by Alan Sokal
75 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.
76 Madsen and Madsen (1990, p. 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, p. 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, pp. 266–72) and Hobsbawm (1993, pp. 63–4) for further discussion of the political implications.
77 Madsen and Madsen (1990, pp. 471–2).
78 Aronowitz (1988b, pp. 292–3) 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, p. 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 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, pp. 158–62, 204–7). 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.
79 Ross (1991, p. 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, p. 91).
80 Oliver (1989, p. 146).
81 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, pp. 154–8.) 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 1970s 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).
82 Irigaray (1985), Hayles (1992). See, however, Schor (1989) for a critique of Irigaray’s undue deference toward conventional (male) science, particularly physics.
83 Thom (1975, 1990), Arnol’d (1992).
84 Concerning the Cartesian/Baconian metaphysics, Robert Markley (1991, p. 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.
85 Capra (1988, p. 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, chs. 6 and 9), Shiva (1990), Best (1991), Haraway (1991, 1994) Mathews (1991), Morin (1992), Santos (1992) and Wright (1992).
86 Markley (1992, p. 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.
87 See Wallerstein (1993, pp. 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.
88 Aronowitz (1988b, p. 344).
89 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.
[Laca
n (1977, pp. 28–9), seminar originally given in 1959.]
For further reflections on irrationality in modern mathematics, see Solomon (1988, p. 76) and Bloor (1991, pp. 122–5).
90 See e.g. Aronowitz (1994) and the discussion following it.
91 Markley (1992, p. 271).
92 Markley (1992, p. 271). Along parallel lines, Donna Haraway (1991, pp. 191–2) 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).
93 Aronowitz (1988b, p. 351). Although this observation appeared in 1988, it is all the more true today.
94 Freire (1970), Aronowitz and Giroux (1991, 1993).
95 For an example in the context of the Sandinista revolution, see Sokal (1987).
96 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, ch. 5) and Patai and Koertge (1994).
97 Trebilcot (1988), Hamill (1994).
98 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, ch. 2) and Gross and Levitt (1994, pp. 203–14).
99 Merchant (1980, 1992), Berman (1981), Callicott (1989, chs 6 and 9), Mathews (1991), Wright (1992), Plumwood (1993a), Ross (1994).
100 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”.
101 A very recent but important contribution to the philosophy of mathematics can be found in the work of Deleuze and Guattari (1994, ch. 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. 117–18]
But science, unlike philosophy, cannot cope with infinite speeds:
... it is by slowing down that matter, as well as the scientific thought above 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. 118–19, 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 (2/3 and 4/6) or are left undetermined (a = 2b) but requires one of the variables to be at a higher power (y2/x = P). [p. 122]
(Note that the English translation inadvertently writes y2/x = P, an amusing error that thoroughly mangles the logic of the argument.)
Surprisingly for a technical philosophical work, this book (Qu’est 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.
102 Aronowitz (1988b, p. 346). For a vicious right-wing attack on this proposition, see Gross and Levitt (1994, pp. 52–4). 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, pp. 45–6) and Bloor (1991, p. 182); and concerning the syllogism, see also Woolgar (1988, pp. 47–8) and Bloor (1991, pp. 131–5). For an analysis of the social images underlying mathematical conceptions of infinity, see Harding (1986, p. 50). For a demonstration of the social contextuality of mathematical statements, see Woolgar (1988, p. 43) and Bloor (1991, pp. 107–30).
103 Campbell and Campbell-Wright (1995, p. 135). See Merchant (1980) for a detailed analysis of the themes of control and domination in Western mathematics and science.
104 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 p0, p1, p2, ... be the respective probabilities that a man has 0, 1, 2, ... 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 1960s. Sadly, the French left – especially but by no means solely the PCF – has traditionally been as enthusiastic for nuclear energy as the right (s
ee Touraine et al. 1980).
105 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).
106 Kosko (1993).
107 Fuzzy systems theory has been heavily developed by transnational corporations – first in Japan and later elsewhere – to solve practical problems of efficiency in labordisplacing automation.
108 Thom (1975, 1990), Arnol’d (1992).
109 An interesting start is made by Schubert (1989).
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110 Lodge (1984, p. 152), italics in the original.
111 This is ten trillion trillion (1025) times smaller than an atom.
112 See, for example, de Gennes (1976).
113 For an amusing attempt, by a postmodernist author who does know some physics, to come up with something Derrida’s words could conceivably have meant that might make sense, see Plotnitsky (1997). The trouble is that Plotnitsky comes up with at least two alternative technical interpretations of Derrida’s phrase ‘the Einsteinian constant’, without providing any convincing evidence that Derrida intended (or even understood) either of them.
114 ‘Manifold’ is a geometrical concept that generalizes the notion of surface to spaces of more than two dimensions.
115 See p. 42 above for a brief explanation of the axiom of choice.
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* This article was submitted to Social Text following the publication of the parody, but was rejected on the grounds that it did not meet their intellectual standards. It was published in Dissent 43(4), pp. 93–9 (Fall 1996) and, in slightly different form, in Philosophy and Literature 20(2), pp. 338–46 (October 1996). See also the critical comment by Social Text co-founder Stanley Aronowitz (1997) and the reply by Sokal (1997b).