The next section reminds me strongly of Richard Rorty's thesis that philosophy has long been driven by the metaphor of concepts in a mind mirroring objects in nature. The goal of philosophy then becomes to make sure that the mirror is flat and polished and provide an accurate reflection of reality. If we could reach the perfect mirror, every existing thing in the world would have a 1-to-1 mapping to clear concepts in our mind. Call it representationalism.
Naturally, this idea is batshit insane. Taken to its limit, this line of thinking leads to something along the lines of Funes El Memorioso not to a chimp who has spent most of its phylogenetic life running from tigers, grubbing for food, and conning the other chimps. Whatever mirroring we actually do has always served a purpose. Nevertheless, the habit of thinking is deeply deeply engrained.
It's immediately clear why repetition creates problems in a bijective universe. I only need one concept to talk about two ducks that are exactly the same. Accordingly, repetition must mean there's something wrong with the mirror, a glitch in the Matrix. The section from pages 11-15 proposes 4 things that could have gone wrong with mirror.
The first two things that can go wrong actually seem to me like different flavors of the same problem.
The mirror metaphor comes along with some implicit baggage that may not be immediately apparent. Since we automatically assume that our thoughts and concepts have some sort of logical structure, the idea of the mirror means that the world must share this same structure. Right at the root of logic is the tree metaphor (wacka wacka). We assume our concepts are organized in a natural phylogenetic sort of tree: living thing>animal>human>John>John-on-the-Tuesday-after-the-4th-of-July-puking-on-the-flag. Since the real world is completely determined down to the tiniest detail in each and every moment, the tree is theoretically infinitely precise, which, by virtue of the fact that our mind mirrors it perfectly, means that we should have a distinct concept for each and every one of the leaves. Obviously, in practice, we find it useful to use the structure of the tree to talk about "big" concepts like "human" or "John" as if they were one thing, even though in reality, they are an infinite collection of things that resemble one another because they all branch off from the tree at a particular point. ("Extension=1" here is just Deleuze's way of describing perfect correspondence -- nothing left out, nothing left over, and nothing mapped to more than one thing)
Every logical limitation of the comprehension of a concept endows it with an extension greater than 1, in principle infinite, and thus of a generality such that no existing individual can correspond to it hic et nunc
and later:
Let us suppose that a concept, taken at a particular moment when its comprehension is finite, is forcibly assigned a place in space and time - that is, an existence corresponding normally to the extension = 1. We would say, then, that a genus or species passes into existence hic et nunc without any augmentation of comprehension. There is a rift between that extension = 1 imposed upon the concept and the extension - - that its weak comprehension demands in principle. The result will be a 'discrete extension' - that is, a pullulation of individuals absolutely identical in respect of their concept, and participation in the same singularity in existence (the paradox of doubles or twins).
The subtle distinction between these two cases comes in whether you're looking at the problem of mistaking a branch for a leaf as a logical or a linguistic one. You could argue that your conceptual mirror really is perfect, but that sometimes its easier to use a sort of logical shorthand that grabs a whole branch at once in order to get something done; but you could always go back to the individual concepts. Or you could think a little more concretely about how we actually use concepts and observe that we don't transmit them directly via mindspeak but use words. Those words are real existing things that work just like the logical concept, summarizing and manipulating an infinity of leaves as one unit.
... the existence of words, which are in a sense linguistic atoms, cannot be doubted. Words possess a comprehension which is necessarily finite, since they are by nature the objects of a merely nominal definition. We have here a reason why the comprehension of the concept cannot extend to infinity: we define a word by only a finite number of words. Nevertheless, speech and writing, from which words are inseparable, give them an existence hic et nunc; a genus thereby passes into existence as such
Looked at this second way, the fact that we often use the same grunt "duck" to refer to one or another actual duck means that there has been a true repetition in nature (namely, us grunting "duck") rather than just a theoretical, logical repetition in our head (which could always in principle be eliminated).
This phenomenon of discrete extension implies a natural blockage of the concept, different in kind from a logical blockage: it forms a true repetition in existence rather than an order of resemblance in thought.
As I said, two flavors of the same problem. In both cases a neighborhood has been substituted for a precise address.
The third way that the mirror can break down is what Deleuze refers to (following Kant) as the "paradox of symmetrical objects". Consider your left and right hand. As mirror images, we only need one concept to correspond to the two of them. This isn't the same sort of approximate neighborhood problem we saw in the first two cases. We can specify all the way down to the concept of my particular hand at a particular time, etc ... in other words we can increases the detail of the specification indefinitely, and yet the end product is still duplicated in fact but not in concept.
You may be tempted to argue here that my two hands are actually not exact mirror images. This is certainly true, but hold off on pushing this particular objection until you read page 20. In the meantime, consider things that as far as we know are exact mirror image duplicates, like, say, the positron and the electron. The symmetry is not left/right but plus/minus in this case, but the point is identical. The two are conceptually the same, but differ only in sign. Remember all those times you screwed up a physics exam because you got a sign wrong somewhere? But it was just a freakin' sign! You basically understood it, right? This sort of symmetry is pervasive in nature at many levels, which means that the whole works is sort of shockingly redundant.
I'm not really completely clear on how this is related to memory. I do see how this means that the concept of a left and right is not "in" nature but is more "of" or "about" nature. Does nature "know" that there has been this symmetrical mirroring? It's more like these two parts exist alongside one another without being aware that there's been a duplication. The only way they can be put back together, or understood as the same concept, is by a third party mind who stands outside nature and provides the link between them. I suspect that the connection to memory may be completed by another bit of unexamined baggage that got smuggled in with the Mirror of Nature metaphor. Somebody has to compare the contents of the mirror and the contents of the world to make sure they match. This implies a faculty of recognition, or better, re-cognition.
The fourth and final thing that can go wrong with the mirror seems to me another version of this meta-problem. What happens if you are unable to remember that this is the second time you're seeing something? You may still be able to know that two forms are the same, but do you know that you know? In other words, can you do something with that knowledge? (I wonder idly whether this is a little like some mental syndrome where someone is able to look into a mirror and look at themselves and see that the person in the mirror is the same as the real person, but be unable to recognize that the person in question is in fact them).
Take an individual notion or a particular representation with infinite comprehension, endowed with memory but lacking self-consciousness. The comprehensive representation is indeed in-itself, the memory is there, embracing all the particularity of an act, a scene, an event or a being. What is missing, however, for a determinate natural reason, is the for-itself of consciousness or recognition. What is missing in the memory is remembrance - or rather, the working through of memory.
Not "knowing that you know" dooms you to rediscover the same thing again and again. You may have memories, but not be sure that they are "your" memories. At first this might sound impossible, but spend five minutes reading Freud and you'll realize that it happens to you every day of your life. We're constantly unconsciously repeating the reaction we had to some earlier event. It's not that we're not capable of distinguishing that past event from the present one, it's that we don't know that we're capable of it. Here, nature repeats itself because we don't know (to infinity) that the mirror is perfect.
When the consciousness of knowledge or the working through of memory is missing, the knowledge in itself is only the repetition of its object: it is played, that is to say repeated, enacted instead of being known.