by Alan Sokal
Topology is not ‘made to guide us’ in structure. This structure is it – as retroaction of the chain order of which language consists.
Structure is the aspherical concealed in the articulation of language insofar as an effect of subject takes hold of it.
It is clear that, as far as meaning is concerned, this ‘takes hold of it’ of the sub-sentence – pseudo-modal – reverberates from the object itself which it wraps, as verb, in its grammatical subject, and that there is a false effect of meaning, a resonance of the imaginary induced by the topology, according to whether the effect of subject makes a whirlwind of asphere [sic] or the subjective of this effect ‘reflects’ itself from it.
Here one must distinguish the ambiguity that inscribes itself from the meaning, that is, from the loop of the cut, and the suggestion of hole, that is, of structure, which makes sense of this ambiguity.
(Lacan 1973, p. 40)
[Because Lacan’s language is so obscure, we reproduce the original French text:]
La topologie n’est pas ‘faite pour nous guider’ dans la structure. Cette structure, elle l’est – comme rétroaction de l’ordre de chaîne dont consiste le langage.
La structure, c’est l’asphérique recelé dans l’articulation langagière en tant qu’un effet de sujet s’en saisit.
Il est clair que, quant à la signification, ce ‘s’en saisit’ de la sous-phrase, pseudo-modale, se répercute de l’objet même que comme verbe il enveloppe dans son sujet grammatical, et qu’il y a faux effet de sens, résonance de l’imaginaire induit de la topologie, selon que l’effet de sujet fait tourbillon d’asphère ou que le subjectif de cet effet s’en ‘réfléchit’.
Il y a ici à distinguer l’ambiguïté qui s’inscrit de la signification, soit de la boucle de la coupure, et la suggestion de trou, c’est-à-dire de structure, qui de cette ambiguïté fait sens.
(Lacan 1973, p. 40)
If we leave aside Lacan’s mystifications, the relationship between topology and structure is easy to understand, but it depends upon what one means by ‘structure’. If this term is understood broadly – that is, as including linguistic and social structures as well as mathematical structures – then it clearly cannot be reduced to the purely mathematical notion of ‘topology’. If, on the other hand, one understands ‘structure’ in its strictly mathematical sense, then one sees easily that topology is one type of structure, but that there exist many others: order structure, group structure, vector-space structure, manifold structure, etc.
21 If the last two sentences have a meaning, they have, in any case, nothing to do with geometry.
22 Compactness is an important technical concept in topology, but rather difficult to explain. Suffice it to say that in the nineteenth century, mathematicians (Cauchy, Weierstrass and others) put mathematical analysis on a solid basis by giving a precise meaning to the concept of limit. These limits were initially used for sequences of real numbers, but it was slowly realized that the notion of limit should be extended to spaces of functions (for example, to study differential or integral equations). Topology was born circa 1900 in part through these studies. Now, among topological spaces one may distinguish a subclass called compact spaces, namely those in which every sequence of elements possesses a subsequence having a limit. (Here we have simplified somewhat, by limiting ourselves to metric spaces.) Another definition (which can be proven to be equivalent to the first one) relies on the intersection properties of infinite collections of closed sets. In the special case of subsets of finite-dimensional Euclidean spaces, a set is compact if and only if it is closed and bounded. Let us emphasize that all the italicized words above are technical terms having very precise definitions (which in general are based on a long chain of other definitions and theorems).
23 See note 19 above.
24 In this sentence, Lacan gives an incorrect definition of open set and a meaningless ‘definition’ of limit. But these are minor points compared to the overall confusion of the discourse.
25 This paragraph is pure pedantry. Obviously, if a set is finite, one can, in principle, ‘count’ it and ‘order’ it. All the discussions in mathematics concerning countability (see note 38 below) or the possibility of ordering sets are motivated by infinite sets.
26 See note 19 above.
27 A number is called irrational if it cannot be written as a ratio of two integers: for example, the square root of two, or π. (By contrast, zero is an integer, hence unavoidably a rational number.) The imaginary numbers, on the other hand, are introduced as solutions of polynomial equations that have no solutions among the real numbers: for example, x2 + 1 = 0, one of whose solutions is denoted i = √ –1 and the other –i.
28 For an exegesis of Lacan’s ‘algorithm’ that is almost as ridiculous as the original text, see Nancy and Lacoue-Labarthe (1992, part I, chapter 2).
29 This last sentence may be a rather confused allusion to a technical procedure used in mathematical logic to define the natural numbers in terms of sets: 0 is identified with the empty set Ø (i.e. the set having no element); then 1 is identified with the set {Ø} (i.e. the set having Ø as its sole element); then 2 is identified with the set {Ø, {Ø}} (i.e. the set having the two elements Ø and {Ø}); and so forth.
30 The paradox to which Lacan is alluding here is due to Bertrand Russell (1872–1970). Let us begin by observing that most ‘normal’ sets do not contain themselves as an element: for example, the set of all chairs is not itself a chair, the set of all whole numbers is not a whole number, etc. On the other hand, some sets do apparently contain themselves as an element: for example, the set of all abstract ideas is itself an abstract idea, the set of all sets is a set, etc. Consider now the set of all sets that do not contain themselves as an element. Does it contain itself? If the answer is yes, then it cannot belong to the sets of all sets that do not contain themselves, and therefore the answer should be no. But if the answer is no, then it must belong to the set of all sets that do not contain themselves, and the answer should be yes. To escape from this paradox, logicians have replaced the naive concept of set by a variety of axiomatic theories.
31 This is perhaps an allusion to a different (though related) paradox, due to Georg Cantor (1845–1918), concerning the nonexistence of the ‘set of all sets’.
32 See e.g. Miller (1977/78) and Ragland-Sullivan (1990) for worshipful commentary on Lacan’s mathematical logic.
33 Because Lacan’s language is so obscure and frequently ungrammatical, we have reproduced the complete French text following our best attempt at a translation.
34 In mathematical logic, the symbol ∀x means ‘for all x’, and the symbol ∃x means ‘there exists at least one x such that’; they are called the ‘universal quantifier’ and the ‘existential quantifier’, respectively. Further down in the text, Lacan writes Ax and Ex to denote the same concepts.
35 Just so. The bar denotes negation (‘it is false that’) and can thus be applied only to complete propositions, not to isolated quantifiers such as Ex or Ax. One might think that here Lacan means and – which would in fact be logically equivalent to his starting propositions Ax·Φx and – but he makes clear that this banal rewriting is not his intention. Anyone is free to introduce a new notation, but he then has the obligation of explaining its meaning.
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36 One of Kristeva’s commentators, Toril Moi, explains the context:
In 1966 Paris witnessed not only the publication of Jacques Lacan’s Écrits and Michel Foucault’s Les Mots et les choses, but also the arrival of a young linguist from Bulgaria. At the age of 25, Julia Kristeva ... took the Left Bank by storm. ... Kristeva’s linguistic research was soon to lead to the publication of two important books, Le Texte du roman and Séméiotiké, and to culminate with the publication of her massive doctoral thesis, La Révolution du langage poétique, in 1974. This theoretical production earned her a chair in linguistics at the University of Paris VII. (Moi 1986, p. 1)
37 Here
Kristeva seems to be appealing implicitly to the ‘Sapir-Whorf thesis’ in linguistics, that is, grosso modo, the idea that our language radically conditions our view of the world. This thesis is nowadays sharply criticized by some linguists: see, for example, Pinker (1995, pp. 57–67).
38 The ‘power of the continuum’ is a concept belonging to the mathematical theory of infinite sets, which was developed by Georg Cantor and other mathematicians starting in the 1870s. It turns out that there are many different ‘sizes’ (or cardinalities) of infinite sets. Some infinite sets are termed countable (or denumerable): for example, the set of all positive integers (1, 2, 3, ...) or, more generally, any set whose elements can be put into one-to-one correspondence with the set of all positive integers. On the other hand, Cantor proved in 1873 that there does not exist a one-to-one correspondence between the integers and the set of all real numbers. Therefore, the real numbers are in a certain sense ‘more numerous’ than the integers: they are said to have the cardinality (or power) of the continuum, as do all those sets that can be put in one-to-one correspondence with them. Let us remark (what is at first surprising) that one can establish a one-to-one correspondence between the real numbers and the real numbers contained in an interval: for example, those numbers between 0 and 1, or those between 0 and 2, etc. More generally, every infinite set can be put into one-to-one correspondence with some of its proper subsets.
39 Translation ours. A slightly different translation of this excerpt and the next one can be found in Kristeva (1980, pp. 70–72).
40 In mathematics, the word ‘transfinite’ is more or less synonymous with ‘infinite’. It is used most commonly to characterize a ‘cardinal number’ or an ‘ordinal number’.
41 As we saw in note 38 above, there exist infinite sets of different ‘sizes’ (called cardinals). The smallest infinite cardinal, called ‘countable’ (or ‘denumerable’), is the one corresponding to the set of all positive integers. A larger cardinal, called the ‘cardinal of the continuum’, is the one corresponding to the set of all real numbers. The continuum hypothesis (CH), introduced by Cantor in the late nineteenth century, asserts that there is no ‘intermediate’ cardinal between the countable and the continuum. The generalized continuum hypothesis (GCH) is an extension of this idea to vastly larger infinite sets. In 1964, Cohen proved that the CH (as well as the GCH) is independent of the other axioms of set theory, in the sense that neither it nor its negation is provable using those axioms.
42 This is a technical result of Gödel-Bernays set theory (one of the versions of axiomatic set theory). Kristeva does not explain its relevance for poetic language. Let us note in passing that to precede such a technical statement by the expression ‘as one knows’ (on le sait) is a typical example of intellectual terrorism.
43 It is rather improbable that Lautréamont (1846–1870) could have ‘consciously practiced’ a theorem of Gödel-Bernays set theory (developed between 1937 and 1940) or even of set theory tout court (developed after 1870 by Cantor and others).
44 Gödel, in his famous article (1931), proved two principal theorems concerning the incompleteness of certain formal systems (complex enough to encode elementary arithmetic) in mathematical logic. Gödel’s first theorem exhibits a proposition that is neither provable nor refutable in the given formal system, provided that this system is consistent. (One may nevertheless see, using reasoning that cannot be formalized within the system, that this proposition is true.) Gödel’s second theorem asserts that, if the system is consistent, it is impossible to prove this property by means that can be formalized within the system itself.
On the other hand, it is very easy to invent inconsistent (i.e. self-contradictory) systems of axioms; and, when a system is inconsistent, there always exists a proof of this inconsistency by means formalized within the system: although this proof may sometimes be difficult to find, it exists, almost by virtue of the definition of the world ‘inconsistent’.
For an excellent introduction to Gödel’s theorem, see Nagel and Newman (1958).
45 See note 31 above. It must be emphasized that no problem occurs for finite sets, such as the set of individuals in a society.
46 Nicolas Bourbaki is the pseudonym of a group of prominent French mathematicians who, since the late 1930s, have published about thirty volumes of their series, Elements of Mathematics. But, despite the title, these books are far from elementary. Whether or not Kristeva has read Bourbaki, this reference has no function other than to impress the reader.
47 The space C0(R3) is composed of all the real-valued continuous functions, defined on R3, that ‘tend to zero at infinity’. But, in the precise definition of this concept, Kristeva should have said: (a) ∣F(X)∣ instead of F(X); (b) ‘exceeds 1/n’ instead of ‘exceeds n’; and (c) ‘containing all the continuous functions F on R3 such that’ instead of ‘where for every continuous function F on R3’.
48 This malapropism probably arises from a combination of two mistakes: on the one hand, it seems that Kristeva has confused predicate logic with propositional logic; and on the other hand, she or her editors have apparently inserted the typographical error ‘proportional’ (proportionnelle) in place of ‘propositional’ (propositionnelle).
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49 There are, of course, many other sources of the relativist Zeitgeist, from Romanticism to Heidegger, but we shall not deal with them here.
50 With, of course, many nuances about the meaning of the word ‘objective’, which are reflected, for instance, in the opposition between such doctrines as realism, conventionalism and positivism. Nevertheless, few scientists would be ready to admit that the whole of scientific discourse is a mere social construction. As one of us wrote, we have no desire to be the Emily Post of quantum field theory (Sokal 1996c, p. 94, reproduced here in Appendix C) – Emily Post is the author of a classic American manual of social etiquette.
51 Limiting ourselves to the natural sciences and taking most of the examples from our own field, physics. We shall not deal with the delicate question of the scientificity of the various social sciences.
52 Bertrand Russell (1948, p. 196) tells the following amusing story: ‘I once received a letter from an eminent logician, Mrs Christine Ladd Franklin, saying that she was a solipsist, and was surprised that there were not others’. We learned this reference from Devitt (1997, p. 64).
53 To claim this does not mean that we claim to have an entirely satisfactory answer to the question of how such a correspondence between objects and perceptions is established.
54 This hypothesis receives a deeper explanation with the subsequent development of science, in particular of the biological theory of evolution. Clearly, the possession of sensory organs that reflect more or less faithfully the outside world (or, at least, some important aspects of it) confers an evolutionary advantage. Let us stress that this argument does not refute radical scepticism, but it does increase the coherence of the anti-sceptical worldview.
55 For example: Water appears to us as a continuous fluid, but chemical and physical experiments teach us that it is made of atoms.
56 Throughout this chapter, we stress the methodological continuity between scientific knowledge and everyday knowledge. This is, in our view, the proper way to respond to various sceptical challenges and to dispel the confusions generated by radical interpretations of correct philosophical ideas such as the underdetermination of theories by data. But it would be naive to push this connection too far. Science – particularly fundamental physics – introduces concepts that are hard to grasp intuitively or to connect directly to common-sense notions. (For example: forces acting instantaneously throughout the universe in Newtonian mechanics, electromagnetic fields ‘vibrating’ in vacuum in Maxwell’s theory, curved space-time in Einstein’s general relativity.) And it is in discussions about the meaning of these theoretical concepts that various brands of realists and anti-realists (e.g., instrumentalists, pragmatists) tend to part company. Relativists sometimes tend to fal
l back on instrumentalist positions when challenged, but there is a profound difference between the two attitudes. Instrumentalists may want to claim either that we have no way of knowing whether ‘unobservable’ theoretical entities really exist, or that their meaning is defined solely through measurable quantities; but this does not imply that they regard such entities as ‘subjective’ in the sense that their meaning would be significantly influenced by extra-scientific factors (such as the personality of the individual scientist or the social characteristics of the group to which she belongs). Indeed, instrumentalists may regard our scientific theories as, quite simply, the most satisfactory way that the human mind, with its inherent biological limitations, is capable of understanding the world.
57 Expressed in a well-defined unit which is unimportant for the present discussion.
58 See Kinoshita (1995) for the theory, and Van Dyck et al. (1987) for the experiment. Crane (1968) provides a non-technical introduction to this problem.
59 Subject, of course, to many nuances on the precise meaning of the phrases ‘approximately true’ and ‘objective knowledge of the natural world’, which are reflected in the diverse versions of realism and anti-realism (see note 56 above). For these debates, see for example Leplin (1984).
60 It is also by proceeding on a case-by-case basis that one can appreciate the immensity of the gulf separating the sciences from the pseudo-sciences.
61 We hasten to add – as if this should even be necessary – that we harbour no illusions about the behaviour of real-life police forces, which are by no means always and exclusively dedicated to finding the truth. We employ this example solely to illustrate the abstract epistemological question in a simple concrete context, namely: Suppose that one does wish to find the truth about a practical matter (such as who committed a murder); how would one go about it? For an extreme example of this misreading – in which we are compared to former Los Angeles Detective Mark Fuhrman (of O.J. Simpson fame) and his infamous Brooklyn counterparts – see Robbins (1998).