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Prior to the advent of civilization, the human condition was defined by nature. Evolutionary biologist call this initial human condition the environment of evolutionary adaptedness (or EEA). The biosphere of the Earth, with all its diverse flora and fauna, was the predominant fact of human experience. Very little that human beings did could have an effect on the human condition beyond the most immediate effects an individual might cause in the environment, such as gathering or hunting for food. Nothing was changed by the passage of human beings through an environment that was, for them, their home. Human beings had to conform themselves to this world or die.

The life of early human communities was defined by nature, not by human activity.

The life of early human communities was defined by nature, not by human activity.

Since the advent of civilization, it has been civilization and not nature that determines the human condition. As one civilization has succeeded another, and, more importantly, as one kind of civilization has succeeded another kind of civilization — which latter happens far less frequently, since like kinds of civilization tend to succeed each other except when this process of civilizational succession is preempted by the emergence of an historical anomaly on the order of the initial emergence of civilization itself — the overwhelming fact of human experience has been shaped by civilization and the products of civilization, rather than by nature. This transformation from being shaped by nature to being shaped by civilization is what makes the passage from hunter-gatherer nomadism to settled agrarian civilization such a radical discontinuity in human experience.

This transformation has been gradual. In the earliest period of human civilizations, an entire civilization might grow up from nothing, spread regionally, assimilating local peoples not previously included in the project of civilization, and then die out, all without coming into contact with another civilization. The growth of human civilization has meant a gradual and steady increase in the density of human populations. It has already been thousands of years since a civilization could flourish and fail without encountering another civilization. It has been, moreover, hundreds of years since all human communities were bound together through networks of trade and communication.

Civilization is now continuous across the surface of the planet. The world-city — Doxiadis’ Ecumenopolis, which I discussed in Civilization and the Technium — is already an accomplished fact (though it is called by another name, or no name at all). We retain our green spaces and our nature reserves, but all human communities ultimately are contiguous with each other, and there is no direction that you can go on the surface of the Earth without encountering another human community.

The civilization of the present, which I call industrial-technological civilization, is as distinct from the agricultural civilization (which I call agrarian-ecclesiastical civilization) that preceded it as agricultural civilization was distinct from the nomadic hunter-gatherer paradigm that preceded it in turn. In other words, the emergence of industrialization interpolated a discontinuity in the human condition on the order of the emergence of civilization itself. One of the aspects of industrial-technological civilization that distinguishes it from earlier agricultural civilization is the effective regimentation and indeed rigorization of the human condition.

The emergence of organized human activity, which corresponds to the emergence of the species itself, and which is therefore to be found in hunter-gatherer nomadism as much as in agrarian or industrial civilization, meant the emergence of institutions. At first, these institutions were as unsystematic and implicit as everything else in human experience. When civilizations began to abut each other in the agrarian era, it became necessary to make these institutions explicit and to formulate them in codes of law and regulation. At first, this codification itself was unsystematic. It was the emergence of industrialization that forced human civilizations to make its institutions not only explicit, but also systematic.

This process of systematization and rigorization is most clearly seen in the most abstract realms of thought. In the nineteenth century, when industrialization was beginning to transform the world, we see at the same time a revolution in mathematics that went beyond all the earlier history of mathematics. While Euclid famously systematized geometry in classical antiquity, it was not until the nineteenth century that mathematical thought grew to a point of sophistication that outstripped and exceeded Euclid.

From classical antiquity up to industrialization, it was frequently thought, and frequently asserted, that Euclid was the perfection of human reason in mathematics and that Aristotle was the perfection of human reason in logic, and there was simply nothing more to be done in the these fields beyond learning to repeat the lessons of the masters of antiquity. In the nineteenth century, during the period of rapid industrialization, people began to think about mathematics and logic in a way that was more sophisticated and subtle than even the great achievements of Euclid and Aristotle. Separately, yet almost simultaneously, three different mathematicians (Bolyai, Lobachevski, and Riemann) formulated systems of non-Euclidean geometry. Similarly revolutionary work transformed logic from its Aristotelian syllogistic origins into what is now called mathematical logic, the result of the work of George Boole, Frege, Peano, Russell, Whitehead, and many others.

At the same time that geometry and logic were being transformed, the rest of mathematics was also being profoundly transformed. Many of these transformational forces have roots that go back hundreds of years in history. This is also true of the industrial revolution itself. The growth of European society as a result of state competition within the European peninsula, the explicit formulation of legal codes and the gradual departure from a strictly peasant subsistence economy, the similarly gradual yet steady spread of technology in the form of windmills and watermills, ready to be powered by steam when the steam engine was invented, are all developments that anticipate and point to the industrial revolution. But the point here is that the anticipations did not come to fruition until the nineteenth century.

And so with mathematics. Newton and Leibniz independently invented the calculus, but it was left on unsure foundations for centuries, and Descartes had made the calculus possible by the earlier innovation of analytical geometry. These developments anticipated and pointed to the rigorization of mathematics, but the development did not come to fruition until the nineteenth century. The fruition is sometimes called the arithmetization of analysis, and involved the substitution of the limit method for fluxions in Newton and infinitesimals in Leibniz. This rigorous formulation of the calculus made possible engineering in its contemporary form, and rigorous engineering made it possible to bring the most advanced science of the day to the practical problems of industry. Intrinsically arithmetical realities could now be given a rigorous mathematical exposition.

Historians of mathematics and industrialization would probably cringe at my potted sketch of history, but here it is in sententious outline:

● Rigorization of mathematics also called the arithmetization of analysis

● Mathematization of science

● Scientific systematization of technology

● Technological rationalization of industry

I have discussed part of this cycle in my writings on industrial-technological civilization and the disruption of the industrial-technological cycle. The origins of this cycle involve the additional steps that made the cycle possible, and much of the additional steps are those that made logic, mathematics, and science rigorous in the nineteenth century.

The reader should also keep in mind the parallel rigorization of social institutions that occurred, including the transformation of the social sciences after the model of the hard sciences. Economics, which is particularly central to the considerations of industrial-technological civilization, has been completely transformed into a technical, mathematicized science.

With the rigorization of social institutions, and especially the economic institutions that shape human life from cradle to grave, it has been inevitable that the human condition itself should be made rigorous. Foucault was instrumental in pointing out salient aspects of this, which he called biopower, and which, I suggest, will eventually issues in technopower.

I am not suggesting this this has been a desirable, pleasant, or welcome development. On the contrary, industrial-technological civilization is beset in its most advanced quarters by a persistent apocalypticism and declensionism as industrialized populations fantasize about the end of the social regime that has come to control almost every aspect of life.

I wrote about the social dissatisfaction that issues in apocalypticism in Fear of the Future. I’ve been thinking more about this recently, and I hope to return to this theme when I can formulate my thoughts with the appropriate degree of rigor. I am seeking a definitive formulation of apocalypticism and how it is related to industrialization.

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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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