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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

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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