Intellectual Impostures

by Alan Sokal

  Irigaray continues with a bizarre mélange of fluids, psychoanalysis and mathematical logic:

  Certainly the emphasis has increasingly shifted from the definition of terms to the analysis of relations among terms (Frege’s theory is one example among many). This has even led to the recognition of a semantics of incomplete beings: functional symbols.

  But, beyond the fact that the indeterminacy thus allowed in the proposition is subject to a general implication of the formal type – the variable is such only within the limits of the identity of (the) form(s) of syntax – a preponderant role is left to the symbol of universality – to the universal quantifier – whose modalities of recourse to the geometric still have to be examined.

  Thus the ‘all’ – of x, but also of the system – has already prescribed the ‘not-all’ of each particular relation established, and that ‘all’ is such only by a definition of extension that cannot get along without projection onto a given space-map, whose between(s) will be given their value(s) on the basis of punctual frames of reference.139

  The ‘place’ thus turns out to have been in some way planned and punctuated for the purpose of calculating each ‘all,’ but also the ‘all’ of the system. Unless it is allowed to extend to infinity, which rules out in advance any determination of value for either the variables or their relations.

  But where does that place – of discourse – find its ‘greater-than-all’ in order to be able to form(alize) itself in this way? To systematize itself? And won’t that greater than ‘all’ come back from its denegation – from its forclusion? – in modes that are still theo-logical [sic]? Whose relation to the feminine ‘not-all’ remains to be articulated: God or feminine pleasure.

  While she waits for these divine rediscoveries, awoman [sic] serves (only) as a projective map for the purpose of guaranteeing the totality of the system – the excess factor of its ‘greater than all’; she serves as a geometric prop for evaluating the ‘all’ of the extension of each of its ‘concepts’ including those that are still undetermined, serves as fixed and congealed intervals between their definitions in ‘language,’ and as the possibility of establishing individual relationships among these concepts.

  (Irigaray 1985a, pp. 107–8)

  A bit further down, Irigaray returns to fluid mechanics:

  What is left uninterpreted in the economy of fluids – the resistances brought to bear upon solids, for example – is in the end given over to God. Overlooking the properties of real fluids – internal frictions, pressures, movements, and so on, that is, their specific dynamics – leads to giving the real back to God, as only the idealizable characteristics of fluids are included in their mathematicization.

  Or again: considerations of pure mathematics have precluded the analysis of fluids except in terms of laminated planes, solenoid movements (of a current privileging the relation to an axis), spring-points, well-points, whirlwind-points, which have only an approximate relation to reality. Leaving some remainder. Up to infinity: the center of these ‘movements’ corresponding to zero supposes in them an infinite speed, which is physically unacceptable. Certainly these ‘theoretical’ fluids have enabled the technical – also mathematical – form of analysis to progress, while losing a certain relationship to the reality of bodies in the process.

  What consequences does this have for ‘science’ and psychoanalytic practice?

  (Irigaray 1985a, p. 109)

  In this passage, Irigaray shows that she does not understand the role of approximations and idealizations in science. First of all, the Navier–Stokes equations are approximations that are valid only on a macroscopic (or at least supra-atomic) scale, because they treat the fluid as a continuum and neglect its molecular structure. And since these equations are themselves very hard to solve, mathematicians try to study them first in idealized situations or through more-or-less controlled approximations. But the fact that, for example, the speed is infinite at the center of a vortex means only that the approximation ought not be taken too seriously near that point – as was obvious from the start, since the approach is in any case valid only on scales much larger than that of molecules. In any case, nothing is ‘given over to God’; there are, quite simply, scientific problems left for future generations.

  Finally, it is hard to see what relation, besides a purely metaphorical one, fluid mechanics could have with psychoanalysis. Suppose that tomorrow someone were to come up with a satisfactory theory of turbulence. In what way would (or should) that affect our theories of human psychology?

  One could continue quoting Irigaray, but the reader is probably lost (so are we). She concludes her essay with some words of consolation:

  And if, by chance, you were to have the impression of not having yet understood everything, then perhaps you would do well to leave your ears half-open for what is in such close touch with itself that it confounds your discretion.

  (Irigaray 1985a, p. 118)

  All in all, Irigaray fails to understand the nature of the physical and mathematical problems arising in fluid mechanics. Her discourse is based solely on vague analogies that, moreover, mix up the theory of real fluids with its already analogical use in psychoanalysis. Irigaray seems to be aware of this problem, as she answers it as follows:

  And if anyone objects that the question, put this way, relies too heavily on metaphors, it is easy to reply that the question in fact impugns the privilege granted to metaphor (a quasi solid) over metonymy (which is much more closely allied to fluids).

  (Irigaray 1985a, pp. 109–10)

  Alas, this reply reminds us of the old Jewish joke: ‘Why does a Jew always answer a question with a question?’ ‘And why shouldn’t a Jew answer a question with a question?’

  Mathematics and Logic

  As we have seen, Irigaray has a penchant for reducing problems in the physical sciences to games of mathematical formalization or even of language. Unfortunately, her knowledge of mathematical logic is as superficial as her knowledge of physics. An illustration of this can be found in her famous essay ‘Is the Subject of Science Sexed?’ After a rather idiosyncratic sketch of the scientific method, Irigaray goes on to write:

  These characteristics reveal an isomorphism in man’s sexual Imaginary, an isomorphism which must remain rigourously masked. ‘Our subjective experiences and our beliefs can never justify any utterance,’ affirms the epistemologist of the sciences.

  You must add that all of these discoveries must be expressed in a language that is well-written, meaning reasonable, that is:

  – expressed in symbols or letters, interchangeable with proper nouns, that refer only to an intra-theoretical object, thus to no character or object from the real or from reality. The scholar enters into a fictional universe that is incomprehensible to those who do not participate in it.

  (Irigaray 1985b, p. 312; Irigaray 1987a, p. 73)

  Here again one encounters Irigaray’s misunderstandings concerning the role of mathematical formalism in science. It is not true that all the concepts of a scientific theory ‘refer only to an intra-theoretical object’. Quite the contrary, at least some of the theory’s concepts must correspond to something in the real world, for otherwise the theory would have no empirical consequences whatsoever (and thus not be scientific). Consequently, the scientist’s universe is not populated solely by fictions. Finally, neither the real world nor the scientific theories that explain it are completely incomprehensible to non-experts; in many cases, there exist good popular or semi-popular expositions.

  The remainder of Irigaray’s text is both pedantic and unwittingly comic:

  – the formative signs for terms and for predicates are:

  + : or definition of a new term;140

  = : which indicates a property by equivalence and substitution (belonging to a whole or to a world);

  ∈: signifying belonging to an object type

  – the quantifiers (and not qualifiers) are:

  ≥ ≤;

  the universal quantifier;
r />   the existential quantifier submitted, as its name indicates, to the quantitative.

  According to the semantics of incomplete beings (Frege), functional symbols are variables found at the boundary of the identity of syntactic forms and the dominant role is given to the universality symbol or universal quantifier.

  – the connectors are:

  – negation: P or not P;141

  – conjunction: P or Q;142

  – disjunction: P or Q;

  – implication: P implies Q;

  – equivalence: P equals Q;

  There is then no sign:

  – of difference other than the quantitative;

  – of reciprocity (other than within a common property or a common whole);

  – of exchange;

  – of fluidity.

  (Irigaray 1985b, pp. 312–3; Irigaray 1987a, pp. 73–4)

  To begin with, Irigaray has confused the concept of ‘quantification’ in logic with the word’s everyday meaning (i.e., making something quantitative or numerical); in actual fact, there is no relationship between these two concepts. The quantifiers in logic are ‘for all’ (universal quantifier) and ‘there exists’ (existential quantifier). For example, the sentence ‘x likes chocolate’ is a statement about a certain individual x; the universal quantifier transforms it into the statement ‘for all x [belonging to some set assumed known], x likes chocolate’, while the existential quantifier transforms it into ‘there exists at least one x [belonging to some set assumed known] such that x likes chocolate’. This clearly has nothing to do with numbers, and Irigaray’s purported opposition between ‘quantifiers’ and ‘qualifiers’ is meaningless.

  Besides, the inequality signs ‘≥’ (greater than or equal to) and ‘≤’ (less than or equal to) are not quantifiers. They relate to quantification in the ordinary sense of the word, not in the sense of quantifiers in logic.

  Moreover, no ‘dominant role’ is granted to the universal quantifier. Quite the contrary, there is a perfect symmetry between the universal and existential quantifiers, and any proposition using one of them can be transformed into a logically equivalent proposition using the other (at least in classical logic, which is Irigaray’s supposed subject).143 This is an elementary fact, taught in every introductory logic course; it is surprising that Irigaray, who speaks so much about mathematical logic, should fail to know it.

  Finally, her assertion that there is no sign (or, what is more relevant, no concept) of difference other than the quantitative is false. In mathematics, there are many types of objects other than numbers – for example, sets, functions, groups, topological spaces, etc. – and, when talking about two such objects, one may of course say that they are identical or different. The standard equality sign (=) is used to indicate that they are identical, and the standard inequality sign (≠) to indicate that they are different.

  Later in the same essay, Irigaray claims to unmask also the sexist biases at the heart of ‘pure’ mathematics:

  – the mathematical sciences, in the theory of wholes [théorie des ensembles], concern themselves with closed and open spaces, with the infinitely big and the infinitely small.144 They concern themselves very little with the question of the partially open, with wholes that are not clearly delineated [ensembles flous], with any analysis of the problem of borders [bords], of the passage between, of fluctuations occurring between the thresholds of specific wholes. Even if topology suggests these questions, it emphasizes what closes rather than what resists all circularity.

  (Irigaray 1985b, p. 315; Irigaray 1987a, pp. 76–7145)

  Irigaray’s phrases are vague: ‘the partially open’, ‘the passage between’, ‘fluctuations between the thresholds of specific wholes’ – what exactly is she talking about? For what it’s worth, the ‘problem’ of boundaries [bords], far from being neglected, has been at the center of algebraic topology since its inception a century ago,146 and ‘manifolds with boundary’ [variétés à bord] have been actively studied in differential geometry for at least 50 years. And, last but not least, what does all this have to do with feminism?

  We were therefore quite surprised to find this passage quoted in a recent book devoted to the teaching of mathematics. The author is a prominent American feminist pedagogue of mathematics, whose goal – which we share wholeheartedly – is to attract more young women to scientific careers. She quotes approvingly this text of Irigaray and continues by saying:

  In the context provided by Irigaray we can see an opposition between the linear time of mathematics problems of related rates, distance formulas, and linear acceleration versus the dominant experiential cyclical time of the menstrual body. Is it obvious to the female mind-body that intervals have endpoints, that parabolas neatly divide the plane, and, indeed, that the linear mathematics of schooling describes the world of experience in intuitively obvious ways?147

  (Damarin 1995, p. 252)

  This theory is startling, to say the least: Does the author really believe that menstruation makes it more difficult for young women to understand elementary notions of geometry? This view is uncannily reminiscent of the Victorian gentlemen who held that women, with their delicate reproductive organs, are unsuited to rational thought and to science. With friends like these, the feminist cause hardly needs enemies.148

  One finds similar ideas in Irigaray’s own writings. Indeed, her scientific confusions are linked to, and are taken to provide support for, more general philosophical considerations of a vaguely relativist nature. Starting from the idea that science is ‘masculine’, Irigaray rejects ‘the belief in a truth independent of the subject’ [la croyance en une vérité indépendante du sujet] and advises women not to

  accept or subscribe to the existence of a neutral, universal science, to which women should painfully gain access and with which they then torture themselves and taunt other women, transforming science into a new superego.

  (Irigaray 1993, p. 203)

  These claims are clearly very debatable. To be sure, they are accompanied by more nuanced assertions, for example: ‘Truth is always the product of some man or woman. This does not mean that truth contains no objectivity’; and ‘All truth is partially relative.’149 The problem is to know exactly what Irigaray wants to say and how she intends to resolve these contradictions.

  The roots of the tree of science may be bitter, but its fruits are sweet. To say that women should shun a universal science amounts to infantilizing them. To link rationality and objectivity to the male, and emotion and subjectivity to the female, is to repeat the most blatant sexist stereotypes. Speaking of the female ‘sexual economy’ from puberty through menopause, Irigaray writes:

  But every stage in this development has its own temporality, which is possibly cyclic and linked to cosmic rhythms. If women have felt so terribly threatened by the accident at Chernobyl, that is because of the irreducible relation of their bodies to the universe.

  (Irigaray 1993, p. 200)150

  Here Irigaray falls straight into mysticism. Cosmic rhythms, relation to the universe – what on earth is she talking about? To reduce women to their sexuality, menstrual cycles and rhythms (cosmic or not) is to attack everything the feminist movement has fought for during the last three decades. Simone de Beauvoir must be turning in her grave.

  6

  BRUNO LATOUR

  The sociologist of science Bruno Latour is well known for his book Science in Action, which we have analysed briefly in Chapter 4. Much less well known, however, is his semiotic analysis of the theory of relativity, in which ‘Einstein’s text is read as a contribution to the sociology of delegation’ (Latour 1988, p. 3). In this chapter, we shall examine Latour’s interpretation of relativity and show that it illustrates perfectly the problems encountered by a sociologist who aims to analyse the content of a scientific theory he does not understand very well.

  Latour considers his article as a contribution to, and extension of, the strong programme in the sociology of science, which asserts that ‘the con
tent of any science is social through and through’ (p. 3). According to Latour, the strong programme has had ‘some degree of success in the empirical sciences’ but less in the mathematical sciences (p. 3). He complains that previous social analyses of Einstein’s theory of relativity have ‘shunned the technical aspects of his theory’ and failed to give any ‘indication of how relativity theory itself could be said to be social’ (pp. 4–5, italics in the original). Latour sets himself the ambitious task of vindicating this latter idea, which he proposes to do by redefining the concept of ‘social’ (pp. 4–5). For the sake of brevity, we won’t enter into the sociological conclusions Latour purports to draw from his study of relativity, but shall simply point out that his argument is undermined by several fundamental misunderstandings about the theory of relativity itself.151

  Latour bases his analysis of the theory of relativity on a semiotic reading of Einstein’s semi-popular book, Relativity: The Special and the General Theory (1920). After a survey of semiotic notions such as ‘shifting in’ and ‘shifting out’ of narrators, Latour attempts to apply these notions to Einstein’s special theory of relativity. But, in so doing, he misunderstands the concept of ‘frame of reference’ in physics. A brief digression is therefore in order.

  In physics, a frame of reference is a scheme for assigning spatial and temporal coordinates (x,y,z,t) to ‘events’. For example, an event in New York City can be located by saying that it takes place at the corner of 6th Avenue (x) and 42nd Street (y), 30 metres above ground level (z), at noon on 1 May 1998 (t). In general, a frame of reference can be visualized as a rigid rectangular framework of metre sticks and clocks, which together allow coordinates of ‘where’ and ‘when’ to be assigned to any event.