At about the same time, in the social and psychological sciences Jacques Lacan pointed out the key role played by differential topology:
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. ^{57,58} |
^{57} Lacan (1970, 192-193), lecture given in 1966. For an in-depth analysis of Lacan's use of ideas from mathematical topology, see Juranville (1984, chap. VII), Granon-Lafont (1985,1990), Vappereau (1985) and Nasio (1987,1992); a brief summary is given by Leupin (1991). See Hayles (1990, 80) for an intriguing connection between Lacanian topology and chaos theory; unfortunately she does not pursue it. See also Zizek (1991, 38-39, 45-47) for some further homologies between Lacanian theory and contemporary physics. Lacan also made extensive use of concepts from set-theoretic number theory: see e.g. Miller (1977/78) and Ragland-Sullivan (1990).
^{58} In bourgeois social psychology, topological ideas had been employed by Kurt Lewin as early as the 1930's, but this work foundered for two reasons: first, because of its individualist ideological preconceptions; and second, because it relied on old-fashioned point-set topology rather than modern differential topology and catastrophe theory. Regarding the second point, see Back (1992).
^{59} Althusser (1993, 50): "Il suffit, à cette fin, reconnaitre que Lacan confère enfin à la pensée de Freud, les concepts scientifiques quélle exige". This famous essay on "Freud and Lacan" was first published in 1964, before Lacan's work had reached its highest level of mathematical rigor. It was reprinted in English translation in 1969 (New Left Review).
^{60} Miller (1977/78, especially pp. 24-25). This article has become quite influential in film theory: see e.g. Jameson (1982, 27-28) and the references cited there. As Strathausen (1994, 69) indicates, Miller's article is tough going for the reader not well versed in the mathematics of set theory. But it is well worth the effort. For a gentle introduction to set theory, see Bourbaki (1970).
^{61} Dean (1993, especially pp. 107-108).
^{62} Homology theory is one of the two main branches of the mathematical field called algebraic topology . For an excellent introduction to homology theory, see Munkres (1984); or for a more popular account, see Eilenberg and Steenrod (1952). A fully relativistic homology theory is discussed e.g. in Eilenberg and Moore (1965). For a dialectical approach to homology theory and its dual, cohomology theory, see Massey (1978). For a cybernetic approach to homology, see Saludes i Closa (1984).
^{63} For the relation of homology to cuts, see Hirsch (1976, 205-208); and for an application to collective movements in quantum field theory, see Caracciolo et al. (1993, especially app. A.1).
^{64} Jones (1985).
^{65} Witten (1989).
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