21 November 2013
When Frank Drake first formulated the eponymously-named Drake equation the number of planetary systems in the universe (the second term in the Drake equation, fp) was an unknown among other unknowns. Now we are rapidly approaching a scientifically-based quantification of this once unknown number. We now know that planetary systems are common, and moreover that planetary systems with smallish, rocky planets in the habitable zones of stars are relatively common. (Cf., e.g., Earth-Like Worlds “Very Common”)
As soon as we reached a level of technological and scientific expertise that made the search for exoplanets practical, we began to find them. The most recent exoplanet discoveries, and the recent announcement that planets and planetary system are common, are primarily due to the NASA Kepler mission. According to the NASA website, the Kepler mission was…
“…specifically designed to survey a portion of our region of the Milky Way galaxy to discover dozens of Earth-size planets in or near the habitable zone and determine how many of the billions of stars in our galaxy have such planets.”
In this, the Kepler mission has been wildly successful. But in order to get to the point at which our civilization could conceive, design, build, and operate the Kepler mission we had to pass through thousands of years of development, and before our civilization developed to its current state of technological prowess, it took terrestrial biology billions of years of development to arrive at organisms capable of creating a civilization that could develop to this level.
Contrast the experience of Kepler’s exoplanet search with the experience of SETI, the search for extraterrestrial intelligence. What did not happen as soon as we began searching for SETI signals? We did not immediately begin hearing a whole range of intelligent extraterrestrial signals, which would have been a result parallel to the immediate successes of the exoplanet search (immediate, that is, in the technological zone of proximal development). Both Kepler and SETI are searches of the sky. The Kepler mission gave nearly immediate results; Frank Drake conducted the first SETI study in 1960. Drake found only an eerie silence, and ever since we have only heard an eerie silence. Once the technological threshold of exoplanet search was reached, the search immediately discovered its object, but once the technological threshold of SETI was reached, the search revealed nothing.
Please understand that, in making this observation, I am in no sense criticizing SETI efforts; I am not saying that SETI is a waste of effort, or a waste of money; I am not saying that SETI is wrongheaded or that it is not a science. On the contrary, I think SETI is interesting and important, and that includes the fact that SETI has found only an eerie silence — this is in itself important and interesting. We have discovered radio silence, except for natural sources. This tells us something about the universe. If there were a truly predatory peer civilization in our region of the Milky Way, it would be expected that they would go to the trouble to broadcast their presence to the universe, in hope of luring unsuspecting peer civilizations. Like Odysseus having himself strapped to the mast of his ship so that he could hear the song of the Sirens while his crew rowed on oblivious, their ears stopped with wax, we would have to listen to such signals restraining ourselves from rushing toward that fatal lure.
What we now know, as a result of SETI’s discovery of the eerie silence, is that METI (messaging extraterrestrial intelligence) beacons are not common. If METI beacons were common in the Milky Way, we would have heard them by now. There may yet be METI beacons, but they are not the first thing that you hear when you begin a SETI program (unlike looking for exoplanets and finding them as soon as you have the capability of looking). If METI beacons exist, they are rare and difficult to find. I think we can go further than this, and assert with some degree of confidence that there is no alien “super-civilization” in our galactic neighborhood constructing vast mega-engineering projects and pumping out high-power EM spectrum emissions that would be easily detectable by any technological civilization that suddenly had the idea to begin listening for such signals.
I wrote above that SETI and exoplanet searches are sensitive to a technological threshold. We passed the SETI threshold in the 1960s, and we have passed the exoplanet search threshold in the first decade of the twenty-first century. There is a further technological threshold, which is also an economic threshold — the ability to detect the unintentional EM spectrum radiation “leakage” from technological civilizations that have not had the interest or the resources to establish a METI beacon, but which, like us, are radiating EM spectrum signals as an epiphenomenal expression of our industrial-technological civilization. I say that this is also an economic threshold, as James Benford and colleagues have taken pains to point out the expense associated with establishing a METI beacon. (This is something I discussed in my Centauri Dreams post SETI, METI, and Existential Risk; James Benford responded on Centauri Dreams with James Benford: Comments on METI; my post on Centauri Dreams, along with responses from Benford and from David Brin, received quite a few comments, so if the reader is interested, it is worthwhile to follow the links and read the ensuing discussion.)
If METI is “shouting to the galaxy” (as James Benford put it), then the unintentional leakage of EM spectrum radiation of industrial-technological civilization is not shouting to the galaxy but rather whispering to the cosmos, and in order to be able to hear a whisper we must listen intently — holding our breath and putting a hand to our ear. Whether or not we choose to listen intently for whispers from the cosmos, we have not yet reached the developmental stage of civilization in which this is practical, though we seem to be moving in that direction. If we should continue the trajectory of our technological development — which, as I see it, entails both increasing automation and routine travel between Earth and space — such an effort will be within our grasp within the coming century.
Advanced industrial-technological civilizations will, by definition, know much more than we know. Their science will be commensurate with their technology and their engineering, since their civilization, if it is an industrial-technological peer civilization (and in so far as industrial-technological civilization is defined by the STEM cycle, which I believe to be the case), will experience the advance of science joined inseparably to the advance of technology and engineering. What would they do with this epistemic advantage? Such an epistemic advantage presents the possibility of SETI and METI asymmetry. We have an asymmetrical advantage over civilizations at an earlier stage of development, as older industrial-technological civilizations would have an asymmetrical advantage over us, with the ability to find us while concealing themselves.
The developmental direction of industrial-technological civilization as defined by the STEM cycle means that any advanced industrial-technological civilization will be “backward compatible” with earlier forms of technological communication. We might not (yet) be able to build a quantum entanglement transmitter in order to communicate instantaneously over cosmic distances (even though we can conceive the possibility), but an advanced peer civilization will be able to listen for our EM spectrum radiation leakage, in the same way that we today could continue to look for signs of ETI compatible with earlier stages of industrial-technological civilization. Karl Friedrich Gauss suggested geometrical shapes laid out in wheat in the wastes of Siberia to get the attention of extraterrestrials, while Joseph von Littrow suggested trenches filled with burning oil in the Sahara. Interesting in this context, although our civilization had the technology to pursue these methods, no one undertook them on a large scale.
When, in the future, we have the ability to image the surface of exoplanets with large extraterrestrial telescopes, we could look for such attempted signals within the capability of less developed civilizations to produce, such as those suggested by Gauss and Littrow. But when it comes to advanced peer civilizations, we don’t have the knowledge to know what to look for. The more advanced the civilization, the farther it lies beyond our civilizational zone of proximal development (ZPD), but the more advanced a civilization the earlier it would have to have its origins in the history of the universe, and at some point in the development of the universe (going backward in time to the origins of the universe) it would not be possible for an industrial-technological civilization to emerge because if we go far enough back in time, the elements necessary to an industrial-technological civilization do not yet exist. So there seems to be a window of development in the history of the universe for the emergence of industrial-technological civilizations. This strikes me as a non-anthropocentric way of expressing one formulation of the anthropic cosmological principle (and an idea worth developing further, since I have been searching for a formulation of the anthropic cosmological principle worthy of the name).
In an optimistic assessment of our place in the universe, we could hope that any substantially more advanced civilization could serve as the “more knowledgeable other” (MKO) that would facilitate our progress through the civilizational zone of proximal development.
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27 May 2013
Addendum on a Future Astropolitics:
Civilization Shaped by Structures of the Universe
In my previous post on astropolitics, The Fundamental Theorem of Astropolitics, I gave a generalization of my earlier definition of geopolitics (which was, “geography constrains human agency”) as the following:
Human agency is constrained by the structure of space.
Upon reflection I have realized that, while this definition is good as far as it goes, it doesn’t go far enough. The primary defect of this formulation is that it is formulated exclusively in terms of constraints upon human agency, which is to say, it focuses on the ways in which human agency is limited or even negated.
In formulating either geopolitics or astropolitics in terms of the limitations on human agency, the ways in which geography or the structure of space facilitates human agency gets lost, and this function of facilitation is no less significant than the function of limitation that follows from the lay of the land or the structure of space.
Another weakness in formulating geopolitics or astropolitics in terms of constraint and limitation is that it implies that, were it only not for the limitations placed upon human agency by outside forces, that this agency would be boundless and infinite. In other words, focusing on limitation and constraint suggests, in a very subtle way, what I have called the political conception of history, i.e., history understood primarily in terms of human agency. While this is an empowering way of viewing history, it cannot be considered any more or less accurate than other other conceptions of history I have outlined — the cataclysmic, eschatological, and naturalistic conceptions (for a review of these conceptions cf. The Naturalistic Conception of History) — and it is likely to be misleading.
When I spoke at the 2012 100YSS event one of the central ideas of my talk was the ways in which the structure of spacetime will govern the expansion of civilization on an interstellar scale, and even beyond this the ways in which human beings (or any other finite being exploring the cosmos) can use the apparent limitations imposed upon us by relativity, the finite velocity of light, and the structure of space itself to facilitate the growth of civilization. (I called my talk, “The Large Scale Structure of Spacefaring Civilization.”)
We have come to see the velocity of light as a barrier to human/organic exploration of the cosmos, but this is a profound misconception. In response to this misconception, those contemplating the possibility of interstellar civilization either are looking for ways to avoid relativistic effects, such as the use of the Alcubierre drive (if only it can be made to work), and those who think that interstellar travel is simply impossible or can only be accomplished at very slow rates of expansion, such as the rates of speed at which the Voyager spacecraft are slowly making their way outside our local solar system (i.e., by way of generational ships and long term human preservation or reconstitution).
What I want to suggest is that relativity is our friend. The finite velocity of light and the phenomenon of time dilation can and will be used by human beings to facilitate interstellar travel. Anyone who has studied these matters carefully will know what I am talking about here, but the popular misconceptions are so prevalent that one must pause to mention them. It is often stated that if we sent out an interstellar mission traveling at a rate that involved relativistic effects, that we could only hope for our distant descendents to arrive; that no one would live to see another solar system. In fact, time would continue to pass on Earth, but the closer a starship can approximate the speed of light, even while never reaching that limiting velocity, the slower time passes on board, so that even very long interstellar voyages can be accomplished within life spans typical by contemporary standards.
Carl Sagan discussed this at some length in his book and television series Cosmos, in which he talked about a starship that could accelerate at one gravity. We can think of the 1G starship as the breakthrough technology that will open up our galaxy to exploration and settlement. Already we can accelerate a spaceship at well more than 1G, although we cannot maintain this acceleration for extended periods of time, so I regard attaining this acceleration for extended periods of time to be a merely technical problem, and not an insuperable “physics” problem. (Some people will disagree with me on this point.) Sagan pointed out that with the humble technology of a 1G starship we could circumnavigate the known universe in a typical human life span. By the time we finished this journey, however, billions of years would have passed.
It is easy to lose sight of this possibility when discussing space flight, and our limited capabilities today, but looking at the ability of industrial-technological civilization to continue delivering exponential technological development, we should not consider this technology to be long out of our reach. That is why I call it a “humble” technology. It doesn’t require breaking the known laws of physics, and it doesn’t require an engineering breakthrough on the level of the Alcubierre drive (though I should mention that I still hold out hope for the development of the Alcubierre drive).
Once we allow ourselves to think in these terms, and to imagine as a real possibility human exploration of the cosmos, even limited to contemporary life spans (which are likely to be lengthened considerably in the coming century), what one comes to realize is not the unattainability of the velocity of light, but really how slow the speed of light is in relation to the size of the cosmos. Light is almost pokey in its progress, since it would take light about 93 billion years to traverse the known universe. The age of the universe seems incomprehensibly ancient, but really, when you think about it in cosmological terms, 13.7 billion years isn’t all that much. We’re only really getting started here on this universe bit. And the size the universe? Again, it seems incomprehensible vast, but if we adjust our perspective, it is well within the limits of human comprehension if we will only take the time and the trouble to systematically expand and extend out understanding.
We can spend our time contemplating the littleness of man in the cosmos, or we can work to attain a perspective commensurate with the universe. It is true that we are indeed very small at present, and it has been the tradition of human thought to meditate upon our insignificance, our smallness before the universe, our manifold weaknesses, our miserably short existence, and the sorrows of the human condition — in short, it has been the tradition to meditate on what Hume called the “monkish virtues.” While we do not think of modern thought in this way, once we pause to put matters in context, we see the degree to which this tradition still retains its power over our minds.
Here is how Hume formulated the “monkish virtues”:
Celibacy, fasting, penance, mortification, self-denial, humility, silence, solitude, and the whole train of monkish virtues; for what reason are they everywhere rejected by men of sense, but because they serve to no manner of purpose; neither advance a man’s fortune in the world, nor render him a more valuable member of society; neither qualify him for the entertainment of company, nor increase his power of self-enjoyment? We observe, on the contrary, that they cross all these desirable ends; stupify the understanding and harden the heart, obscure the fancy and sour the temper.
David Hume, An Enquiry Concerning the Principles of Morals, 1777, Section IX, Conclusion, Part I
To Hume’s litany of the reasons to reject the monkish virtues we might also add that this is no way to go about building a civilization. This is to think in terms of constraints. But we must also think in terms of possibilities, and if we are ever to construct the spacefaring civilization that we can now clearly conceptualize, we will have to think more in terms of possibilities and less in terms of limitations. As central to the creation of a spacefaring civilization as the technological developments is the conceptual revolution that needs to be sustained, and as ambitious and as megalomaniac as this sounds, we must formulate and inculcate a human perspective that takes the human role in the cosmos for granted. We must learn to think on a cosmological scale.
For those who wonder at the hubris of what I am saying, the punishment of our pride will come about in due course, for no grand enterprise (and there is no grander enterprise than the expansion of civilization) is without reversals, but we cannot begin this enterprise by thinking only in terms of what we cannot do. We would never get off the ground — literally, we would never pass beyond planet-bound civilization to transplanetary civilization — if we thought only in terms of the meagerness of our abilities.
While some limitations are unambiguously limiting, others can be seen as a constraint or an opportunity depending upon one’s perspective. This is true of the structure of the universe and time dilation, which is built into relativistic physics. Future civilization will not try to defy this structure of spacetime (by trying to do something impossible according to physics), but will exploit this structure of spacetime in order to expand civilization in unexpected and unprecedented ways.
Astrophysics will shape interstellar civilization. The development of civilization will follow the availability of matter and energy; both matter and energy are found in and around the vicinity of stars, stars are collected in galaxies, and galaxies are found in clusters. Civilization will follow this same structure, from stars to galaxies to clusters, and civilization will do so because this is where the matter and energy at to be found.
Matter shapes the structure of spacetime; in seeking the resources of matter and energy, civilization will find itself in those regions of the universe shaped by the presence of matter. Matter, moreover, is convertible with energy, and vice versa. Civilization seeks matter in part in order to convert it into energy in order to power the industries of industrial-technological civilization. At some future time civilization may also seek energy in order to convert it into matter.
Civilization as we have known it has sought to expand itself in space, but time dilation will allow civilization also to expand in time. Given the breakthrough technology of a 1G starship, civilization will not only move outward in space, but also later in time. While time on Earth may be considered the baseline, a fleet of starships with enough capacity to carry a sufficient portion of terrestrial civilization to establish this civilization at a new center, will carry that civilization to a later time commensurate with the distance traveled outward. Because of time dilation, relatively little time will have passed on the voyage, even while a great deal of time will have passed on Earth.
In other words, while separated by years and lightyears, it will still be essentially the same civilization. From an omnipresent perspective — what might be called the “view from nowhere” (to borrow a phrase from Thomas Nagel) — we can see that these are temporally separated instances of one an the same civilization are. I call this a temporally distributed civilization. (This was one of the central points of my 2012 100YSS talk.) Given this structure of a temporally distributed civilization, there is quite literally no going home again. It would always be possible to migrate outward, and into later times (though essentially into the same civilization displaced later in time), but back would mean going into far future civilization that no longer resembled the civilization one had left behind.
Civilization conceived and executed on this cosmological scale, integral with the largest astrophysical processes, would leave observable traces. In Transcendent Man, the film about Ray Kurzweil, Kurzweil talks about looking up into the night sky and seeking signs of an alien technological singularity. Others have thought to search the skies for mega-engineering projects, looking for the astronomical markers of Dyson spheres or the use of a black hole as a source of energy.
Nothing definitive has yet been seen in the night sky. There does not appear to have been any civilization of cosmological scale that has preceded us — though there may be one out there, only now coming to maturity and not yet visible to us. Or maybe there is nothing out there. For many, the lack of evidence of a civilization of cosmological scale is proof (not definitive, deductive proof, but incremental, inductive proof, not leading to certainty, but to likelihood and probability) that there is no such civilization, nor can there be such a civilization.
Some dreamers who reject the possibility of interstellar travel but want to know something of the other inhabitants who might be out there in the Milky Way and beyond resign themselves to the quietism of SETI, sitting in a room monitoring instruments, hoping to catch a glimpse of alien intelligence from signals among the stars. This model of interstellar exchange is presented as practicable, and therefore something that a person might reasonably believe in, even if it departs from the Buck Rogers model of flying around and visiting other planets, all the while with a trusty sidearm on one’s hip.
I know that there are a great many people who maintain that there will never be any interstellar civilization, therefore no interaction between multiple interstellar civilizations, therefore no interstellar exchanges of any significance — whether for trade or war or culture or otherwise — because of the distances involved and the energy levels that would be required. I do not think that this is an insuperable problem, because in large measure the problem is our own perspective and the human tendency to sabotage our own efforts. Such habits of thought and action are valuable for a planet-bound civilization, but would be crippling for a transplanetary civilization.
I, on the other hand, view the large scale structure of interstellar civilization as an inevitable (or nearly inevitable) outcome of the continued expansion of industrial-technological civilization, in accordance with the Industrial-Technological Thesis that defines technological progress as intrinsic to this form of civilization. The only event that would derail the eventual realization of interstellar civilization is if civilization itself were to be derailed — hence my concern with existential risk.
A theoretical astropolitics would furnish the conceptual infrastructure for any future interstellar trade, interstellar war, or even interstellar “cultural exchanges” (as they were delicately called during the Cold War). And, as should be apparent from the foregoing, it seems clear that, as long as our industrial-technological civilization continues in its present trajectory of development, all of this will come to pass in the fullness of time.
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21 May 2013
The Human Future after Geopolitics:
The Large Scale Structure of Political Societies
Some time ago in The Fundamental Theorem of Geopolitical Thought I formulated just such a theorem as follows: Human agency is constrained by geography. While geopolitics must remain central to understanding contemporaneous political thought, this will not always be so. The time will come when we will, of necessity, pass beyond geopolitics.
In many posts in which I have discussed the extraterrestrialization of terrestrial civilization (cf. e.g., Addendum on Extraterrestrialization and The Farther Reaches of Civilization) and the advent of Copernican civilization (cf. e.g., Civilization and the Technium and Earth Science, Planetary Science, Space Science) I have clearly implied that, as civilization expands off the surface of the earth, the political life of man will be forced to change in order to keep pace with these events, much as human societies have been forced to change rapidly as a result of the industrial revolution and its consequences. It does not matter how desperately those heavily-invested in the present global order will resist this change: the change will come if industrial-technological civilization continues its trajectory and does not succumb to existential risks.
If the political structure of extraterrestrialized civilization will be described by a future science of astropolitics, the fundamental theorem of astropolitics can be formulated as concisely as my fundamental theorem of geopolitics, and it would be formulated thus:
Human agency is constrained by the structure of space.
This is a straightforward generalization of my fundamental theorem of geopolitics, and as that theorem can be summarized as geography matters, the fundamental theorem of astropolitics can be similarly summarized as space matters.
The generalization of the scope of human agency from geography to the structure of space itself suggests that we also ought to generalize beyond the human, since by the time earth-originating civilization is an extraterrestrial civilization human beings will have become transhuman or post-human, and in the fullness of time homo sapiens will be followed by successor species. Thus…
Human and human-successor agency is constrained by the structure of space.
However, since this formulation of the fundamental theorem of astropolitics would hold for any peer civilization, there is no reason to limit the formulation to human beings, human successors, or earth-originating life. Thus…
Any conscious agency is constrained by the structure of space.
It is even superfluous to mention the qualification of “conscious” agency, since any naturalistic agency whatsoever is and will be constrained by the structure of space (supernatural agencies as comprehended in eschatological conceptions of history would presumably not be constrained by space). However, since our concern at present is to understand the large scale structure of political societies, we are concerned with those agents that represent peer industrial-technological civilizations that might establish (or have already established) a (peer) civilization beyond the surface of their homeworld.
Despite the many different formulations that might be given to the fundamental theorem of astropolitics, depending on the degree of generalization to be embodied in the formulation, all of these generalizations are intuitively continuous with the fundamental theorem of geopolitics, as well they ought to be. The geographical and topographical features that are central to geopolitical thought are the local structures of space corresponding to the human epistemic and perceptual order of magnitude. When the growth of civilization forces the parallel expansion of human epistemic and perceptual orders of magnitude, the structure of space itself will concern us more than the local mountain ranges, rivers, and deserts that now shape our terrestrial strategic thought.
The structural similarity between the fundamental theorem of geopolitics and the fundamental theorem of astropolitics masks the profound transformation of human political life that will come about in the event that human civilization expands to the degree that astropolitical thought will better describe strategic agency than geopolitical thought. A robust, self-sustaining human presence off the surface of the earth will impact human political societies so dramatically that it will eventually mean the end of the nation-state system. Such a change in human political thought will develop over more than a century, and will probably require two or three centuries to be fully assimilated throughout human civilization.
In my Political Economy of Globalization I attempted to describe the peculiar form of dishonesty that is employed in political thought that is to be found when our political ideas do not keep up with actual political developments:
…not every political entity that has a seat at the table at the United Nations conforms to the paradigm of the nation-state; some are more state, others more nation, yet others falling under neither category. Feudal monarchies rub elbows with republics and city-states, none of them representing any genuine national aspirations of a people or peoples for self-determination.
If the United Nations had existed in the eighteenth century, the Ottoman Empire would have been a member; if the United Nations had existed in the nineteenth century the Austro-Hungarian Empire would have been a member state. These empires are long since dissolved, but we can easily imagine that had the UN been in existence at the time of their dissolution these events would have been characterized in apocalyptic terms and attended with much hand wringing.
And if the dissolution of individual nation-states causes the level of distress one sees in the international system, it should be apparent that the end of the nation-state system itself will be viewed by some as a catastrophe of unparalleled proportions. However, it will take some time for the change to be noticed, which I also noted in my Political Economy of Globalization:
In the distant future, there will be, of course, political entities that will be called states. But the modern nation-state, eponymously defined in terms of nationhood, but in fact defined in terms of territorial sovereignty, cannot survive in its present form to be among the political entities of the future. Perhaps the new political entities will be called nation-states, as a holdover from our own time, but they will not have the character of nation-states any more than the Ottoman Empire had the character of a nation-state. While the latter was an identifiable state, to be sure, it was not a nation-state.
Conventional contemporary political and social science scarcely ever questions the role of the nation-state in human affairs (as though it were a permanent feature of civilization, which it is not), but we are under no obligation to allow these conventional limitations upon political imagination constrain our own formulations. It is enough to be constrained by the structure of space; there is no need to voluntarily burden oneself with additional constraints.
But we must unquestionably begin with the nation-state as the source of our present political situation, because all that follows in the future from the present situation will follow from the familiar nation-state system and the political thought of our time that privileges the nation-state system. The human, all-too-human scale of the nation-state system is the political parallel of the human, all-too-human scale of the geographical and topographical obstacles that are the present boundaries to human agency.
There is story I can’t resist repeating here about practical geopolitics, which is what military operations in the age of the nation-state represent. It is, in fact, a story within a story, as related by Hermann von Kuhl of Alfred von Schlieffen:
“He lived exclusively for his work and his great tasks. I remember how we once travelled through the night from Berlin to Insterburg, where the great staff ride was to begin. General Schheffen travelled with his aide-de-camp. In the morning the train left Königsberg and entered the Pregel valley, which was basking prettily in the rays of the rising sun. Up to then not a word had been spoken on the journey. Daringly the A.D.C. tried to open a conversation and pointed to the pleasant scene. ‘An insignificant obstacle,’ said the Graf — and conversational demands until Insterburg were therewith met.”
THE SCHLIEFFEN PLAN: Critique of a Myth, GERHARD RITTER, Foreword by B. H. LIDDELL HART, OSWALD WOLFF (PUBLISHERS) LIMITED, London, W.i, 1958, p. 99
Schlieffen’s single-minded focus on geographical features as exclusively representing opportunities or obstacles for campaigning — features that for others might represent aesthetics objects, or any kind of object significant in human experience — demonstrates geopolitical thought as at once practical and abstract. It is possible for geopolitics to be practical and abstract at the same time because the abstractions it considers are features like “insignficant obstacle,” while it takes no account of features such as “pleasant scene.” Astropolitics will be practical and abstract in the same way, although its objects will not be objects of ordinary human experience such as “insignificant obstacle” or “pleasant scene.”
The magnification of the scale of human concerns in astropolitics will not merely involve a larger canvas for human ambition, but will also introduce complexities not represented at the geopolitical scale. On the level of ordinary human experience time and space can be treated in isolation from each other, so that we have history and geography as abstract conceptions; at the higher energy levels, greater distances, higher speeds, and greater gravitational influences of a much-expanded spacefaring civilization, space and time will of necessity be treated together as space-time.
After I first formulated my fundamental theorem on geopolitical thought I followed it with two additional principles, the second law of geopolitics…
The scope of human agency defines a center, beyond which lies a periphery in which human agency is marginal.
Human agency is essentially a temporal agency.
As I had summarized the fundamental theorem of geopolitical thought as geography matters, I summarized the third law of geopolitical thought as history matters. As we have seen above, the large scale structure of the universe must be understood in terms of space-time, meaning that we cannot isolate cosmological geography from cosmological history. History and geography on a cosmological scale are even more intimately bound up in each other than they are on the human, all-too-human scale of terrestrial politics.
This suggests a further generalization of the fundamental theorem of astropolitics:
Human agency (or any conscious agency) is constrained by space-time.
History and geography have always been intimately tied together, and his, of course, is one of the great lessons of geopolitics, that geography shapes history. It is also true, has been true, that history shapes geography, but the forces by which the history of life on earth have shaped geography have occurred on a timescale that is not apparent to human perception.
In a future political science of astropolitics, we will have a history that reflects the large scale structure of the cosmos, and a large scale structure of the cosmos that reflects the history of the universe. While human agency (or other conscious agents) has not yet acted on a scale to have shaped the initial 13.7 billion years of cosmic history, if our civilization or its successor institutions should endure, its history could well shape the large scale structure of space-time.
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27 April 2012
The thesis that epistemic space is primarily shaped and structured by geometrical intuition may be equated with Bergson’s exposition of the spatialization of the intellect. Bergson devoted much of his philosophical career to a critique of the same. Bergson’s exposition of spatialization is presented in terms of a sweeping generality as the spatialization of time, but a narrower conception of spatialization in terms of the spatialization of consciousness or of human thought follows from and constitutes a special case of spatialization.
One might well ask, in response to Bergson, how we might think of things in non-spatial terms, and the answer to this question is quite long indeed, and would take us quite far afield. Now, there is nothing wrong with going quite far afield, especially in philosophy, and much can be learned from the excursion.
There is a famous passage in Wittgenstein’s Tractatus Logico-Philosophicus about “logical space,” at once penetrating and obscure (like much in the Tractatus), and much has been read into this by other philosophers (again, like much in the Tractatus). Here is section 1.13:
“The facts in logical space are the world.”
And here is section 3.42:
“Although a proposition may only determine one place in logical space, the whole logical space must already be given by it. (Otherwise denial, the logical sum, the logical product, etc., would always introduce new elements — in co-ordination.) (The logical scaffolding round the picture determines the logical space. The proposition reaches through the whole logical space.)”
I will not attempt an exposition of these passages; I quote them here only to give the reader of flavor of Wittgenstein’s . Clearly the early Wittgenstein of the Tractatus approached the world synchronically, and a synchronic perspective easily yields itself to spatial expression, which Wittgenstein makes explicit in his formulations in terms of logical space. And here is one more quote from Wittgenstein’s Tractatus, from section 2.013:
“Every thing is, as it were, in a space of possible atomic facts. I can think of this space as empty, but not of the thing without the space.”
I find this particularly interesting because it is, essentially, a Kantian argument. I discussed just this argument of Kant’s in Kantian Non-Constructivism. It was a vertiginous leap of non-constructive thought for the proto-constructivist Kant to argue that he could imagine empty space, but not spatial objects without the space, and it is equally non-constructive for Wittgenstein to make the same assertion. But it gives us some insight into Wittgenstein’s thinking.
Understanding the space of atomic facts as logical space, we can see that logical space is driven by logical necessity to relentlessly expand until it becomes a kind of Parmenidean sphere of logical totality. This vision of logical space realizes virtually every concern Bergson had for the falsification of experience given the spatialization of the intellect. The early Wittgenstein represents the logical intellect at its furthest reach, and Wittgenstein does not disappoint on this score.
While Wittgenstein abandoned this kind of static logical totality in this later thought, others were there to pick up the torch and carry it in their own directions. An interesting example of this is Donald Davidson’s exposition of logical geography:
“…I am happy to admit that much of the interest in logical form comes from an interest in logical geography: to give the logical form of a sentence is to give its logical location in the totality of sentences, to describe it in a way that explicitly determines what sentences it entails and what sentences it is entailed by. The location must be given relative to a specific deductive theory; so logical form itself is relative to a theory.”
Donald Davidson, Essays on Actions and Events, pp. 139-140
In a more thorough exposition (someday, perhaps), I would also discuss Frege’s exposition of concepts in terms of spatial areas, and investigate the relationship between Frege and Wittgenstein in the light of their shared equation of logic and space. (I might even call this the principle of spatial-logical equivalence, which principle would be the key that would unlock the relationship between epistemic space and geometrical intuition.)
Certainly the language of spatiality is well-suited to an exposition of human thought — whether it is uniquely suited is an essentialist question. But we must ask at this point if human thought is specially suited to a spatial exposition, or if a spatial exposition is especially suited for an exposition of human thought. It is a question of priority — which came first, the amenability of spatiality to the mind, or the amenability of the mind to spatiality? Which came first, the chicken or the egg? Is the mind essentially spatial, or is space essentially intellectual? (The latter position might be assimilated to Kantianism.)
From the perspective of natural history, recent thought on human origins has shifted from the idea of a “smart ape” to the idea of a “bipedal ape,” the latter with hands now free to grasp and to manipulate the environment. Before this, before human beings were human, our ancestors lived in trees where spatial depth perception was crucial to survival, hence our binocular vision from two eyes placed side by side in the front of the face. Color vision additional made it possible to identify the ripeness of fruit hanging in the trees. In other words, we are a visual species from way back, predating even our minds in their present form.
With this observation it becomes obvious that the human mind emerged and evolved under strongly visual selection pressure. Moreover, visual selection pressure means spatial selection pressure, so it is no wonder that the categories native to the human mind are intrinsically spatial. Those primates with the keenest ability to process spatial information in the form of visual stimuli would have had a differential survival and reproductive advantage. This is not accidental, but follows from our natural history.
But now I have mentioned “natural history” again, and I pause. Temporal selection pressure has been no less prevasive than spatial selection pressure. All life is a race against time to survive as long as possible while producing as many viable offspring as possible. Here we come back to Bergson again. Why does the intellect spatialize, when time is as pervasive and as inescapable as space in human experience?
With this question ringing in our ears, and the notable examples of philosophical logical-spatial equivalence mentioned above, why should we not have (parallel to Wittgenstein’s exposition of logical space) logical time and (parallel to Davidson’s exposition of logical geography) logical history?
To think through the idea of logical history is so foreign that is sounds strange even to say it: logical time? Logical history? These are not phrases with intuitive self-evidence. At least, they have very little intuitive self-evidence for the spatializing intellect. But in fact a re-formulation of Davidson’s logical geography in temporal-historical terms works quite well:
…the logical form of a sentence is to give its logical position in the elapsed sequence of sentences, to describe it in a way that explicitly determines what are following sentences it entails and what previous sentences it is entailed by…
Perhaps I ought to make the effort to think things through temporally in the same way that I have previously described how I make the effort to think things through selectively when I catch myself thinking in teleological terms.
In the meantime, it seems that our geometrical intuition is a faculty of mind refined by the same forces that have selected us for our remarkable physical performance. And as with our physical performance, which is rendered instinctive, second nature, and unconscious simply through our ordinary interaction with the world (all the things we must do anyway in order to survive), our geometrical intuition is often so subtle and so unconsciously sophisticated that we do not even notice it until we are presented with some Gordian knot that forces us to think explicitly in spatial terms. Faced with such a problem, we create sciences like topology, but before we have created such a science we already have an intellect strangely suited to the formulation of such a science. And, as I have written elsewhere, we have no science of time. We have science-like measurements of time, and time as a concept in scientific theories, but no scientific theory of time as such.
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Fractals and Geometrical Intuition
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6 November 2011
Third in a Series
In two previous posts in this series on theoretical geopolitics I have identified the following two principles:
●The Fundamental Theorem of Geopolitical Thought: Human agency is constrained by geography.
●The Second Law of Geopolitical Thought: The scope of human agency defines a center, beyond which lies a periphery in which human agency is marginal.
These first two principles of geopolitical thought as I have formulated them together yield centers of human agency (i.e., centers of power) at each level of metaphysical ecology. These forces have decomposed the world into geographical regions and ultimately yielded the geographically defined nation-state of the present age. The territories of hunter-gatherer bands, the city-states and empires of antiquity, the kingdoms of the middle ages, and the nation-states of today are all expressions of the geographical constraint of human agency.
The next step beyond geography is something I have already discussed at some length in The Second Law of Geopolitical Thought, and which I will now make explicit a in third principle:
●The Third Law of Geopolitical Thought: Human agency is essentially a temporal agency.
If the fundamental theorem of geopolitical thought tells us that geography matters, then the third law of geopolitical thought tells us that history matters.
As I noted above, I have already discussed the temporality of human agency in the context of ecological temporality in The Second Law of Geopolitical Thought, though when I formulated the second law I was primarily thinking in geo-spatial terms formulated in metaphysical ecology rather than in historico-temporal terms formulated in ecological temporality.
Human agency has both geographical (spatial) and historical (temporal) aspects, so that it would be sufficient simply to understand human agency in its full dimensions to appreciate this, but since the first principle of theoretical geopolitics, that human agency is constrained by geography, explicitly reminds us of the geographical dimension of human agency, it is appropriate that we should be similarly explicitly reminded of the historical dimension of human agency in a principle reserved for that purpose.
As I have observed on several occasions, I consider metaphysical ecology and metaphysical history to be alternative formulations of the same state of affairs, so that in the same spirit the third law of geopolitical thought is to be regarded as an alternative formulation of the Fundamental Theorem of Geopolitical Thought. I can make this alternative formulation more explicit by rephrasing the third principle as human agency is constrained by history. In this form the third principle of theoretical geopolitics closely approximates a famous line from Marx that I have quoted (with approval) many times:
“Men make their own history, but they do not make it as they please; they do not make it under self-selected circumstances, but under circumstances existing already, given and transmitted from the past.”
Karl Marx, The Eighteenth Brumaire of Louis Napoleon, first paragraph
Human agency is constrained both by geography and history, and these geographical and historical constraints define the scope of human agency, as invoked in the second principle of theoretical geopolitics, viz. The scope of human agency defines a center, beyond which lies a periphery in which human agency is marginal.
We can even substitute, salva veritate, the explicit invocation of geographical and historical constraints for the formulation in terms of human agency, so that the second principle of theoretical geopolitics reads like this: human agency constrained by geography and history defines a center, beyond which lies a periphery in which human agency is marginal (or non-existent).
The virtue of this latter formulation lies in the immediacy with which we can see that there are both geographical and historical centers and peripheries. In the simplest model of geopolitics, there would be only one center and one periphery. This center would be both a geographical and historical center, and all that lies outside that center would constitute the geographical and historical periphery.
One way to imagine this simplest model would be through a thought experiment: suppose that Western history consisted only of classical antiquity, and that the history of the ancient world was followed by no further achievements of Western civilization. In this case, the high point of the development of the Roman Empire would constitute both the geographical and historical center of Western history, and we could refine the geographical center to be the city of Rome, and the historical center to be, say, 180 AD, at which point Gibbon commenced his history. Outside these centers, all else is peripheral, and the farther from the center one moves, the more peripheral events become.
We don’t even have to do that thought experiment as a counter-factual exercise if we only confine our scope to classical antiquity. In other words, we can simply say that Rome in 180 AD was the geographical and historical center of classical antiquity. Non-Westerners reading this can perform similar thought experiments by substituting for Rome, say, the Persian Empire or the Chinese Empire or the Mogul Empire (though I suspect that many from India would not regard the Muslim Moguls as constituting the center of Indian history). Muslims might like to consider the Abbasid Dynasty as the historical and geographical center of pre-modern Islamic civilization. All of these identifications are obviously problematic, but all of them also, I think, have something to teach us in this context.
In a more complex and subtle model of geopolitics, we need to recognize that there are multiple centers and multiple peripheries and overlap and intersect (like Wittgensteinian family resemblances). We also need to recognize that geographical and historical centers can be offset, that is to say, the “center” of a people’s history may be distinct from the “center” of a people’s geography.
Our own history once again can supply us with examples of a more subtle account of theoretical geopolitics. Classical antiquity had numerous centers both in terms of history and geography: besides Rome there is of course Athens under Pericles, and on the far periphery of Rome there was Parthia under the Arsacids, on another periphery there were the Germans under Arminius, and also the later kingdoms of Egypt.
There is something a little artificial about making a separation between historical and geographical centers, because human agency is also spatio-temporal: that is to say, it consists of actions that take place in both space and time, and because human action is spatio-temporal centers of geography and history are usually aligned, even if they may be offset in some cases.
This is especially true of non-settled peoples. What was the center of Viking history? Viking voyages. The Vikings had their settlements in Norway and Iceland, and their trading outposts and even places they returned to time and again to rain and loot (like the British Isles), but the center of Viking civilization was in the act of voyaging, and voyaging is in equal parts spatial and temporal, geographical and historical. If we consider the nomadic Sami people of the far north of the Scandinavian peninsula, the center of their world is the annual migration of the reindeer. This is a recurring event, and so the historical center is very different from peoples who have abandoned this ancient hunter-gatherer modus vivendi. This suggests the possibility of recurring geographical and historical centers. This is an interesting idea that I will perhaps take up at another time.
Although human agency is constrained by both space and time, and integrates the two in spatio-temporal action, because of the particular properties of space and time, human agency is differently constrained by space than it is by time and vice versa. Time involves far more stringent constraints because we cannot move freely in time in the way that we can (ideally) move freely in space. Of course, we cannot even move absolutely freely in space, which is the whole point of geopolitics. We are even more tightly constrained by time.
Given that ideal freedom of movement in space is constrained by topography and the limitations of human agency, so that actual freedom of movement in a geographical context is far less than the ideal spatial freedom of geography, this is a lesson to us in regard to historical constraints. The “ideal” freedom of movement in time is nearly non-existent. We can, to a very limited extent distribute our activities in time, and we can chose when to begin and end certain activities, but most of time is beyond our control even if we were to appeal to Newton’s “Absolute, true and mathematical time,” which, “of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration.”
If the parallelism of space and time holds so far that there exists a parallelism also between the relations of pure space to actual space and pure time to actual time, then the recognition that real world spatial constraints are far more limiting than ideal spatial constraints suggests that real world temporal constraints are far more limiting that ideal temporal constraints. In this case, if the parallelism holds, history would be a far more rigorous constraint upon human agency than geography, in which case we ought to be thinking in terms of temporal politics instead of geopolitics.
This is an interesting idea that requires separate consideration. Perhaps it needs to be a separate theorem of theoretical geopolitics. I will have to think about this a little more. So, for the time being, I will let it rest there.
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19 March 2011
In fully industrialized societies space and time are commodities as is almost everything else. Space and time are bought and sold, traded on the open market. Moreover, the commodification of space and time have profound implications for human lives necessarily lived in space and time. It is often considered the height of dehumanization and depersonalization to assign a dollar value to human life (not to mention unspeakably vulgar), and while this is sometimes done for legal purposes, because of its inflammatory character our institutions often make an effort to avoid putting a price on life. However, there is little or no hesitation to putting a price on time and space, and this has the direct effect of regimenting human life according to monetary values.
For the vast majority of individuals living in industrialized societies, where you live is an economic decision rather than an existential decision. In short, you live where you can afford to live. Only for the wealthiest of the wealthy is there true choice of where to live, and for them the sky is the limit. Take a look at the Saturday edition of the Financial Times sometime and you will see the kind of places advertised for the kind of people who have a chalet in Switzerland, an apartment in New York, a flat in London, and a vacation house on Cyprus. It is money that buys access to these spaces; more money means more space in quantitative terms and more spaces in numerical, iterative terms.
Before the industrialization of Occidental civilization, where you lived was an existential choice — or, rather, an existential consequence of your circumstances, most of which are not chosen. Most often, it was an existential default position, but where you lived had much more to do with your identity than with your income. In fact, you probably wouldn’t have had any kind of income at all, in the formal economic sense, just as you probably would not have had a job in the formal economic sense. But you would have had a home, and this home would have been rooted in traditions that disappeared deep into the mists of time. During the same period, lending money at interest was forbidden, and as we all know, interest is the price of time.
Today, with rent as the price of space and interest as the price of time, the mortgage symbolizes the commodification of space and time (I wrote about this previously in Addendum on Non-Transient Spaces), and the commodification of space and time are the inevitable long term result of industrialization — at least, industrialization according to the particular path of development it took in Occidental civilization. It is entirely possible that industrial development might have followed a different paradigm, and of course Marxism represented a different paradigm of commodification under industrialization, though it was a vision fated to remain unrealized. Once the consequences of the industrialization of space and time are grasped in their fullness, it clarifies certain developments that otherwise seem inscrutable.
Many Americans hold in their imagination the image of ideal and idyllic small town life, and many are the social critiques that focus on the departure of present-day life in the US from the fictionalized vision of what life ought to be like (this is a particular case of the primitivist fallacy). One of the central features of this vision is the front porch. It has been pointed out that before the age of the automobile, the carriage house was hidden behind the main house. Now garages are often the most prominent feature of a house as seen from the street. And, of course, the relation of the house to the street is central.
So pervasive have the critiques of building based on the convenience of the automobile become that a great many experiments in designing residential areas that depart from the emergent order of the automobile have been developed. Several of these are literally within walking distance of my office in Beaverton. (For readers unfamiliar with the area, Beaverton is a western suburb of Portland, the largest city in Oregon. Nike’s world headquarters is right around the corner from my office, and Intel has a large presence — and a large payroll — in the area.) One can see the many forms of experimentation just by walking around. All the pictures for this post were taken by me, today, during a walk of an hour and twenty minutes’ duration.
Before the last recession, which hit housing particularly hard (not surprisingly, as the recession was precipitated by the crash of the sub-prime real estate lending market), there was plenty of money available for experimentation in the housing market. Especially in this area of Beaverton, any connection to the housing market prior to 2008 was virtually a license to print money — whether real estate speculator, developer, builder, or real estate agent, all made lots of money, and for a time it looked like the good times would continue to roll, so a lot of the profits were plowed right back into real estate developments.
Some of the late-comers to the party built projects that never took off and still sit idle and empty. There is a brick development on the corner of Baseline and 170th (not pictured here) that was another experiment in high-density mixed residential and commercial development (definitely part of the TOD paradigm criticized by the Cascade Policy Institution in their report The Mythical World of Transit Oriented Development) that I pass every day. It looks a bit sad, hosting a Subway sandwich shop, a minimart, and a nail salon, but most of it is empty, and several hopeful businesses that opened here early on left with their signs still in the window.
In New Wine in Old Bottles I quoted this from Jane Jacobs’ classic work The Life and Death of Great American Cities:
“As for really new ideas of any kind — no matter how ultimately profitable or otherwise successful some of them might prove to be — there is no leeway for such chancy trial, error and experimentation in the high-overhead economy of new construction. Old ideas can sometimes use new buildings. New ideas must use old buildings.”
It is ironic that in some cases this will occur because of bankruptcy. New buildings are built near the end of a construction boom, they either sit empty or the businesses that initially occupy them fail in the recession that follows the boom, and sometimes the owners of the structures themselves go bankrupt. Eventually the properties find their way on to the market again, but the whole process takes years, and it may be years before the economy begins generating new business opportunities and new businesses again. By this time the once-new buildings for old ideas have become old buildings possibly affordable by new ideas.
The multi-family common-wall houses in Steele Park have front porches. The single-family houses in The Village at Waterhouse (pictured above) have front porches also, and here they have even built an alley and put the garages behind the houses. But it doesn’t matter how many front porches you build. People aren’t sitting on their front porches, sipping lemonade and talking to their neighbors strolling past. I know that this sounds simple-minded to the point of absurdity, but one must understand the intuitive motivation behind the careful planning rhetoric. (The situation is strangely similar to politicians in the US who hold apocalyptic religious views but who know that talking about these explicitly in public would go over badly, so one is left to infer their views from otherwise inscrutable policy positions.)
People don’t sit on their front porches sipping lemonade because they are living where they are living for economic reasons, not existential reasons. If they want to pass the time pleasantly, they will drive to the place where they prefer to pass their time, but where it is impracticable to live for any of a number of reasons. Furthermore, the schedules of the families who live in these hopeful developments are also dictated by economics, so that the soccer moms are driving their kids to after-school activities wherever these are located, and these locations are again determined economically, not existentially.
What do you get when you build “affordable” housing? You get residents who can “afford” to live in “affordable” houses. Again, this is an economic decision; it has nothing to do with a home hallowed by ancestors and time immemorial, nor has it anything to do with finding your true center, finding your place in the world. An ancestral home or finding one’s true center are existential forms of housing, not economic forms of housing. And this is not to suggest any criticism of affordable housing. If anything, affordable housing is more in need now than ever before. Even following the real estate crash of the last recession, housing prices represent the largest single expenditure in a typical family’s budget.
Despite all the experimentation in housing, including unintentional experimentation, most of the recent developments I have seen do not seem to be accomplishing what they set out to accomplish. Unintended consequences are far more significant than intended consequences, and residents exapt the good intentions of architects and planners, living not according to the design and the plan, but rather according to the dictates and imperatives of life in industrialized society.
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28 January 2011
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.
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.
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
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31 October 2010
Over the past few days I’ve posted several strictly theoretical pieces that have touched on geometrical intuition and what I have elsewhere called thinking against the grain — A Question for Philosophically Inclined Mathematicians, Fractals and the Banach-Tarski Paradox, and A visceral feeling for epsilon zero.
Not long previously, in my post commemorating the passing of Benoît Mandelbrot, I discussed the rehabilitation of geometrical intuition in the wake of Mandelbrot’s work. The late nineteenth and early twentieth century work in the foundations of mathematics largely made the progress that it did by consciously forswearing geometrical intuition and seeking instead logically rigorous foundations that made no appeal to our ability to visualize or conceive particular spatial relationships. Mandelbrot said that, “The eye had been banished out of science. The eye had been excommunicated.” He was right, but the logically motivated foundationalists were right also: we are misled by geometrical intuition at least as often as we are led rightly by it.
Geometrical intuition, while it suffered during a period of relative neglect, was never entirely banished, never excommunicated to the extent of being beyond rehabilitation. Even Gödel, who formulated his paradoxical theorems employing the formal machinery of arithmetization, therefore deeply indebted to the implicit critique of geometrical intuition, wrote: “I only wanted to show that an innate Euclidean geometrical intuition which refers to reality and is a priori valid is logically possible and compatible with the existence of non-Euclidean geometry and with relativity theory.” (Collected Papers, Vol. III, p. 255) This is, of course, to damn geometrical intuition by way of faint praise, but being damned by faint praise is not the same as being condemned (or excommunicated). Geometrical intuition was down, but not out.
As Gödel observed, even non-Euclidean geometries are compatible with Euclidean geometrical intuition. When non-Euclidean geometries were first formulated by Bolyai, Lobachevski, and Riemann (I suppose I should mention Gauss too), they were interpreted as a death-blow to geometrical intuition, but it became apparent as these discoveries were integrated into the body of mathematical knowledge that what the non-Euclidean geometries had done was not to falsify geometrical intuition by way of counter-example, but to extend geometrical intuition through further (and unexpected) examples. The development of mathematics here exhibits not Aristotelian logic but Hegelian dialectical logic: Euclidean geometry was the thesis, non-Euclidean geometry was the antithesis, and contemporary geometry, incorporating all of these discoveries, is the synthesis.
Bertrand Russell, who was central in the philosophical struggle to find rigorous logical formulations for mathematical theories that had previously rested on geometrical intuition, wrote: “A logical theory may be tested by its capacity for dealing with puzzles, and it is a wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science.” (from the famous “On Denoting” paper) Though Russell thought of this as a test of logical theories, it is also a wholesome plan to stock the mind with counter-intuitive geometrical examples. Non-Euclidean geometry greatly contributed to the expansion and extrapolation of geometrical intuition by providing novel examples toward which intuition can expand.
In the interest of offering exercises and examples for geometrical intuition, In Fractals and the Banach-Tarski Paradox I suggested the construction of a fractal by raising a cube on each side of a cube. I realized that if instead of raising a cube we sink a cube inside, it would make for an interesting pattern. With a cube of the length of 3, six cubes indented into this cube, each of length 1, would meet the other interior cubes at a single line.
If we continue this iteration the smaller cubes inside (in the same proportion) would continue to meet along a single line. Iterated to infinity, I suspect that this would look interesting. I’m sure it’s already been done, but I don’t know the literature well enough to cite its previous incarnations.
The two dimensional version of this fractal looks like a square version of the well-known Sierpinski triangle, and the pattern of fractal division is quite similar.
One particularly interesting counter-intuitive curiosity is the ability to construct a figure of infinite length starting with an area of finite volume. If we take a finite square, cut it in half, and put the halves end-to-end, and then cut one of the halves again, and again put them end-to-end, and iterate this process to infinity (as with a fractal construction, though this is not a fractal), we take the original finite volume and stretch it out to an infinite length.
With a little cleverness we can make this infinite line constructed from a finite volume extend infinitely in both directions by cutting up the square and distributing it differently. Notice that, with these constructions, the area remains exactly the same, unlike Banach-Tarski constructions in which additional space is “extracted” from a mathematical continuum (which could be of any dimension).
Thinking of these above two constructions, it occurred to me that we might construct an interesting fractal from the second infinite line of finite area. This is unusual, because fractals usually aren’t constructed from rearranging areas in quite this way, but it is doable. We could take the middle third of each segment, cut it into three pieces, and assemble a “U” shaped construction in the middle of the segment. This process can be iterated with every segment, and the result would be a line that is infinite two over: it would be infinite in extent, and it would be infinite between any two arbitrary points. This constitutes another sense in which we might construct an infinite fractal.
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Fractals and Geometrical Intuition
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29 October 2010
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.
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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Fractals and Geometrical Intuition
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