by Alan Sokal
22 Bohr (1928), cited in Jammer (1974, p. 90).
23 Bell (1987, especially chs 10 and 16). See also Maudlin (1994, ch. 1) for a clear account presupposing no specialized knowledge beyond high-school algebra.
24 Greenberger et al. (1989, 1990), Mermin (1990, 1993).
25 Aronowitz (1988b, p. 331) has made a provocative observation concerning nonlinear causality in quantum mechanics and its relation to the social construction of time:
Linear causality assumes that the relation of cause and effect can be expressed as a function of temporal succession. Owing to recent developments in quantum mechanics, we can postulate that it is possible to know the effects of absent causes; that is, speaking metaphorically, effects may anticipate causes so that our perception of them may precede the physical occurrence of a “cause.” The hypothesis that challenges our conventional conception of linear time and causality and that asserts the possibility of time’s reversal also raises the question of the degree to which the concept of “time’s arrow” is inherent in all scientific theory. If these experiments are successful, the conclusions about the way time as “clock-time” has been constituted historically will be open to question. We will have “proved” by means of experiment what has long been suspected by philosophers, literary and social critics: that time is, in part, a conventional construction, its segmentation into hours and minutes a product of the need for industrial discipline, for rational organization of social labor in the early bourgeois epoch.
The theoretical analyses of Greenberger et al. (1989, 1990) and Mermin (1990, 1993) a provide striking example of this phenomenon; see Maudlin (1994) for a detailed analysis of the implications for concepts of causality and temporality. An experimental test, extending the work of Aspect et al. (1982), will likely be forthcoming within the next few years.
26 Bohm (1980). The intimate relations between quantum mechanics and the mindbody problem are discussed in Goldstein (1983, chs. 7 and 8).
27 Among the voluminous literature, the book by Capra (1975) can be recommended for its scientific accuracy and its accessibility to non-specialists. In addition, the book by Sheldrake (1981), while occasionally speculative, is in general sound. For a sympathetic but critical analysis of New Age theories, see Ross (1991, ch. 1). For a critique of Capra’s work from a Third World perspective, see Alvares (1992, ch. 6).
28 Bohr (1963, p. 2), emphasis in Bohr’s original.
29 Newtonian atomism treats particles as hyperseparated in space and time, backgrounding their interconnectedness (Plumwood 1993a, p. 125); indeed, “the only ‘force’ allowed within the mechanistic framework is that of kinetic energy – the energy of motion by contact – all other purported forces, including action at a distance, being regarded as occult” (Mathews 1991, p.17). For critical analyses of the Newtonian mechanistic worldview, see Weil (1968, especially ch. 1), Merchant (1980), Berman (1981), Keller (1985, chs 2 and 3), Mathews (1991, ch. 1) and Plumwood (1993a, ch. 5).
30 According to the traditional textbook account, special relativity is concerned with the coordinate transformations relating two frames of reference in uniform relative motion. But this is a misleading oversimplification, as Latour (1988) has pointed out:
How can one decide whether an observation made in a train about the behaviour of a falling stone can be made to coincide with the observation made of the same falling stone from the embankment? If there are only one, or even two, frames of reference, no solution can be found since the man in the train claims he observes a straight line and the man on the embankment a parabola. ... Einstein’s solution is to consider three actors: one in the train, one on the embankment and a third one, the author [enunciator] or one of its representants, who tries to superimpose the coded observations sent back by the two others ... [W]ithout the enunciator’s position (hidden in Einstein’s account), and without the notion of centres of calculation, Einstein’s own technical argument is ununderstandable ... [pp. 10–11 and 35, emphasis in original]
In the end, as Latour wittily but accurately observes, special relativity boils down to the proposition that
more frames of reference with less privilege can be accessed, reduced, accumulated and combined, observers can be delegated to a few more places in the infinitely large (the cosmos) and the infinitely small (electrons), and the readings they send will be understandable. His [Einstein’s] book could well be titled: ‘New Instructions for Bringing Back Long-Distance Scientific Travellers’. [pp. 22–3]
Latour’s critical analysis of Einstein’s logic provides an eminently accessible introduction to special relativity for non-scientists.
31 Minkowski (1908), translated in Lorentz et al. (1952, p. 75).
32 It goes without saying that special relativity proposes new concepts not only of space and time but also of mechanics. In special relativity, as Virilio (1991, p. 136) has noted, “the dromospheric space, space-speed, is physically described by what is called the ‘logistic equation,’ the result of the product of the mass displaced by the speed of its displacement, MxV.” This radical alteration of the Newtonian formula has profound consequences, particularly in the quantum theory; see Lorentz et al. (1952) and Weinberg (1992) for further discussion.
33 Steven Best (1991, p. 225) has put his finger on the crux of the difficulty, which is that “unlike the linear equations used in Newtonian and even quantum mechanics, non-linear equations do [not] have the simple additive property whereby chains of solutions can be constructed out of simple independent parts”. For this reason, the strategies of atomization, reductionism and context-stripping that underlie the Newtonian scientific methodology simply do not work in general relativity.
34 Gödel (1949). For a summary of recent work in this area, see ’t Hooft (1993).
35 These new notions of space, time and causality are in part foreshadowed already in special relativity. Thus, Alexander Argyros (1991, p. 137) has noted that
in a universe dominated by photons, gravitons, and neutrinos, that is, in the very early universe, the theory of special relativity suggests that any distinction between before and after is impossible. For a particle traveling at the speed of light, or one traversing a distance that is in the order of the Planck length, all events are simultaneous.
However, I cannot agree with Argyros’ conclusion that Derridean deconstruction is therefore inapplicable to the hermeneutics of early-universe cosmology: Argyros’ argument to this effect is based on an impermissibly totalizing use of special relativity (in technical terms, “light-cone coordinates”) in a context where general relativity is inescapable. (For a similar but less innocent error, see Note 40 below.)
36 Jean-François Lyotard (1989, pp. 5-6) has pointed out that not only general relativity, but also modern elementary-particle physics, imposes new notions of time:
In contemporary physics and astrophysics ... a particle has a sort of elementary memory and consequently a temporal filter. This is why contemporary physicists tend to think that time emanates from matter itself, and that it is not an entity outside or inside the universe whose function it would be to gather all different times into universal history. It is only in certain regions that such – only partial – syntheses could be detected. There would on this view be areas of determinism where complexity is increasing.
Furthermore, Michel Serres (1992, pp. 89-91) has noted that chaos theory (Gleick 1987) and percolation theory (Stauffer 1985) have contested the traditional linear concept of time:
Time does not always flow along a line ... or a plane, but along an extraordinarily complex manifold, as if it showed stopping points, ruptures, sinks [puits], funnels of overwhelming acceleration [cheminées d’accélération foudroyante], rips, lacunae, all sown randomly ... Time flows in a turbulent and chaotic manner; it percolates. [Translation mine. Note that in the theory of dynamical systems, “puits” is a technical term meaning “sink”, i.e. the opposite of “source”.]
These multiple insights into the nature of time, provided by differen
t branches of physics, are a further illustration of the complementarity principle.
37 General relativity can arguably be read as corroborating the Nietzschean deconstruction of causality (see e.g. Culler 1982, pp. 86–8), although some relativists find this interpretation problematic. In quantum mechanics, by contrast, this phenomenon is rather firmly established (see Note 25 above).
38 General relativity is also, of course, the starting point for contemporary astrophysics and physical cosmology. See Mathews (1991, pp. 59–90, 109–16, 142–63) for a detailed analysis of the connections between general relativity (and its generalizations called “geometrodynamics”) and an ecological worldview. For an astrophysicist’s speculations along similar lines, see Primack and Abrams (1995).
39 Discussion to Derrida (1970, pp. 265–6).
40 Derrida (1970, p. 267). Right-wing critics Gross and Levitt (1994, p. 79) have ridiculed this statement, willfully misinterpreting it as an assertion about special relativity, in which the Einsteinian constant c (the speed of light in vacuum) is of course constant. No reader conversant with modern physics – except an ideologically biased one – could fail to understand Derrida’s unequivocal reference to general relativity.
41 Luce Irigaray (1987, pp. 77–8) has pointed out that the contradictions between quantum theory and field theory are in fact the culmination of a historical process that began with Newtonian mechanics:
The Newtonian break has ushered scientific enterprise into a world where sense perception is worth little, a world which can lead to the annihilation of the very stakes of physics’ object: the matter (whatever the predicates) of the universe and of the bodies that constitute it. In this very science, moreover [d’ailleurs], cleavages exist: quantum theory/field theory, mechanics of solids/dynamics of fluids, for example. But the imperceptibility of the matter under study often brings with it the paradoxical privilege of solidity in discoveries and a delay, even an abandoning of the analysis of the infinity [l’in-fini] of the fields of force.
I have here corrected the translation of “d’ailleurs”, which means “moreover” or “besides” (not “however”).
42 Wheeler (1964).
43 Isham (1991, sec. 3.1.4).
44 Green, Schwarz and Witten (1987).
45 Ashtekar, Rovelli and Smolin (1992), Smolin (1992).
46 Sheldrake (1981, 1991), Briggs and Peat (1984, ch. 4), Granero-Porati and Porati (1984), Kazarinoff (1985), Schiffmann (1989), Psarev (1990), Brooks and Castor (1990), Heinonen, Kilpeläinen and Martio (1992), Rensing (1993). For an in-depth treatment of the mathematical background to this theory, see Thom (1975, 1990); and for a brief but insightful analysis of the philosophical underpinnings of this and related approaches, see Ross (1991, pp. 40–2, 253n).
47 Waddington (1965), Corner (1966), Gierer et al. (1978).
48 Some early workers thought that the morphogenetic field might be related to the electromagnetic field, but it is now understood that this is merely a suggestive analogy: see Sheldrake (1981, pp. 77, 90) for a clear exposition. Note also point (b) below.
49 Boulware and Deser (1975).
50 For another example of the “turf” effect, see Chomsky (1979, pp. 6–7).
51 To be fair to the high-energy-physics establishment, I should mention that there is also an honest intellectual reason for their opposition to this theory: inasmuch as it posits a subquantum interaction linking patterns throughout the universe, it is, in physicists’ terminology, a “non-local field theory”. Now, the history of classical theoretical physics since the early 1800s, from Maxwell’s electrodynamics to Einstein’s general relativity, can be read in a very deep sense as a trend away from action-at-a-distance theories and towards local field theories: in technical terms, theories expressible by partial differential equations (Einstein and Infeld 1961, Hayles 1984). So a non-local field theory definitely goes against the grain. On the other hand, as Bell (1987) and others have convincingly argued, the key property of quantum mechanics is precisely its non-locality, as expressed in Bell’s theorem and its generalizations (see Notes 23 and 24 above). Therefore, a non-local field theory, although jarring to physicists’ classical intuition, is not only natural but in fact preferred (and possibly even mandatory?) in the quantum context. This is why classical general relativity is a local field theory, while quantum gravity (whether string, weave or morphogenetic field) is inherently non-local.
52 Differential topology is the branch of mathematics concerned with those properties of surfaces (and higher-dimensional manifolds) that are unaffected by smooth deformations. The properties it studies are therefore primarily qualitative rather than quantitative, and its methods are holistic rather than Cartesian.
53 Alvarez-Gaumé (1985). The alert reader will notice that anomalies in “normal science” are the usual harbinger of a future paradigm shift (Kuhn 1970).
54 Kosterlitz and Thouless (1973). The flowering of the theory of phase transitions in the 1970s probably reflects an increased emphasis on discontinuity and rupture in the wider culture: see Note 81 below.
55 Green, Schwarz and Witten (1987).
56 A typical such book is Nash and Sen (1983).
57 Lacan (1970, pp. 192–3), lecture given in 1966. For an in-depth analysis of Lacan’s use of ideas from mathematical topology, see Juranville (1984, ch. VII), Granon-La-font (1985, 1990), Vappereau (1985) and Nasio (1987, 1992); a brief summary is given by Leupin (1991). See Hayles (1990, p. 80) for an intriguing connection between Lacanian topology and chaos theory; unfortunately she does not pursue it. See also Žižek (1991, pp. 38–9, 45–7) 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 1930s, 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, p. 50): “Il suffit, à cette fin, de reconnaître que Lacan confère enfin à la pensée de Freud, les concepts scientifiques qu’elle 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 New Left Review (Althusser 1969).
60 Miller (1977/78, especially pp. 24–5). This article has become quite influential in film theory: see e.g. Jameson (1982, pp. 27–8) and the references cited there. As Strathausen (1994, p. 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. 107–8).
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, pp. 205–8); 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).
66 James (1971, pp. 271–2). It is, however, worth noting that the space RP3 is homeomorphic to the group SO(3) of rotational symmetries of conventional three-dimensional Euclidean space. Thus, some aspects of three-dimensional Euclidicity are preserved (albeit in modified form) in the postmodern physics, just as some aspects of Newtonian mechanics were preserved in modified f
orm in Einsteinian physics.
67 Kosko (1993). See also Johnson (1977, pp. 481–2) for an analysis of Derrida’s and Lacan’s efforts toward transcending the Euclidean spatial logic.
68 Along related lines, Eve Seguin (1994, p. 61) has noted that “logic says nothing about the world and attributes to the world properties that are but constructs of theoretical thought. This explains why physics since Einstein has relied on alternative logics, such as trivalent logic which rejects the principle of the excluded middle.” A pioneering (and unjustly forgotten) work in this direction, likewise inspired by quantum mechanics, is Lupasco (1951). See also Plumwood (1993b, pp. 453–9) for a specifically feminist perspective on nonclasssical logics. For a critical analysis of one nonclassical logic (“boundary logic”) and its relation to the ideology of cyberspace, see Markley (1994).
69 Irigaray (1987, pp. 76–7), essay originally appeared in French in 1982. Irigaray’s phrase “théorie des ensembles” can also be rendered as “theory of sets”, and “bords” is usually translated in the mathematical context as “boundaries”. Her phrase “ensembles flous” may refer to the new mathematical field of “fuzzy sets” (Kaufmann 1973, Kosko 1993).
70 See e.g. Hamza (1990), McAvity and Osborn (1991), Alexander, Berg and Bishop (1993) and the references cited therein.
71 Green, Schwarz and Witten (1987).
72 Hamber (1992), Nabutosky and Ben-Av (1993), Kontsevich (1994).
73 In the history of mathematics there has been a long-standing dialectic between the development of its “pure” and “applied” branches (Struik 1987). Of course, the “applications” traditionally privileged in this context have been those profitable to capitalists or useful to their military forces: for example, number theory has been developed largely for its applications in cryptography (Loxton 1990). See also Hardy (1967, pp. 120–1, 131–2).
74 The equal representation of all boundary conditions is also suggested by Chew’s bootstrap theory of “subatomic democracy”: see Chew (1977) for an introduction, and see Morris (1988) and Markley (1992) for philosophical analysis.