Page 82: Deleuze writes here on the experience of madness. Within our Pascal example, two numbers are generating (via the fibonacci sequence) a GCD. The GCD is tethered to the two numbers so that the two numbers can potentialy be factored in an orderly fashion. Should the GCD break away on its own, it is effectively groundless and has no crystalline structure connected to it by which it could become organized. This is no problem for small numbers, but you could see how if a person has a psychotic episode (e.g. schizophrenia), it would be akin to having a madman (i.e. the GCD) walk around in your brain, saying anything, doing anything, no discipline, no structure, perverse morals, inability to weigh costs and benefits, etc. One visualization that I use to see this is let's say you have a hideout inside a mountain. The interior of this structure is filled with tunnels and diagonal elevators (your crystalline structure). Should an event of madness occur, the mountain gets inverted inside out, such that all these structures are pushed to the outside, to the surface. Now, the only means to get to the top of the mountain (or anywhere else) is to follow the fibonacci spirals that were inherent inside the structure that have now become manifest on the outside -- effectively following a path around around and around the mountain, slowly gaining elevation, until at last you are back at the top. When the mountain is turned inside-out, it appears as if there are lots of gaping holes in the body. Rather than structuring the structure, the crystalline meshwork is on the outside -- the gaps, the breaks, are clearly evident. (cf. p.87) In this situation, everything is mixed up, such that in the psyche, schizophrenia dominates. When Deleuze says, on Artaud, that "For him [Artaud], there is not, there is no longer, any surface," (86) he means that the psyche has lost any semblance of consistency -- the forces of the depths of the soul rise to the surface indiscriminately; hence the loss of sense: "In this collapse of the surface, the entire world loses its meaning." (87) The corruption of sense results in indecipherable language, streams of language more like animal "howls" than spoken words. (89) The sound of phonetic language has become entirely dissociated from its meaning(s). Within our diagram, this would be all the non-red dots of the Pascal diagram, the non-factors of the GCD; there is a pattern, but it is almost impossible to decipher because it would mean decoding the inverse of a crystalline image (i.e. if there are very few red dots and many non-red dots, it is like decoding language through the gaps in speech, through the breaks). Deleuze speaks of this madness-event as nonsense "engulfing" sense. (91) We can get a clear visualization of that if we think of the mountain structure turning inside out such that the non-red dots are on the outside.
In this chapter, Deleuze opposes surfaces to poles of depth, or mixtures of depth. Again, we should read here, "the crystalline" and "the organic", respectively, from his other works.
"Psychoanalysis must have geometric dimensions....Psychoanalysis is the psychoanalysis of sense." (92) I think this is the takeaway for New Pioneerism...that to perform cultural analysis, as one might perform psychoanalysis, one must analyze geometrically, structurally.
Deleuze states his preference for Artaud over Carroll (93). The point being that from an understanding of the depths, one could always have the building blocks of the surface. Carroll's examination of the surface is epiphenomenal. In the two levels of the house, Artaud sits on the ground floor (cf. The Fold). Deleuze states that Artaud made his discoveries through suffering. He was creating a gap (an aleatory point, a differentiator) in himself through pain. The same thing could be said about the discoveries of Nietzsche. This reminds me of how George Soros says that his greatest discoveries and wins in investing happen to coincide with severe back pain.
Sunday, March 27, 2011
Wednesday, March 23, 2011
Twelveth Series of Paradox
Axiom: "A mental Void" = "Aion" (74)
Page 75: Deleuze equates "good sense" with bifurcation, dialectics, the powers of division and negation, the ability to separate the wheat from the chaff, etc. We said earlier that this "formula" would be akin to cutting the Pascal distribution in half. Deleuze states that good sense takes us from the "most to the least differentiated"; i.e. by cutting a large distribution several times, you eventually come up with a large host of small factors of a number. What Deleuze then points out is that there is always an undifferentiated remainder which results as a product of these bifurcations (e.g. you must always take one path at the fork in the road); this remainder can be likened to the repressed, perhaps, or Deleuze uses the example of a hill that adjoins a open space used for an enclosure; either way, it has some kind of undifferentiated potential energy.
Page 77: Deleuze speaks of the "opposite" of good sense, which would be a "recreation". This is the key to sense -- that it arises organically, like a witty joke, an epiphany showing a sense of perversion -- of taking a structure and twisting it, giving it an extra arm, etc. This is in opposition to "good sense" or logic by division or negation.
Page 77: Deleuze states that Chronos goes from past to future "only to the degree that presents follow one another inside partial worlds or partial systems." In our example, this is akin to the decomposition from one large number to a smaller "world" or number by means of factorization (i.e. bifurcation). These groups of partial, decomposed groups are arranged in nested form.
Page 77: "To the oriented line of the present, which 'regularizes' in an individual system each singular point which it takes in, the line of the Aion is opposed." The Pascal triangles are "organ"-ized, oriented, nested, as opposed to the moment of Aion, which is the moment of difference, the differentiator. Chronos is "regularized" insofar as it has been disciplined. In this way, Chronos are the organs, and Aion is the surface, the skin that is being constantly stetched. In the process of stretching and folding skin, you get new organs; hence the outside becomes the inside. These organs, or Pascal triangles, or "functions", are what Deleuze denotes as "common sense".
Page 80: Humpty Dumpty's pure intensity comes up again: "he is uniquely made of shifting and 'disconcerting' singularities.
Page 80: "One does not grow without the other shrinking." Explaining this statement within the Pascal triangle visualization is difficult. I believe it means that if you have two triangles that have come into intersection, they generate a GCD which is larger than either; hence, they both shrink. Or could it mean that the GCD moves over to one side or the other, thereby shrinking the other (i.e. taking its common factors with it)?
"Nonsense has a internal and original relation to sense." (81)
Page 75: Deleuze equates "good sense" with bifurcation, dialectics, the powers of division and negation, the ability to separate the wheat from the chaff, etc. We said earlier that this "formula" would be akin to cutting the Pascal distribution in half. Deleuze states that good sense takes us from the "most to the least differentiated"; i.e. by cutting a large distribution several times, you eventually come up with a large host of small factors of a number. What Deleuze then points out is that there is always an undifferentiated remainder which results as a product of these bifurcations (e.g. you must always take one path at the fork in the road); this remainder can be likened to the repressed, perhaps, or Deleuze uses the example of a hill that adjoins a open space used for an enclosure; either way, it has some kind of undifferentiated potential energy.
Page 77: Deleuze speaks of the "opposite" of good sense, which would be a "recreation". This is the key to sense -- that it arises organically, like a witty joke, an epiphany showing a sense of perversion -- of taking a structure and twisting it, giving it an extra arm, etc. This is in opposition to "good sense" or logic by division or negation.
Page 77: Deleuze states that Chronos goes from past to future "only to the degree that presents follow one another inside partial worlds or partial systems." In our example, this is akin to the decomposition from one large number to a smaller "world" or number by means of factorization (i.e. bifurcation). These groups of partial, decomposed groups are arranged in nested form.
Page 77: "To the oriented line of the present, which 'regularizes' in an individual system each singular point which it takes in, the line of the Aion is opposed." The Pascal triangles are "organ"-ized, oriented, nested, as opposed to the moment of Aion, which is the moment of difference, the differentiator. Chronos is "regularized" insofar as it has been disciplined. In this way, Chronos are the organs, and Aion is the surface, the skin that is being constantly stetched. In the process of stretching and folding skin, you get new organs; hence the outside becomes the inside. These organs, or Pascal triangles, or "functions", are what Deleuze denotes as "common sense".
Page 80: Humpty Dumpty's pure intensity comes up again: "he is uniquely made of shifting and 'disconcerting' singularities.
Page 80: "One does not grow without the other shrinking." Explaining this statement within the Pascal triangle visualization is difficult. I believe it means that if you have two triangles that have come into intersection, they generate a GCD which is larger than either; hence, they both shrink. Or could it mean that the GCD moves over to one side or the other, thereby shrinking the other (i.e. taking its common factors with it)?
"Nonsense has a internal and original relation to sense." (81)
Monday, March 21, 2011
Eleventh Series of Nonsense
Axiom: Deleuze states that at the intersection of two series, "word=x and thing=x". (66).
See this post for background on nonsense.
Page 67: Good stuff on the "sense" of the onomatopoeia. These words as well as nonsensical words are "absurd" "paradoxes" as they are a class that contain themselves as a member; that is, they declare their own sense. The visual analogy here would be a circle, a ring, or a donut. The nonsensical word (e.g. "Phlizz") is caught in a loop of self-procreation, an elemental form of non-identity ("What is it? It's nothing."). Even terms that are "devoid of signification" have "a sense." (70) The remainder of difference between the two series in the nonsensical word has dwindled to such a small number, it cannot be pinned down such that it can be excised and the structure can invert or dissipate. This would be akin, on a different level, to the "nonsensical" phenomenon of not being able to get a "terrible" song out of one's head. There seems to be no "cure" for this type of event save to let the structure exhaust itself (i.e. time passes). The visual representation of this would be a Mobius strip or Mobius donut where the twist has shrunk down to minute size, so small that the structure looks like a plain donut or circle. If we have to use a method like parallax or triangulation to detect that tiny twist in the in the Mobius donut, one could see how this could take a long time.
Axiom: Sense is always a surface effect. (70). Imagine a machine that can determine the shape of a balloon from the inside. Sense is the production of a surface that then wraps the balloon almost perfectly.
Nonsense is the divided element, the blind spot, that both produces sense and is perceived in the interpretation and evaluation of sense or surface effects.
Page 71: Reference to the "blind spot".
"Structure is in fact a machine for the production of incorporeal sense." (71)
Page 72: The ideal game is not one that is perfect, but one that produces the incorporeal or as Deleuze puts it, "surface effects". But one could say, in the tradition of Leibniz, that every moment, every event, is in itself perfect: "God has chosen the most perfect world." (Discourse on Metaphysics) This would be true in our examples if every event conforms to to the structure of the fibonacci sequence, whose proportions are generally held as the model of perfection.
"What is bureaucratic in these fantastic machines...?" -- determining the logic, the order, the "bureaucracy" of the singularities in the series (e.g. the distribution of the red dots in the Pascal triangle). (72)
Page 73: It's interesting that Deleuze says that "today's task" is to "produce sense" -- nothing less than a philosophical call to arms -- yet the production of sense cannot be forced, there is no formula by which to produce it. Instead, it arises organically, out of a blind spot, under favorable conditions.
See this post for background on nonsense.
Page 67: Good stuff on the "sense" of the onomatopoeia. These words as well as nonsensical words are "absurd" "paradoxes" as they are a class that contain themselves as a member; that is, they declare their own sense. The visual analogy here would be a circle, a ring, or a donut. The nonsensical word (e.g. "Phlizz") is caught in a loop of self-procreation, an elemental form of non-identity ("What is it? It's nothing."). Even terms that are "devoid of signification" have "a sense." (70) The remainder of difference between the two series in the nonsensical word has dwindled to such a small number, it cannot be pinned down such that it can be excised and the structure can invert or dissipate. This would be akin, on a different level, to the "nonsensical" phenomenon of not being able to get a "terrible" song out of one's head. There seems to be no "cure" for this type of event save to let the structure exhaust itself (i.e. time passes). The visual representation of this would be a Mobius strip or Mobius donut where the twist has shrunk down to minute size, so small that the structure looks like a plain donut or circle. If we have to use a method like parallax or triangulation to detect that tiny twist in the in the Mobius donut, one could see how this could take a long time.
Axiom: Sense is always a surface effect. (70). Imagine a machine that can determine the shape of a balloon from the inside. Sense is the production of a surface that then wraps the balloon almost perfectly.
Nonsense is the divided element, the blind spot, that both produces sense and is perceived in the interpretation and evaluation of sense or surface effects.
Page 71: Reference to the "blind spot".
"Structure is in fact a machine for the production of incorporeal sense." (71)
Page 72: The ideal game is not one that is perfect, but one that produces the incorporeal or as Deleuze puts it, "surface effects". But one could say, in the tradition of Leibniz, that every moment, every event, is in itself perfect: "God has chosen the most perfect world." (Discourse on Metaphysics) This would be true in our examples if every event conforms to to the structure of the fibonacci sequence, whose proportions are generally held as the model of perfection.
"What is bureaucratic in these fantastic machines...?" -- determining the logic, the order, the "bureaucracy" of the singularities in the series (e.g. the distribution of the red dots in the Pascal triangle). (72)
Page 73: It's interesting that Deleuze says that "today's task" is to "produce sense" -- nothing less than a philosophical call to arms -- yet the production of sense cannot be forced, there is no formula by which to produce it. Instead, it arises organically, out of a blind spot, under favorable conditions.
Tuesday, March 15, 2011
Wednesday, March 9, 2011
Tenth Series of the Ideal Game
"Each throw [of the dice] is itself a series, but in a time much smaller than the minimum of continuous, thinkable time; and, to this serial minimum, a distribution of singularities corresponds." (59) A time "much smaller than continuous, thinkable time" is synonymous with becoming, an ontological state of infinite indeterminedness; it is a die that has not stopped spinning. The distribution of virtual points the cast of the die could take is arranged into a probabalistic distribution. In our earlier example, we used Pascal's triangle as an example of this kind of distribution (e.g. getting X number of heads from flipping a coin Y times).
At the same time, the set of all dicethrows makes up an ontological world which is "greater than the maximum of continuous, thinkable time." (59) In this way, all of our lives belong to a universal dicethrow which is infinite, i.e. the probability distribution of the World and everything in it and everything not in it. This would be a probability which accounts for all the other probabilities in the system.
"Each throw operates a distribution of singularities, a constellation." (60) See the previous post on crystallization, snowflakes, and Pascal's triangle. When Deleuze says constellation, he is being literal.
"This is a nomadic and non-sedentary distribution." (60) Nomads are a big part of Deleuze (and Guattari's) theories. They imply that that a point may stay in one place (like a Bedouin), but the landscape shifts all around him/her (the sand dunes shifting in the night). Any singularity is enclosed by a constantly shifting, complex meshwork of other singularities "communicating" via "resonance". (60) Imagine a seascape of shifting tuning forks; you tap one, and all the others with complementary frequencies begin to vibrate and hum. This is the short version of how I believe the brain works and transmits signals. The brain emits waves. And these waves are produced how?
The ideal game "can only be thought of as nonsense" (60). Here is one of our first important introductions to nonsense. Paradoxically, ironically, sense produces nonsense. It is the organs of sense which produce such stupid non-games as the caucus race (cf. Carroll). Nonsense is a distribution of singularities that is difficult to interpret due to its complexity. Deleuze's point here is that even in the simplest distributions, there is an element of nonsense, of paradox. Even the simplest words that an infant learns to speak have multiple facets. And if we go back to the other chapters, we remember that this is due to the paradoxical element, either the letter without content, or the letter that is constantly shifting place, impossible to pin down. If we were to diagram this in our Pascal triangle, we could imagine a spiral. This spiral is a form, no doubt, but it has a hole in the center; ultimately, there is an aleatory gap constructing it. Interestingly, the golden spiral's proportions conform to the fibonacci sequence, which earlier we stated could be termed "the organic". So sense produces a crystallization, but in our interpretation and evaluation of bodies, what we see is nonsense, the organic, the "empty square", the spiral with a hole in the middle, etc. This paradox, that we should produce sense, but interpret nonsense, is "the reality of thought." (60)
Deleuze opposes the operation of sense to the player who attempts to dominate the game by "dividing it". What does this mean? Just as we said that an event results in a singularity, a bifurcation, it is possible that the game-player can learn that division is the means by which to produce events, and use this fact in order to dominate games. This type of forced division is akin to negation, an effective logical division (i.e. "this is yes, and this is no", "this is the positive space in the image, this is the negative space."). This dialectical behavior is very Hegelian, and much of what Deleuze writes, including this, is a reaction to that. In our Pascal diagram, this form of negation may be akin to cutting a distribution in half in order to decrease its size so as better to factor it. It would be akin to a large number that you want to factor and you throw small primes at it in order to cut it down. This is much different from the affirmation that comes from seeing a pattern in the distribution, looking for the fibonacci spirals, or the constellations of points, and allowing the number that will precipitate the division of the greatest common divisor to arise organically through sense ("static ontological genesis"). The ability to divide a distribution in a forced way amounts to nothing other than a formula, and often produces contradiction results because in order to make the number smaller and more easy to manage, you may be pulling out one of the key pillars that allows the structure to stand up and "make sense" in a different context.
Page 61: Deleuze cites Borges' Babylonian lottery: "the lottery is an intensification of chance." Throughout Deleuze's other work, intensity is an important piece of jargon. Here, he is describing the fact that chance, the dicethrow, subsists in every moment of every little game that we and everything around us are involved in -- it is the DNA of life -- so a lottery is but an intensification of chance, of piling chance upon a chance, of chance to the chance exponent. Chance, or becoming, being unmeasurable in terms of magnitude, or affirmative or negative (and he brings up the Tortoise and the Hare here, akin to Occam's razor), does have this property of intensity. Later, he will describe intensity as how fast a molecule spins, as when heat is added to water molecules in a pot. From the Stanford Encyclopedia of Philosophy, "intensity, [Deleuze] argues, constitutes the genetic condition of extensive space." Chance, "infinitely subdivisible", is Aion, unlimited time moving in both directions from the present at once. You can divide a fibonacci spiral infinitely by the golden ratio, and you still end up with the same self-similar structure.
Page 61: To imagine Deleuze's conception of past, present, and future, let us take our Pascal's triangle and graph it. Any one horizontal slice of it looks like a normal curve due to the normal distribution of probabilities. If you plot all the slices, mirror it across the Z-axis, and bend it a little, you get Gabriel's Horn. Deleuze's argument is that the present is inside this horn, the mixtures, the bodies mixed up, Chronos. At the edges, unlimited and defining, is Aion, the indivisible past and future. The movement of Aion forces the present (Chronos), to expand and contract. So if you imagine Gabriel's horn as an upside-down cone, at its unlimited tip, it is inscribing the present. The horn itself is like one big filter or pachinko machine. Deleuze's point on page 62 is that at the tip of the horn (this is referred to as the memory cone in his work on Henri Bergson), its capacity to become unlimited means that it can be stretched out for a long long time. Imagine Gabriel's horn with an indivisible tip that is two miles long. Negotiating the interior of that almost volumeless portion would be akin to Borges' "labyrinth made of the single straight line which is invisible and everlasting." (62) We said earlier that factoring the number that this cone represents depends upon gauging the distance between horizontal slices in Pascal's triangle (e.g. via a method like parallax). If the triangle has collapsed into a long almost-empty line, performing this task might seem impossible.
Page 62: Imagine two Gabriel horns interlocking. This is one version of the present, of becoming. If one stretches out this present infinitely, as we spoke of earlier, we get a new Aion, a new perspective of past and future by which to organize another "present". In this way, past, present, and future interact. So no matter how long the tip of Gabriel's horn gets, there is always a thread that leads to another Gabriel's horn. In this way, becoming becomes "knotted"as "interlocking presents". Each one of these knots, these interactions between two horns (later, Deleuze will call them "machines"), results in a singularity, one of our red dots in the earlier Pascal diagram. When you "unfold" Chronos, you get Aion, the unlimited time of becoming. (62)
"The past and future indicate only the relative difference between two presents." (62) Once again, a reference to difference, read as derivatives, here.
Page 63: Deleuze differentiates here between two presents, the living present, which would be more akin to typical, pulsed time, time measured by a clock, Chronos, and Aion, unpulsed, unlimited time. What he is saying is that there is an equivalency between the two expressions of time; this differentiator between the two modalities of time, of course, is sense. "The event...retains an eternal truth upon the line of the Aion." (63) In Nietzsche and Philosophy, Deleuze spend a long time working on his eternal return, and I believe that is what he is trying to get at here. In the production of the event, folded-up Chronos dimensions get unfolded into straight lines. One way to visualize this is to think of the difference between the senses of sight and sound. Sound is distributed along straight lines -- i.e. a word is a series of frequencies coming one after another (hence, an MP3 can be expressed as consecutive 1s and 0s at the binary level of computer language). Sight on the other hand, is the interpretation of many wavelengths at once. We see not only one color frequency, but many simultaneously in what we interpret as a 3-dimensional universe due to stereostropic vision. Within the Pascal diagram, those 'snowflakes' or constellations can be unfurled into straight lines as well (imagine untwisting a line running upwards through a tornado). The 'unfurling' of each gyre is an event synonymous with an eternal return -- sense is spinning down and decoding a tornado of constellations so as to spit out a perfect crystalline structure that traces a new dimension. As Deleuze says, "The singular of each event are distributed over this line, always in relation to the aleatory point which subdives them ad infinitum.' (64) Hence, Aion takes the numerical relations embedded between two Pascal triangles and distributes them proportionally along a straight line according to their GCD. For instance, if one number is 7, and the other 5, Aion incarnates a new singularity every 35 points. The Aion, besides being a force of bifurcation, is a source of eternal truth, in every multiple of 35, both 7 and 5 are "eternally true".
Within our example with the Pascal triangle, the moment of decoding or eternal return is when one triangle intersects with another, generating a GCD which end up factoring both triangles -- hence, the splitting or bifurcation we referred to earlier.
Page 63-64: Deleuze says that the event is smaller than the minimum of continuous thinkable time because there is a gap in the Aion between past and future. Aion is fundamentally divided ("The Aion is the straight line traced by the aleatory point' (64)) because it is in this gap that we find the difference between two machines, two Pascal triangles, at "the border of two tables" (64) -- the gap is the differentiating function which decodes both series, leaving them both changed. What is the stuff of Aion? It is the golden ratio, it is the Pascal triangle turned on its side so that the series sum to the fibonacci numbers. Hence the sequence can be as small as one, and self replicate to contain the GCD of the largest numbers. See: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html#alli. The Aion is the "unique cast" (64) cast of the die because it contain all chance within it; this unique cast is synonymous with univocality his other works.
The Aion subdivides while being procreative at the same time. It generates a GCD, while returning the non-common factors. For instance, 10 and 12 would return 60 as a GCD, and return 5 and 6 as the non-common factors. Hence, the event both divides and creates.
Deleuze says the Aion appears as an "empty square on one side and as a supernumerary object on another." (65) We read this as two sides of the same coin, as mentioned previously. The fibonacci spiral which extends the interaction of two numbers into a new dimensions is the supernumerary, and it is dependent upon pulling a number out of one or both of the triangles (the empty square, the constellation missing a star, etc.). In other writings of Deleuze, these themes are evoked as the n-1 and n+1, which is borrowed from statistical degrees of freedom.
At the same time, the set of all dicethrows makes up an ontological world which is "greater than the maximum of continuous, thinkable time." (59) In this way, all of our lives belong to a universal dicethrow which is infinite, i.e. the probability distribution of the World and everything in it and everything not in it. This would be a probability which accounts for all the other probabilities in the system.
"Each throw operates a distribution of singularities, a constellation." (60) See the previous post on crystallization, snowflakes, and Pascal's triangle. When Deleuze says constellation, he is being literal.
"This is a nomadic and non-sedentary distribution." (60) Nomads are a big part of Deleuze (and Guattari's) theories. They imply that that a point may stay in one place (like a Bedouin), but the landscape shifts all around him/her (the sand dunes shifting in the night). Any singularity is enclosed by a constantly shifting, complex meshwork of other singularities "communicating" via "resonance". (60) Imagine a seascape of shifting tuning forks; you tap one, and all the others with complementary frequencies begin to vibrate and hum. This is the short version of how I believe the brain works and transmits signals. The brain emits waves. And these waves are produced how?
The ideal game "can only be thought of as nonsense" (60). Here is one of our first important introductions to nonsense. Paradoxically, ironically, sense produces nonsense. It is the organs of sense which produce such stupid non-games as the caucus race (cf. Carroll). Nonsense is a distribution of singularities that is difficult to interpret due to its complexity. Deleuze's point here is that even in the simplest distributions, there is an element of nonsense, of paradox. Even the simplest words that an infant learns to speak have multiple facets. And if we go back to the other chapters, we remember that this is due to the paradoxical element, either the letter without content, or the letter that is constantly shifting place, impossible to pin down. If we were to diagram this in our Pascal triangle, we could imagine a spiral. This spiral is a form, no doubt, but it has a hole in the center; ultimately, there is an aleatory gap constructing it. Interestingly, the golden spiral's proportions conform to the fibonacci sequence, which earlier we stated could be termed "the organic". So sense produces a crystallization, but in our interpretation and evaluation of bodies, what we see is nonsense, the organic, the "empty square", the spiral with a hole in the middle, etc. This paradox, that we should produce sense, but interpret nonsense, is "the reality of thought." (60)
Deleuze opposes the operation of sense to the player who attempts to dominate the game by "dividing it". What does this mean? Just as we said that an event results in a singularity, a bifurcation, it is possible that the game-player can learn that division is the means by which to produce events, and use this fact in order to dominate games. This type of forced division is akin to negation, an effective logical division (i.e. "this is yes, and this is no", "this is the positive space in the image, this is the negative space."). This dialectical behavior is very Hegelian, and much of what Deleuze writes, including this, is a reaction to that. In our Pascal diagram, this form of negation may be akin to cutting a distribution in half in order to decrease its size so as better to factor it. It would be akin to a large number that you want to factor and you throw small primes at it in order to cut it down. This is much different from the affirmation that comes from seeing a pattern in the distribution, looking for the fibonacci spirals, or the constellations of points, and allowing the number that will precipitate the division of the greatest common divisor to arise organically through sense ("static ontological genesis"). The ability to divide a distribution in a forced way amounts to nothing other than a formula, and often produces contradiction results because in order to make the number smaller and more easy to manage, you may be pulling out one of the key pillars that allows the structure to stand up and "make sense" in a different context.
Page 61: Deleuze cites Borges' Babylonian lottery: "the lottery is an intensification of chance." Throughout Deleuze's other work, intensity is an important piece of jargon. Here, he is describing the fact that chance, the dicethrow, subsists in every moment of every little game that we and everything around us are involved in -- it is the DNA of life -- so a lottery is but an intensification of chance, of piling chance upon a chance, of chance to the chance exponent. Chance, or becoming, being unmeasurable in terms of magnitude, or affirmative or negative (and he brings up the Tortoise and the Hare here, akin to Occam's razor), does have this property of intensity. Later, he will describe intensity as how fast a molecule spins, as when heat is added to water molecules in a pot. From the Stanford Encyclopedia of Philosophy, "intensity, [Deleuze] argues, constitutes the genetic condition of extensive space." Chance, "infinitely subdivisible", is Aion, unlimited time moving in both directions from the present at once. You can divide a fibonacci spiral infinitely by the golden ratio, and you still end up with the same self-similar structure.
Page 61: To imagine Deleuze's conception of past, present, and future, let us take our Pascal's triangle and graph it. Any one horizontal slice of it looks like a normal curve due to the normal distribution of probabilities. If you plot all the slices, mirror it across the Z-axis, and bend it a little, you get Gabriel's Horn. Deleuze's argument is that the present is inside this horn, the mixtures, the bodies mixed up, Chronos. At the edges, unlimited and defining, is Aion, the indivisible past and future. The movement of Aion forces the present (Chronos), to expand and contract. So if you imagine Gabriel's horn as an upside-down cone, at its unlimited tip, it is inscribing the present. The horn itself is like one big filter or pachinko machine. Deleuze's point on page 62 is that at the tip of the horn (this is referred to as the memory cone in his work on Henri Bergson), its capacity to become unlimited means that it can be stretched out for a long long time. Imagine Gabriel's horn with an indivisible tip that is two miles long. Negotiating the interior of that almost volumeless portion would be akin to Borges' "labyrinth made of the single straight line which is invisible and everlasting." (62) We said earlier that factoring the number that this cone represents depends upon gauging the distance between horizontal slices in Pascal's triangle (e.g. via a method like parallax). If the triangle has collapsed into a long almost-empty line, performing this task might seem impossible.
Page 62: Imagine two Gabriel horns interlocking. This is one version of the present, of becoming. If one stretches out this present infinitely, as we spoke of earlier, we get a new Aion, a new perspective of past and future by which to organize another "present". In this way, past, present, and future interact. So no matter how long the tip of Gabriel's horn gets, there is always a thread that leads to another Gabriel's horn. In this way, becoming becomes "knotted"as "interlocking presents". Each one of these knots, these interactions between two horns (later, Deleuze will call them "machines"), results in a singularity, one of our red dots in the earlier Pascal diagram. When you "unfold" Chronos, you get Aion, the unlimited time of becoming. (62)
"The past and future indicate only the relative difference between two presents." (62) Once again, a reference to difference, read as derivatives, here.
Page 63: Deleuze differentiates here between two presents, the living present, which would be more akin to typical, pulsed time, time measured by a clock, Chronos, and Aion, unpulsed, unlimited time. What he is saying is that there is an equivalency between the two expressions of time; this differentiator between the two modalities of time, of course, is sense. "The event...retains an eternal truth upon the line of the Aion." (63) In Nietzsche and Philosophy, Deleuze spend a long time working on his eternal return, and I believe that is what he is trying to get at here. In the production of the event, folded-up Chronos dimensions get unfolded into straight lines. One way to visualize this is to think of the difference between the senses of sight and sound. Sound is distributed along straight lines -- i.e. a word is a series of frequencies coming one after another (hence, an MP3 can be expressed as consecutive 1s and 0s at the binary level of computer language). Sight on the other hand, is the interpretation of many wavelengths at once. We see not only one color frequency, but many simultaneously in what we interpret as a 3-dimensional universe due to stereostropic vision. Within the Pascal diagram, those 'snowflakes' or constellations can be unfurled into straight lines as well (imagine untwisting a line running upwards through a tornado). The 'unfurling' of each gyre is an event synonymous with an eternal return -- sense is spinning down and decoding a tornado of constellations so as to spit out a perfect crystalline structure that traces a new dimension. As Deleuze says, "The singular of each event are distributed over this line, always in relation to the aleatory point which subdives them ad infinitum.' (64) Hence, Aion takes the numerical relations embedded between two Pascal triangles and distributes them proportionally along a straight line according to their GCD. For instance, if one number is 7, and the other 5, Aion incarnates a new singularity every 35 points. The Aion, besides being a force of bifurcation, is a source of eternal truth, in every multiple of 35, both 7 and 5 are "eternally true".
Within our example with the Pascal triangle, the moment of decoding or eternal return is when one triangle intersects with another, generating a GCD which end up factoring both triangles -- hence, the splitting or bifurcation we referred to earlier.
Page 63-64: Deleuze says that the event is smaller than the minimum of continuous thinkable time because there is a gap in the Aion between past and future. Aion is fundamentally divided ("The Aion is the straight line traced by the aleatory point' (64)) because it is in this gap that we find the difference between two machines, two Pascal triangles, at "the border of two tables" (64) -- the gap is the differentiating function which decodes both series, leaving them both changed. What is the stuff of Aion? It is the golden ratio, it is the Pascal triangle turned on its side so that the series sum to the fibonacci numbers. Hence the sequence can be as small as one, and self replicate to contain the GCD of the largest numbers. See: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html#alli. The Aion is the "unique cast" (64) cast of the die because it contain all chance within it; this unique cast is synonymous with univocality his other works.
The Aion subdivides while being procreative at the same time. It generates a GCD, while returning the non-common factors. For instance, 10 and 12 would return 60 as a GCD, and return 5 and 6 as the non-common factors. Hence, the event both divides and creates.
Deleuze says the Aion appears as an "empty square on one side and as a supernumerary object on another." (65) We read this as two sides of the same coin, as mentioned previously. The fibonacci spiral which extends the interaction of two numbers into a new dimensions is the supernumerary, and it is dependent upon pulling a number out of one or both of the triangles (the empty square, the constellation missing a star, etc.). In other writings of Deleuze, these themes are evoked as the n-1 and n+1, which is borrowed from statistical degrees of freedom.
Thursday, March 3, 2011
Ninth Series of the Problematic
"Singularities are turning points and points of inflection." (52) OK, to summarize, we have two series coming into conjunction. I am representing those two series by two numbers. At the point of intersection, there is becoming, a unlimited, undefinable asymptote to both that is infinitely stretched. It is at this point that the two series are immanent to one another. Here, Deleuze is saying that this intersection or singularity can actually be represented by an inflection point. This is key, because earlier, we said that the event, sense, is a differentiator, effectively mapping a derivative of two functions (sense is the "paradoxical agent" (53)). The inflection point would be where this derivative changes sign. Deleuze, following Lacan, represents this moment of change diagramatically -- "bottlenecks, knots, foyers, and centers." (52) Language is effectively a folded surface due to the "stitching" (cf. Lacan) of these two series. I was recently watching The King's Speech with Colin Firth. This movie is about stammering, knots in the speech production function of King George VI. These knots are nothing but examples of the way series can jumble themselves through interaction. If the knot is so hard (as was the case with King George), the only thing that can seem to release the words is productive force -- rolling around on the floor, swearing profusely and loudly, etc. Within Jacques Lacan's work, this inflection point or turning back is akin to his point de capiton, or "quilting point". Sense has the power to arrange thought and language in such a way that it functions as a button on a couch holds stuffing in place. Also, in chaos theory, a singularity has a specific description of one of these points of bifurcation. If you reverse the time on a bifurcation, you get a convergence. In this chaos diagram, we would be talking about one of the points where the diagram bifurcates or divides:
http://en.wikipedia.org/wiki/File:LogisticMap_BifurcationDiagram.png
This document on chaos theory in Deleuze is very good, but may be a little confusing:
http://www.protevi.com/john/DG/PDF/Remarks_on_Complexity_Theory.pdf.
"Singularity is neutral." (52) Singularities do not distinguish between right and wrong, affirmative and negative. They simply result in the production of sense. They can be contrasted only with a sea of nonsense.
So if we go back to the Pascal's triangle we cited earlier, each one of those red dots is a "knot", a singularity that resulted when two series of numbers intersected and it was found that that that dot "resonates" (or is divisible) with the greatest common divisor of the two series. These dots were emitted by the intersection of the two series. If you think of a broken zipper, the zipper goes up and down, locking and unlocking the teeth. Except in our diagram, the event, the differentiator, leaves the numbers fundamentally changed, folded, knotted, etc.
All history is a recording of the series of all events or singularities. (53) All the world is but n zippers moving in n directions, locking and unlocked n teeth.
Page 53: Deleuze states that events are points of crystallization. Because Pascal's triangle is a discrete representation of probability (i.e. if you have X coins and flip them Y times, what is the probability you will get Z heads?), we can see the red dots in the Pascal's triangle as the crystallization of points of probability. You can see how by changing the number in the diagram, you can make different snowflake-looking patterns. In Deleuze's two books on film, The Time Image and The Movement Image, he spend a lot of time opposing the crystalline to the organic. I believe that if we can conceive of the crystalline as these snow-flake looking diagrams, the organic can also be represented by Pascal's triangle, albeit in a different way. If you look at Pascal's triangle from the side, you get numbers summing to fibonacci numbers: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html#pascal
The fibonacci numbers have been demonstrated in many natural systems, including nautilus shells, limb growth of trees, and the shape of apple cores. Fibonacci series can best be summarized by "take the last two numbers and add them together to get the next one." This growth rate has been called the "golden section". I believe these series of fibonacci numbers in Pascal's triangle can be termed the organic, and through their intersection, we get singularities, the patterns, the snowflakes, the discrete, expressed probabilities or points. The "unlimited Aion" (53), the force of becoming, I believe is the force of this organic, the "Infinitive" (53), the fibonacci sequence in our diagram.
Page 54: Deleuze introduces "problems" and "problematics" to speak of events. What is a knot but a problem that can be unknotted by a "decoder", by the production of an event that renders sense to what was previously obscure? In this way, "the event by itself is problematic and problematizing." (54) Deleuze uses problematic to speak of the event, but he could just as well say "probabalistic", thereby referring to the Pascal's triangle visualization. He does speak of "distributions" (56), which denote the same statistical connotation.
Deleuze take a little bit of tangent here to speak of derivatives and integrals. If sense is the differentiator, and time can move forwards or backwards, then we must be able to speak of the integration of a surface as well. When two series come together, the differentiator produces a greatest common divisor (GCD). If we were to work backwards and factor that GCD, we might use integration to do so (i.e. finding the area between two curves, cf. http://en.wikipedia.org/wiki/Fermat_primality_test). The definition of the solution to a problem can be visualized as the mapping of the space beneath a indistinguishable horizon line (i.e. integration). (54)
Page 55: Direct relation here between mathematical shapes and relations to "psychological and moral character". To get a grasp of this, watch this great documentary on Daniel Tammet, a savant who through his synesthesia, can see numbers as if they were shapes, allowing him to recite 22,514 digits of Pi by seeing the string of numbers as a line, a moving, self-differentiating line. His recitation of Pi is as if he were reciting all the dips and crags in a mountainous landscape. For him, numbers have certain meanings for him -- nine is "blue", eleven is "friendly" and five is "loud". We must think of not only numbers, but all language and thought diagramatically and structurally.
http://en.wikipedia.org/wiki/File:LogisticMap_BifurcationDiagram.png
This document on chaos theory in Deleuze is very good, but may be a little confusing:
http://www.protevi.com/john/DG/PDF/Remarks_on_Complexity_Theory.pdf.
"Singularity is neutral." (52) Singularities do not distinguish between right and wrong, affirmative and negative. They simply result in the production of sense. They can be contrasted only with a sea of nonsense.
So if we go back to the Pascal's triangle we cited earlier, each one of those red dots is a "knot", a singularity that resulted when two series of numbers intersected and it was found that that that dot "resonates" (or is divisible) with the greatest common divisor of the two series. These dots were emitted by the intersection of the two series. If you think of a broken zipper, the zipper goes up and down, locking and unlocking the teeth. Except in our diagram, the event, the differentiator, leaves the numbers fundamentally changed, folded, knotted, etc.
All history is a recording of the series of all events or singularities. (53) All the world is but n zippers moving in n directions, locking and unlocked n teeth.
Page 53: Deleuze states that events are points of crystallization. Because Pascal's triangle is a discrete representation of probability (i.e. if you have X coins and flip them Y times, what is the probability you will get Z heads?), we can see the red dots in the Pascal's triangle as the crystallization of points of probability. You can see how by changing the number in the diagram, you can make different snowflake-looking patterns. In Deleuze's two books on film, The Time Image and The Movement Image, he spend a lot of time opposing the crystalline to the organic. I believe that if we can conceive of the crystalline as these snow-flake looking diagrams, the organic can also be represented by Pascal's triangle, albeit in a different way. If you look at Pascal's triangle from the side, you get numbers summing to fibonacci numbers: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html#pascal
The fibonacci numbers have been demonstrated in many natural systems, including nautilus shells, limb growth of trees, and the shape of apple cores. Fibonacci series can best be summarized by "take the last two numbers and add them together to get the next one." This growth rate has been called the "golden section". I believe these series of fibonacci numbers in Pascal's triangle can be termed the organic, and through their intersection, we get singularities, the patterns, the snowflakes, the discrete, expressed probabilities or points. The "unlimited Aion" (53), the force of becoming, I believe is the force of this organic, the "Infinitive" (53), the fibonacci sequence in our diagram.
Page 54: Deleuze introduces "problems" and "problematics" to speak of events. What is a knot but a problem that can be unknotted by a "decoder", by the production of an event that renders sense to what was previously obscure? In this way, "the event by itself is problematic and problematizing." (54) Deleuze uses problematic to speak of the event, but he could just as well say "probabalistic", thereby referring to the Pascal's triangle visualization. He does speak of "distributions" (56), which denote the same statistical connotation.
Deleuze take a little bit of tangent here to speak of derivatives and integrals. If sense is the differentiator, and time can move forwards or backwards, then we must be able to speak of the integration of a surface as well. When two series come together, the differentiator produces a greatest common divisor (GCD). If we were to work backwards and factor that GCD, we might use integration to do so (i.e. finding the area between two curves, cf. http://en.wikipedia.org/wiki/Fermat_primality_test). The definition of the solution to a problem can be visualized as the mapping of the space beneath a indistinguishable horizon line (i.e. integration). (54)
Page 55: Direct relation here between mathematical shapes and relations to "psychological and moral character". To get a grasp of this, watch this great documentary on Daniel Tammet, a savant who through his synesthesia, can see numbers as if they were shapes, allowing him to recite 22,514 digits of Pi by seeing the string of numbers as a line, a moving, self-differentiating line. His recitation of Pi is as if he were reciting all the dips and crags in a mountainous landscape. For him, numbers have certain meanings for him -- nine is "blue", eleven is "friendly" and five is "loud". We must think of not only numbers, but all language and thought diagramatically and structurally.
Wednesday, March 2, 2011
Eighth Series of Structure
Deleuze citing Cosmos, by Witold Gombrowicz: "There are always too many signifying signs." (48) Great book, if you have the chance to read it.
Page 49: Revolution lies in the asignifying gap between series. It is unexpected and comes out of a blind spot. It perfectly reverses expectations and is only explainable in hindsight, retroactively.
Page 50: Deleuze recapitulates what he has previously said about series, excesses and lacks, and the object without a place, etc. He uses these criteria as a basis for a "structure". Later, in other books, these structures become "machines". Here, as in the Fourth Series, he implies derivatives, "differential calculus," as an orienting frame for the operation of sense.
If you can, imagine two series which are referenced by two numbers; their two distributions are the distribution of numbers evenly divisible by the reference number. When these two series converge, they create a greatest common divisor, which is the paradoxical element differentiating the two series. This is an example:
http://www.its.caltech.edu/~mamikon/PasFastC.html
If you type in a number, the Pascal's triangle highlights all the cells containing numbers evenly divisible by the that number. Intersect one triangle of one number with another, and you would get something like a greatest common divisor. Each highlighted point in each triangle is what Deleuze calls here a "singularity". A structure would be the intersection of two of these triangles.
Axiom: "There is no structure without series." (51)
Page 49: Revolution lies in the asignifying gap between series. It is unexpected and comes out of a blind spot. It perfectly reverses expectations and is only explainable in hindsight, retroactively.
Page 50: Deleuze recapitulates what he has previously said about series, excesses and lacks, and the object without a place, etc. He uses these criteria as a basis for a "structure". Later, in other books, these structures become "machines". Here, as in the Fourth Series, he implies derivatives, "differential calculus," as an orienting frame for the operation of sense.
If you can, imagine two series which are referenced by two numbers; their two distributions are the distribution of numbers evenly divisible by the reference number. When these two series converge, they create a greatest common divisor, which is the paradoxical element differentiating the two series. This is an example:
http://www.its.caltech.edu/~mamikon/PasFastC.html
If you type in a number, the Pascal's triangle highlights all the cells containing numbers evenly divisible by the that number. Intersect one triangle of one number with another, and you would get something like a greatest common divisor. Each highlighted point in each triangle is what Deleuze calls here a "singularity". A structure would be the intersection of two of these triangles.
Axiom: "There is no structure without series." (51)
Tuesday, March 1, 2011
Seventh Series of Esoteric Words
In this chapter, Deleuze applies what we have learned about serialization and the point of relative displacement to Lewis Carroll explicitly. He says that throughout Carroll's work there are words that function in the same way as the MacGuffin or purloined letter discussed in the last chapter: "In principle, it is the empty square, the empty shelf, the blank word." (44) These are words like Phlizz, Snark, Azzigoom Pudding, etc. These words are a "switch" between two or more series, a means of passage. (47) These words, especially the portmanteau words, may seem synthetic, but they are in fact often disjunctive; they can be broken down, deconstructed, re-ordered, and reconnoitered. (46) They may appear born from nothing, but in fact, they have an inherent archaeology, a genealogy.
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