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.