As I suspected earlier, the tangent into the Sophist got out of hand. But since we now know that Non-Being = Difference = Possibility, we are in a position to deliver the last part of the promised thesis and relate this type of being to questions and problems. This is the last link in Deleuze's exposition of Plato, and takes us right up to the Heideggerean interlude that begins on pg. 64.
There's actually not much explicit connection between non-being and problems in the Sophist. The closest relation would be the approach we saw to the definition of the philosopher as someone who understands how the Forms blend.
Stranger: Shall we not say that the division of things by classes and the avoidance of the belief that the same class is another, or another the same, belongs to the science of dialectic?
Theaetetus: Yes, we shall.
Stranger: Then he who is able to do this has a clear perception of one form or idea extending entirely through many individuals each of which lies apart, and of many forms differing from one another but included in one greater form, and again of one form evolved by the union of many wholes, and of many forms entirely apart and separate. This is the knowledge and ability to distinguish by classes how individual things can or cannot be associated with one another.
I say this simply because it appears to be describing a pragmatic, case-by-case problem solving type of approach, in contrast to Aristotle or Hegel's scheme which always seem to give the same general answer to every particular question. Here, the only general answer is, "well, it's a complicated mix". Door Openers and Door Closers.
There is also the more implicit connection that the Sophist shares with almost every Platonic dialogue -- it is structured according to the problem, "what is X?". What is the Sophist, the Statesman, the Lover, the Just City? In fact, all of Plato's Ideas are approached as problems investigated by Socratic irony and dialectical method.
Irony consists in treating things and beings as so many responses to hidden questions, so many cases for problems yet to be resolved. We recall that Plato defined the dialectic as proceeding by 'problems', by means of which one attains the pure grounding principle - that is, the principle which measures the problems as such and distributes the corresponding solutions.
At first this reads as just general commentary on Plato's style. In fact, I would say that the whole connection between Plato and problems stays fairly abstract at this point in D&R. It is the first time Deleuze has mentioned 'problems', and he hasn't really given us a definition here. All we really know at this point is that he sees problems as ontological and not merely epistemological.
Neither the problem nor the question is a subjective determination marking a moment of insufficiency in knowledge. Problematic structure is part of objects themselves, allowing them to be grasped as signs, just as the questioning or problematising instance is a part of knowledge allowing its positivity and its specificity to be grasped in the act of learning.
I'm sure we'll see the concept of a problem get fleshed out as we go, but all the work on Ideas and Not-Being has already put us in a position to read between the lines. We saw that the Forms were mythical, other-worldly principles that nevertheless structured a whole chain of real differences between things. The chain was ordered by degree of participation in the Idea -- level 0 for the original true Being of the Idea itself, level 1 for the 'best' instantiation or copy of it, ... , down to level N for something that is so unlike the original that it was a 'fake' version of it, one mixed up with everything else with no claim to the throne. So we see already that the Forms are directly related to the notion of possibility. They are inseparable from the different possible copies of themselves. They relate those differences to one another and bind them all together. One universal Idea is already inherently many possible different things arranged in a particular order. Said more simply, the Forms structure a possibility space. They aren't a given point in a pre-existing space, some point that happens to be different from our current point. They lay out the structure of the whole space of possibilities. They are the singularities in a differential field.
But wait. We discovered in the Sophist that even those things at the 'end' of the chain, those Nth derivative 'fakes' and 'semblances' that blend together many Forms in an impure mixture, also have a type of Being, the type of Being we established as non-Being. But this non-Being was not the same as the negative or the opposite of Being. In fact, by the grand finale of the Sophist, we're confusing him with the Philosopher, and finding we can't rely on contradiction to separate the images and fakes from the real thing. This is why they invented 'scare quotes'. You know something weird is going on when Plato starts talking about "divine fakes":
Theaetetus: Tell me again how each part is distinguished.
Stranger:We know that we and all the other animals, and fire, water, and their kindred elements, out of which natural objects are formed, are one and all the very offspring and creations of God, do we not?
Theaetetus: Yes.
Stranger: And corresponding to each and all of these there are images, not the things themselves, which are also made by superhuman skill.
Theaetetus: What are they?
Stranger:The appearances in dreams, and those that arise by day and are said to be spontaneous—a shadow when a dark object interrupts the firelight, or when twofold light, from the objects themselves and from outside, meets on smooth and bright surfaces and causes upon our senses an effect the reverse of our ordinary sight, thus producing an image.
Theaetetus: Yes, these are two works of divine creation, the thing itself and the corresponding image in each case.
The way everything is muddled together in non-Being at the limit of participation in an Idea mirrors the way everything is muddled together at the end of the Sophist. But this chaos of the Being of non-Being is in fact the very problem we began with!
... an indefinite representing multiplicity which must be eliminated in order to bring to light the Idea which constitutes a pure line of descent. The search for gold provides the model for this process of division.
The mixtures we find in the world present a real problem, not just for our knowledge, but in themselves -- how could they have been constructed from blending together many Forms? Given this point that we see, what spaces interacted to make it possible? In a sense, this is the essence of philosophy. We're not asking the scientific question: "why is this world like it is?" We're asking: "how is a world like this possible, and so what other worlds are possible?"
Being (what Plato calls the Idea) 'corresponds' to the essence of the problem or the question as such. It is as though there were an 'opening', a 'gap', an ontological 'fold' which relates being and the question to one another. In this relation, being is difference itself. Being is also non-being, but non-being is not the being of the negative; rather, it is the being of the problematic, the being of problem and question. Difference is not the negative; on the contrary, non-being is Difference: heteron, not enantion. For this reason non-being should rather be written (non)-being or, better still, ?-being. In this sense, it turns out that the infinitive, the esse, designates less a proposition than the interrogation to which the proposition is supposed to respond. This (non)-being is the differential element in which affirmation, as multiple affirmation, finds the principle of its genesis.
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