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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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Tuesday


Henri-Louis Bergson, 18 October 1859 to 04 January 1941, philosopher and time and duration.

Henri-Louis Bergson, 18 October 1859 to 04 January 1941, philosopher of Dionysian time and duration.

In the early twentieth century Henri Bergson was a name to conjure with. He was an intellectual celebrity not unlike, say, Foucault before his death: both men could pack a hall with excited Parisians eager to hear the intellectual developments of the most advanced mind of France. Bergson was a man of sharp, angular features, a large bulbous forehead, and deeply set eyes, the overall effect of which reminds one not a little of Count Orlock played by Max Schreck in Murnau’s Nosferatu.

Nosferatu was the bête noire of all that belongs to the light of day, and he himself belongs to Chaos and Old Night.

Nosferatu was the bête noire of all that belongs to the light of day, and he himself belongs to Chaos and Old Night.

As a philosopher who made crowds swoon, he inevitably attracted the enmity of other philosophers, Bertrand Russell especially, for whom Bergson was his bête noire. I can imagine that Russell might have chuckled at the idea of Bergson as Count Orlock. But for Russell the chuckle would have been mixed with a sense of disquiet because Bergson represented to him much that Murnau’s symphony of horror represented to its audience: the irruption of the irrational within an ordered world, the rejection of reason in favor of Dionysian indulgence, the mind subordinated to natural forces in their most horrific appearance (not unlike Pentheus in The Bacchae). For Russell, Bergson represented the forces of Chaos and Old Night let loose upon the world (in an intellectualized form), just as an early cinema-goer might have seen the story of Nosferatu as Chaos and Old Night let loose upon the world (in a dramatically cinematic form).

The young Bertrand Russell rarely passed up an opportunity to criticize Bergson.

The Apollonian young Bertrand Russell rarely passed up an opportunity to criticize Bergson.

Bergson is no longer a name with which to conjure, but when he is remembered, one of the themes for which he is remembered is that of the spatialization of time. For Bergson, intellectual activity cannot reconcile itself to time as it is actually experienced, so that it must create surrogates for time, and it does so, according to Bergson, by assimilating time to space. The mind creates images and representations of time that employ the constructions of geometry. So it is that we represent the continuity of historical time by a line cut by dates. This manner of representing history is so common we never think twice about it.

A time line of events in the life of Henry VIII.

A spatialized and schematized time line of events in the life of Henry VIII.

Are we forced to choose between Russell and Bergson? Both have valid points to make. While I am sympathetic to Russell’s rationalism, I think that Bergson had a point in his critique of spatialization, but Bergson did not go far enough with this idea. Not only has there been a spatialization of time, there has also been a temporalization of space. We see this in the contemporary world in the prevalence of what I call transient spaces: spaced designed to pass through but not spaces in which to abide. Airports, laundromats, bus stations, and sidewalks are all transient spaces. The social consequences of industrialization that have forced us to abide by the regime of the calendar and the time clock by the very fact of quantifying time into discrete regions and apportioning them according to a schedule also forces us to wait. The waiting room ought to be recognized as one of the central symbols of our age; the waiting room is par excellence the temporalization of space.

waiting room

The modeling of real world phenomena by quantifiable means — be these phenomena spatial or temporal — involves at least two known unknowns: finite precision errors and finite dimensional errors. The former (finite precision errors) occur when decimal expansions are arbitrarily cut off at, say, six decimal places or eight decimal places or whatever the model demands. Our finite computing systems cannot calculate real numbers with infinite decimal expansions, so they must be terminated at some point or we cannot even begin our attempt at modeling. The latter (finite dimension errors) occur when a continuum of possibilities must be broken down into discrete units. A rainbow is a continuous gradation of color, but for the sake of our conceptual schematism we break it down into red, orange, yellow, green, blue, indigo, and violet (some recitations leave out the indigo). The familiar rainbow color schematism involves finite dimensional errors.

Some familiar artefacts of life lend themselves to geometrical exposition, so that their spatialization contributes to our understanding of them.

Some familiar artefacts of life lend themselves to geometrical exposition, so that their spatialization contributes to our understanding of them.

Ordinary experience is overwhelmingly continuous and only occasionally quantized. The application of flow charts that map the epistemic spaces of our lives to matters of experience involve countless finite dimensional errors. We accept these errors as the price of systematically extrapolating our knowledge, but we ought always to employ such extrapolations with caution. Just as the map is not the territory, so too the extrapolation is not the knowledge itself. Every flow chart and every algorithm embody and reify finite dimension errors. Since it is in their nature to do so, we do not think of these as errors, but should we confuse the map of life with the territory of life we would be compromised.

Instructions: an algorithm for the safe use of a chain saw. Can all aspects of life be similarly and as successfully schematized?

Instructions: an algorithm for the safe use of a chain saw. Can all aspects of life be similarly and as successfully schematized?

Conceptual schematisms are routines of the mind, and we all know how easily we slip into routines. A routine, whether a habit of body or of mind, is an algorithm for life, a finite decision procedure by which those individuals who would otherwise be without purpose determine their course of action and thus manage to fill the vacant hours of the clock. It is a recipe for life, to be sure, but it is not a recipe for anything other than mediocrity in life, and perhaps a guarantee of it.

A recipe is an algorithm for the production of food stuffs, a finite sequence of instructions intended to secure a predictable result.

A recipe is an algorithm for the production of food stuffs, a finite sequence of instructions intended to secure a predictable result.

The mapping of time as an epistemic space, as in a flow chart, is not without consequences. A distinctive feature of algorithms is their finitude. The mapping of life’s paths with discrete, finite alternatives limits options to a few pre-determined alternatives. Any individual of ordinary critical capacity, capable thinking for themselves, will quickly reject any such attempt to limit their options in life, but many among us are unable to see beyond the roles embodied in society. Sartre called this the spirit of seriousness. Finite dimensional errors represent the spirit of seriousness necessary to the practice of science.

There are a number of humorous twitter algorithms floating around the internet at present, but behind the humor is the implied tension of reducing a human activity to a rule.

There are a number of humorous twitter algorithms floating around the internet at present, but behind the humor is the implied tension of reducing a human activity to a rule.

If, as I have argued elsewhere, freedom is a form of infinity, subordinating our lives to a finite, algorithmic regime not only results in an inauthentic life, it robs us of the freedom that makes us human. Without our freedom, we become automatons. And there seems to be an intuitive understanding of the danger that industrialized society poses in terms of regimenting life to the point of transforming life into a hollow, mechanistic exercise. We discussed this at some length in Fear of the Future.

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A Theory of Maps

26 February 2009


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From Epistemic Space to Abject Technology

Recently in Epistemic Space I wrote that, “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.” Not long after posting that I came to realize that the special kind of knowledge represented by maps is a function of maps lying at the intersection of hardware technologies and social technologies.

What do I mean when I say that a map lies at the intersection of hardware technologies and social technologies? I mean that a map is a little bit of both, and not precisely one or the other. In other words, a map is abject technology. A map is both a fact and an artifact. People use a map as a tool in order to find their way; they also use a map as a basis for constructing and extending their knowledge base, and to organize space and time. And for a group of elites to organize space and time is no abstract exercise. It involves who remains “free”, who is in prison, who can travel where, where you can build what kind of structure, where you can camp, where you can ride a motorcycle and whether or not you have to wear a helmet. And in so far as we cooperate with these principles of organization, we realize in fact a social technology conceived in the minds of political elites.

Ontogenic Mapping Conventions

Maps are more than colored outlines of nation-states and street indices. Flow charts, also mentioned in Epistemic Space, are maps of time and of process. Clocks, calenders, and schedules are also maps of time. Organizational charts are maps of the structure of a given organization. Politicians speak of a “roadmap” when then refer to any sequence of steps intended to achieve certain results, i.e., an algorithm.

A map or a schedule or a calendar (all of which below we will simply call maps) is what philosophers call a convention. For philosophers, a convention is not a bunch of drunk doctors at a Holiday Inn, it is an agreement among a number of people to do things in the same way. It carries the connotation that doing things in a given way may not represent any great truth (although, then again, it may), and it may not reflect the natural order of things (again, it also may do so), but it may nevertheless be an orderly and systematic way to go about things.

Maps, understood in the broadest sense, create facts. A map can become a model by which we construct our lives, governing when and how we move, and how we distribute our time. In order to describe conventions such as maps that create, or contribute to the creation of, facts I would like to co-opt a term usually used in biology : ontogenic, the adjectival form of ontogeny (The origin and development of an individual organism from embryo to adult; also called ontogenesis). That is to say, maps are ontogenic; maps generate facts; maps create truths. There is even a sense in which we can say that a map projects facts and truths.

Degrees of Ontogenesis

We can distinguish between levels of human participation in the constitution of facts. Some facts are more given and less created, others more created and less given. I am skeptical that we have access to the purely created or the purely given. Some facts which we participate in creating are nevertheless opaque, but I don’t think that this is always the case. Among created facts are airline and train schedules, or maps of all kinds, for example, and though they may often appear unreasonable to those of us who don’t like getting up at 5 am to catch a plane, for those who master the logistics behind transportation even the details can be made transparent. Certainly the facts of transportation schedules, which we play a large role in creating, are more transparent than the facts of the weather, which we play a very small role in creating.

It would be awkward to call a bus schedule a “fact,” although it is closely connected with many facts. A schedule is, rather, a convention — a map in time — and facts follow from a decision to order things in a certain way. Some of the facts which follow from conventions actually follow logically: if a subway is to start at 6 in the morning and stop at each station every five minutes, then we can conclude that it will stop at each station at 6:05, 6:10, etc. However, many of the facts involve intensional contexts, i.e., they involve further human agency. A schedule cannot guarantee that the subway will stop at 6:05, as the subway employees may not care about being on time. In this case it follows logically that if the subway has not arrived by 6:05 it is late, but little else follows logically. This additional human involvement makes conventions continually subject to human agency and variable in light of the possibility of interpretation.

As for Time, so for Space

All that we have observed in regard to time and the establishment of temporal conventions in schedules applies, mutatis mutandis, to space and the establishment of spatial conventions in maps. There are levels of human participation in maps, and the greater our level of participation, the greater the ontogenic role of the map. We have rather less control over a topographical chart, and rather more control over a street map. We can physically change the facts on the ground that the street map is supposed to depict. We can also physically change geographical features, though we are perhaps less likely to do so. But we may be motivated, in some cases, to call a stream a seasonal trickle or to call a seasonal trickle a class-one stream if there are legal reasons for doing so (which, in logging country, may govern whether or not you can log and how much of a buffer you will need to leave on each side of the stream to protect riparian habitat).

The existence of ontogenic technologies such as maps calls into question the simplistic division of philosophical theories of the world into the constructive and the non-constructive, the anti-realist and the realist, conventionalism and Platonism, because the recognition of a category of the ontogenic is a recognition of things that are partially made and partially given, but not wholly made or wholly given. Given that our world is as it is, that it is a compromise through and through, and that little in it comfortably falls under a clearcut category (except the category of the abject), we ought to openly recognize this in our conceptual scheme. There are stubborn facts that are not social constructions, and there are social constructions that cannot be reduced to simply facts. But more common than either stubborn facts of social constructions are pliant facts and impersonal constructions.

The Map Makes the Territory

While we know that “the map is not the territory” guarantees that any and all maps must be, at least to some extent, abstract, because of the ontogenic nature of maps we can, however, say that the map makes the territory. Because two nation-states agree on a boundary, they build fences and towers and keep guards with guns and dogs on their side of the agreed line. This is the concrete realization of a formal convention, specifically, the formal convention of mapping.

There is an amusing and oft-related story that Louis XV, after having financed a geographical expedition to map his kingdom the more accurately, discovered that his kingdom was rather smaller than he had supposed, and he remarked that he had lost more territory to his cartographers than he had ever gained in conquest. That surveyors, cartographers, geographers, and astronomers have been representatives of the crown, and their findings given the force of law, is one manifestation of the Scientific Revolution.

The territorial nation-state — hence the entire nation-state system of the contemporary global political order — cannot exist without precisely defined geographical borders. By the same token, the territorial principle in law is meaningless without a defined territory within which the national law is to be enforced. Thus the nation-state system, and every particular nation-state, rests upon the regime of cartography, which is to say that it rests upon a convention. And for that reason it is also as much to say that a nation-state is the relic of an abject technology; in other words, the nation-state is an abject institution.

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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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