In “Science and Truth” there is a reference to Gödel’s theorem of incompleteness (Lacan, 1965: 861). Gödel was a mathematical logician who invented his theorem in 1931. It is applied to formal systems and asserts that those containing” minimum of arithmetic are incomplete and inconsistent. The arithmetic they contain is logical according to Peano’ s axioms.
Lacan is applying this theorem to the subject of science. I will try to show that this is the point in the Écrits at which a formalisation of psychoanalysis begins as a system which contains a minimum of arithmetic.
Gödel’s proof his conclusion is left for whomsoever desires to work through it. Only the conclusion is stated as follows: In any formal language A there exists a statement S such that if A is consistent, neither S nor its negative can be proved in A. The propositions of A cannot be proved by reference to A. It does not mean that S is false but undecidable. The axioms of A are not just incomplete but incompletable since the addition of an axiom does not block the emergence of another statement S’ that cannot be decided. His theorem of incompleteness entails that the consistency of A cannot be proved by any means within A.
Let A be the Other which is the concept of the unconscia-us. In “Science and Truth”, Lacan is not applying the theorem to the concept of the unconscious but to the subject of the unconscious. Nevertheless, Gödel’s theorem asserts that there is a lack in the Other, that the Other is incomplete and inconsistent, which is why Lacan writes it as the barred Other: A.
A statement S in the field of the Other cannot be guaranteed as true. Take a construction or any form of interpretation which attempts to complete the Other and make it consistent, that is, to fill out the lack in the Other. Such an intelpretation is neither true nor false, but undecidable; it is a proposition of the Other and cannot be proved by reference to the Other. Whatever fresh knowledge follows in the wake of an interpretation is not an indication of its truth since this knowledge is also a proposition of the Other. Neither the analysand’s ‘yes’ nor his ‘no’ are signs of the truth or falsity of an interpretation. Freud says in “Constructions in Analysis” that what is important is what comes indirectly (Freud, 1937d). The analysand says “no” and somewhere else says “yes.” That would be an interpretation that keeps the subject divided between the true and the false, which does not suture the subject.
The subject of science is being made the focus of logic. Modem logic sutures the subject of science. Gödel’s theorem, says Lacan, demonstrates that the suture has failed.
Logic makes a decision on what is true and false. Aristotelian logic makes it in natural language. Modern logic creates an artificial language, that is, a formal system, in which the decision is made.
The subject of natural language is described by Lacan in “Science and Truth” as the speaking suhject of linguistics in which the subject is determined as meaning in a battery of signifiers (Lacan, 1965: 860). This is not the subject of science. The subject in psychoanalysis, however, is the subject of science (Lacan, 1965: 858). The subject of science cannot be, then, a subject of natural language. It seems to me that this can be taken as the point at which a formalization is beginning.
Modern logic attempts to reveal the structure of science ostensibly. Lacan says it sutures, not science, but the subject of science. Science does not say true or false; the subject does. In a philosophy of science called logical empiricism theoretical terms are made dependent on observation terms. The truth of the observation terms must be guaranteed in theoretical terms. The subject of science must always be true. The subject who says “no” and somewhere else says ‘yes’ is divided between the true and the false (see Miller, 1994). Suturing this division makes the subject true. Gödel’s theorem confIrms the existence of the division. The subject is a logical inconsistency. The fIrst step in this formalisation asserts that the subject is undecidable, which is an indication that the subject contains arithmetic.
In the clinic it is an empirical fact that the subject speaks a natural language. On the other hand. the division of the subject is not an empirical fact but the effect of a reduction which may take a long time to accomplish (Lacan, 1965: 855). This reduction has to do with the shrinkage of knowledge since the subject is also described by Lacan as the result of the rejection of knowledge (Lacan, 1965: 856). The reduction is the direction of the treatment to a decompleted and inconsistent Other. the effect of which is the subject of science.
The subject is divided, Lacan argues, between truth and knowledge (Lacan, 1965: 856). If knowledge shrinks, it is contingent. In logic truth is necessary. It is not, however, the subject thal is necessarily true. According to Gödel’s theorem, it is a logical inconsistency.
There is a formalisation of both ends of the fantasy; beginning with the subject, arithmetic is introduced into the system, and, therefore, Gödel’s theorem is asserted. In the paper that precedes the one under consideration, Lacan begins by stating that the drive as constructed by Freud is prohibited to psychologising thought which supposes a moral in nature (Lacan, 1964: 851). Here is a good reason for formalisation. The basis of psychologising thought is natural language. The drive cannot enter natural language.
In his introduction to the Foundarions of Arithmetic, Frege says that his method goes against psychologising thought. Formalisation constitutes a reduction that may take a long time of psychologising thought. Knowledge shrinks, and the subject encounters the truth of the drive. The drive divides the subject and desire (Lacan, 1964: 853). The truth of desire seems to account for the logical inconsistency of the subject. Such is the structure of fantasy, according to Lacan.
Also in the article which precedes “Science and Truth” is Lacan’s account of the point at which an analyst is made (Lacan, 1964: 854). At the end of analysis the drive has something to do with the emergence of the desire of the analyst. This is not enlarged upon by Lacan here, though elsewhere he states that it is also the desire of the analyst which has been operating in the accomplishment of the analysis. It must be the desire to create a language in which the subject can say the truth. Formalisatbn is the expression of the relation of the desire of the analyst to the truth. It seems to me that the desire of the analyst is structured by Gödel’s theorem, and the form of interpretation must be affected by it. Without it, there will be no concept of the sometimes long reduction to the division of the subject, to the point of a manque à savoir, a want-to-know, since the subject is a lack outside knowledge. It seems to be a reduction to the first axiom of Peano: zero is a number. A lack in the foundations is just this zero; a painful emptiness that will make the subject inconsistent between the true and the false, and make it desire to find an Other that is complete and consistent.
Gödel’s Incompleteness Theorem
The opening paragraph of Kurt Gödel’s 1931 paper:
The development of mathematics in the direction of greater precision has led to large areas of it being formalized, so that proofs can be carried out according to a few mechanical rules. The most comprehensive formal systems to date are, on the one hand, the Principia Mathematica of Whitehead and Russell and, on the other, the Zermelo-Fraenkel system of axiomatic set theory. Both systems are so extensive that all methods of proof used in mathematics today can be formalized in them; i.e., can be reduced to a few axioms and rules of inference It would seem reasonable, therefore, to surmise that these axioms and rules of inference are sufficient to decide all mathematical questions which can be formulated in the system concerned. In what follows it will be shown that this is not the case, but rather that, in both of the cited systems, there exist relatively simple problems of theory of ordinary whole numbers which cannot be decided on the basis of the axioms.
Kurt Gödel, “Über unentscheidbare Sätze del Principia Mathematica und verwandter Systeme I”, Monatshefte für Mathematik und Physik 38: 173-98. Composite translation, cited in Raymond Smullyan, The Lady or the Tiger?, Harmondsworth: Penguin, 1982, p. 163.
