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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 being 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.

The Banach-Tarski paradox is paradoxical at least in part because it does not seem to, “carve nature at the joints.” This violation of our geometrical intuition comes about as a result of the development of other intuitions, and it is ultimately the clash of intuitions that is paradoxical. Kant famously maintained that there can be no conflict among moral duties; parallel to this, it might be taken as a postulate of natural reason that there can be no conflict among intellectual intuitions. While this principle has not be explicitly formulated to my knowledge, it is an assumption pervasively present in our reasoning (that is to say, it is an intuition about our intiutions). Paradoxes as telling as the Banach-Tarski paradox (or, for that matter, most of the results of set theory) remind us of the limitations of our intuitions in addition to reminding us of the limitations of our geometrical intuition.

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Fractals and Geometrical Intuition

1. Benoît Mandelbrot, R.I.P.

2. A Question for Philosophically Inclined Mathematicians

3. Fractals and the Banach-Tarski Paradox

4. A visceral feeling for epsilon zero

5. Adventures in Geometrical Intuition

6. A Note on Fractals and Banach-Tarski Extraction

7. Geometrical Intuition and Epistemic Space

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Epistemic Space

19 February 2009


Wittgenstein, on the left, wrote one of the masterpieces of twentieth century philosophy, the Tractatus Logico-Philosophicus

Wittgenstein, on the left, wrote one of the masterpieces of twentieth century philosophy, the Tractatus Logico-Philosophicus

Ludwig Wittgenstein, in his Tractatus Logico-Philosophicus, introduced the concept of logical space. This does not play a large role in the Tractatus, but a few other philosophers have found it to be of interest and have fleshed out the concept. Donald Davidson formulated an analogous conception of 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.” (Essays on Actions and Events, “Criticism, Comment, and Defence”, p. 140)

wittgenstein_davidson

Ultimately, all space is logical space, and all geography is logical geography, or, rather, these categories of logical space and logical geography are the most formal and abstract formulations of space as it is conceived by the intellect as an ideal form of order. Wittgenstein and Davidson present to us the most idealized and refined formulations of concepts that we employ daily in our ordinary lives in a less refined and less ideal form. But if we are to come to a theoretical understanding of space, we must master the abstract and formal conceptions. Geopolitics is ultimately incomprehensible without logical geography.

Foucault sought to map the spaces of knowledge, and called his chair that of "History of systems of thought"

Foucault sought to map the spaces of knowledge, and called his chair that of "History of systems of thought"

Our knowledge is laid out in epistemic space, so that our epistēmē (as Foucault called it; ἐπιστήμη in the original Greek) governs not only how we see and understand the world, but also how we move through it and how we construct our lives within the world, for the world is a world in space defined epistemically, that is to say, defined in terms of our knowledge.

The Tabula Peutingeriana

The Tabula Peutingeriana: this is a lesson in how differently ancient and modern peoples see (and construct) the space in which they live.

On 27 November 2007, in celebration of its inclusion in UNESCO’s Memory of the World Register, the Tabula Peutingeriana was displayed in Vienna. I should have liked to have seen this. It is a medieval copy of an ancient Roman road map. The copy is quite large, like many medieval maps, though quite long and thin, about seven meters by thirty-four centimeters. The Mediterranean Sea is stretched out like a river in this elongated space. The original is thought to date from some time in the fifth century AD.

The map mosaic at Madaba, Jordan.

The map mosaic at Madaba, Jordan.

One of the few maps to date from antiquity is perhaps a hundred or so years later than the original for the Tabula Peutingeriana, and this is the map mosaic at Mādabā, Jordan. While damaged, it survives in part because of the robust character of mosaics. Colored stone and glass set in concrete survives the centuries much better than parchment or papyrus. This map mosaic, like the Tabula Peutingeriana, and indeed as with all maps, there is a surprising combination of practical detail and ideological schematism. A map is a practice of political ideology.

tp-roma

A detail from the Tabula Peutingeriana, showing the city of Rome as an Emperor, with globe and sceptre, seated on a throne.

As strange as the Tabula Peutingeriana looks to modern eyes, stranger still is the map of the world by Mahmud al-Kashgari. The Tabula Peutingeriana seems stretched and distorted, but it is still recognizably a map. The map of al-Kashgari might not be recognized as a map by the modern, western eye. Its schematism of a circle within a square contrasts with the schematism of “T in O” maps mentioned below, but perhaps as intriguingly, mirror the structure of Hagia Sophia, the great church built under the rule of Justinian, but which became the model of mosques the world over after Constantinople was taken by the Turks.

A map of the world by Mahmud al-Kashgari from his Diwan Lugat at-Turk, believed to date from 1072 AD.

A map of the world by Mahmud al-Kashgari from his Diwan Lugat at-Turk, believed to date from 1072 AD.

As with most maps of late antiquity and later, Jerusalem is shown at the center of the world in the Mādabā map, a Christian-era map, whereas the focus of the Roman map was Rome itself, represented by a crowned man sitting on a throne (on the far right of the larger section of the Tabula Peutingeriana pictured above, and shown in detail immediately above). And, as we know, all roads lead to Rome. The Via Appia Antiqua is shown radiating from Rome at about 4 o’clock. That the Roman map was a road map is a sign of the role that communications networks played in Roman administration.

A medieval "T" map, also called a "T-O" map or a "T and O" map or a "T in O" map.

A medieval "T" map, also called a "T-O" map or a "T and O" map or a "T in O" map.

Even more schematic, and nearly devoid of practical detail, is the medieval “T in O” map: the very name describes its structure. In many of these maps Jerusalem in prominently in the center with Asia on top, Europe to the lower left of the “T” and Africa to the lower right of the “T”. Such a construction of the world is purely about expressing the relation of the major divisions of the world to its center, positioning the human world within the divine cosmos — marking one’s place within the totality, to borrow a term of the Davidson quote above.

The Thomas Digges chart of a Copernican solar system from 1576.

The Thomas Digges chart of a Copernican solar system from 1576.

Even as maps became more scientifically sophisticated after the scientific revolution, they remain highly schematic and their purpose is often to show the interrelation of major epistemic divisions so that man can know his place in the world. The Thomas Digges Copernican solar system (shown above) is more sophisticated than a medieval “T in O” map, but similarly schematic in conception. A map orders the world for us, and in so ordering our world, orders our lives.

Scientific realism has produced its own abstract and schematic maps, not unlike the dramatic poster art of so-called socialist realism.

Scientific realism has produced its own abstract and schematic maps, with striking color contrasts and bold graphic motifs not unlike the dramatic poster art of so-called socialist realism.

Some of the most advanced scientific maps of our time continue to be as schematic as maps of the past, highly specialized depictions of the state of our knowledge, and such that can only be meaningfully interpreted and understood by an adept of the culture so formulated. One of the most famous scientific images of our time is that of the cosmic microwave background radiation, which showed very subtle differences in the background radiation. This slight departure from a purely homogeneous background radiation is the oldest evidence we have of the natural history of the universe. Here time is shown unfolded across deep space, mapped, as it were. The order mapped in space overflows into an order in time.

Maps of the Roman Empire are inherently problematic, as almost all have been formulated according to a paradigm derived from the territorial nation-state.

Maps of the Roman Empire are inherently problematic, as almost all have been formulated according to a paradigm derived from the territorial nation-state: at the heart of the empire is not a particular territory, but the Mediterranean Sea.

The maps we draw of the migration and distribution of species, with their long, sinuous lines demarcating broad swathes of territory, are redolent of the maps historians attempt to draw for past political entities, with their long, curving lines across deserts, steppe, and forest where the territorial sovereignty of any political entity would be questionable, especially before the age of the territorially defined nation-state.

Attempts to map bird populations and migrations always remind me of attempts to map pre-nation-state political entities: both the world of the past and the avian world are alien to us.

Attempts to map bird populations and migrations always remind me of attempts to map pre-nation-state political entities: both the world of the past and the avian world are alien to us.

A map represents a special kind of knowledge, and indeed a special approach to knowledge — a bird’s eye view of knowledge in which epistemic space is plotted out in a scheme that is both abstract and synthetic, at once intuitive and non-constructive. We are all familiar with the saying that, “The map is not the territory” (credited to Alfred Korzybski), which emphasizes the abstract and schematic character of maps. Like Magritte’s picture of a pipe, which is self-evidently not a pipe and yet recognizably a pipe, a map represents, and as a representation it assumes and presupposes certain principles of representation. Maps, thus, are texts inscribed in a symbolic language.

process-flowchart

In our bureaucratized industrial society, we live by flow charts, which are transparently maps of epistemic space. In this way we see at a glance our life mapped out, the paths we will take, the choices we must make, and even the choices that lead to other choices leave us within the well-worn schema of life reduced to an algorithm.

Medieval maps were often highly "realistic" in their use of projection and perspective, yet highly "realistic" in the one-to-one correspondence observed between objects and their representation.

Medieval maps were often highly "unrealistic" in their use of projection and perspective, yet highly "realistic" in the one-to-one correspondence observed between objects and their representation.

Today one commonly hears others say “It’s just semantics” as though semantics don’t matter, and one could equally well imagine someone saying, in the same dismissive vein, “It’s just syntactics” as though it doesn’t matter what language you happen to be speaking. But it does matter. A perspicuous symbolism can be the difference between getting your meaning across or failing to do so. In Principia Mathematica Russell and Whitehead wrote, “The terseness of the symbolism enables a whole proposition to be represented to the eyesight as one whole, or at most in two or three parts divided where the natural breaks, represented in the symbolism, occur.” (Whitehead and Russell, Principia Mathematica to *56, London and New York: Cambridge University Press, 1978, p. 2) And this was the same sort of thing that Wittgenstein was trying to do in his Tractatus, and in doing so found himself explicitly formulating a doctrine of logical space.

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The Forma Urbis Romae was a map of ancient Rome carved into marble slabs and on display in the Templum Pacis.

The Forma Urbis Romae was a map of ancient Rome carved into marble slabs, affixed to a wall, and (in antiquity) permanently on display in the Templum Pacis.

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