Lacan is one of the most famous and controversial psychoanalysts. I shall not discuss his work on psychoanalysis; but one finds many mathematical notions in his writings. Here are some examples: at a conference held in Baltimore (USA) in 1966, Lacan said:
This diagram [the Möbius strip] can be considered the basis of a sort of essential inscription at the origin, in the knot which constitutes the subject. This goes much further than you may think at first, because you can search for the sort of surface able to receive such inscriptions. You can perhaps see that the sphere, that old symbol for totality, is unsuitable. A torus, a Klein bottle, a cross-cut surface, are able to receive such a cut. And this diversity is very important as it explains many things about the structure of mental disease. If one can symbolize the subject by this fundamental cut, in the same way one can show that a cut on a torus corresponds to the neurotic subject, and on a cross-cut surface to another sort of mental disease. [Lacan (1970), pp. 192-193]
Lacan does not give any reason to think that the rather abstract geometrical notions which he mentions explain ``many things about the structure of mental disease." Of course, one might think that this is just an analogy. Well, even so: what purpose would these analogies fill? But here is a dialogue following Lacan's lecture:
HARRY WOOLF: May I ask if this fundamental arithmetic and this topology are not in themselves a myth or merely at best an analogy for an explanation of the life of the mind?
JACQUES LACAN: Analogy to what? ``S'' designates something which can be written exactly as this S. And I have said that the ``S'' which designates the subject is instrument, matter, to symbolize a loss. A loss that you experience as a subject (and myself also). In other words, this gap between one thing which has marked meanings and this other thing which is my actual discourse that I try to put in the place where you are, you as not another subject but as peoplethat are able to understand me. Where is the analogon? Either this loss exists or it doesn't exist. If it exists it is only possible to designate the loss by a system of symbols. In any case, the loss does not exist before this symbolization indicates its place. It is not an analogy. It is really in some part of the realities, this sort of torus. This torus really exists and it is exactly the structure of the neurotic. It is not an analogon; it is not even an abstraction, because an abstraction is some sort of diminution of reality, and I think it is reality itself. [Lacan (1970), pp. 195-196]
So, when he is asked explicitly whether it is an analogy, Lacan denies it. Of course, to say that the torus is ``reality itself", makes no sense, even if one were speaking about physics, where mathematics can be applied in a precise way.
Lacan used also notions of (point set) topology:
In this space of jouissance, to take something bounded, closed, is a location, and to speak about it is a topology. ...What does the most recent development of topology allow us to put forward concerning the location of the Other, of this sex as Other, as absolute Other? I will put forward the notion of compactness. Nothing is more compact than a fracture; clearly, the intersection of everything that closes being admitted as existing on an infinite number of sets, it follows that the intersection implies this infinite number. It is the very definition of compactness.[Lacan (1975)]
Here Lacan uses several words that enter into the mathematical definition of compactness (intersection, closed etc...) without paying any attention to their meaning. His ``definition" makes no sense whatsoever. And, of course, no argument is given that could conceivably justify a relationship between compactness and ``jouissance".
In other texts, Lacan ``develops" the role of imaginary numbers:
Thus, by calculating that signification according to the algebraic method used here, namely
...Thus the erectile organ comes to symbolize the place of jouissance, not in itself, or even in the form of an image, but as a part lacking in the desired image: that is why it is equivalent to the of the signification produced above, of the jouissance that it restores by the coefficient of its statement to the function of the lack of signifier -1. [Lacan (1971); seminar held in 1960.]
Clearly, the square root of -1 looks deep and mysterious to people who have not studied mathematics. But the relation between and jouissance is even more mysterious, at least for us.
In the works of Lacan, one finds many other abuses, e.g. on mathematical logic, physics and knot theory. It seems reasonable to assume that, far from providing honest and useful analogies, these references allowed Lacan to impress his non-mathematical audience with a superficial erudition and to put a varnish of scientificity on his discourse.