N.Pepperell over at Rough Theory has written a truly terrific post musing on some of my recent attempts to work out Lacan’s logic of fantasy. Apart from the fact that it responds to things that I’ve recently written and therefore affords me narcissistic gratification and provides some evidence that I exist, I think what I like most about this post is the way that its both generous in its reading while also remaining critical in a productive way. Responding to some of my comments about objet a and the remainder, N.P. writes,

Sinthome then relates the persistence of this “remainder” to the possibility for critique, arguing, if I’m understanding correctly, that the remainder retains the residue of a presymbolic realm from which the symbolic realm is necessarily constructed. The symbolic realm – including fantasy as desire expressed in symbolic form – therefore necessarily drags along in its wake its own “outside”.

I’d like to suggest that there’s another way of understanding Lacan’s concept of the remainder that doesn’t resort to treating it as a sort of pre-symbolic residue. Rather than treating the remainder as a residue of the pre-symbolic that resists symbolic integration, remainder could be taken in a much more literal mathematical sense as the result of an operation. Suppose we take a simple act of division such as the division of 3 by 5. Our solution is 1.666666667. Here there’s something that escapes the operation, something that is left over when 3 is subjected to 5. Lacan often liked to liken objet a or the remainder to the golden ratio and irrational numbers. He develops this comparison or analogy in detail beginning with the unpublished Seminar 14, The Logic of Fantasy, and makes passing allusion to it in Seminar 20, On Feminine Sexuality, The Limits of Love and Knowledge, when he remarks that,

If there is something in my Ecrits that shows that my fine orientation, since it is of that fine orientation that I try to convince you, is not such a recent development, it is the fact that right after the war, where nothing obviously seemed to promise a pretty future, I wrote “Logical Time and the Assertion of Anticipated Certainty.” One can quite easily read therein– if one writes and not only if one has a good ear –that it is already little a that thetisizes the function of hast. In that article, I highlighted the fact that something like intersubjectivity can lead to a salutary solution. But what warrants a closer look is what each of the subjects sustains, not insofar as he is one among others, but insofar as he is, in relation to the two others, what is at stake in their thinking. Each intervenes in this ternary only as the objet a that he is in the gaze of the others.

In other words, there are three of them, but in reality, there are two plus a. This two plus a, from the standpoint of a, can be reduced, not to the two others, but to a One plus a. You know, moreover, that I have already used these functions to try to represent to you the inadequacy of the relationship between the One and teh Other, and that I have already provided as a basis for this little a, the irrational number known as the golden number. It is insofar as, starting from little a, the two others are taken as One plus a, that what can lead to an exit in haste functions. (48-9)

I cannot get into a careful analysis of this dense passage at present, as my mind is mush and it would require a close commentary on Plato’s various dialectics of the One and the Other in the Parmenides, along with a discussion of certain elements of set theory. Perhaps Bobo or Austin are up to this work. I do give an extremely simplified version of what Lacan is referring to with respect to logical time in a comment replying to Anon, where I discuss the intersubjectivity at stake in mowing my lawn. In addition to this, the Japanese analyst Shingu Kazushige has written a very nice book meditating on this enigmatic line entitled Being Irrational: Lacan, the Objet a, and the Golden Mean. What is interesting about this metaphor of objet a as an irrational number or the golden ratio is that it evokes the notion of a twist, distortion, or ripple in the symbolic that isn’t a hold-over from a mythological pre-symbolic past (how could such a past fail to be mythological, given that we can only approach the world through language?), and that results from operations in the symbolic itself. Perhaps the “cash value” of this concept would be that it offers the possibility of a form of resistance immanent to the symbolic itself… Which is to say, that it shows the manner in which the symbolic is unable to produce closure.