by Alan Sokal
116 See, for example, Brunet (1931) and Dobbs and Jacob (1995).
117 Or more precisely: There is a vast body of extremely convincing astronomical evidence in support of the belief that the planets and comets do move (to a very high degree of approximation, though not exactly) as predicted by Newtonian mechanics; and if this belief is correct, then it is the fact of this motion (and not merely our belief in it) that forms part of the explanation of why the eighteenth-century European scientific community came to believe in the truth of Newtonian mechanics. Please note that all our assertions of fact – including ‘today in New York it’s raining’ – should be glossed in this way.
118 For what it’s worth, these decisions can presumably be justified on Bayesian grounds, using our prior experience of the probability of finding elephants in lecture halls, of the incidence of psychosis, of the reliability of our own visual and auditory perceptions, and so forth.
119 Latour (1987). For a more detailed analysis of Science in Action, see Amsterdamska (1990). For a critical analysis of the later theses of Latour’s school (as well as of other trends in sociology of science), see Gingras (1995).
120 Re (b), the ‘homely example’ in Gross and Levitt (1994, pp. 57–8) makes the point clearly.
121 The nuclear reactions that power the sun are expected to emit copious quantities of the subatomic particle called the neutrino. By combining current theories of solar structure, nuclear physics and elementary-particle physics, it is possible to obtain quantitative predictions for the flux and energy distribution of the solar neutrinos. Since the late 1960s, experimental physicists, beginning with the pioneering work of Raymond Davis, have been attempting to detect the solar neutrinos and measure their flux. The solar neutrinos have in fact been detected; but their flux appears to be less than one-third of the theoretical prediction. Astrophysicists and elementary-particle physicists are actively trying to determine whether the discrepancy arises from experimental error or theoretical error, and if the latter, whether the failure is in the solar models or in the elementary-particle models. For an introductory overview, see Bahcall (1990).
122 See, for example, Bahcall et al. (1996).
123 An even more extreme example of this confusion appears in a recent article by Latour in La Recherche, a French monthly magazine devoted to the popularization of science (Latour 1998). Here Latour discusses what he interprets as the discovery in 1976, by French scientists working on the mummy of the pharaoh Ramses II, that his death (circa 1213 BC) was due to tuberculosis. Latour asks: ‘How could he pass away due to a bacillus discovered by Robert Koch in 1882?’ Latour notes, correctly, that it would be an anachronism to assert that Ramses II was killed by machine-gun fire or died from the stress provoked by a stock-market crash. But then, Latour wonders, why isn’t death from tuberculosis likewise an anachronism? He goes so far as to assert that ‘Before Koch, the bacillus has no real existence.’ He dismisses the common-sense notion that Koch discovered a pre-existing bacillus as ‘having only the appearance of common sense’. Of course, in the rest of the article, Latour gives no argument to justify these radical claims and provides no genuine alternative to the common-sense answer. He simply stresses the obvious fact that, in order to discover the cause of Ramses’ death, a sophisticated analysis in Parisian laboratories was needed. But unless Latour is putting forward the truly radical claim that nothing we discover ever existed prior to its ‘discovery’ – in particular, that no murderer is a murderer, in the sense that he committed a crime before the police ‘discovered’ him to be a murderer – he needs to explain what is special about bacilli, and this he has utterly failed to do. The result is that Latour is saying nothing clear, and the article oscillates between extreme banalities and blatant falsehoods.
124 The principle applies with particular force when such a sociologist is studying contemporary science, because in this case there is no other scientific community besides the one under study who could provide such an independent assessment. By contrast, for studies of the distant past, one can take advantage of what subsequent scientists learned, including the results from experiments going beyond those originally performed. See note 88 above.
125 Nor would Steve Fuller, who asserts that ‘STS [Science and Technology Studies] practitioners employ methods that enable them to fathom both the “inner workings” and the “outer character” of science without having to be expert in the fields they study.’ (Fuller 1993, p. xii)
126 See Chapter 6 below.
127 The so-called Sapir-Whorf thesis in linguistics appears to have played an important role in this evolution: see note 37 above. Note also that Feyerabend, in his autobiography (1995, pp. 151–2), disowned the radical-relativist use of the Sapir-Whorf thesis that he had made in Against Method (Feyerabend 1975, chapter 17).
128 The book’s senior author is Gérard Fourez, a philosopher of science who is very influential (at least in Belgium) in pedagogical matters, and whose book La Construction des sciences (1992) has been translated into several languages.
129 Note that this appears in a text that is supposed to enlighten high-school teachers.
130 Or, worse, minimising the importance of facts, not by giving any argument, but simply by ignoring them in favour of consensus beliefs. Indeed, the definitions in this book systematically conflate facts, information, objectivity and rationality with – or reduce them to – intersubjective agreement. Moreover, a similar pattern is found in Fourez’s La Construction des sciences (1992). For example (p. 37): ‘To be “objective” means to follow instituted rules. ... Being “objective” is not the opposite of being “subjective”: rather, it is to be subjective in a certain way. But it is not to be individually subjective since one will follow socially instituted rules...’. This is highly misleading: following rules does not ensure objectivity in the usual sense (people who blindly repeat religious or political slogans certainly follow ‘socially instituted rules’, but they can hardly be called objective) and people can be objective while breaking many rules (e.g., Galileo).
131 Note also that defining ‘fact’ as ‘there is hardly any controversy ...’ runs into a logical problem: Is the absence of controversy itself a fact? And if so, how to define it? By the absence of controversy about the assertion that there is no controversy? Obviously, Fourez and his colleagues are using in the social sciences a naively realist epistemology that they implicitly reject for the natural sciences. See pp. 77–8 above for an analogous inconsistency in Feyerabend.
132 That is, the scientific view and the one based on traditional Vedic ideas. [Note added by us.]
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133 For good introductions to special and general relativity, see, for example, Einstein (1960 [1920]), Mermin (1989) and Sartori (1996).
134 During the 1920s, the astronomer Edwin Hubble discovered that the galaxies are moving away from the Earth, at speeds that are proportional to their distance from the Earth. Between 1927 and 1931, various physicists proposed explanations of this expansion within the framework of Einstein’s general relativity (without making the Earth a privileged centre of observation) as arising from an initial cosmic ‘explosion’; this theory was later nicknamed the ‘Big Bang’. But, though the Big Bang hypothesis explains the observed expansion in a very natural way, it is not the only possible theory: towards the end of the 1940s, the astrophysicists Hoyle, Bondi and Gold proposed the alternative theory of the ‘Steady State Universe’, according to which there is a general expansion without a primeval explosion (but with the continuous creation of new matter). However, in 1965, the physicists Penzias and Wilson discovered (by accident!) the cosmic microwave background radiation, whose spectrum and almost-isotropy turned out to be in complete agreement with the prediction based on general relativity for a ‘residue’ from the Big Bang. In part because of this observation, but also for many other reasons, the Big Bang theory is today almost universally accepted among astrophysicists, though there is a lively debate on the details.
For a non-technical introduction to the Big Bang theory and the observational data supporting it, see Weinberg (1977), Silk (1989) and Rees (1997).
The ‘Reaves’ to whom Irigaray refers is presumably Hubert Reeves, a Canadian astrophysicist living in France who has written several popular books on cosmology and astrophysics.
135 Except in the last millionth of a billionth of a billionth of a billionth of a billionth of a second, when quantum gravitational effects become important.
136 Hayles’ argument begins with an explanation of the important conceptual differences between linear differential equations and the nonlinear ones arising in fluid mechanics. It’s a respectable attempt at scientific journalism, albeit marred by a few errors (e.g. she confuses feedback with nonlinearity, and she asserts that Euler’s equation is linear). From this point on, however, her argument deteriorates into a caricature of postmodern lit-crit. Seeking to trace the historical development of fluid mechanics in the period 1650–1750, she claims to identify ‘a pair of hierarchical dichotomies [what else?!] in which the first term is privileged at he expense of the second: continuity versus repture, and conservation versus dissipation.’ (Hayles 1992, p. 22) There follows a rather confused discussion of the conceptual foundations of differential calculus, an imaginative (to say the least) exegesis of the ‘subliminal gender identifications’ in early hydraulics, and a Freudian analysis of thermodynamics ‘from heat death to jouissance’. Hayles concludes by asserting a radically relativist thesis:
Despite their names, conservation laws are not inevitable facts of nature but constructions that foreground some experiences and marginalize others. ... Almost without exception, conservation laws were formulated, developed, and experimentally tested by men. If conservation laws represent particular emphases and not inevitable facts, then people living in different kinds of bodies and identifying with different gender constructions might well have arrived at different models for flow.
(Hayles 1992, pp. 31–2)
But she gives no argument to support her claim that the laws of conservation of energy and momentum, for example, might be other than ‘inevitable facts of nature’; nor does she give the slightest hint of what kinds of ‘different models for flow’ might have been arrived at by ‘people living in different kinds of bodies’.
137 Hayles, who is in general favorable to Irigaray, notes that:
From talking with several applied mathematicians and fluid mechanicists about Irigaray’s claim, I can testify that they unanimously conclude she does not know the first thing about their disciplines. In their view, her argument is not to be taken seriously.
There is evidence to support this view. In a footnote to the chapter’s first page, Irigaray airily advises the reader ‘to consult some texts on solid and fluid mechanics’ without bothering to mention any. The lack of mathematical detail in her argument forces one to wonder whether she has followed this advice herself. Nowhere does she mention a name or date that would enable one to connect her argument with a specific theory of fluids, much less to trace debates between opposing theories.
(Hayles 1992, p. 17)
138 For a non-technical explanation of the concept of linearity (applied to an equation), see p. 133 below.
139 The three preceding paragraphs, which supposedly concern mathematical logic, are devoid of meaning, with one exception: the assertion that ‘a preponderant role is left to ... the universal quantifier’ is meaningful and false (see note 143 below).
140 As we all learned in primary school, the symbol ‘+’ denotes the addition of two numbers. We are at a loss to explain how Irigaray got the idea that it indicates the ‘definition of a new term’.
141 We apologize to the reader for our pedantry: the negation of a proposition P is not ‘P or not P’, but simply ‘not P’.
142 This is presumably a typographical error; it occurs also in the original French and was overlooked by the translator. The conjunction of two propositions is, of course, ‘P and Q’.
143 To see this, let P(x) be any statement about an individual x. The proposition ‘for all x, P(x)’ is equivalent to ‘there does not exist x such that P(x) is false’. Similarly, the proposition ‘there exists at least one x such that P(x)’ is equivalent to ‘it is false that, for all x, P(x) is false’.
144 In actual fact, set theory (théorie des ensembles) studies the properties of ‘bare’ sets, that is, sets without any topological or geometrical structure. The questions alluded to here by Irigaray belong rather to topology, geometry and analysis.
145 Let us remark that the published English translation, quoted above, contains several errors. Théorie des ensembles is ‘set theory’, not ‘theory of wholes’. Ensembles flous presumably refers to the mathematical theory of ‘fuzzy sets’. Bords is best translated in the mathematical context as ‘boundaries’.
146 See, for example, Dieudonné (1989).
147 Let us remark that, in this passage, the word ‘linear’ is used three times, inappropriately and with three different apparent intended meanings. See p. 133 below for a discussion of abuses of the word ‘linear’.
148 Nor is this an isolated case. Hayles concludes her article on fluid mechanics by saying that
the experiences articulated in this essay are shaped by the struggle to remain within the bounds of rational discourse while still questioning some of its major premises. Whereas the flow of the argument has been female and feminist, the channel into which it has been directed is male and masculinist. (Hayles 1992, p. 40)
Hayles thus appears to accept, without the slightest hint of self-consciousness, the identification of ‘rational discourse’ with ‘male and masculinist’.
149 Irigaray (1993, p. 203).
150 For some even more shocking statements in the same vein, see Irigaray (1987b, pp. 106–8).
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151 Let us nevertheless quote the physicist Huth (1998), who has also made a critical analysis of Latour’s article: ‘In this article, the meanings of the terms “society” and “abstraction” are so stretched to fit his interpretation of relativity that they lose any semblance of common meaning, and shed no new light on the theory itself.’
152 For a good introduction to the theory of relativity, see, for example, Einstein (1960 [1920]), Mermin (1989) or Sartori (1996).
153 Indeed, by interpreting the collision of two protons with respect to the frame of reference attached to one of them, one can learn important things about the internal structure of protons.
154 Let us note in passing that Latour copies these equations incorrectly (Latour, 1988, p. 18, Figure 8). It should be v/c2 instead of v2/c2 in the numerator of the last equation.
155 Mermin (1997b) points out, correctly, that certain technical arguments in the theory of relativity involve comparing three (or more) frames of reference. But this has nothing to do with Latour’s purported ‘third frame that collects the information sent by the two others’.
156 Let us note that Latour, like Lacan (see pp. 18–19), insists here on the literal validity of a comparison that could, at best, be taken as a vague metaphor.
157 This notion arises in Latour’s sociology.
158 Mermin (1997b).
159 Mermin doesn’t go that far: he concedes that ‘there are, to be sure, many obscure statements that appear to be about the physics of relativity, which may well be misconstruals of elementary technical points.’ (Mermin 1997b, p. 13)
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160 Numerous examples of such texts are cited in Sokal’s parody (see Appendix A).
161 See also Bricmont (1995a) for a detailed study of confusions concerning the ‘arrow of time’.
162 Lyotard (1984, chapter 13).
163 In each cubic centimetre of air, there are approximately 2.7 x 1019 (= 27 billion billion) molecules.
164 Perrin (1990 [1913], pp. xii–xiv).
165 Ordinary (smooth) geometric objects can be classified according to their dimension, whic
h is always a whole number: for example, the dimension of a straight line or a smooth curve is equal to 1, while that of a plane or a smooth surface is equal to 2. By contrast, fractal objects are more complicated, and need to be assigned several distinct ‘dimensions’ to describe different aspects of their geometry. Thus, while the ‘topological dimension’ of any geometrical object (smooth or not) is always a whole number, the ‘Hausdorff dimension’ of a fractal object is in general not a whole number.
166 However, some physicists and mathematicians believe that the media hype surrounding these two theories has vastly exceeded their scientific accomplishments: see, for example, Zahler and Sussmann (1977), Sussmann and Zahler (1978), Kadanoff (1986) and Arnol’d (1992).
167 These are technical concepts from differential calculus: a function is called continuous if (here we are oversimplifying a bit) its graph can be drawn without taking the pencil off the paper, while a function is called differentiable if, at each point of its graph, there exists a unique tangent line. Let us note in passing that every differentiable function is automatically continuous, and that catastrophe theory is based on very beautiful mathematics concerning (ironically for Lyotard) differentiable functions.
168 ‘Non-rectifiable’ is another technical term from differential calculus; it applies to certain non-differentiable curves.
169 See also Bouveresse (1984, pp. 125–30) for a critique along similar lines.
170 With one small qualification: Metatheorems in mathematical logic, such as Gödel’s theorem or independence theorems in set theory, have a logical status that is slightly different from that of conventional mathematical theorems. It should, however, be emphasized that these rarefied branches of the foundations of mathematics have very little impact on the bulk of mathematical research and almost no impact on the natural sciences.