Concept and Form:
The Cahiers pour l’Analyse
and Contemporary French Thought

As a branch of science, mathematics poses a set of philosophical problems that are distinct from those raised by other natural sciences. Where debates within the philosophy of biology or physics, for example, often turn on questions of method or evidence in scientific practice, a ‘philosophy of mathematics’ has as its primary concern a question that is at once epistemological and ontological: what is the status of a mathematical object? Mathematical objects are no doubt abstract, but are they real? Are they mere tools, figments of thought that help us understand the world? Or do mathematical objects have an ontological status that transcends their epistemological function in thought? If the truth of mathematics is indifferent to its source in human thought, then what is the status or role of the subject that thinks mathematically within mathematical discourse itself? These much-debated questions remained crucial for the Cahiers pour l’Analyse because they had a special bearing on the journal’s primary concerns: the capacity of a formalised or abstract thought to break with ideology and articulate the real through science, and the status of the subject within science generally.

The emergence of mathematics in human history is as much a theoretical problem as a historical one.¹ The history of mathematics is sometimes described as a progressive sequence of abstraction that began with the most basic experience of counting or enumerating. But ‘counting’ is itself a deceptively complex phenomenon because it raises the question of whether the discrete entities articulated in ‘counting’ are the result of a count or its condition. The philosopher of mathematics Jean-Toussaint Desanti, a thinker with loose connections to the editors of the Cahiers, captured the essence of this problem in the introduction to his inquiry into the nature of ‘mathematical idealities’: ‘Where does the mathematician speak from? Where does he come from? He is not from the Heavens, since what he says is never in its entirety already said. He is not from the Earth, which holds us to other discourses: we “encounter” some rocks and some trees. But three rocks, two trees? Never. In order to see them, some operation is already necessary.’²

A crisis of foundation has thus historically lain at the heart of mathematics itself, a crisis which takes the form of a ‘chicken-egg’ argument between our intuitive experience of the world and the capacity of thought to articulate and determine the mathematical structure of the world. The apparent instability of the foundations of the mathematics became obvious with the development of the calculus and the use of real numbers to measure physical processes like the acceleration of a falling body over time. Attempts to resolve this crisis came to a head in the last third of the nineteenth century, punctuated by Georg Cantor’s invention of transfinite mathematics (CpA 10.3) and the efforts of Gottlob Frege to ground the foundations of mathematics in logic. These developments, and the ulterior efforts to deal with their consequences in European philosophy, shaped the context for the Cahiers’ own engagements with the field.

According to a pre-Cantorian (e.g. Kantian) perspective, the ground of the mathematical knowledge itself remains fundamentally intuitive. An intuitive notion of space remains primary with respect to geometric or arithmetic demonstration. Indeed, the philosophy of Kant makes a virtue of this fact, showing the science of mathematics to be grounded in the ‘intuitions’ of space and time. Following the lead of the anti-Kantian philosopher and mathematician Bernard Bolzano, who called for the full ‘arithmetisation of analysis’ in his efforts toward a doctrine of science, many mathematicians of the nineteenth century found this concession to ‘intuition’ intolerable. Cantor’s demonstration of distinct (i.e. differently-sized) infinite numbers exceeds any reference to sensory intuition, and his work provoked new controversies, notably dividing formalists (e.g. David Hilbert) from intuitionists (e.g. Luitzen Brouwer). For Hilbert, mathematics proceeded through pure inference on the basis of a set of internally consistent but otherwise ‘arbitrary’ axioms; its truth content was not reliant upon a prior intuition, but on a rigorous sequence of determinations. For intuitionists, as the name suggests, legitimate mathematical results could not be proved independently of some reference back to the elementary intuitive ground of mathematics (e.g. an experience of what Brouwer called ‘two-ity’).

Formalism was a dominant influence on the key ‘figure’ of twentieth-century French mathematics, Nicolas Bourbaki, the pseudonym for a shifting collective of mathematicians who sought to produce a set of comprehensive works on mathematics grounded in set theory. Bourbaki, in turn, influenced the mathematical perspective of the Cahiers, exemplified especially by Alain Badiou (CpA 9.8, 10.8). By contrast, Jean Ladrière’s contribution was more influenced by intuitionism (CpA 10.6).

Apart from the strictly mathematical developments in the wake of set theory, the question of mathematics’ relation to logic was also of crucial importance for the Cahiers. Frege’s effort to ground mathematics in logic was taken up in particular by Bertrand Russell and Alfred North Whitehead in their Principia Mathematica, and subsequently abandoned. The relation of mathematics to an anterior logic, and the relation of that logic to an anterior logic - in a word, the relation between the ascending hierarchy of stratification and the reflexivity of suture - would be a major site of contestation in the Cahiers, culminating in the final issue on formalisation (CpA 10).

Finally, the Cercle d’Épistémologie’s proclivity for mathematics had a more immediate set of influences in French epistemology, above all in the writings of Gaston Bachelard, Georges Canguilhem, Jean Cavaillès, and Alexandre Koyré. Each of these thinkers (though Canguilhem less so than the others) treated mathematics as the paradigmatic model of modern rationalism. For Cavaillès, the capacity of mathematical thought to determine its own objects was a primary concern in an age when the ‘objects’ of mathematical physics could never be experienced apart from the scientific sequences in which they were embedded. In his work La Philosophie du non, Bachelard equally noted the specificity of modern ‘microphysics’, and affirmed its capacity to overcome naïve realism by providing a mathematical articulation, not of entities, but of processes. Modern physics made good on a promise already latent in Newtonian mechanics, in which it was ‘the necessity of understanding the process or becoming [devenir] that rationalise[d] the realism of being […]. We must pass from the realism of things to a realism of laws.’³ This transmutation, in which mathematics ceases to describe fixed ‘things’ or ‘entities’ and comes to describe intrinsically open-ended, and hence ‘imperfect’ processes, also lay at the heart of Alexandre Koyré’s praise of Galileo in From the Closed World to the Infinite Universe (cf. epistemological breaks). Koyré’s theses concerning ‘Galileanism’ were crucial for the Cahiers. Jean-Claude Milner describes this impact as follows: ‘Something is in play, a genuine overturning with regard to what came before. For the first time in history, mathematical entities do not serve to think the eternal; they serve to think the passage’.⁴

In the Cahiers pour l’Analyse

Mathematics figures in the first issue of the journal in Yves Duroux and Jacques-Alain Miller’s engagements with Frege’s Foundations of Arithmetic, though in each instance it is the attempt to ground mathematics in logic, rather than mathematics per se, that is the primary concern. In ‘Psychologie et logique’ (CpA 1.2), Duroux analyses the attempt to construct a logical account of number in Frege’s work. Duroux’s main point is to show how Frege avoids collapsing into psychologism or empiricism by developing a concept of number that is a matter of logical construction and consequence rather than experience. Duroux’s presentation of Frege provides crucial background to Miller’s effort to enumerate a ‘logic of the signifier’ grounded in Frege’s own Foundations of Arithmetic in ‘La Suture: Éléments de la logique du signifiant’ (CpA 1.3). (For more information see logic).

In ‘Le point du signifiant’ (CpA 3.5) Jean-Claude Milner will read something like the ‘logic of the signifier’ in Plato’s Sophist, discerning a generative ‘non-being’ correlative to the concept of ‘not-identical-with-itself’ that accounts for the relationship between ontology and number in Plato’s discourse. For Milner, the ‘vacillation’ of the concept of ‘non-being’ between function and term plays a role not unlike the subject in Miller’s ‘logic of the signifier’, at once extending a logical series and occupying a place within it. For Milner, Plato’s ‘logic’ was an instructive case precisely because it lacked a concept of zero, and consequently a viable concept of number. ‘When Plato reasons […] he always reasons with reference to numbers that can have a geometrical representation, and this is linked to the fact that he had no concept of zero’.

The ninth issue of the Cahiers is titled ‘Généalogie des sciences’ and it begins with an exchange between the Cercle d’Épistémologie and Michel Foucault, whose recently published books (The History of Madness, The Birth of the Clinic, and especially The Order of Things [Les Mots et les choses]) had established him as an authority on the topic. Foucault discusses mathematics specifically in his ‘Réponse’ (CpA 9.2). Mathematics is a science of such a type that ‘every episode of its historical development can be taken up again in the interior of its deductive system; its history can, in fact, be described as a movement of lateral extension, then of repetition and generalization at a higher level, such that each moment appears either as a special region or a definite degree of formalisation’ (CpA 9.2:36; trans. 327-28). Foucault mentions as an example the generalisation of Cartesian algebra by Joseph-Louis Lagrange, Niels Henrik Abel, and Evariste Galois. Mathematics provides a ‘limit-case’ for an analysis of science because ‘the history of mathematics is always at the point of passing the limit of epistemological description’ (CpA 9.2:36). By contrast, ‘sciences’ like psychology and sociology are always at the limit of a ‘genetic description’, i.e. a description that accounts for their historical emergence as discourses. For Foucault, it would be an error to make either of these cases a ‘model’ for a genealogy of science because they occupy extremes (hence Foucault’s concern for sciences like biology or medicine, somewhere between the abstraction of mathematics and the ‘human sciences’). ‘All sciences (even ones as highly formalised as mathematics) presuppose a space of historicity that does not coincide with the interaction of its forms’ (CpA 9.2:36/328). Although one should not go so far as to encourage a ‘doxological illusion’, based on the actuality of opinions (rather than analysis of the choices and ‘strategic possibilities of conceptual games’), it is equally necessary to avoid the ‘formalising illusion’.

In the book which further develops his response to the Cercle in CpA 9.2, The Archaeology of Knowledge, Foucault says that mathematics is the only discursive practice to cross all thresholds at the same time. Its epistemological threshold, for instance, is crossed at the same time as its threshold of formalization. But again, for Foucault that means it is exceptional with respect to the general history of the sciences:

Mathematics [is] the only discursive practice to have crossed at one and the same time the thresholds of positivity, epistemologisation, scientificity, and formalisation. The very possibility of its existence implied that which, in all other sciences, remains dispersed throughout history, should be given at the outset: its original positivity was to constitute an already formalised discursive practice (even if other formalisations were to be used later). Hence the fact that their establishment is both so enigmatic (so little accessible to analysis, so confined within the form of the absolute beginning) and so valid (since it is valid both as an origin and as a foundation); hence the fact that in the first gesture of the first mathematician one saw the constitution of an ideality that has been deployed throughout history, and has been questioned only to be repeated and purified.⁵

Mathematics is the main subject of another contribution to this volume, Alain Badiou’s ‘La Subversion infinitésimale’ (CpA 9.8). The main question at issue in this, the first of Badiou’s two contributions to the Cahiers, is the abruptly ‘subversive’ (i.e. revolutionary) power of scientific formalisation, its capacity to interrupt the ideological categories of continuity, quality and temporality. Badiou affirms this power as essential to the operations of mathematical logic in general, and to Abraham Robinson’s ‘non-standard’ theorisation of ‘infinitely small’ quantities in particular. The first half of Badiou’s article makes a general point about the determination, in any numerical system, of an empty place for a ‘number’ that is excluded as impossible within the limits of that system. The second half of the article explores the process whereby Robinson’s non-standard analysis allows, in the domain of the standard number system, infinitesimal or ‘infinitely small’ numbers to come to occupy such an initially ‘impossible’ place, and thus render the impossible possible.

The subversive power of mathematics and thoroughgoing formalisation against all ideological models of intuition (or reflexivity) is also the dominant theme of Badiou’s ‘Marque et manque: À propos du zero’ (CpA 10.8), which develops a critique of Miller’s ‘La Suture’ (CpA 1.3). Mobilising productive stratification against recursive suture, Badiou provides an interpretation of Gödel’s undecidability theorems that emphasises the foreclosed nature of well-formed logical syntax once it establishes itself in its rupture from ‘ill-formed’ expressions. Badiou includes two paeans to mathematics and its capacity for breaking with ideology in the conclusion to his main text. The first is Spinoza’s observation in the appendix to Book I of the Ethics: ‘Man would never have ventured beyond illusion had it not been for this surprising fact: mathematics’. The second comes from Lautréamont’s Maldoror: ‘O austere mathematics, I have not forgotten you, since your wise lessons, sweeter than honey, filtered into my heart like a refreshing wave […]. Without you in my struggle against man, I would perhaps have been defeated’ (CpA 10.8:163-4).

Mathematics dominate much of the final issue of the Cahiers, titled ‘Formalisation’. It includes translations of texts central to the history of formalisation, including articles by George Boole (CpA 10.2) and Georg Cantor (CpA 10.3). Critical interjections into this history are reproduced as well, such as Kurt Gödel’s critique of the mathematical logic of Bertrand Russell (CpA 10.4; CpA 10.5). Jean Ladrière provides an interpretation of the Löwenheim-Skolem theorem that emphasises the essential role of intuition in all ‘acts’ of formalisation (CpA 10.6). The final article of the journal, a piece on Wittgenstein by Jacques Bouveresse, provides an ironic coda to the enterprise, offering a critique that describes philosophical attempts to ground mathematics as a ‘malady’, yet nonetheless celebrates mathematics’ utility and ability to persist in its indifference to philosophy (CpA 10.9).

Select bibliography

• Bachelard, Gaston. La Philosophie du non. Paris: PUF, 1940.
• Badiou, Alain. L’Etre et l’événement. Paris: Seuil, 1988. Being and Event, trans. Oliver Feltham. London: Continuum, 2005.
• ---. Le Nombre et les nombres. Paris: Seuil, 1990. Number and Numbers, trans. Robin Mackay. London: Polity, 2008.
• Barbut, Marc. ‘Sur le sens du mot structure en mathématiques’. Les Temps Modernes 246 (1966). ‘On the Meaning of the Word “Structure” in Mathematics’, trans. Susan Gray, in Michael Lane, ed. Structuralism: A Reader. London: Jonathan Cape, 1970.
• Bouligand, Georges. Aspects de la mathématisation. Paris: Conférences du Palais de la Découverte, 1958.
• Castel, Robert. ‘Méthode structurale et idéologies structuralistes’. Critique, 210 (1964).
• Cavaillès, Jean.. Sur la logique et la théorie de la science [1942], 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.
• Desanti, Jean-Toussaint. Les Idéalites mathématiques. Paris: Seuil, 1968.
• Foucault, Michel. ‘Réponse au Cercle d’Épistémologie’. CpA 9.3. ‘On the Archaeology of the Sciences: Response to the Epistemology Circle’, in Essential Works vol. 2, Aesthetics, Method and Epistemology, ed. James D. Faubion. New York: New Press, 1998. 297-334.
• ---. L’Archéologie du savoir. Paris: Gallimard, 1969. The Archaeology of Knowledge, trans. Alan Sheridan. New York: Pantheon, 1972. Some chapters of the book are online at http://www.marxists.org/reference/subject/philosophy/works/fr/foucault.htm.
• Grattan-Guinness, Ivor. The Rainbow of Mathematics: A History of the Mathematical Sciences. New York: W.W. Norton, 2000.
• ---. The Search for Mathematical Roots, 1870-1940. Princeton: Princeton University Press, 2001.
• Koyré, Alexandre. From the Closed World to the Infinite Universe. Baltimore: Johns Hopkins University Press, 1957.
• Van Heijenoort, Jean. From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879-1831. Cambridge, MA: Harvard University Press, 1967.