The conflict between on the one hand a biological naturalism or essentialism and on the other hand a social constructivism is a conflict or an opposition that is as old as psychoanalysis itself. It can even be claimed that it is a pre-Freudian conflict as the necessity of using the uncommon concept of drive made psychoanalysis resistant to a reduction to either biologism or culturalism. Nevertheless there have been spokesmen for both biologism and culturalism from the beginning of psychoanalysis.

In Lacanian psychoanalysis, following Freud, you find a defense of the concept of sex as “real”, not in the sense of a biological determinism but in the sense of that which resists symbolization, that which creates ail impasse, that which is nevertheless not pre- symbolic but intrinsically connected to the symbolic and the imaginary. In the late Lacan the real is never “pre-” or ontologically prior to the symbolic. Instead of wearying everybody with this constant “it is not… ,” Lacanian formalism serves the purpose of presenting this impasse, the inconsistency of the symbolic.

The seemingly naive question of why it was necessary for Lacan to express or present his thinking in pseudo-mathematical terms, his so-called “mathemes”, why he was obsessed with mathematics in the widest sense of the word—logic, algebra, and topology—can find an answer in the question about the impasse of sex. It shall be argued that the turn to mathematics is an effort to think against the negation and disavowal of sex which both biological determinism and sex conceived as gender—as a social construction—represent.

In the following interpretation of the formulas of sexuation from the Encore Seminar [1] in connection with some points concerning negations in intuitionistic logic it shall be argued that this particular formalization is exemplary for the necessity of formalization. The interpretation can never replace the logical and algebraic terms, the mathemes. The “prosaic” interpretation or translation of these is always just one possible interpretation among others of an impasse that can only be presented adequately in mathematical terms. Consequently Lacan could call mathematics “the science of the real”.

This justifies or can at least explain the fact that one does not find an interpretation of the formulas of sexuation in the Encore Seminar or the previous unpublished Seminar …*ou pire*, only short hints at an interpretation. Their enigmatic status provokes different interpretations, and they thereby reveal the fundamental precondition for thinking, that is, transference. Thought or knowledge is supposed to be condensed in those formulas and each from his point of view develops an interpretation. [2]

I will start with a brief summary of some basic points concerning the way sexual difference is thought in psychoanalysis in order to show how the traditional conception of sexual difference are different variations on the same theme: the negation or disavowal of sex.

According to Freud castration is a threat as far as the boy and the man are concerned, a fait accompli for the girl and the woman. Sexual difference is defined by the question whether one has or has not a penis. The woman’s body is defined by what you do not see, defined by what she lacks. According to Freud sexual difference is not a question of one sex having something—a penis—and the other sex having something else—clitoris, labia, and vagina—that could be considered to be worth just as much. This would be a comforting democratic fantasy about equality between the sexes. Sexual difference concerns the fact that woman is represented by her lack of the masculine attribute par excellence. She is not complete.

To Lacan, however, castration is not a question whether the subject has or has not what he calls the phallus [3] as no one can “have”, possess the phallus. Therefore you cannot distinguish between the sexes as far as castration is concerned as all, irrespective of sex, are subject to castration.

The definition of the subject as lack is a point made elsehwehere in the history of philosophy, but the originality of Lacan’s point is that it turns the issue of castration away from the questions of whether or not the subject has the phallus toward the issue of whether the Other does not have it. It is the Other who lacks, it is the Other who is incomplete, written is the Other who is castrated.

A traditional evolutionary or ontogenetic exposition of the question goes as follows: the child discovers that the (phallic) mother lacks something, that she acts according to laws the child does not understand, that according to the law her desire has to be directed elsewhere, “the father”, “work”, “what ever” demands her attention, her desire. Lacan’s famous thesis, “desire is the desire of the Other” means that my desire is not “mine” but the question of what the Other desires, the question of the lack of the Other, that the subject is the question about what it is for the Other. When for instance the Other is an enigma you ask the question: this is what he says, but what does he mean, and what does he want from me, if anything? What am I to him? An enigma is an “enonciation” without an enoncé, a statement without a meaning. Any statement can therefore represent an enigma.

The name for what is lacking in the Other is the phallus. Phallus is the signifier of the lack of signifier, the signifier of castration. The phallus is the name of the object of desire, but as it is not possible to decide what is the object of the desire of the Other, the phallus is the name for the fact that desire has no object that can be represented as such, but is the well known metonymic “slide” from signifier to signifier, it is the signifier of lack as such.

**Sex as a fantasy**

For Lacan, however, the case is that man can only be a universal via the fantasy of the woman. The way woman and sexual difference has usually been thought of is fantasmatic in the sense that these are ways in which it can be covered up that man is not whole neither as a man nor united with a woman. The well-known points concerning the representation of sexual difference are all variations on the same theme, ways to cover up the lack and the fact that there exists no sexual relationship. This lack is covered up by representing sexual difference as a symmetrical relationship that ascribes the lack to the woman—man represents the well known clichés: rationality, activity, culture. The woman incarnates the reverse side of these clichés, irrationality, passivity, intuition, nature. This has been shown and repeated over and over again. Sexual difference is thought of as complementary elements as in Aristophanes’ speech in Plato’s *Symposium* where man is cut into two parts, a female part and a male part, and thereafter the visible result: the two sexes that desire to be reunited, that united will form a whole. The lack has been covered up, “the twain become one flesh” as it is called in The Old Testament.

The basic point concerning sexual difference is that the two poles are not symmetrical and equal not only in the sense that the male pole is more culturally appreciated but also in the sense that the two poles are not both original – as Aristophanes’ speech could imply. One pole is the original; one pole is ontologically prior to the other. If we once again start with the beginning of our culture, with nothing less than Genesis, we learn that Eve has been taken from or built from the famous rib of Adam. The woman is derived from man, man is the cliché and woman is a deviation from this form and norm. Therefore we can talk about the human being as being identical with man, man as man, man as universal.

These are all too well known points. The attempt to represent woman must inevitably be a definition in relation to man, man and woman as opposites. Femininity is within this horizon necessarily the seamy side of man. Woman is only as a result of the comparison to man. There can be no such thing as an independent definition of woman, a definition that does not imply a definition of man.

Sexual difference is a contradictory one, which means in logic that if one proposition is true, its negation must be false, also called “Law of Excluded Middle,” p or not-p. If, however, sexual difference was thought of as a contrary relationship there would be a third possibility. The traditional example goes: all swans are white, which is contrary to ‘no swans are white’. There is a third possibility: ’some swans are black’, so both propositions can be false. This means for our present problem concerning sexual difference that if everybody is castrated contrary to ‘nobody is castrated’, there would be a third possibility: some have a vagina, labia, clitoris. And sexual difference would not be a question of castration, but a question of having a penis in contrast to having a vagina, labia, clitoris. The psychoanalytic point is simply that this would be a fantasy of completeness, a fantasy that denies castration. For Lacan there is only one thing to be said about woman which is “that she does not exist” and “that nothing can be said about the Woman”. But what does “exist” mean here?

The woman as a symptom

Lacan also states that, “there is no sexual relationship”. In the seminar … ou pire (unpublished from 1971-72) the female position is defined as “that which is not included in the phallic function without, however, being its negation”. And this is what the formulas of sexuation from the seminar *Encore* present.

♂♀

∃x ¬(Φx) ¬(∃x) ¬(Φx) ∀x Φx ¬(∀x) Φx

On the left, male side it says ∃x ¬(Φx), which means that there exists someone, Φx, for whom the law of castration does not count. And then ∀x Φx, which means that it applies to everybody, ∀x, that they are subject to the law of castration. On the male side we have the two contradictory propositions. The crucial point is firstly that both propositions cannot be true and secondly that the particular negation of the phallic function, ∃x ¬(Φx) is not the woman. The woman cannot and should not be thought of as a negation of man, as the particular in contradiction to the universal. Not to be defined by castration is rather a position that consists of the limitless power and the limitless access to and enjoyment (*jouissance*) of all women. The mythical primal father in Freud’s Totem and Taboo, is not subject to the law against incest and thereby castration. He is the law by being the exception to the rule or the law. In other words, this last modern myth is like all myths, an answer to a structural and logical necessity, a logical impasse and not just a fairy tale. The universal is defined by its limit, in our case if ‘men’ are a set that is defined by castration it must imply in all its banality that there is someone or something that has produced this law by which it can be decided whether or not an element belongs to the set, is inside or outside, and consequently this ’something’ or ’someone’ cannot be defined by the trait of being castrated. It is the exception that proves the rule. In this sense the universal quantifier presupposes the particular negation and the dead primal father fills the place of the exception. It is no doubt a fairy tale, but nevertheless a necessary one. If it is true that everybody is subject to castration it cannot also be true that there exists one who is not subject to castration. The idea must nevertheless be thought of in the form of a negation, the particular negation is a logical consequence. In Freud this is thought of as a creation myth in order to explain how the fundamental law against incest is constituted and is constitutive for culture in general.

On the right side, the female side it says ¬(∃x) ¬(Φx). There exists no one who is not castrated. The question is why this is not identical with the male side ∀x Φx, why two negations do not as in classical logic negate each other, not, not-p is identical with p. Lacan receives assistance from intuitionistic logic which dreams of a mathematics free of negations. [4] The reason for this dream is rather metaphysical than mathematical as it concerns the question of the nature of the mathematical object. Platonism, or realism, is based on the assumption that mathematical objects exist independently of mathematics and mathematicians.

For intuitionists, however, mathematical objects exist only by virtue of being what they call “mentally constructed”. “To exist” means “to be a mental construction”, that is without reference to the metaphysical questions regarding the nature of the constructed objects, such as whether these objects exist independently of our knowledge of them. The basis for intuitionists is consequently that we can only assert a proposition p if we can prove p. That we do not have a proof of the negation of p does not imply that we have proven p. The object has not been constructed wherefore we cannot say that it has been proven to be true. Intuitionistic logic does not recognize the principle of the “Law of Excluded Middle.” If we assume a proposition not-p and this leads to contradictions, we have indirectly proven the proposition p. But intuitionists do not recognize p or not-p as a universally valid inference. It is a rule that is not valid concerning infinite sets. Heyting’s example is a sequence of prime numbers. A prime number is a number that can only be divided by 1 and itself (1,2,3,5,7,11,13,17,19, etc.) The definition of a natural number can be,

1) “k is the greatest prime such that k-l is also a prime, or k=1 if such a number does not exist.”

2) “1 is the greatest prime such that l-2 is also a prime, or 1= 1 if such a number does not exist.”

The difference between 1) and 2) is that k can be calculated. k=3, as 2 is the only even prime number, all following prime numbers-l must necessarily be even numbers and therefore not prime numbers. The definition of k is therefore finite, since k can be defined. The point is that we do not possess a general method for calculating l, as it is not known whether the sequence of pairs of twin primes p, p+2 is or is not finite. Therefore intuitionists will reject 2) as a definition of a natural number. It cannot be decided whether the sequence of twin prime numbers is or is not finite (1+3, 3 + 5, 5+7, 11+13, 17+19, etc.) We can only take one prime number at a time and add 2 infinitely. There is no proof of or general method for deciding whether this set of prime numbers is or is not finite. It transcends verification. Therefore l-2 cannot intuitionistically be recognized as being a natural number. l-2 cannot be defined as there exists no general method for calculating it.

The difference between Cantor’s and the intuitionists’ definition of a set concerns the difference of how the law is determined. Following Cantor the law which decides which elements can be comprised as a set is there in advance. This law does not concern itself whether there are actually any elements that can be subsumed under the definition of this law. Therefore you can operate with the empty set, Ø. A set is any collection of objects that can be thought of as One, as a whole. The set corresponds to the platonic Idea. As far as the intuitionists are concerned, however, it makes no sense to talk about a law or a set irrespective of its being possible to decide if an element belongs to the set and if there are any elements that can be said to belong to the set at all. As far as the intuitionists are concerned the empty set is an “empty formal postulate”. The set is a law that decides whether this or that predicate can designate this or that element. “Cantorians” can operate with “the set of the eternal feminine” independently of and unconcerned with whether there actually exist any elements that can be attributed to the predicate “the eternal feminine”. Intuitionists, however, cannot operate with the set of “women” without deciding the question whether there exist any elements in the set. For an intuitionist the intension of a set is identical with its extension. The set of “women” can be determined by a law that designates the predicate “delicate” and “maternal”. Thereafter it must be decided one by one whether this or that element belongs to the set of “delicate” and “maternal” and thus can be said to belong to the set of “women”.

The elimination of the double negation, on the ground that two negations negate each other, is not a valid inference in intuitionistic logic. You cannot assert not-, not-p → p, but you can assert p → not-, not-p, as a proof of p must result in that you cannot get a proof of p’s negation. It must be noted that it is not shown how you move between the formulas – you can infer from the left side to the right side, but not the other way round. The formulas do not indicate rules of inference but must be interpreted as propositions about elements.

The contradictory sentence on the right side says “not all women are subject to castration” and corresponds to “there exists someone who is not subjected to castration” on the left side, that is, Freud’s primal father or “the name of the father” in Lacan. But, again, here you cannot talk about a proposition that can be shown to be either true or false. Therefore Cathrine Millot can note (following Lacan) in her essays on transsexualism, that ¬(∀x) Φx, the idea of Woman with a capital letter is one of the names of the father. The Woman = The primal father. Consequently you can only be a woman if you are not a real woman.

“Nothing can be said about the woman,” says Lacan. No one exists who is not determined by castration, but hereby you have stated nothing about the woman as distinct from man, and nothing has been stated that can be judged to be either true or false. The right side of the formulas says only that women are also subjected to the law of castration but it says nothing about what they are as distinct from men. In other words, woman does not form a well-defined set where you can decide in advance whether this or that element belongs to the set. The question of castration is undecided. It transcends verification. The woman is not a universal. The formulas serve the purpose of presenting the idea that there both is and is not a difference between men and women, and this difference cannot be decided.

Thus the difference between Freud and Lacan is that Freud was a platonist or a Cantorian—the set of women was decided by a law called lack of penis—and Lacan was an intuitionist. But this difference only holds until Freud’s old age when he became a skeptic. Both Freud and Lacan reached the same conclusion: as far as the question of women was concerned they did not have much to contribute. For Freud, femininity stayed as the much-quoted “dark continent” when you leave out Freud’s youthful shortcuts in three Themes of Sexuality, where femininity fits masculinity like a glove. In his old age he became a skeptic and appealed to the women among his pupils and colleagues for an answer to his doubt. In his late article from “New Introductory Lectures on Psychoanalysis” it is stated that “the little girl is a little man” in the pre-oedipal phase. In the less popular and more subtle article “Female Sexuality”, this idea of “a complete parallelism” has been given up. The famous passage goes: *Aile Erwartungen eines glatten Parallelismus zwischen mannlicher und weiblicher Sexualentwicklung haben wir ja langst aufgegeben. Die Einsicht in die praodipale Vorzeit des Madchens wirkt a1s Uberraschung, ahnlich wie auf anderem Gebiet, die Aufdeckung der minoisch-mykenischen Kultur hinter der griechischen. Alles auf dem Gebiet dieser ersten Mutterbindung erschien mir so schwa analytisch zu erfassen, so a1tersgrau, schattenhaft, kaum wiederbelebbar, als ob es einer besonders unerbittlichen Verdrangung erlegen ware.* [5]

Comparably Lacan’s notations are precise and elegant ways in which the same point about the impasse of sex can be presented. Apparently the woman fits man like a glove, as being his negation, seemingly there is a relation, a symmetrical relationship, *ein glatter Parallelismus*. But a closer inspection of the formulas shows that apparently the same thing is stated on both sides (with and without negations). If we follow the rules of inference in classical logic as far as negations are concerned, there is no difference between men and women; all are castrated. But if we follow intuitionistic logic there is the small but crucial difference between the male and female side of the formulas, insofar as on the feminine side nothing has been proposed that can be shown to be either true or false.

When Lacan says, “there is no sexual relationship”, relationship means a symmetrical, complementary relationship as for instance the interior in relation to the exterior, culture-nature, rationality-intuition, sense-sensibility, logic-epic. The phallus is indeed what marks the sexual difference, but not that which can represent sexual difference. “Nothing can be said about the woman”, there is no feminine essence, the woman is not a universal. It is rather sex that disturbs the universal man. Consequently the point is not that instead of having just one universal, man, we must have two universals, man and woman, for the sake of completion. The point is rather that sex is the constant contestation of the universal “man”. Phrased in the terminology of psychoanalysis: castration, the pointing out of incompletion, of lack.

Woman—that does not exist – has been the name of that which has been expelled, exiled from the symbolic—but also that which “keeps coming back”, “keeps returning” and with a disruptive effect on the symbolic and the imaginary in the guise of that which cannot be without friction, “that which keeps returning to the same place”. In short, a symptom. A symptom is “something” that is neither a thing nor nothing. It raises a question and implies a lack of knowledge. Its place is in speech. It is something to be heard by the Other. A fantasy, however, does not raise but rather answers a question. It is an answer to an impossibility, an answer that covers up a lack and prevents further questioning. In this sense all representations of femininity are fantasies.

To conclude: sex and woman have to be thought of in “the real”. That something “does not exist” (woman) and “is not” (the sexual relationship) does not mean that it is “nothing” but rather that no signifier can represent it, that it belongs to the real. To exist means to be represented by a signifier.

The logical notations serve the purpose of talking about a difference that cannot be determined, a presentation of difference that does not end up in imaginary complementarity. The interpretation of the formulas of sexuation above is one of many examples of the fact that Lacan’s formalizations are not only to be considered a short hand for that which could be expressed adequately in ordinary prose. The formalizations are not just supposed to be redundant. If that were the case an exposition in ordinary prose would be able to replace the formal notations. The contrary is the case: the formalizations serve the purpose of presenting and talking about an experience of a paradox, an impasse, an impossibility. With the help of the formulas it becomes possible both to recognize an impossibility as an impossibility without covering it up in imaginary representations or historically variable constructions and to know what kind of impossibility we are talking about.

It is in this sense that psychoanalysis can be reduced to neither a biological nor a constructivist, historicist perspective. Both poles of this opposition negate the impasse, negate “the real” of sexual difference.

**Notes**

[1] Jacques Lacan, *Le Seeminaire, Livre XX, Encore,* Paris: Seuil, 1975.

[2] For another interpretation of the formulas in connection with the antinomies of reason in Kant’s* Critique of Pure Reason*, see Joan Copjec *Read My Desire*, chapter 8 ”Sex and the Euthanasia of Reason,” The MIT Press, Cambridge: Massachusetts, 1994. See also Catherine Millot’s Horsexe, *Essai sur le transsexualisme*, Point Hors Ligne: Paris 1983.

[3] With the exception of the article on “Fetichism” Freud writes the noun “phallus”, more the adjective as in the “phallic” phase and the “phallic” mother.

[4] Heyting, A., *Intuitionism. An Introduction*, Amsterdam 1956.

[5] *GWXIV*, p.519.

Art: Hellen Van Meene, Untitled, C-print, 2008.