"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.
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