge, settled societies emerge, human beings live in much greater density and organized state societies emerge. This occurred between 10,000 and 15,000 years ago, i.e., about a tenth of the total history of our species. Once again, it looks like we idled along for a long time without much happening, and then — Bang! — suddenly things started happening with much greater rapidity. As the Cambrian explosion saw life passing a threshold of complexity, the agricultural revolution saw human societies pass a threshold of complexity.

Now we move in even closer to history, approaching to the point where we look not at hundreds of thousands or tens of thousands of years, but only at hundreds of years. We are now considering a far shorter portion of time than that between the Cambrian explosion and the agricultural revolution. History at this level of magnification reveals to us another period of exponential growth, this time the growth represented by the Industrial Revolution. Just over two hundred years ago, beginning in England, spreading to Europe, and eventually making its way even today to Asia and Africa in the twenty-first century, societies that had had a stable form for thousands of years began to change much more rapidly. The Industrial Revolution uprooted stable societies and replaced them with something radically different. But this process, rather than taking hundreds or thousands of years, tends to transform traditional, stable societies within a period of decades, turning an acculturation to absence of change into a way of thinking when individuals expect to see dramatic changes within their lifetime and we say that “change is the only constant.”

Move in closer to history once again and look only at the last few decades. The oldest societies that had experienced the industrial revolution developed a pattern of stability. It is stable growth to be sure — the populations of industrialized nation-states have grown so accustomed to increasing standards of living that they rebel when the economy does not grow several percentage points per year — but it is a kind of stability within the chaotic growth and breakneck change that is the industrial revolution. Just in the past few decades even this stable growth has been given another jolt forward. The twin developments of high speed global transportation (the passenger jet) and high speed global communication (telecommunications and the internet) mean that human lives are changing at an accelerated rate of growth — yet another growth revolution in which a threshold has been crossed that allows growth across a number of other sectors.

What more could follow in this fractal structure of exponential growth? Will we need to consider the changes that will take place in human life — and, more generally, in life on earth — at a level of months, weeks, days, hours, or seconds? While I have written several posts that were highly skeptical of the so-called technological singularity (and I retract nothing that I have said in these posts), this is about the only thing that I can imagine that could once again spike the growth chart and produce yet another exponential growth curve, this time on an even shorter time scale corresponding to an extrapolation of the fractal structure of previous growth revolutions.

Human beings, however, live at a particular level of time consciousness and historical consciousness. We cannot perceive the vast periods of time studied by cosmology, though we can come to understand them through science, and we would not be able to perceive a fractal structure of exponential growth that disappeared into ever smaller periods of time. Thus one possibility is that something like the technological singularity could occur, but it would just as rapidly disappear from our view. We would go on devoting an hour to a leisurely lunch, even while at far higher magnifications of time further revolutions of exponential growth were going on unseen by us. We might come to understand these smaller periods of time through science, but they would mean as little to us in the present as the ultimate fate of the cosmos as contemplated by cosmology.

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Tuesday


This sketch by Hans Holbein of the family of Sir Thomas Moore (the painting has been lost) shows Moore at the center of the household, presiding like an emperor, while the women of the house are on their knees before him.

In yesterday’s Personal Dystopias I argued that the same kind of thinking that produces dystopian results from utopian intentions when practiced on a visionary scale is responsible for dystopian results from utopian intentions on a personal, local, and even private scale. Thinking more about this, another obvious example is despotism: there are large scale despotisms and small scale despotisms. The traditional patriarchal family structure found throughout much of the world is often modeled on despotic rule within the context of a single family. This is no longer acceptable in the industrialized world as it was in the recent past, but it played a significant role as recently as what I called the conformist patriarchy of mid-twentieth century America in The Agricultural Paradigm.

I realized today that these are examples of what Deleuze called “micropolitics.” I also realized, in contrasting the idea of micropolitics with the implied idea of macropolitics, that these are political instantiations of the microcosm / macrocosm theme, an ancient fractal theme in Western thought in which the large is mirrored in the small and the small is mirrored in the large. From this perspective, it would be interesting to engage in a detailed and through analysis of political paradigms seeking their parallels in private and domestic life, at the same time as seeking parallels of private and domestic life in systems of political and social organization.

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If George Washington is the Father of his Country, does that make us all one, big happy family?

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