Geometrical Intuition and Epistemic Space

Friday

---

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.

. . . . .

. . . . .

. . . . .

Addendum on the Origins of Time

Thursday

---

Yesterday’s longish post The Origins of Time occupied me for quite some time. Parts of it appeared in fragmentary form on my Tumblr blog, Grand Strategy Annex, in the posts on The Experience of Innocence and Innocence and Time Consciousness. I also made notes and occasional sketches in my notebooks as I was working on these ideas.

Below is one sketch that I made last summer in order to try to sum up the idea of the construction of ecological temporality in a way that would appeal to geometrical intuition.

While this drawing is too schematic and too simple to be quite true, it nevertheless has a certain value, as all abstractions have a certain value. And that’s what this is: a sketch of an abstraction.

This is an attempt to delineate the large scale structures of space and time from the standpoint not of physics or cosmology (which is how we are accustomed to seeing exposition of the large scale structure of space and time) but from a philosophical perspective. What I was trying to show with this image was how time has its origins in micro-temporal interactions, and is predominately a temporality of micro-temporality until larger structures emerge along with the larger temporal structures entailed by these larger structures. As larger structures emerge, micro-temporality becomes less central to the way the world works, and the less comprehensive forms of temporality fall away as the center of cosmological history migrates to the larger structures.

In my closing speculation of yesterday’s The Origin of Time I suggested that the ultimate telos of civilization is for humanistic temporality and cosmological temporality to merge, and if this should come to pass, it would come to pass at the farthest reach of metaphysical temporality. This is …

I have also incorporated in the drawing above what should have been obvious to me earlier, which is to abbreviate metaphysical temporality as meta-temporality (the same thing can be done with metaphysical ecology rendered as meta-ecology). The abbreviation of “metaphyscial” to “meta-” is then readily assimilated to the familiar ecological levels of mirco-, meso-, exo-, and macro-, to which we now add meta-.

An interesting lesson to take from the relation of this drawing to my ideas about ecological temporality and the origins of time is that an image can express an abstraction as readily as can words, though we do not ordinarily think of pictures, sketches, videos, illustrations, and so forth as abstractions. Indeed, we typically think of images as giving concrete embodiment to an idea that was difficult to grasp on the basis of a text alone. But this is not so. Illustrations are not easy to understand because of their concreteness; illustrations are easy to understand because of the role of geometrical intuition in human thought.

Vision plays a disproportionate role in human knowledge. We know that, for other species, the relative contribution of the senses constitutes a different mix in each case. For dogs, smell plays a very large role; for bats and dolphins, hearing plays a disproportionate role; perhaps eagles are in a similar boat with us, relying as they do on particularly keen eyesight to detect prey on the ground from flying altitude.

We don’t even have electro-receptors like a shark or pits like a pit viper, so we can’t know what it is like to be a shark or a viper (to borrow a phrase from the famous Thomas Nagel essay, What is it like to be a bat?). Since we have ears and noses we can at least make a guess as to what it is like to live a life in which these senses play a disproportionate role in experience.

While we can augment our senses with instrumentation, we are more or less stuck with the cognitive architecture that evolved under selection pressures directly bearing upon those senses crucial to our physical survival and reproduction. Because the ancestors of human beings took the path of relying on our vision — probably binocular stereoscopic vision for swinging through the branches of trees and color vision for distinguishing the ripeness of fruit — we have a cognitive architecture that is heavily integrated with visual processing power.

So, we have the minds we have, and while we have learned to help them along a bit with languages and ideas, the apple doesn’t fall far from the tree. I take it that this is one reason that Wittgenstein said Nothing contrasts with the form of the world.

The form of our world is a visual world, and in a visual world geometrical intuition counts for a lot. And since geometrical intuition counts for a lot, geometrical abstractions — i.e., images that illustrate abstractions — also count for a lot.

. . . . .

. . . . .

. . . . .

The Third Law of Geopolitical Thought

Sunday

---

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.

The simplest model of center and 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.

A more realistic model of multiple centers and overlapping peripheries. Beyond this spatial model, one ought also to imagine multiple centers and overlapping peripheries in time.

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.

. . . . .

. . . . .

. . . . .

Industrialized Space and Time

Saturday

---

A typical house of the Automobile Age, the most prominent feature of which is its large, protruding garage with direct access to the street. One can go from the climate-controlled comfort of one's vehicle to the climate-controlled comfort of one's home without ever being exposed to the conditions of the neighborhood.

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.

Steele Park, an early example of Transit Oriented Development (TOD), built close to the Elmonica Max station (less than five minutes' walk).

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.

A Max train at the Elmonica Max station. Portland has a better mass transit system than many US cities, and the Max light rail system is the centerpiece of this development.

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.

Steele Park as seen from across Baseline Road. Despite high hopes for this early attempt at development centered on mass transit, Steele Park is not an attractive place.

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.

Immediately across from Steele Park, on the south side of Baseline Road, and even closer to the Elmonica Max station, this development is far more attractive than Steele Park. It is, of course, incredibly fake and phony, but where are fake and phony people to live if not in fake and phony housing developments?

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.

Just down Baseline and around the corner from the above pictured developments, the Baseline Station Condominiums show a little more imagination and are neither quite as grim as Steele Park nor quite as phony as the Elmonica Station development, though they represent yet another experiment in transit oriented development.

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.

TOD aims to encourage residents to use their cars less, to walk more, and to increase housing density in the vicinity of readily available mass transit opportunities. Suburban areas around US cities traditionally had low, single-story development, typically single-family homes, so that to go up to two or three stories is a novelty.

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.

The Village at Waterhouse is another experiment that seeks to recover the social values of American society that were swept away by industrialization.

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 170^th (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.

Every house at The Village at Waterhouse has a front porch, but you don't see anyone sitting on the front porch sipping lemonade. It is a nice vision, but it takes more than a particular form of building to make it happen.

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.

This is the alley behind the houses at The Village at Waterhouse, so that cars would access houses from behind and garages would not protrude from the front of the house and dominate the street.

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.)

At The Village at Waterhouse they even have a little “village green” in the center, presumably where the local militia would turn out to practice drill.

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.

The default development paradigm where experimentation has not been consciously pursued has converged upon large houses closely spaced on small lots, with very little room between them. This follows logically from the cost of space and the desire to maximize personal living space within these parameters.

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.

Here's another conscious attempt at experimentation along 173rd, with cars parking behind and doors that open directly out onto the sidewalk. No one ever uses these doors, defeating the purpose of building them.

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.

. . . . .

It rained on my walk today, and I got wet. As you can see, I forgot my hat.

. . . . .

. . . . .

A Note on Fractals and Banach-Tarski Extraction

Friday

---

Stephan Banach and Alfred Tarski

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

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

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

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

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

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

. . . . .

. . . . .

. . . . .

Fractals and the Banach-Tarski Paradox

Friday

---

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

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

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

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

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

Hilbert's version of a space filling curve.

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

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

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

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

. . . . .

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

. . . . .

. . . . .

. . . . .

A Question for Philosophically Inclined Mathematicians

Friday

---

Given the astonishing yet demonstrable consequence of the Banach-Tarski paradox, it is the sort of thing that one’s mind returns to on a regular basis in order to savor the intellectual satisfaction of it. The unnamed author of the Layman’s Guide to the Banach-Tarski Paradox explains the paradox thus:

The paradox states that it is possible to take a solid sphere (a “ball”), cut it up into a finite number of pieces, rearrange them using only rotations and translations, and re-assemble them into two identical copies of the original sphere. In other words, you’ve doubled the volume of the original sphere.

The whole of the entry at Wolfram Mathworld runs as follows:

First stated in 1924, the Banach-Tarski paradox states that it is possible to decompose a ball into six pieces which can be reassembled by rigid motions to form two balls of the same size as the original. The number of pieces was subsequently reduced to five by Robinson (1947), although the pieces are extremely complicated. (Five pieces are minimal, although four pieces are sufficient as long as the single point at the center is neglected.) A generalization of this theorem is that any two bodies in R³ that do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other (i.e., they are equidecomposable).

The above-mentioned Layman’s Guide to the Banach-Tarski Paradox attempts to provide an intuitive gloss on this surprising result of set theory (making use of the axiom of choice, or some equivalent assumption), and concludes with this revealing comment:

In fact, if you think about it, this is not any stranger than how we managed to duplicate the set of all integers, by splitting it up into two halves, and renaming the members in each half so they each become identical to the original set again. It is only logical that we can continually extract more volume out of an infinitely dense, mathematical sphere S.

Before I read this today, I’d never come across such a clear and concise exposition of the Banach-Tarski paradox, and in provides food for thought. Can we pursue this extraction of volume in something like a process of transfinite recursion, arriving at some geometrical equivalent of ε₀? This is an interesting question, but it isn’t the question that I started out thinking about as suitable for the philosophically inclined mathematician.

When I was thinking about the Banach-Tarski paradox today, I began wondering if a sufficiently generalized formulation of the paradox could be applied to ontology on the whole, so that we might demonstrate (perhaps not with the rigor of mathematics, but as best as anything can be demonstrated in ontology) that the world entire might be decomposed into a finite number of pieces and then reassembled into two or more identical worlds.

With the intuitive gloss quoted above, we can say that this is a possibility in so far as the world is ontologically infinitely dense. What might this mean? What would it be for the world to be ontologically dense in the way that infinite sets are infinitely dense? Well, this kind of question goes far beyond intuition, and therefore lands us in the open-texture of language that can accommodate novel uses but which has no “natural” meaning one way or the other. The open-texture of even our formal languages makes it like a quicksand: if you don’t have some kind of solid connection to solid ground, you are likely to flail away until you go under. It is precisely for this reason that Kant sought a critique of reason, so that reason would not go beyond its proper bounds, which are (as Strawson put it) the bounds of sense.

But as wary as we should be of unprecedented usages, we should also welcome them as opportunities to transcend intuitions ultimately rooted in the very soil from which we sprang. I have on many occasions in this forum argued that our ideas are ultimately derived from the landscape in which we live, by way of the way of life that is imposed upon us by the landscape. But we are not limited to that which our origins bequeathed to us. We have the power to transcend our mundane origins, and if it comes at the cost of occasional confusion and disorientation, so be it.

So I suggest that while there is no “right” answer to whether the world can be considered ontologically infinitely dense, we can give an answer to the question, and we can in fact make a rational and coherent case for our answer if only we will force ourselves to make the effort of thinking unfamiliar thoughts — always a salutary intellectual exercise.

Is the world, then, ontologically infinitely dense? Is the world everywhere continuous, so that it is truly describable by a classical theory like general relativity? Or is the world ultimately grainy, so that it must be described by a non-classical theory like quantum mechanics? At an even more abstract level, can the beings of the world be said to have any density if we do not restrict beings to spatio-temporal beings, so that our ontology is sufficiently general to embrace both the spatio-temporal and the non-spatio-temporal? This is again, as discussed above, a matter of establishing a rationally defensible convention.

I have no answer to this question at present. One ought not to expect ontological mysteries to yield themselves to a few minutes of casual thought. I will return to this, and think about it again. Someday — not likely someday soon, but someday nonetheless — I may hit upon a way of thinking about the problem that does justice to the question of the infinite density of beings in the world.

I do not think that this is quite as outlandish as it sounds. Two of the most common idioms one finds in contemporary analytical philosophy, when such philosophers choose not to speak in a technical idiom, are those of “the furniture of the universe” and of “Carving nature at its joints.” These are both wonderfully expressive phrases, and moreover they seem to point to a conception of the world as essentially discrete. In other words, they suggest an ultimate ontological discontinuity. If this could be followed up rigorously, we could answer the above question in the negative, but the very fact that we might possibly answer the question in the negative says two important things, 1) that the question can, at least in some ways, be meaningful, and therefore as philosophically significant and worthy of our attention, and 2) if a question can possibly be answered the negative, it is likely that a reasonably coherent case could also be made for answering the question in the affirmative.

. . . . .

. . . . .