Heisenberg (1958), Bohr (1963).

Kuhn (1970), Feyerabend (1975), Latour (1987), Aronowitz (1988b), Bloor (1991).

Merchant (1980), Keller (1985), Harding (1986,1991), Haraway (1989,1991), Best (1991).

Aronowitz (1988b, especially chaps. 9 and 12).

Ross (1991, introduction and chap. 1).

Irigaray (1985), Hayles (1992).
Harding (1986, especially chaps. 2 and 10); Harding (1991, especially chap. 4).

For a sampling of views, see Jammer (1974), Bell (1987), Albert (1992), Dürr, Goldstein and Zanghí (1992), Weinberg (1992, chap. IV), Coleman (1993), diary Maudlin (1994), Bricmont (1994).

Heisenberg (1958, 15, 28-29), emphasis in Heisenberg's original. See also Overstreet (1980), Craige (1982), Hayles (1984), Greenberg (1990), Booker (1990) and Porter (1990) for examples of cross-fertilization of ideas between relativistic quantum theory and literary criticism.

Unfortunately, Heisenberg's uncertainty principle has frequently been misinterpreted by amateur philosophers. As Gilles Deleuze and Félix Guattari (1994, 129-130) lucidly point out,
in quantum physics, Heisenberg's demon does not express the impossibility of measuring both the speed and the position of a particle on the grounds of a subjective interference of the measure with the measured, but it measures exactly an objective state of affairs that leaves the respective position of two of its particles outside of the field of its actualization, the number of independent variables being reduced and the values of the coordinates having the same probability. ...Perspectivism, or scientific relativism, is never relative to a subject: it constitutes not a relativity of truth but, on the contrary, a truth of the relative, that is to say, of variables whose cases it orders according to the values it extracts from them in its system of coordinates ...

Bohr (1928), cited in Pais (1991, 314).
Aronowitz (1988b, 251-256).

See also Porush (1989) for a fascinating account of how a second group of scientists and engineers - cyberneticists - contrived, with considerable success, to subvert the most revolutionary implications of quantum physics. The main limitation of Porush's critique is that it remains solely on a cultural and philosophical plane; his conclusions would be immeasurably strengthened by an analysis of economic and political factors. (For example, Porush fails to mention that engineer-cyberneticist Claude Shannon worked for the then-telephone monopoly AT&T.) A careful analysis would show, I think, that the victory of cybernetics over quantum physics in the 1940's and 50's can be explained in large part by the centrality of cybernetics to the ongoing capitalist drive for automation of industrial production, compared to the marginal industrial relevance of quantum mechanics.

Pais (1991, 23). Aronowitz (1981, 28) has noted that wave-particle duality renders the ``will to totality in modern science'' severely problematic:
The differences within physics between wave and particle theories of matter, the indeterminacy principle discovered by Heisenberg, Einstein's relativity theory, all are accommodations to the impossibility of arriving at a unified field theory, one in which the ``anomaly'' of difference for a theory which posits identity may be resolved without challenging the presuppositions of science itself.
For further development of these ideas, see Aronowitz (1988a, 524-525, 533).

Heisenberg (1958, 40-41).
Bohr (1934), cited in Jammer (1974, 102). Bohr's analysis of the complementarity principle also led him to a social outlook which was, for its time and place, notably progressive. Consider the following excerpt from a 1938 lecture (Bohr 1958, 30):
I may perhaps here remind you of the extent to which in certain societies the roles of men and women are reversed, not only regarding domestic and social duties but also regarding behaviour and mentality. Even if many of us, in such a situation, might perhaps at first shrink from admitting the possibility that it is entirely a caprice of fate that the people concerned have their specific culture and not ours, and we not theirs instead of our own, it is clear that even the slightest suspicion in this respect implies a betrayal of the national complacency inherent in any human culture resting in itself.

Froula (1985).
Honner (1994).
Plotnitsky (1994). This impressive work also explains the intimate connections with Gödel's proof of the incompleteness of formal systems and with Skolem's construction of nonstandard models of arithmetic, as well as with Bataille's general economy. For further discussion of Bataille's physics, see Hochroth (1995).

Numerous other examples could be adduced. For instance, Barbara Johnson (1989, 12) makes no specific reference to quantum physics; but her description of deconstruction is an eerily exact summary of the complementarity principle:
Instead of a simple ``either/or'' structure, deconstruction attempts to elaborate a discourse that says neither ``either/or'', nor ``both/and'' nor even ``neither/nor'', while at the same time not totally abandoning these logics either.
See also McCarthy (1992) for a thought-provoking analysis that raises disturbing questions about the ``complicity'' between (nonrelativistic) quantum physics and deconstruction.

Permit me in this regard a personal recollection: Fifteen years ago, when I was a graduate student, my research in relativistic quantum field theory led me to an approach which I called ``de[con]structive quantum field theory'' (Sokal 1982). Of course, at that time I was completely ignorant of Jacques Derrida's work on deconstruction in philosophy and literary theory. In retrospect, however, there is a striking affinity: my work can be read as an exploration of how the orthodox discourse (e.g. Itzykson and Zuber 1980) on scalar quantum field theory in four-dimensional space-time (in technical terms, ``renormalized perturbation theory'' for the tex2html_wrap_inline1385 theory) can be seen to assert its own unreliability and thereby to undermine its own affirmations. Since then, my work has shifted to other questions, mostly connected with phase transitions; but subtle homologies between the two fields can be discerned, notably the theme of discontinuity (see Notes 22 and 81 below). For further examples of deconstruction in quantum field theory, see Merz and Knorr Cetina (1994). line.

Bohr (1928), cited in Jammer (1974, 90).

Bell (1987, especially chaps. 10 and 16). See also Maudlin (1994, chap. 1) for a clear account presupposing no specialized knowledge beyond high-school algebra.

Greenberger et al. (1989,1990), Mermin (1990,1993).

Aronowitz (1988b, 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) provide a 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.

Bohm (1980). The intimate relations between quantum mechanics and the mind-body problem are discussed in Goldstein (1983, chaps. 7 and 8). line.

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, chap. 1). For a critique of Capra's work from a Third World perspective, see Alvares (1992, chap. 6).

Bohr (1963, 2), emphasis in Bohr's original.

Newtonian atomism treats particles as hyperseparated in space and time, backgrounding their interconnectedness (Plumwood 1993a, 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, 17). For critical analyses of the Newtonian mechanistic worldview, see Weil (1968, especially chap. 1), Merchant (1980), Berman (1981), Keller (1985, chaps. 2 and 3), Mathews (1991, chap. 1) and Plumwood (1993a, chap. 5).

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. 22-23]
Latour's critical analysis of Einstein's logic provides an eminently accessible introduction to special relativity for non-scientists.

Minkowski (1908), translated in Lorentz et al. (1952, 75).

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, 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.

Steven Best (1991, 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.

Gödel (1949). For a summary of recent work in this area, see 't Hooft (1993).
These new notions of space, time and causality are in part foreshadowed already in special relativity. Thus, Alexander Argyros (1991, 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.)

Jean-François Lyotard (1989, 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, 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 different branches of physics, are a further illustration of the complementarity principle.

General relativity can arguably be read as corroborating the Nietzschean deconstruction of causality (see e.g. Culler 1982, 86-88), although some relativists find this interpretation problematic. In quantum mechanics, by contrast, this phenomenon is rather firmly established (see Note 25 above).

General relativity is also, of course, the starting point for contemporary astrophysics and physical cosmology. See Mathews (1991, 59-90, 109-116, 142-163) 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).

Discussion to Derrida (1970, 265-266).
Derrida (1970, 267).

Right-wing critics Gross and Levitt (1994, 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.

Luce Irigaray (1987, 77-78) 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'').

Wheeler (1964).

Isham (1991, sec. 3.1.4).

Green, Schwarz and Witten (1987).
Ashtekar, Rovelli and Smolin (1992), Smolin (1992).

Sheldrake (1981,1991), Briggs and Peat (1984, chap. 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, 40-42, 253n).

Waddington (1965), Corner (1966), Gierer et al. (1978).

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, 77, 90) for a clear exposition. Note also point (b) below.

Boulware and Deser (1975).
For another example of the ``turf'' effect, see Chomsky (1979, 6-7).

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 1800's, 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.

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.

Alvarez-Gaumé (1985). The alert reader will notice that anomalies in ``normal science'' are the usual harbinger of a future paradigm shift (Kuhn 1970).

Kosterlitz and Thouless (1973). The flowering of the theory of phase transitions in the 1970's probably reflects an increased emphasis on discontinuity and rupture in the wider culture: see Note 81 below.

Green, Schwarz and Witten (1987).

A typical such book is Nash and Sen (1983). ``typology''!!!
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).

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).

Althusser (1993, 50): ``Il suffit, à cette fin, 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 1969 (New Left Review).

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).

Dean (1993, especially pp. 107-108).

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).

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).

Jones (1985).
Witten (1989).
James (1971, 271-272). It is, however, worth noting that the space tex2html_wrap_inline1407 is homeomorphic to the group tex2html_wrap_inline1409 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 form in Einsteinian physics.

Kosko (1993). See also Johnson (1977, 481-482) for an analysis of Derrida's and Lacan's efforts toward transcending the Euclidean spatial logic.

Along related lines, Eve Seguin (1994, 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, 453-459) for a specifically feminist perspective on nonclassical logics. For a critical analysis of one nonclassical logic (``boundary logic'') and its relation to the ideology of cyberspace, see Markley (1994).

Irigaray (1987, 76-77), 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).

See e.g. Hamza (1990), McAvity and Osborn (1991), Alexander, Berg and Bishop (1993) and the references cited therein.

Green, Schwarz and Witten (1987).

Hamber (1992), Nabutosky and Ben-Av (1993), Kontsevich (1994).

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, 120-121, 131-132).

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.

Among the large body of works from a diversity of politically progressive perspectives, the books by Merchant (1980), Keller (1985), Harding (1986), Aronowitz (1988b), Haraway (1991) and Ross (1991) have been especially influential. See also the references cited below.

Madsen and Madsen (1990, 471). The main limitation of the Madsen-Madsen analysis is that it is essentially apolitical; and it hardly needs to be pointed out that disputes over what is true can have a profound effect on, and are in turn profoundly affected by, disputes over political projects. Thus, Markley (1992, 270) makes a point similar to that of Madsen-Madsen, but rightly situates it in its political context:
Radical critiques of science that seek to escape the constraints of deterministic dialectics must also give over narrowly conceived debates about realism and truth to investigate what kind of realities - political realities - might be engendered by a dialogical bootstrapping. Within a dialogically agitated environment, debates about reality become, in practical terms, irrelevant. ``Reality,'' finally, is a historical construct.
See Markley (1992, 266-272) and Hobsbawm (1993, 63-64) for further discussion of the political implications.

Madsen and Madsen (1990, 471-472).
Aronowitz (1988b, 292-293) makes a slightly different, but equally cogent, criticism of quantum chromodynamics (the currently hegemonic theory representing nucleons as permanently bound states of quarks and gluons): drawing on the work of Pickering (1984), he notes that
in his [Pickering's] account, quarks are the name assigned to (absent) phenomena that cohere with particle rather than field theories, which, in each case, offer different, although equally plausible, explanations for the same (inferred) observation. That the majority of the scientific community chose one over another is a function of scientists' preference for the tradition rather than the validity of explanation. However, Pickering does not reach back far enough into the history of physics to find the basis of the research tradition from which the quark explanation emanates. It may not be found inside the tradition but in the ideology of science, in the differences behind field versus particle theories, simple versus complex explanations, the bias toward certainty rather than indeterminateness.
Along very similar lines, Markley (1992, 269) observes that physicists' preference for quantum chromodynamics over Chew's bootstrap theory of ``subatomic democracy'' (Chew 1977) is a result of ideology rather than data:
It is not surprising, in this regard, that bootstrap theory has fallen into relative disfavor among physicists seeking a GUT (Grand Unified Theory) or TOE (Theory of Everything) to explain the structure of the universe. Comprehensive theories that explain ``everything'' are products of the privileging of coherence and order in western science. The choice between bootstrap theory and theories of everything that confronts physicists does not have to do primarily with the truth-value offered by these accounts of available data but with the narrative structures - indeterminate or deterministic - into which these data are placed and by which they are interpreted.
Unfortunately, the vast majority of physicists are not yet aware of these incisive critiques of one of their most fervently-held dogmas. For another critique of the hidden ideology of contemporary particle physics, see Kroker et al. (1989, 158-162, 204-207). The style of this critique is rather too Baudrillardian for my staid taste, but the content is (except for a few minor inaccuracies) right on target.

Ross (1991, 29). For an amusing example of how this modest demand has driven right-wing scientists into fits of apoplexy (``frighteningly Stalinist'' is the chosen epithet), see Gross and Levitt (1994, 91).

Oliver (1989, 146).
While chaos theory has been deeply studied by cultural analysts - see e.g. Hayles (1990,1991), Argyros (1991), Best (1991), Young (1991,1992), Assad (1993) among many others - the theory of phase transitions has passed largely unremarked. (One exception is the discussion of the renormalization group in Hayles (1990, 154-158).) This is a pity, because discontinuity and the emergence of multiple scales are central features in this theory; and it would be interesting to know how the development of these themes in the 1970's and afterwards is connected to trends in the wider culture. I therefore suggest this theory as a fruitful field for future research by cultural analysts. Some theorems on discontinuity which may be relevant to this analysis can be found in Van Enter, Fernández and Sokal (1993).

Irigaray (1985), Hayles (1992). See, however, Schor (1989) for a critique of Irigaray's undue deference toward conventional (male) science, particularly physics.

Thom (1975,1990), Arnol'd (1992).
Concerning the Cartesian/Baconian metaphysics, Robert Markley (1991, 6) has observed that
Narratives of scientific progress depend upon imposing binary oppositions - true/false, right/wrong - on theoretical and experimental knowledge, privileging meaning over noise, metonymy over metaphor, monological authority over dialogical contention. ...[T]hese attempts to fix nature are ideologically coercive as well as descriptively limited. They focus attention only on the small range of phenomena - say, linear dynamics - which seem to offer easy, often idealized ways of modeling and interpreting humankind's relationship to the universe.
While this observation is informed primarily by chaos theory - and secondarily by nonrelativistic quantum mechanics - it in fact summarizes beautifully the radical challenge to modernist metaphysics posed by quantum gravity.

Capra (1988, 145). One caveat: I have strong reservations about Capra's use here of the word ``cyclical'', which if interpreted too literally could promote a politically regressive quietism. For further analyses of these issues, see Bohm (1980), Merchant (1980,1992), Berman (1981), Prigogine and Stengers (1984), Bowen (1985), Griffin (1988), Kitchener (1988), Callicott (1989, chaps. 6 and 9), Shiva (1990), Best (1991), Haraway (1991,1994), Mathews (1991), Morin (1992), Santos (1992) and Wright (1992).

Markley (1992, 264). A minor quibble: It is not clear to me that complex number theory, which is a new and still quite speculative branch of mathematical physics, ought to be accorded the same epistemological status as the three firmly established sciences cited by Markley.

See Wallerstein (1993, 17-20) for an incisive and closely analogous account of how the postmodern physics is beginning to borrow ideas from the historical social sciences; and see Santos (1989,1992) for a more detailed development.

Aronowitz (1988b, 344).
At this point, the traditional scientist's response is that work not conforming to the evidentiary standards of conventional science is fundamentally irrational, i.e. logically flawed and therefore not worthy of credence. But this refutation is insufficient: for, as Porush (1993) has lucidly observed, modern mathematics and physics have themselves admitted a powerful ``intrusion of the irrational'' in quantum mechanics and Gödel's theorem - although, understandably, like the Pythagoreans 24 centuries ago, modernist scientists have attempted to exorcise this unwanted irrational element as best they could. Porush makes a powerful plea for a ``post-rational epistemology'' that would retain the best of conventional Western science while validating alternative ways of knowing. Note also that Jacques Lacan, from a quite different starting point, came long ago to a similar appreciation of the inevitable role of irrationality in modern mathematics:
If you'll permit me to use one of those formulas which come to me as I write my notes, human life could be defined as a calculus in which zero was irrational. This formula is just an image, a mathematical metaphor. When I say ``irrational,'' I'm referring not to some unfathomable emotional state but precisely to what is called an imaginary number. The square root of minus one doesn't correspond to anything that is subject to our intuition, anything real - in the mathematical sense of the term - and yet, it must be conserved, along with its full function.
[Lacan (1977, 28-29), seminar originally given in 1959.] For further reflections on irrationality in modern mathematics, see Solomon (1988, 76) and Bloor (1991, 122-125).

See e.g. Aronowitz (1994) and the discussion following it.

Markley (1992, 271).
Markley (1992, 271). Along parallel lines, Donna Haraway (1991, 191-192) has argued eloquently for a democratic science comprising ``partial, locatable, critical knowledges sustaining the possibility of webs of connections called solidarity in politics and shared conversations in epistemology'' and founded on ``a doctrine and practice of objectivity that privileges contestation, deconstruction, passionate construction, webbed connections, and hope for transformation of systems of knowledge and ways of seeing.'' These ideas are further developed in Haraway (1994) and Doyle (1994).

Aronowitz (1988b, 351). Although this observation appeared in 1988, it is all the more true today.

Freire (1970), Aronowitz and Giroux (1991,1993).

For an example in the context of the Sandinista revolution, see Sokal (1987).

Merchant (1980), Easlea (1981), Keller (1985,1992), Harding (1986,1991),

Haraway (1989,1991), Plumwood (1993a). See Wylie et al. (1990) for an extensive bibliography. The feminist critique of science has, not surprisingly, been the object of a bitter right-wing counterattack. For a sampling, see Levin (1988), Haack (1992,1993), Sommers (1994),

Gross and Levitt (1994, chap. 5) and Patai and Koertge (1994).

Trebilcot (1988), Hamill (1994).

Ezeabasili (1977), Van Sertima (1983), Frye (1987), Sardar (1988), Adams (1990), Nandy (1990), Alvares (1992), Harding (1994). As with the feminist critique, the multiculturalist perspective has been ridiculed by right-wing critics, with a condescension that in some cases borders on racism. See e.g. Ortiz de Montellano (1991), Martel (1991/92), Hughes (1993, chap. 2) and Gross and Levitt (1994, 203-214).

Merchant (1980,1992), Berman (1981), Callicott (1989, chaps. 6 and 9), Mathews (1991), Wright (1992), Plumwood (1993a), Ross (1994).

See Wojciehowski (1991) for a deconstruction of Galileo's rhetoric, in particular his claim that the mathematico-scientific method can lead to direct and reliable knowledge of ``reality''.

A very recent but important contribution to the philosophy of mathematics can be found in the work of Deleuze and Guattari (1994, chap. 5). Here they introduce the philosophically fruitful notion of a ``functive'' [Fr. fonctif], which is neither a function [Fr. fonction] nor a functional [Fr. fonctionnelle] but rather a more basic conceptual entity:
The object of science is not concepts but rather functions that are presented as propositions in discursive systems. The elements of functions are called functives. [p. 117]
This apparently simple idea has surprisingly subtle and far-reaching consequences; its elucidation requires a detour into chaos theory (see also Rosenberg 1993 and Canning 1994):
...the first difference between science and philosophy is their respective attitudes toward chaos. Chaos is defined not so much by its disorder as by the infinite speed with which every form taking shape in it vanishes. It is a void that is not a nothingness but a virtual, containing all possible particles and drawing out all possible forms, which spring up only to disappear immediately, without consistency or reference, without consequence. Chaos is an infinite speed of birth and disappearance. [pp. 117-118]
But science, unlike philosophy, cannot cope with infinite speeds:
...it is by slowing down that matter, as well as the scientific thought able to penetrate it [sic] with propositions, is actualized. A function is a Slow-motion. Of course, science constantly advances accelerations, not only in catalysis but in particle accelerators and expansions that move galaxies apart. However, the primordial slowing down is not for these phenomena a zero-instant with which they break but rather a condition coextensive with their whole development. To slow down is to set a limit in chaos to which all speeds are subject, so that they form a variable determined as abscissa, at the same time as the limit forms a universal constant that cannot be gone beyond (for example, a maximum degree of contraction). The first functives are therefore the limit and the variable, and reference is a relationship between values of the variable or, more profoundly, the relationship of the variable, as abscissa of speeds, with the limit. [pp. 118-119, emphasis mine]
A rather intricate further analysis (too lengthy to quote here) leads to a conclusion of profound methodological importance for those sciences based on mathematical modelling:
The respective independence of variables appears in mathematics when one of them is at a higher power than the first. That is why Hegel shows that variability in the function is not confined to values that can be changed ( tex2html_wrap_inline1415 and tex2html_wrap_inline1417 ) or are left undetermined (a=2b) but requires one of the variables to be at a higher power ( tex2html_wrap_inline1423 ). [p. 122]
(Note that the English translation inadvertently writes tex2html_wrap_inline1423 , an amusing error that thoroughly mangles the logic of the argument.) Surprisingly for a technical philosophical work, this book (Qu'est-ce que la philosophie?) was a best-seller in France in 1991. It has recently appeared in English translation, but is, alas, unlikely to compete successfully with Rush Limbaugh and Howard Stern for the best-seller lists in this country.

Aronowitz (1988b, 346). For a vicious right-wing attack on this proposition, see Gross and Levitt (1994, 52-54). See Ginzberg (1989), Cope-Kasten (1989), Nye (1990) and Plumwood (1993b) for lucid feminist critiques of conventional (masculinist) mathematical logic, in particular the modus ponens and the syllogism. Concerning the modus ponens, see also Woolgar (1988, 45-46) and Bloor (1991, 182); and concerning the syllogism, see also Woolgar (1988, 47-48) and Bloor (1991, 131-135). For an analysis of the social images underlying mathematical conceptions of infinity, see Harding (1986, 50). For a demonstration of the social contextuality of mathematical statements, see Woolgar (1988, 43) and Bloor (1991, 107-130).

Campbell and Campbell-Wright (1993, 11). See Merchant (1980) for a detailed analysis of the themes of control and domination in Western mathematics and science.

Let me mention in passing two other examples of sexism and militarism in mathematics that to my knowledge have not been noticed previously: The first concerns the theory of branching processes, which arose in Victorian England from the ``problem of the extinction of families'', and which now plays a key role inter alia in the analysis of nuclear chain reactions (Harris 1963). In the seminal (and this sexist word is apt) paper on the subject, Francis Galton and the Reverend H.W. Watson wrote (1874):
The decay of the families of men who occupied conspicuous positions in past times has been a subject of frequent research, and has given rise to various conjectures ...The instances are very numerous in which surnames that were once common have since become scarce or have wholly disappeared. The tendency is universal, and, in explanation of it, the conclusion has hastily been drawn that a rise in physical comfort and intellectual capacity is necessarily accompanied by a diminution in `fertility' ... Let tex2html_wrap_inline1425 be the respective probabilities that a man has tex2html_wrap_inline1427 sons, let each son have the same probability of sons of his own, and so on. What is the probability that the male line is extinct after r generations, and more generally what is the probability for any given number of descendants in the male line in any given generation?
One cannot fail to be charmed by the quaint implication that human males reproduce asexually; nevertheless, the classism, social-Darwinism and sexism in this passage are obvious. The second example is Laurent Schwartz's 1973 book on Radon Measures. While technically quite interesting, this work is imbued, as its title makes plain, with the pro-nuclear-energy worldview that has been characteristic of French science since the early 1960's. Sadly, the French left - especially but by no means solely the PCF - has traditionally been as enthusiastic for nuclear energy as the right (see Touraine et al. 1980).

Just as liberal feminists are frequently content with a minimal agenda of legal and social equality for women and ``pro-choice'', so liberal (and even some socialist) mathematicians are often content to work within the hegemonic Zermelo-Fraenkel framework (which, reflecting its nineteenth-century liberal origins, already incorporates the axiom of equality) supplemented only by the axiom of choice. But this framework is grossly insufficient for a liberatory mathematics, as was proven long ago by Cohen (1966).

Kosko (1993).
Fuzzy systems theory has been heavily developed by

transnational corporations - first in Japan and later elsewhere - to solve practical problems of efficiency in labor-displacing automation.

Thom (1975,1990), Arnol'd (1992).
An interesting start is made by Schubert (1989).

Readers are cautioned not to infer my views on any subject except insofar as they are set forth in this Afterword. In particular, the fact that I have parodied an extreme or ambiguously stated version of an idea does not exclude that I may agree with a more nuanced or precisely stated version of the same idea.

For example: ``linear'', ``nonlinear'', ``local'', ``global'', ``multidimensional'', ``relative'', ``frame of reference'', ``field'', ``anomaly'', ``chaos'', ``catastrophe'', ``logic'', ``irrational'', ``imaginary'', ``complex'', ``real'', ``equality'', ``choice''.

By the way, anyone who believes that the laws of physics are mere social conventions is invited to try transgressing those conventions from the windows of my apartment. I live on the twenty-first floor. (P.S. I am aware that this wisecrack is unfair to the more sophisticated relativist philosophers of science, who will concede that empirical statements can be objectively true - e.g. the fall from my window to the pavement will take approximately 2.5 seconds - but claim that the theoretical explanations of those empirical statements are more-or-less arbitrary social constructions. I think that also this view is largely wrong, but that is a much longer discussion.)

The natural sciences have little to fear, at least in the short run, from postmodernist silliness; it is, above all, history and the social sciences - and leftist politics - that suffer when verbal game-playing displaces the rigorous analysis of social realities. Nevertheless, because of the limitations of my own expertise, my analysis here will be restricted to the natural sciences (and indeed primarily to the physical sciences). While the basic epistemology of inquiry ought to be roughly the same for the natural and social sciences, I am of course perfectly aware that many special (and very difficult) methodological issues arise in the social sciences from the fact that the objects of inquiry are human beings (including their subjective states of mind); that these objects of inquiry have intentions (including in some cases the concealment of evidence or the placement of deliberately self-serving evidence); that the evidence is expressed (usually) in human language whose meaning may be ambiguous; that the meaning of conceptual categories (e.g. childhood, masculinity, femininity, family, economics, etc.) changes over time; that the goal of historical inquiry is not just facts but interpretation, etc. So by no means do I claim that my comments about physics should apply directly to history and the social sciences - that would be absurd. To say that ``physical reality is a social and linguistic construct'' is just plain silly, but to say that ``social reality is a social and linguistic construct'' is virtually a tautology.

Ryan (1992).
Hobsbawm (1993, 63).
Andreski (1972, 90).
Computers existed prior to solid-state technology, but they were unwieldy and slow. The 486 PC sitting today on the literary theorist's desk is roughly 1000 times more powerful than the room-sized vacuum-tube computer IBM 704 from 1954 (see e.g. Williams 1985).

I certainly don't exclude the possibility that present theories in any of these subjects might be erroneous. But critics wishing to make such a case would have to provide not only historical evidence of the claimed cultural influence, but also scientific evidence that the theory in question is in fact erroneous. (The same evidentiary standards of course apply to past erroneous theories; but in this case the scientists may have already performed the second task, relieving the cultural critic of the need to do so from scratch.)
Ross (1991, 25-26); also in Ross (1992, 535-536).

Ross (1991, 26); also in Ross (1992, 535). In the discussion following this paper, Ross (1992, 549) expressed further (and quite justified) misgivings:
I'm quite skeptical of the ``anything goes'' spirit that is often the prevailing climate of relativism around postmodernism. ...Much of the postmodernist debate has been devoted to grappling with the philosophical or cultural limits to the grand narratives of the Enlightenment. If you think about ecological questions in this light, however, then you are talking about ``real'' physical, or material, limits to our resources for encouraging social growth. And postmodernism, as we know, has been loath to address the ``real,'' except to announce its banishment.
U.S. Bureau of the Census (1975, 47, 55; 1994, 87). In 1900 the mean life expectancy at birth was 47.3 years (47.6 years for whites, and a shocking 33.0 years for ``Negro and other''). In 1995 it is 76.3 years (77.0 years for whites, 70.3 years for blacks). I am aware that this assertion is likely to be misinterpreted, so let me engage in some pre-emptive clarification. I am not claiming that all of the increase in life expectancy is due to advances in scientific medicine. A large fraction (possibly the dominant part) of the increase - especially in the first three decades of the twentieth century - is due to the general improvement in the standards of housing, nutrition and public sanitation (the latter two informed by improved scientific understanding of the etiology of infectious and dietary-deficiency diseases). [For reviews of the evidence, see e.g. Holland et al. (1991).] But - without discounting the role of social struggles in these improvements, particularly as concerns the narrowing of the racial gap - the underlying and overwhelming cause of these improvements is quite obviously the vast increase in the material standard of living over the past century, by more than a factor of five (U.S. Bureau of the Census 1975, 224-225; 1994, 451). And this increase is quite obviously the direct result of science, as embodied in technology.
Ross (1991, 26); also in Ross (1992, 536).
By the way, intelligent non-scientists seriously interested in the conceptual problems raised by quantum mechanics need no longer rely on the vulgarizations (in both senses) published by Heisenberg, Bohr and sundry physicists and New Age authors. The little book of Albert (1992) provides an impressively serious and intellectually honest account of quantum mechanics and the philosophical issues it raises - yet it requires no more mathematical background than a modicum of high-school algebra, and does not require any prior knowledge of physics. The main requirement is a willingness to think slowly and clearly.
Snow (1963, 20-21). One significant change has taken place since C.P. Snow's time: while humanist intellectuals' ignorance about (for example) mass and acceleration remains substantially unchanged, nowadays a significant minority of humanist intellectuals feels entitled to pontificate on these subjects in spite of their ignorance (perhaps trusting that their readers will be equally ignorant). Consider, for example, the following excerpt from a recent book on Rethinking Technologies, edited by the Miami Theory Collective and published by the University of Minnesota Press: ``it now seems appropriate to reconsider the notions of acceleration and deceleration (what physicists call positive and negative speeds)'' (Virilio 1993, 5). The reader who does not find this uproariously funny (as well as depressing) is invited to sit in on the first two weeks of Physics I.
I wasn't joking about that. For anyone who is interested in my views, I would be glad to provide a copy of Sokal (1987). For another sharp critique of the poor teaching of mathematics and science, see (irony of ironies) Gross and Levitt (1994b, 23-28).

Telepathy: Hastings and Hastings (1992, 518), American Institute of Public Opinion poll from June 1990. Concerning ``telepathy, or communication between minds without using the traditional five senses'', 36% ``believe in'', 25% are ``not sure'', and 39% ``do not believe in''. For ``people on this earth are sometimes possessed by the devil'', it is 49-16-35 (!). For ``astrology, or that the position of the stars and planets can affect people's lives'', it is 25-22-53. Mercifully, only 11% believe in channeling (22% are not sure), and 7% in the healing power of pyramids (26% not sure). Creationism: Gallup (1993, 157-159), Gallup poll from June 1993. The exact question was: ``Which of the following statements comes closest to your views on the origin and development of human beings: 1) human beings have developed over millions of years from less advanced forms of life, but God guided this process; 2) human beings have developed over millions of years from less advanced forms of life, but God had no part in this process; 3) God created human beings pretty much in their present form at one time within the last 10,000 years or so?'' The results were 35% developed with God, 11% developed without God, 47% God created in present form, 7% no opinion. A poll from July 1982 (Gallup 1982, 208-214) found almost identical figures, but gave breakdowns by sex, race, education, region, age, income, religion, and community size. Differences by sex, race, region, income and (surprisingly) religion were rather small. By far the largest difference was by education: only 24% of college graduates supported creationism, compared to 49% of high-school graduates and 52% of those with a grade-school education. So maybe the worst science teaching is at the elementary and secondary levels.
See Note 120 above.
Chomsky (1984, 200), lecture delivered in 1969.
Ryan (1992).

Daniel Sleator
Thu Jun 6 15:34:37 EDT 1996