Algebra

Algebra (algèbre) is a branch of mathematics which reduces the solution of problems to manipulations of symbolic expressions. Lacan begins to use algebraic symbols in his work in 1955 (see schema L), in an attempt to formalise psychoanalysis. Three main reasons lie behind this attempt at formalisation:

1. Formalisation is necessary for psychoanalysis to acquire scientific status (see science). Just as Claude Lévi-Strauss uses quasi-mathematical formulae in an attempt to set anthropology on a more scientific footing, Lacan attempts to do the same for psychoanalysis.

2. Formalisation can provide a core of psychoanalytic theory which can be transmitted integrally even to those who have never experienced psychoanalytic treatment. The formulae thus become an essential aspect of the training of psychoanalysts which take their place alongside the training analysis as a medium for the transmission of psychoanalytic knowledge.

3. Formalisation of psychoanalytic theory in terms of algebraic symbols is a means of preventing intuitive understanding, which Lacan regards as an imaginary lure which hinders access to the symbolic. Rather than being understood in an intuitive way, the algebraic symbols are to be used, manipulated and read in various different ways.

Most English translations of Lacan also translate the algebraic symbols which appear in his work. For example, Alan Sheridan, in his translation of Ecrits, renders the symbol A (for Autre) as O (for Other). However, Lacan was opposed to such a practice, as Sheridan himself points out. In this dictionary, in line with Lacan's own preference, the algebraic symbols are left as they are in the original French texts.

The algebraic symbols used by Lacan, which appear principally in the MATHEMES, SCHEMA L and the GRAPH OF DESIRE, are listed below, together with their most common meaning. However, it is important to remember that the symbols do not always refer to the same concept throughout Lacan's work, but are used in different ways as his work develops. The most important example of such a shift in meaning is the use of the symbol a, which is used in radically different ways in the 1950s and in the 1960s. However, even other symbols which are relatively stable in meaning are occasionally used in very different ways; for example, s nearly always designates the signified, but is used in one algorithm to denote the subject supposed to know (see Lacan, 1967). Therefore some caution should be exercised when referring to the following list of equivalences.

A = the big Other A = the barred Other a = (see objet petit a) a' = (see objet petit a) S = 1. (before 1957) the subject 2. (from 1957 on) the signifier 3. (in the schemas of Sade) the raw subject of pleasure S = the barred subject Si = the master signifier S2 = the signifying chain/knowledge s = the signified (in the Saussurean algorithm) S(A) = the signifier of a lack in the Others (A) = the signification of the Other (the messagelsymptom) D = demand d = desire m = the ego (moi) i = the specular image (schema R) i(a) = 1. the specular image (graph of desire) 2. the ideal ego (optical model) I = the ego-ideal (schema R) I(A) = the ego-ideal (graph of desire) H = the real phallus <fi = the symbolic phallus [upper-case phi] 9 = the imaginary phallus [lower-case phi] (-9) = castration [minus phi] S = the symbolic order (schema R) R = the field of reality (schema R) I = the imaginary order (schema R) P = the symbolic father/Name-of-the-Father p = the imaginary father M = the symbolic mother J = jouissance Je = phallic jouissance JA = the jouissance of the other E = the statement e = the enunciation V = the will to enjoy (volontÈ de jouissance)

The typographical details and diacritics are extremely important in Lacanian algebra. The difference between upper- and lower-case symbols, the difference between italicised and non-italicised symbols, the use of the apostrophe, the minus sign, and subscripts; all these details play their part in the algebraic system. For example the upper-case letters usually refer to the symbolic order, whereas the lower-case letters usually refer to the imaginary. The use of the bar is also important, and varies even within the same formula.