In the twentieth century, the project to formalise logic and mathematics led to encounters with a series of paradoxes that exposed internal limits to formalisation. The significance of formalisation and its limitations are analysed in the Cahiers pour l’Analyse.
The increasing emphasis on formalisation that one finds over the course of the Cahiers pour l’Analyse was conditioned first of all by two related contexts, one local, and another more general set of concerns in European thought. The local context was that of French epistemology, wherein figured the works and ideas of Gaston Bachelard, George Canguilhem, Jean Cavaillès, and Alexandre Koyré. The broader context, in which the local phenomenon of French epistemology should also be situated, was provided by developments in logic and mathematics since the middle of the nineteenth century and the conflicts generated by these developments within European philosophy more generally.
Jan Łukasiewicz, in his study of Aristotle’s logic, introduces a helpful distinction between formal and formalist approaches to logic, cited by Jacques Brunschwig in the first article in the tenth volume of the Cahiers (CpA 10.1). All logic, including classical Aristotelian logic, is formal, since logic abstracts from concrete terms or variables and restricts itself to the analysis of the relations between such terms, or the logical ‘places’ that might be occupied by such terms. A logic only becomes truly formalistic when its terms are themselves fixed in abstract symbols and letters, so that all possible ambiguity arising from different definitions of terms is excluded. In Lukasiewicz’s terms, a properly formalist logic ‘requires that the same thought should always be expressed by means of exactly the same series of words ordered in exactly the same manner. When a proof is formed according to this principle, we are able to control its validity on the basis of its external form only, without referring to the meaning of the terms used in the proof’.1 The articles in the Cahiers addressing formalisation interrogate the possibility of complete formalisation in logic, mathematics, science and theory in general, and pose new questions about the relation between formalisation in the sciences and subjectivity.
In the late nineteenth century, Gottlob Frege had sought to establish a thoroughly formalised Begriffschrift, or ‘concept-language’, that would be a new foundation of any logical mode of presentation or argumentation, including mathematics. Although Bertrand Russell soon demonstrated that Frege’s own approach was vitiated by paradox, Frege’s attempt to clarify the logical foundations of mathematics is an important reference point in the first volume of the Cahiers (CpA 1.2, 1.3). In the tenth volume of the Cahiers, devoted entirely to ‘Formalisation’, Alain Badiou analyses later developments in mathematical logic, proposing an epistemology of logic capable of articulating the internal limitations on formalisation discovered by Russell and Gödel (CpA 10.8).
A term of abuse for the Romantics and Hegel, ‘formalisation’ as both a method and a goal was one of the major consequences of the revolutions in mathematics associated with Frege and Georg Cantor. Making good on tendencies in the work of the proto-logicist Bernard Bolzano, Cantor simultaneously established set theory and the modern philosophy of the infinite, renewing the foundations of logic and mathematics at the same time. He deduced the existence of transfinite numbers - variably sized infinite quantities - purely through arithmetical methods, i.e., methods that owed nothing to intuition. The fact that variably sized infinites could not be intuited in the mind was of no concern when their existence could be justified through internally consistent demonstration. What mattered was the formal consistency of a demonstration or proof.2
In the twentieth century, formalism came to be specifically associated (along with realism and intuitionism) with one of the three main approaches to the foundations of mathematics. For formalists like David Hilbert, mathematics was grounded in internally consistent sets of axioms and their consequences, such that logical consistency was the only criterion of truth. The objects described by mathematics were wholly ideal or abstract; as such, their application or extra-mathematical correlate was of no concern. Within formalism ‘one cannot assert that a theorem is true [in the familiar sense of a correspondence with some extra-propositional reality], any more than one can assert that the axioms are true. As statements in pure mathematics, they are neither true nor false, since they talk about undefined terms. All we can say in [formalist] mathematics is that the theorem follows logically from the axioms.’3
Realists, or Platonists, remain equally committed to formalist methods, but they maintain the intrinsic (mind-independent) reality of the mathematical objects they describe. This position is most famously associated with Kurt Gödel who defended the self-evidence or plausibility of mathematical insight as no less certain than that of sense data (CpA 10.5:86/449). The traditional twentieth-century opposition between realist and formalist approaches to the foundations of mathematics comes under renewed scrutiny in the later work of Alain Badiou (notably in Being and Event (1988), where mathematics is equated with ontology as the discourse of being-qua-being).
The third school of post-Cantorian mathematics is intuitionism and is most closely associated with L.E.J. Brouwer. For intuitionists, experience is both the source and the benchmark of mathematical truth. Intuitionists deem true only that which can be effectively ‘constructed’ or ‘intuited’ in the mind - this requirement denies truth, among many other things, to Cantor’s affirmation of trans-finite numbers. A defender of an intuitionist position would be published in the Cahiers issue devoted to formalisation. Jean Ladrière’s exposition of the Löwenheim-Skolem theorem shows how the tendency toward formalisation itself always calls upon on a intuitive moment or element (CpA 10.6) that at once escapes formalisation and spurs its efforts.
Among thinkers in the French epistemological tradition, Jean Cavaillès was the most concerned with the consequences of formalisation in logic and mathematics. Cavaillès’s basic position, as articulated in Sur la logique et la théorie de la science, was that Hilbertian formalism needed to be complemented by an ontology which grounded the status of mathematical objects. After showing the limitations of neo-Kantianism, Brouwerian intuitionism, and the logical positivism of the Vienna Circle in this regard, Cavaillès devoted his attention to an extensive reading of Edmund Husserl’s Formal and Transcendental Logic. His conclusion, however, was that Husserlian phenomenology was compromised by the same fundamental solipsism as Kantianism when it came to the status of mathematical objects. In both instances, these objects only received their status as ‘real objects’ by being referred back to an ahistorical transcendental consciousness which conferred legitimacy upon them insofar as they conformed to predetermined categories, or ‘categorial forms’ in Husserl’s rubric. For Cavaillès this was untenable for two reasons. First, in an era when mathematical physics was producing scientific insights and results that far exceeded an intuitive grasp of their constitutive processes, any attempt to refer the truth content of the mathematical descriptions of these processes to structures manifestly inadequate to them was chimerical. Second, Cavaillès was committed to the fundamentally historical nature of scientific thought and the historical nature of formalisation itself. In other words, the formalisation that took place in discrete mathematical practice was predicated upon an unending process of formalisation at a metahistorical level. As such, reference to a transcendental, i.e., ahistorical, structure would be of no use: ‘the mathematical sequence possesses an internal coherence that cannot be rushed; progression is of the essence, and decisions that neglect this get lost in the void.’4
Cavaillès concluded his inquiry with a claim that would be of fundamental inspiration for the Cahiers: ‘it is not a philosophy of consciousness but a philosophy of the concept that can yield a doctrine of science’.5 The theoretical connection established in Cavaillès’s essay between the limits of intuitionism and the solipsism of phenomenology would also form an essential thread in the Cahiers.
Finally, the editors of the Cahiers drew directly on ideas about formalisation developed within psychoanalytic theory by Jacques Lacan. In his 1953 ‘Rome Discourse’ (‘The Function and Field of Speech and Language in Psychoanalysis’), Lacan had praised ‘the form of mathematicisation’ and ‘formalisation’ more generally, which put ‘within our reach a strict approach to our own field’ (E, 284-5). More specifically, he claimed that mathematics could help to ‘symbolise […] the intersubjective time that structures human action, whose formulas are beginning to be provided by game theory’. As an example of such an approach, he refers to his 1945 essay ‘Logical Time and the Assertion of Anticipated Certainty’ (E, 197-213), where he explains how logical formalism can clarify the science of human action by elucidating ‘the structure of intersubjective time that psychoanalytic conjecture needs to ensure its own rigour’ (E, 287).
Earlier in the same essay, Lacan had categorically declared: ‘Analytic symbolism, I insist, is strictly opposed to analogical thinking’ (E, 263). For Lacan, the symbols which constituted the subject’s discourse must be formalised in their relation to one another; they must not be ‘referred back’ (i.e., analogized) to a pre-existing intuition or memory they ‘resemble’. Moreover, the commitment to formalisation in Lacan’s own thinking is inextricable from his investments in the matheme as a mode of knowledge’s transmission and of its access to the real. This element of Lacan’s project would be extremely important for Badiou’s project after the Cahiers, a project which, more than any other comparable effort in recent French thought, carries out the commitment to formalisation first articulated in the Cahiers pour l’Analyse.6
In the Cahiers pour l’Analyse
In ‘La Science et la vérité’, Lacan makes a case for the value of formalisation for psychoanalytic theory, referring to several of its manifestations in the modern sciences. Game theory [la théorie des jeux] ‘takes advantage of the thoroughly calculable character of a subject strictly reduced to the formula for a matrix of signifying combinations’ (CpA 1.1:12; E, 860). Other sciences hit certain problems that reveal the limits of formalisation, however. In linguistics, the rules of combination in grammar and syntax need to be combined with the distinction between the level of enunciation and that of the statement (CpA 1.1:12; E, 860). Lacan says that it is specifically ‘in the realm of logic that theory’s refractive indices appear in relation to the subject of science’. In the paradoxes and limitations of modern formal logic, Lacan claims that one finds the complex relationship between science and subjectivity expressed in its pure form as a suture: modern logic ‘is indisputably the strictly determined consequence of an attempt to suture the subject of science, and Gödel’s last theorem shows that the attempt fails there, meaning that the subject in question remains the correlate of science, but an antinomic correlate since science turns out to be defined by the deadlocked endeavour to suture the subject’ (CpA 1.1:12; E, 861). Lacan argues that structuralism is currently in the process of introducing a new kind of subjectivity into the ‘human sciences’: a subject that is in a state of ‘internal exclusion from its object’, in the manner of a side of a Moebius strip. He suggests that it is the very encounter with the limits of the formalisation of set theory, through paradoxes such as Russell’s paradox and Gödel’s incompleteness theorems, that provides the key to the introduction of this novel subjective dimension into structuralism.
In his conclusion, Lacan makes an allusive remark concerning the relation between science and subject that will form the basis for Jacques-Alain Miller’s argument in ‘La Suture: Éléments d’une logique du signifiant’ (CpA 1.3): ‘the logical form given [scientific] knowledge includes a mode of communication which sutures the subject it implies’ (CpA 1.1:27-8; E, 877). Miller’s account of Frege’s attempt to logically formalise the foundations of mathematics will develop the implications of Lacan’s claim regarding science itself. But Miller’s goal is also to show how Frege’s own logic points to an anterior ‘logic of the signifier’:
What I am aiming to restore, piecing together indications dispersed through the work of Jacques Lacan, is to be designated the logic of the signifier - it is a general logic in that its functioning is formal in relation to all fields of knowledge including that of psychoanalysis which, in acquiring a specificity there, it governs (CpA 1.3:38).
For Miller, this logic is ‘formal’ insofar as it can be extended to describe ‘fields of knowledge’. Moreover, its formalism is what allows it to be transmitted as a coherent and unique logic, a ‘logic of the signifier’.
André Green’s contribution to Volume 3 of the Cahiers makes the case for a renewed attention to affect following upon Lacan’s emphasis on formalisation (CpA 3.2), In this same issue Jean-Claude Milner’s ‘Le point du signifiant’ (CpA 3.5) returns to themes of Miller’s ‘Suture’ in a reading of Plato’s The Sophist, arguing that Plato’s effort to think dialectically the relation between ‘being’ and ‘non-being’ provides a way to formalise the function of the objet petit a:
[Can we not] formalise in this way the object (a), described as being like the stasis of a fall’s cyclical repetition? It is as if we have a logic capable of situating the formal properties of any term submitted to fissional operation, but not one capable of marking specificities.
In contrast to Frege’s articulation, which reduces the chain to its minimal couple, the interpretation proper to a less summary formalism is perhaps not univocal. What we have broached here under the form of a fissional system, though lacking the ability to specify them further, are the lineaments of a logic of the signifier, and the source of all the mirage-effects its miscognition induces (CpA 3.5:78).
In the question period of Serge Leclaire’s second ‘Compter avec la psychanalyse’ seminar (CpA 3.6), Milner questions whether Leclaire’s account of the structures of the genealogical tree and the cycle of reproduction (CpA 3.6:93), despite portraying ‘non-coinciding cycles’, nevertheless ‘assumes the calm function of the non-cyclical [la fonction calme du non-cyclique]?’ (CpA 3.6:94). This could be subjected to a ‘logical formalisation’ which would help to connect the two structures. Leclaire argues that the body, as the site of drives and desire, would add a ‘constraint’ upon such a formalisation. Milner replies that the ‘signifying concatenations’ treated by psychoanalysis could in principle be reduced to ‘the plurality of elements and the constraints that induce the relations of these elements’. Leclaire replies that this would give no account of the ‘constraint’ on formalisation. Milner replies that ‘the nature of the constraint’ can itself ‘only be formal’ (CpA 3.6:95).7
Michel Foucault criticises an excessive dependence on formalisation in discourse analysis in his ‘Response to the Cercle d’Épistémologie’ (CpA 9.2): ‘One can always establish the semantics and syntax of a scientific discourse. But it is necessary to protect oneself from what one would call the formalising illusion [illusion formalisatrice]: that is, from imagining that these laws of construction are at the same time and with full title the conditions of existence’ (CpA 9.2:37-38; 330, trans. modified). For Foucault, ‘the formalising illusion elides knowledge [savoir]’ in its social, non-epistemic dimensions.8
The final issue of the Cahiers is largely devoted to the subject of ‘Formalisation’, as its title announces. The volume contains key texts from the history of modern logic as well as crucial interventions from members of the Cercle d’Épistémologie and other contemporary thinkers.
The volume opens with a piece by Jacques Brunschwig concerning Aristotle’s Prior Analytics (CpA 10.1). Brunschwig endorses the notion that Aristotle’s logic is ‘formal without being formalistic’ in that it concerns form in its relation to matter (hence it is formal), but not formalistic in that it does not develop a formal symbolic system that can be maximally extended. A French translation of an excerpt from George Boole’s Mathematical Analysis of Logic (1847) fittingly follows Brunschwig’s piece (CpA 10.2). Seeking to move beyond Aristotelian or syllogistic logic, Boole developed a version of algebraic formalisation wherein logical symbols x or y were defined in terms of their conceptual extension, i.e., in terms of the class or group of objects they delimit. His approach anticipates some of the principles of Cantorian set theory, as well as Fregean logic, and allows for a clarification and generalisation of the rules of logical inference.
The next piece is an article of Georg Cantor’s translated by Jean-Claude Milner especially for this issue of the Cahiers pour l’Analyse. ‘Fondements d’une théorie générale des ensembles’ (CpA 10.3) considers some of the philosophical aspects of Cantor’s theory of transfinite numbers and sketches the lineaments of his demonstration of the existence of variably sized infinite quantities. The second half of this piece reproduced in the Cahiers reads as something of a manifesto for formalisation avant la lettre, in which Cantor castigates numerous preceding intuitive notions of the infinite which have prevented a properly mathematical grasp of the transfinite as a determinate domain.
A translation of Bertrand Russell’s ‘Theory of Logical Types’ (CpA 10.4) follows Cantor’s essay and assesses the crucial problem of self-reference and self-membership that confounded early conception. Kurt Gödel’s ‘La logique mathématique de Russell’ (CpA 10.5) considers some of the limitations of Russell’s framework, and provides a robust defence of his own mathematical realism.
As noted above, Jean Ladrière’s assessment of the Löwenheim-Skolem theorem emphasises the role intuition itself plays in all efforts at formalisation (CpA 10.6). Robert Blanché’s ‘Sur le système de connecteurs interpropositionnels’ (CpA 10.7) provides an austere formal classification of the propositional connectors used in classical logic: formal logic allows ordinary-language connectors like ‘and’, ‘or’ and ‘if’ to be articulated more precisely, and provides conventional notation to express negation (¬), implication (→ or ⊃), disjunction (∨) conjunction (∧ or ⋅) and equivalence (↔).
Alain Badiou’s ‘Marque et manque/À propos du zero’ (CpA 10.8) confronts formalised science to suture, and by extension to ideology. Developing a technical account of stratification, inspired by Gödel’s arguments concerning logical syntax, Badiou marshals a critique of Miller’s argument in ‘La Suture’ (CpA 1.3) concerning the function of lack in scientific discourse. Jacques Bouveresse’s piece on Wittgenstein, the last article of the journal, effectively endorses the latter’s line regarding the insolubility of theoretical attempts to ground mathematics in a formalised logic, affirming mathematics’ utility over its consistency (CpA 10.9).
- Badiou, Alain. Le Concept de modèle: Introduction à une épistémologie matérialiste des mathématiques. Paris: Maspero, 1969. The Concept of Model: An Introduction to the Materialist Epistemology of Mathematics, trans. and ed. Zachary Luke Fraser and Tzuchien Tho. Melbourne: re.press, 2007. Online at http://www.re-press.org/book-files/OA_Version_9780980305234_The_Concept_of_Model.pdf.
- ---. L’Etre et l’événement. Paris: Seuil, 1988. Being and Event, trans. Oliver Feltham. London: Continuum, 2005.
- Canguilhem, Georges. Idéologie et rationalité dans l’histoire des sciences de la vie: Nouvelles études d’histoire et de philosophie des sciences. Paris: Vrin, 1977. Ideology and Rationality in the History of the Life Sciences, trans. Arthur Goldhammer. Cambridge, MA: MIT Press, 1988.
- Cavaillès, Jean. Sur la logique et la théorie de la science , prefaces by Gaston Bachelard, Georges Canguilhem and Charles Ehresmann, 2nd edition. Paris: Vrin, 2008. ’On Logic and the Theory of Science’, trans. Theodore Kisiel, in Joseph J. Kockelmans and Theodore J. Kisiel, eds. Phenomenology and the Natural Sciences: Essays and Translations. Evanston: Northwestern University Press, 1970.
- Davis, Philip J., and Hersh, Reuben. The Mathematical Experience. Harmondsworth: Penguin, 1980.
- Foucault, Michel. L’Archéologie du savoir. Paris: Gallimard, 1969. The Archaeology of Knowledge, trans. Alan Sheridan. New York: Pantheon, 1982.
- Grattan-Guinness, Ivor. The Search for Mathematical Roots, 1870-1940. Princeton: Princeton University Press, 2001.
- 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.
- Leclaire, Serge. Psychanalyser. Paris: Seuil, 1969. Psychoanalyzing, trans. Peggy Kamuf. Stanford: Stanford University Press, 1998.
- Łukasiewicz, Jan. Aristotle’s Syllogistic from the Standpoint of Modern Logic, 2nd edition. Oxford: Clarendon Press, 1957.
- Van Heijenoort, Jean. From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879-1831. Cambridge, MA: Harvard University Press, 1967.
1. Jan Lukasiewicz, Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, 16. ↵
2. Cantor’s work would be of fundamental importance for the later philosophy of the Cercle d’Épistémologie’s greatest celebrant of formalisation, Alain Badiou. For Badiou, the independence of formalised procedures from referents entails that formalisation transcends matters of signification. In A Subject to Truth, Peter Hallward writes: ‘Generic formalisation directly presents its terms in such a way as to block their possible re-presentation: that which is accessed through pure formalisation cannot be attained by any other means. The generic can only be rendered in terms that are indifferent to any “specifying” signification, which is to say indifferent to any signification tout court’ (Subject to Truth, xxx). ↵
3. David and Hersh, Mathematical Experience, 340. ↵
4. Jean Cavaillès, Sur la logique, 70/404 (translation modified). ↵
5. Ibid., 78/409 (translation modified). ↵
6. ‘Lacan’s enduring conviction that “the question of the real is commensurable with the question of language” only finds its final principle, Badiou observes, with the process of “mathematisation […]. Lacan holds mathematisation to be the key to any thinkable relation to the real. He never changed his mind on this point.” With far more systematic rigour than his mentor, Badiou will argue in his turn that “the grasp of thought upon the real can be established only by the regulated power of the letter”, a regulation that only mathematics can perfect’ (Hallward, Subject to Truth, 13). ↵
7. Leclaire returns to the problem of formalisation in his 1969 book Psychoanalyzing, where he notes the paradox that a system of reference is needed for psychoanalysis, but that the analyst must simultaneously set it aside in order to carry out the process of psychoanalytic listening. ‘Adapting psychoanalysis to a complete formalisation (which is what the common effort to theorize naturally aims to achieve) would have to exclude from its field, ipso facto, the very possibility of analysis in search of extreme singularity’ (15-16). The ‘always recurring difficulty of psychoanalysis, which no institution will ever be able to resolve, derives from the fact that it is vulnerable, on the one hand, to the degradation of a closed systematization, and, on the other, to the anarchy of intuitive processes’ (16). ↵
8. Georges Canguilhem takes up the analytical framework of Foucault’s L’Archéologie du savoir in his 1977 Idéologie et rationalité dans l’histoire des sciences de la vie. Noting Foucault’s distinction between the thresholds of positivity, epistemologisation, scientificity and formalization, he remarks that ‘it seems to me in any case that, the claims of certain geneticists notwithstanding, none of those disciplines [from the life sciences] has yet crossed the threshold of formalization’ (x). He notes that Foucault’s framework can explain how ‘epistemologisation’ and ‘formalisation’ may be carried out at different speeds by scientists. ↵