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- The first approach is to describe it in terms borrowed from linguistics, using a Saussurean-inspired model of language as a system of signifiers.
- The second approach is to describe it in terms borrowed from mathematics.
The two approaches are complementary, since both are attempts to describe formal systems with precise rules, and both demonstrate the power of the signifier.
Although there is a general shift in Lacan's work from the linguistic approach which predominates in the 1950s to a mathematical approach which predominates in the 1970s, there are traces of the mathematical approach as early as the 1940s. The branches of mathematics which Lacan uses most are algebra and topology, although there are also incursions into set theory and number theory.
"Mathematical formalization is our goal, our ideal."
"The root of the difficulty is that you can only introduce symbols, mathematical or otherwise, by using everyday language, since you have, after all, to explain what you are going to do with them."
Thus Lacan's use of mathematics is not an attempt to escape from the ambiguity of language, but, on the contrary, to produce a way of formalizing psychoanalysis which produces multiple effects of sense without being reducible to a univocal signification. Also, by using mathematics Lacan attempts to prevent all attempts at imaginary intuitive understanding of psychoanalysis.
- Lacan, Jacques. Écrits: A Selection. Trans. Alan Sheridan. London: Tavistock Publications, 1977. pp. 316-18
- Lacan, Jacques. Le Séminaire. Livre XX. Encore, 1972-73. Ed. Jacques-Alain Miller. Paris: Seuil, 1975. p. 108
- Lacan, Jacques. The Seminar. Book VII. The Ethics of Psychoanalysis, 1959-60. Trans. Dennis Porter. London: Routledge, 1992. p. 236
- Lacan, Jacques. Écrits: A Selection. Trans. Alan Sheridan. London: Tavistock Publications, 1977. p.311
- Lacan, Jacques. The Seminar. Book I. Freud's Papers on Technique, 1953-54. Trans. John Forrester. New York: Nortion; Cambridge: Cambridge University Press, 1988. p.2