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.

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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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Grand Strategy Annex

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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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