Let me jump into the text at one of many critical points.
For every speaking being, the cause of its desire is, in terms of structure, strictly equivalent, so to speak, to its bending, that is, to what I have called its division as subject. (114-115)
If each pair of the following set of interpretations is not quite mutually-implying, at least each is linked to the entire set, Borromean-fashion. Perhaps, this is not a defect when it comes to a reading of a text whose central formal inspiration lies in pointing out the nonlocality of judgments of identity and difference in topological structures.
-For Fink's "bending", I suggest that we read "torsion", i.e., the result, on a transcendental surface, or a manifold which is thought, of an alteration in the dimensional space in which it is embedded. Whether in the present Lacanian form, or in Deleuze's meditation on the Riemannian effects of the addition and subtraction of dimensions of thought (N-1 / N+1, the former being particularly a matheme for torsion, in contrast with the permanent drive toward reflective disambiguation through the multiplication of dimensions; but as Deleuze makes admirably clear at points to which he is not entirely faithful, only understandable through the latter, subtraction through reflection) we can recognize the torsion of thought as a form (even an Idea) on which contemporary French philosophy converges with increasing intensity and explicitness. Moreover, it is a pure metamatheme, if one may assign this name to formalisms that require a metamathematical standpoint to be understood, with the caveat, of course, that we recognize, since Godel's fateful mathematization of metamathematics, that every metamatheme is also a matheme, that mathematics is not only a first-order theory, but the locus of a dizzying assortment of potentially reflexive morphsisms.
The prototype of Lacanian torsion is surely the Mobius. The question that faces a philosopher, in contemplating the Mobius, is whether the fatal twist that defines its being affects only an empirical manifold, or the very fabric of thought. That is to say, does an event, contrary to the law of transcendental usage, befall the concepts of identity and difference when the Mobius is twisted, to just the extent that a physical force is exerted on a ring of paper? Contrary to popular usage, it is not a Platonist who is forced to the former deflationary analysis - at least not if the author of the Sophist is a Platonist.
-To be equivalent "in terms of structure", is, on the well-established Lacanian thesis that structure is topology, to be topologically equivalent. We can understand this, at a first pass, in terms of the ordinary distinction between sense and reference. That is, the object-cause of desire (objet petit a) is seen
The payoff of this first translation is that it, rightly, transposes reference out of the zone of the model theory of first-order logic and into mathematics proper (even if mathematics, in the Lacanian text, is limited to topology for the sake of preserving a role for the psychoanalytic supplement.)
-By "subject" I understand that dimension of what is usually called a "mind" that is not reducible to the function of knowledge.
This definition of a subject would call for two further remarks.
First, I invoke that identification that I am proposing to make explicit in these remarks - which is partly supporting of / supported by the Lacanian text (the usual circularity of coherent discourse), but also partly critical of it - is that knowledge in the particular form of first-order theory is fantasy or ideology. Of course, this thesis implies that science, as first order theory, is blind to its own political status, a claim entirely in line with Zizek's fusion of Lacanian topology with Critical Theory. But also, and perhaps more significantly, this identification implies that it is specifically qua first-order theory that science is ideology. Given that FOT's only effective function (like that of Plato's Sophist) is to use the tacit metatheory of mimesis to deceive its subject into thinking that fantasy / ideology has a contentual relation to the real, that it reaches a referent other than itself, all of the actually effective work of science, as something distinct from its self-understanding (the metatheory of first-order theory as FOL-plus model theory), could never have been first-order theory. Maintaining this thesis requires only the recognition that 1) the mathematical content of science is effective, and 2) mathematics cannot be cast without remainder as a first-order theory. This second thesis recasts finally in a clear way what provided mathematics with its fatal attraction for Plato, and also what Kant sought in it under the heading of a
Nevertheless, its source is not dependent on any philosophical system, but on the purely negative pronouncement of Godel's First Incompleteness Theorem, which, if I may use a Lacanian idiom, couldn't care less about knowledge.
Thus it is the matheme, as against the fantasy of knowledge, that is "integrally transmissible", according to Lacan's oracular pronouncement. Or as Badiou puts it perhaps with greater clarity, "The non-ideological content of science is mathematics." And this criterion functions as much as a limit to the power of Critical Theory to displace scientific discourse - or indeed even to hermeneutical philosophy's self-appointed mission, whether in its Heideggerian or Wittgensteinian modes, to interpret the matheme - as it does to validate the criticism of first-order thought. It would be no accident that by virtue of dealing themselves unlimited powers of interpretation, both Heideggerians and Wittgensteinians find it necessary to make ambivalent recourse to an equally sweeping hesychasm. For Heidegger, philosophy's status as "useless but masterful knowing" is finally expressed in Gelassenheit, while for Wittgensteinians, having pointed out that meaning is use, philosophy subsequently "leaves everything as it is." Just as Meno (and Schmitt) would have it, thought's governing moves are made outside the game of thought. But a criterially-based metalogical reflection does not vacillate between omnipotence and impotence in this way. If it is true that the only legitimate target for actual philosophical criticism is a structure which is that of illusion, namely first-order thought, nevertheless, as Lacan never tires of pointing out, the function of this illusion in the real, that is, at the level of its signifier, is perfectly positive. Thus if - as Badiou states and Lacan performs - there is a sense in which first-order theory does indeed capture the content of what can be finitely represented of our everydayness, thus of what can be made into a conscious representation / knowledge in it, then this is quite enough work for philosophy to do.