In modern logic and science, axioms are the primitive propositions of a deductive theory and ‘axiomatics’ is the theory of deductive systems. The axiomatisation of set theory was an important reference in the Cahiers pour l’Analyse.
In his classic study L’axiomatique (1955), Robert Blanché, a contributor to the Cahiers’ volume 10 on ‘Formalisation’, emphasises the modern character of axioms in contrast to the ‘postulates’ one finds in Euclidean geometry and other earlier mathematical models. For Euclid, the ‘postulates’ are like ‘common notions’, and they are effective insofar as they refer to an intuition, as in the case of the postulate which states that parallel lines never touch. This postulate demands assent by calling upon a spatial intuition.
By contrast, modern axiomatics is designed precisely to avoid recourse to intuition. A theory is axiomatised when all of its presuppositions and presumptions are fully out in the open, articulated as such.1 The virtue of axiomatics is a total clarity and transparency of logical operations; its ‘default’, as it were, is that it calls upon no element but its own consistency to maintain its legitimacy. As Blanché notes, ‘even in the formal science par excellence, axiomatised mathematics, it is necessary to resign oneself to the distinction, which was thought to be effaced, between truth and demonstrability’.2 But Blanché equally notes axiomatics’ essential role in scientific progress. He also emphasises several aspects that help account for its appeal to the editors of the Cahiers pour l’Analyse, who were determined to develop accounts of subjectivity and of science that did not rely upon the figure of ‘intuition’ found in Kantianism or phenomenology.
Though typically regarded as a point of departure, axioms themselves are always the result of a ‘process of abstraction’ (76).3 But with abstraction comes an increase in ‘generality’ as well. Axioms allow for the positing of isomorphisms and the thinking of invariants. By abstracting from the concrete and from the domain of intuitive or empirical experience, ‘the axiomatic method has the precise value of revealing isomorphisms between apparently heterogeneous concrete theories, gathering them in the unity of an abstract system’ (46). Indeed, ‘one thinks the multiple in the one’ (77). The discovery of ‘vast intellectual landscapes’ theretofore linked by a few fragments is accomplished through the extrication of ‘the invariant structure’ that axiomatics allows.
Certain features are peculiar to this new ‘general’ thinking however. First, axiomatics begins with an abandonment of the effort to think scientific objects directly, focusing instead, deliberately and wilfully, solely on symbols and their logical relations (62). What this entails is that axiomatics also abandons any concern for the philosophical concept of ‘essence’, a move that has implications for questions of ontology presented by mathematics (80). For Blanché, this is a crucial element of axiomatics; it liberates us from a ‘dogmatism of synthesis’ (107) that insists on an ultimate reconciliation (or indeed synthesis) between reason and experience or the rational and empirical. It is ‘the refusal of any a priori’ that accounts for the scientific progress gained in axiomatics, a progress that is grounded in an incessant tension between the ‘symbolic schema’ and the ‘concrete model’ that never resolves into a synthetic unity, but rather leads to the adjustment of each (108).
In this brief overview, we see several aspects of axiomatics that account for its appeal to the Cercle d’Épistémologie and in particular its resonance with Lacanian psychoanalysis. First, there is the thinking only with symbols, i.e. the position that thought only operates with and has access to chains of signification. Second, there is the abandonment of any reference to ‘essence’, i.e. the ‘real’ itself is only present as a complete absence from the discourse. Third, the ‘truth’ of an axiomatic discourse - despite Blanché’s own misgivings about the use of the term - is grounded in nothing but the discourse itself. Fourth, in its abrogation of ‘synthesis’ as a model or aim for science, in favour of a vision of science as predicated upon tension and stratification, adjustment and breakthrough, axiomatics as such is consistent with the concept of the epistemological break that recurs throughout the Cahiers.
Beyond the most general sense of axiomatics, there was a particular instance of axiomatisation that was crucial for the authors of the Cahiers: the Zermelo-Fraenkel axiomatisation of set theory. After Georg Cantor’s demonstration of transfinite cardinal numbers, and his attempt to ground mathematics in a theory of sets, mathematicians were split as to the viability of his results, especially concerning two related problems. On the one hand, mathematicians were concerned to avoid the set theoretical paradoxes - e.g. ‘the set of all sets who do not include themselves as members’ - permitted by Cantor’s framework. On the other, many were troubled by what Michael Hallett has called Cantor’s ‘ordinal theory of cardinality’, i.e. his belief that ultimately all the transfinite cardinals ‘are capable of being ordinally numbered’, or ‘counted’ up from zero.4 (Cf. Cantor’s ‘Fondements d’une théorie générale des ensembles’ CpA 10.3). Such a position was untenable to intuitionist mathematicians precisely because such a procedure could not be constructed, that is to say, accomplished and intuited, in thought.5
Ernst Zermelo’s response to this quandary was to decouple set theory from intuition altogether by grounding it in axioms. What was decisive in his gesture, however, was that axioms no longer concerned the objects of mathematical or logical operations, i.e. what sets ‘are’, but the method of their presentation and concatenation.6 In other words, the how completely supplants the what. As John van Neumann later put it, ‘in the spirit of the axiomatic method one understands by “set” nothing but an object of which one knows no more and wants to know no more than what follows about it from the postulates’.7 The most controversial of the ZF axioms was the ‘axiom of choice’, which states: ‘If a is a set, all of whose elements are non-empty sets no two of which have any elements in common, then there is a set c which has precisely one element in common with each element of a’.8 This axiom defies all construction because it posits the existence of a rule without providing knowledge of how it might be implemented.
Mathematicians remained divided on the ontological status of the ‘axiom of choice’ even though it allows for much in the way of mathematical demonstration and operation. But it is precisely the rupture with the ontological status of mathematical objects - insofar as this connotes a question of ‘essence’ - that is crucial to axiomatisation. The ‘essence’ of axiomatisation is the formalism of the method itself and its indifference to its objects.9
In the Cahiers pour l’Analyse
In the journal’s first article, ‘La Science et la vérité’ (CpA 1.1), Jacques Lacan suggests that axiomatisation points to the collapse of the distinction between exact and conjectural sciences ‘once conjecture is subject to exact calculation (using probability) and exactness is merely grounded in a formalism separating axioms and compounding law from symbols’ (CpA 1.1:14 E, 863). In the following article, Yves Duroux grounds his reading of Frege in Giuseppe Peano’s axioms of number (CpA 1.2). Jacques-Alain Miller introduces his argument in ‘La Suture: Éléments d’une logique du signifiant’ (CpA 1.3) with reference to the role suture plays in the constitution of an axiomatic system:
The main thread of this analysis will be Gottlob Frege’s argument in Grundlagen der Arithmetik crucial here because it puts into question those terms which in Peano’s axiomatic, adequate for a construction of a theory of natural numbers, are taken as primary - that is, the zero, the number, the successor. This calling into question of the theory, by disintricating, from the axiomatic where the theory is consolidated, the suturing, delivers up this last (CpA 1.3:40/trans. 26).
In volume 9, ‘Généalogie des science’, François Regnault’s ‘Dialectique d’épistémologies’ (CpA 9.4) reads the hypotheses in Plato’s Parmenides concerning the relation of the One to its others, and the absolute to the relative, as a series of ‘axioms’. Regnault’s goal is to track the implications for epistemology, that is, a scientific theory of science, according to varying ‘axiomatic’ foundations.
In the same issue, Thomas Herbert’s ‘Remarques pour une théorie générale des idéologies’ refers to the ‘juridical-political axiomatic’ created through the metaphorical displacement of categories from the economic sphere into the political and then ideological spheres (CpA 9.5:87).
In ‘La Formalisation en linguistique’ (CpA 9.7), Antoine Culioli concludes that any axiomatisation of linguistics must be predicated on a refusal to reduce language to a collection of individual phenomena.
In his ‘La Subversion infinitésimale’ (CpA 9.8), the first of Alain Badiou’s two contributions to the Cahiers, the success with which the mathematician Abraham Robinson proved that infinitesimals are indeed numbers is evidence of the subversive power of a formalised thought. Inspired by Lacan, Badiou contends in this essay that for any given ‘domain of fixed proofs, impossibility characterises the real’ (CpA 9.8:122). A specific and consistent axiomatic system proceeds on the basis of specific exclusions. Not all statements are possible within the system. The variable, then, insofar as it indicates a place that cannot be occupied but that nevertheless can be inscribed or ‘constructed’, figures as the ‘operator of the real for a domain’. This inscription is what was accomplished in Robinson’s theory of infinitesimals.
Volume 10 of the Cahiers is devoted to ‘formalisation’. A text by Jacques Brunschwig on Aristotle (CpA 10.1), together with the writings of George Boole (CpA 10.2) and Georg Cantor (CpA 10.3), provide some of the pre-history of modern axiomatisation. The translation of Bertrand Russell’s ‘Theory of Logical Types’ (CpA 10.4) develops his own ‘axiom of reducibility’, an axiom Russell posited to salvage his ramified, hierarchical theory of types. This axiom stipulates that any higher order function with an object a among its arguments is formally equivalent to a first-order function with a among its arguments. In effect, then, this extra-logical axiom backtracks on the hierarchical order Russell had established to avoid the problem of self-reference in order to account for the identity of qualities expressed at different levels of the hierarchy.
Kurt Gödel justifies affirmation of mathematical axioms in terms of what he calls logical ‘evidence’, on analogy with that played by sensory data in justifying our beliefs in scientific laws. ‘Axioms need not be evident in themselves, but rather their justification lies (exactly as in physics) in the fact that they make it possible for these “sense perceptions” to be deduced; which of course would not exclude that they have a kind of intrinsic plausibility similar to that in physics’ (CpA 10.5:86/449).
The limits of axiomatisation receive their fullest treatment in Jean Ladrière’s ‘Le Théorème de Löwenheim-Skolem’ (CpA 10.6). Ladrière begins with a general discussion of the relation between semantics and syntax as methods of formalisation. A semantic method is one that formalises a theory extrinsically, that is, by describing its construction with reference to a domain of objects to which it might apply. By contrast, a syntactical method of formalisation is intrinsic and axiomatic, in that it departs from axioms internal to the theory itself and obtains all the propositions of the theory from this point of departure. The Löwenheim-Skolem theorem complicates this relation by showing how thoroughgoing axiomatisation on a syntactical level can produce contradictions with the semantic ‘sense’ or meaning of a mathematical proof. This property is paradoxical because it is possible to represent, within axiomatic set theory, celebrated reasoning that proves the existence of non-denumerable sets. Thus through its very act, the axiomatic proof of the denumerability of any model accomplished in the theory effectively contradicts Cantor’s proof of the existence of non-denumerable sets. Working through this impasse, for Ladrière, points to the element of intuition that attends to axiomatic formalisation itself.
Alain Badiou’s ‘Marque et manque: À propos du zéro’ (CpA 10.8) is, among other things, a paean to the virtues of axiomatics as described in Blanché’s volume. The target of Badiou’s critique is Miller’s theory of suture, and the latter’s claim that all discourse, including that of science, is grounded in an inaugural suture wherein a primordial lack is covered over through the establishment of the subject in discourse. For Badiou, science, properly speaking, establishes itself in a rupture with ideology (cf. epistemological break) that results in the exclusion of the subject from science of any errant or ‘sutured’ lack within it. Crucial to Badiou’s argument is the self-identity of the marks that comprise logical or mathematical discourse, including the mark of zero. Badiou revisits Gödel’s undecidability theorems to affirm that ‘undecidability’ and ‘inconsistent’ do not communicate ‘lack’, but rather show simply what is excluded from the domain of well-formed expressions that constitute a science. This domain has no lack within it. If in ‘La Subversion infinitésimale’ (CpA 9.8), Badiou focused on the impasse of formalisation as a site of Lacanian real (a position also articulated in a more phenomenological language in Ladrière’s article, CpA 10.6),10 in ‘Marque et manque’ the emphasis is on the infinite stratification of axiomatised science, a ‘shared delirium’ and ‘psychosis of no subject’ that constitutes ‘an Outside without a blind-spot’ (CpA 10.8:161-2).
In the final article of the Cahiers, Jacques Bouveresse explains Wittgenstein’s treatment of axioms as being like rules of a game that are, consequently, at once devoid of an ontological remit and not in need of any other ground but themselves. Bouveresse cites Wittgenstein: ‘Hilbert’s error lay in his wanting to demonstrate that the axioms “of arithmetic have the same properties as the game [ont les propriétés du jeu], and that’s impossible”’ (CpA 10.9:197). One might consider as an analogy here the fact the ‘rules’ of a basketball game are not the same as a basketball game.
- Badiou, Alain. Being and Event, trans. Oliver Feltham. London: Continuum, 2005.
- Badiou, Alain. The Concept of Model: An Introduction to the Materialist Epistemology of Mathematics, ed. Zachary Luke Fraser and Tzuchien Tho. Melbourne: re.press, 2007.
- ---. Number and Numbers, trans. Robin Mackay. London: Polity, 2008.
- Blanché, Robert. L’Axiomatique. Paris: PUF, 1955.
- Cavaillès, Jean. Méthode axiomatique et formalisme . In Oeuvres completes de philosophie des sciences, ed. Bruno Huisman. Paris: Hermann, 1994.
- Hallett, Michael. Cantorian Set Theory and Limitation of Size. Oxford: Oxford University Press, 1984.
- Hallward, Peter. Badiou: A Subject to Truth. Minneapolis: University of Minnesota Press, 2003.
- Ladrière, Jean. Les Limitations internes des formalismes: Étude sur la signification du théorème de Gödel et des theorems apparentés dans la théorie des fondements des mathématiques. Louvain: E. Nauwalaerts/Paris: Gauthier-Villars, 1957 [reprint, same pagination, with Ladrière’s corrections, Sceaux: Editions Jacques Gabay, 1992].
- Ladrière, Jean. ‘Axiomatic System’. In the 1967 edition of the New Catholic Encyclopedia. Vol. 1, 948-950, 1967.
- Van Heijenoort, Jean. From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879-1831. Cambridge, MA: Harvard University Press, 1967.
1. In an entry on ‘Axiomatic System’ in the 1967 edition of the New Catholic Encyclopedia (Vol. 1, 948-50), Jean Ladrière, a contributor to the Cahiers (CpA 10.6), wrote as follows: ‘A scientific theory is made up of propositions that are in turn composed of terms. One can establish the validity of a proposition by deducing it from other propositions, but it is impossible to proceed in this way to infinity. In the same manner, one can explain the sense of a term by defining it through the use of other terms, but again it is impossible to proceed to infinity. To build a theory it is therefore necessary to start from terms that are accepted without definition and from propositions that are considered as valid without demonstration. To these primitive elements are then added rules of definition, with whose help it is possible to define new terms from the undefined primitive terms or from terms already defined, and rules of deduction, with whose help it is possible to obtain new propositions from the primitive propositions or from propositions already deduced. The primitive propositions are called the axioms of the theory. The propositions that can be educed by means of the rules of deduction are said to be proved or demonstrated. The axioms and the proved propositions are the theorems of the theory’. ↵
2. Robert Blanché, L’Axiomatique, 68. Further references in the main text. ↵
3. Blanché writes: ‘It’s only in books that one begins with an axiomatic system; in the mind of the axiomatician, one winds up there’ (87). ↵
4. Hallett, Cantorian Set Theory, 164; xiv. ↵
5. Jean Cavaillés provided an assessment of intuitionism’s shortcoming on this score that crucially anticipates Alain Badiou’s position in ‘Marque et manque: Á propos du zero’ (CpA 10.8), when he wrote: ‘If abstract thought entails necessity, if mathematical development [devenir] is the appearance of genuine novelty, it is necessary that that the creation be situated in this sensible domain represented by the combinatory space. […] The role of the intellectual or logic is as restrained as possible, e.g. the simple setting of acquired results or adopted conventions, the mind’s fidelity to what it has done. But here again the sensible intervenes: the return to rules is inscribed in the configuration of the sign. A written reasoning cannot deceive, since it would mean excluded figures appearing in the drawing itself’ (Méthode axiomatique et formalisme, 93-4). Cavaillés’s point is that since mathematical reasoning is something that takes place in writing, intuitionists should be satisfied by constructions that take place in writing, seeing that there is no elsewhere of mathematical objects for them to refer to or ‘intuit’. ↵
6. Blanché, 59. ↵
7. Von Neumann, ‘An Axiomatisation of Set Theory’, in Van Heijenoort, ed., From Frege to Gödel, 395. ↵
8. In Bertrand Russell’s illustration: if we have infinitely many pairs of shoes, we don’t need the axiom of choice to pick one shoe from each pair: we can just pick the left shoe, say. But if we have infinitely many pairs of socks and want to pick one sock from each pair, then we need the axiom of choice. ↵
9. Of all of the contributors to the Cahiers none has more vigorously pursued the commitment to the modern figure of axiomatisation than Alain Badiou. As Peter Hallward writes in A Subject to Truth: ‘It is no exaggeration to say that the consequences of its axiomatic foundation determine the whole orientation of Badiou’s later philosophy. The axiom guarantees an original break with the merely given, inherited, or established. It ensures a foundation before or beyond the worldly. Axioms “pose the problem of a ‘non-worldly’ or non-fortuitous existence” [Badiou, “Silence, solipsism, sainteté”, 45]. The axiom is alone adequate to the decision of the radically undecidable. The axiom, ultimately, is the sole condition and exclusive medium of the subject: Badiou will thus claim that true “decisions (nominations, axioms) suppose no subject, since there is no subject other than as the effect of such decisions” [Badiou, “L’Etre, l’événement et la militance”, 19]. The axiom provides a way of describing a radically infinite multiplicity as originating in one wholly incorruptible, wholly unconditional point’ (Subject to Truth, 104-105). ↵
10. Badiou in fact cites Ladrière’s assessment of the Löwenheim-Skolem theorem in Being and Event in a discussion of denumerability and the necessity for the ‘axiom of choice’ to work through an impasse in theorems of reflection (360-1, 495). ↵