## Résumés

La définition, par Jean Cavaillès, des mathématiques comme *pensée effective* ou encore comme *travail effectif* est une composante essentielle de son analyse du « devenir objectif » des mathématiques. Ce concept a deux références : l’effectif en tant que *réalité effective* et l’histoire de la *calculabilité effective*. Dans cet article, j’examine le traitement que propose Cavaillès de deux séquences dans l’histoire mathématique de ce concept : tout d’abord, les débats entre les analystes français à propos de la définissabilité effective, et ensuite les travaux de Gödel, Church et Kleene concernant la calculabilité effective. On verra que la temporalisation par Cavaillès du concept d’effectif informe son rejet de toute théorie *a priori* de la science et sa relativisation corrélative du transcendental, et par suite sa théorie d’un temps logique discontinu.

## Entrées d’index

### Mots-clés :

Cavaillès, Borel, Lebesgue, Church-Kleene, calculabilité effectif, travail mathématique, temps logique, relativisation du transcendental### Keywords:

Cavaillès, Borel, Lebesgue, Church-Kleene, effective calculability, mathematical work, logical time, relativisation of the transcendental## Plan

Haut de page## Texte intégral

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^{ }This paper is a development of a presentation given at FPMW 13, hosted at the Université Côte d’A (...) - 2
^{ }Cavaillès responding to Albert Lautman in “La pensée mathematique”, a transcription of their join (...)

[P]ersonally, I am very reluctant to posit anything that would dominate the effective thought of the mathematician. I see an exigency in the problems themselves.2

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^{ }*OC*226.

1If Jean Cavaillès’ work –incomplete, more a promise than a theory– continues to assert a fascination for philosophers and historians of mathematics it is because he outlined a research programme that would aim to progressively reveal what he designated as the “objectivity, mathematically founded, of mathematical becoming.”3 Cavaillès’ claim for an identity between the objectivity and the historicity of mathematics is what grounds his reputation as a central figure in the tradition of historical epistemology of mathematics. This same claim harbours an implicit definition of mathematics, that is, of its specificity: mathematics is the sole domain of thought whose rational character lies in its transforming its own objective (symbolic, conceptual, practical) conditions in a regulated manner, whence Cavaillès’ characterisation of mathematics as an *experimental* activity. In turn, the singularity of mathematical concepts* *is to be sought not in their generality, but in the specificities of such transformations, i.e. in the way in which mathematical concepts change in history, in their mode of passage.

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^{ }Cf.*OC*85,*OC*100,*OC*106,*OC*180,*OC*187,*OC*540,*OC*628.

2The novelty of Cavaillès’ project thus consists in his having practised a new norm of reading the history of mathematics: to always read the body of already produced mathematical works under the image of *movement –*that is, as indicated above, as a becoming*– *rather than as a completed foundation. Central to the elaboration of this norm in Cavaillès’ truncated *oeuvre* is his enigmatic definition of mathematics as *effective* *thought *[*la pensée* *effective*] or *effective work *[*travail effectif*].4 This designation intermingles philosophical referents –principally via French neo-Kantian reflections on the effective (or the efficacious) as the *actual– *and a precise mathematical or technical history –that is, the development of the concept of *effective calculability*,* *which runs through the distinct phases of the discourse of the “French analysts” or “French empiricists” (principally Émile Borel and Henri Lebesgue), the history of recursive functions, and the formation of the modern notion of computability.

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^{ }Cf. H. B. Sinaceur,*Cavaillès*, Paris, Les Belles Lettres, 2013, p. 86: “L’effectivité hégélienne (...) - 6
^{ }To resume the polemical term that Cavaillès adopts from Léon Brunschvicg.

3A focus on this concept of the effective highlights how, for all that Cavaillès has a reputation as philosopher of *necessity*, he can equally be read as having advanced a philosophy of *actuality *in light of modern mathematics. In this sense, I agree with Hourya Benis Sinaceur that attention to the different modalities of the concept of effectiveness should be a guiding thread for reading Cavaillès.5 However, I would like to suspend the gesture which grounds Sinaceur’s masterful analysis of the role of effectiveness in Cavaillès’ “theory of reason”, which is to view Cavaillès’ use of the concept as a direct extension of the Hegelian (and post-Hegelian) discourse on *Wirklichkeit*. I do this not because I am arguing for any ultimate incompatibility between the Cavaillèsian and Hegelian theories, but because I find it necessary –at least initially– to preserve the autonomy of the referent in Cavaillès’ philosophical lexicon if we are to analyse Cavaillès’ contribution to “mathematical philosophy”.6 With this in mind, I would like to emphasise three points.

4First, the interest of Cavaillès as a philosopher lies in his rejection of the *a priori*, or, to place the emphasis differently, in his developing a novel concept of the *historical a priori*. This is continuous with Cavaillès’ rejection of what he calls “the Kantian theory of mathematics”. Yet, Cavaillès’ relationship to Kantianism in the broad sense, and in particular to the concept of intuition, was not consistent throughout his work. Examining the concept of the effective –and, specifically, the way in which Cavaillès *temporalises* the notion of the effectiveness of mathematics*– *is essential to periodising Cavaillès’ changing views on this matter.

5Second, this rejection of the *a priori *is consistent with a central *methodological *tenet of Cavaillès’ work and the relation it seeks to exhibit between mathematical work and philosophical practice: the relationship between mathematical and philosophical concepts must not be that the former stand to the latter as *examples*, such that mathematics gives specific content to more general philosophical concepts, i.e. as particulars to universals. Rather, the relation between mathematics and philosophy is conceived according to singular points of contact. The concept of the effective is particularly useful for assessing this aspect of Cavaillès’ work in that it is a mathematical referent with which he was intimately familiar and which underwent a profound reformalisation in the course of his working life. This reformalisation forms the background for Cavaillès’ extended confrontation with new work by Kurt Gödel, Alonzo Church and Stephen Kleene around effectiveness and recursion in *Transfini et continu* (written in 1940-1941, hereafter *TC*). The primary aim of this essay is to examine how it is that the encounter with the new formalisation of effective computability informs, first, a shift in Cavaillès’ reading of Borel and Lebesgue, and, second, Cavaillès’ novel critique of the Kantian theory of mathematics in terms of its conflation of *effective process* [*procès effectif*] and *effectuated process* [*procès effectué*].

6Third, the concept of the effective is crucial for understanding the object of Cavaillès’ final work, *Sur la logique et théorie de la science *(written in 1942, hereafter *LTS*). Whereas the doctoral studies are animated by the search for the “central intuition” that would justify the unity of the theories with which they are concerned, *LTS* presents a thoroughly desubjectivised theory of mathematical history under the name of what Cavaillès will come to call *l’enchaînement mathematique *or *l’enchaînement formel. *Cavaillès adopts this notion of formal or mathematical *concatenation* from classical rationalist treatises on deduction (notably those by Descartes, Spinoza, and d’Alembert), but only defines his own use of the term in one instance in *LTS*, in the course of the criticism he levels at Carnap’s project for a general syntax:

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^{ }OC517-518, translation from R. Mackay and K. Peden,On Logic and the Theory of Science, Falmouth (...)More than just an error of prediction, there is a serious failing here, a misrecognition of what is essential in formal concatenation [

l’enchaînement formel], i.e. its progression, which is both necessary and conditioned at every step by the effective [l’effectif]. A simultaneous positing of all possibilities […] contradicts the very notion of a formal system: the very meaning of the term implies that it be generated by successive movements of breaking apart and surpassing. The fact that everything does not be established all at once [d’un seul coup] has nothing to do with history, but is the characteristic of the intelligible […].7

7On the analysis of Cavaillès’ work that I will outline here, the fact that Cavaillès, first, specifies his notion of *concatenation *by stating that its progression is *conditioned by the effective*, and second, states that the result of this conditioning is that “everything cannot be established all at once” provides the key to understanding both his mature theory and his transformation of the rationalist tradition. Thus, while this essay will primarily be concerned with developments in Cavaillès’ work prior to the composition of *LTS*, it should be understood as a contribution to rereading the latter work, which, I will suggest, can be understood as advancing a theory of *logical time*.

# I. The French Analysts in Cavaillès’ Doctoral Dissertations

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^{ }OC14.[T]he analysts, Baire, Borel, and Lebesgue, militant in the concrete theory of sets, which they developed to the point of being new creators, are all empiricists. For them, it is a question of their own work: detached from any speculation, they want to determine which objects and methods they will be allowed to deploy without the risk of running into a contradiction.8

8Although Cavaillès’ doctoral dissertations –*Remarques sur la formation de la théorie abstraite des ensembles *(1938, hereafter *TAE*) and *Méthode* *axiomatique et formalisme *(1938, hereafter *MAF*)– are both concerned with more contemporary (and primarily German) developments at what was then the forefront of set theory and the formalist programme respectively, the earlier work of the French analysts played an important role both in framing the epistemological histories that Cavaillès wrote and in delineating the broader conceptual problematics that he extracted from these histories. Two points should be emphasised.

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^{ }For an overview of some of the vagaries of this nascent “concrete” set theory, cf. T. Murata, “Fr (...) - 10
^{ }The “Five Letters” are initially published as R. Baire, E. Borel, J. Hadamard and H. Lebesgue, “C (...)

9First, the stakes of the “abstract” set theory dealt with in *TAE* are to a significant extent defined in opposition to the informal commitment to “concrete” set theory as espoused by Borel, Lebesgue, et al. Cavaillès had in mind the following epistemological situation: on one hand, the French analysts made crucial progress in the study of real functions, the theory of measure and the theory of integration by using modern set theoretical methods, and even extended set theoretical research in important directions; on the other, they expressed explicit methodological scepticism with respect to the autonomy of set theory as a domain of mathematical research, a scepticism which they attempted to ground by reference to a preference for “concrete” or “applicable” mathematics.9 It is this nascent commitment to the ideal of a “concrete” set theory –initially articulated in terms of an inchoate practical attitude– that Borel and Lebesgue will attempt to specify in the period following the debate of the famous “Five Letters” (exchanged between René Baire, Borel, Jacques Hadamard and Lebesgue in 1905 over the reception of Ernst Zermelo’s Axiom of Choice) by developing (somewhat opposed) quasi-formal notions of the effective.10 The contested distinction between concrete and abstract set theory was initially understood in the French context in terms of a debate around what constituted an admissible definition in mathematics, and an attempt was made to delimit this in terms of *effective* *definability*. As we shall see, this notion of the effectively definable was not adequately formalised in the work of Borel and Lebesgue, but they did set a conceptual problematic that is the background for Cavaillès’ own notion of effective work, insofar as they attempted to specify one modality of the effective (as “actual” or practically efficacious mathematics) by reference to another (the effective as the calculable or as what could be concretely “effectuated” in an unambiguous manner).

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^{ }Cf.*OC*362 and*OC*29. - 12
^{ }I take this as one reason why Cavaillès published*MAF*as the sequel to*TAE*. While*MAF*is Cavaillè (...) - 13
^{ }For an analysis of the history of the relevant results here, cf. G. H. Moore, “Lebesgue’s Measure (...) - 14
^{ }*OC*224. - 15
^{ }*OC*227.

10Second, Cavaillès’ interest in the concept of the effective is connected to the necessity of developing “a definition of mathematical work in general.”11 This demand, which is found in both the final sentence of *TAE* and at the end of the opening of* MAF *(following the treatment of Lebesgue), can be read as the conceptual bridge between the two dissertations: the problems posed by the history of abstract set theory cannot be resolved without an interrogation of the concept of mathematical (or effective) work; in turn, it is the history of the debate around formalism and the axiomatic method that will be the essential material for this latter investigation.12 In order to understand why these problematics are connected, it should be emphasised that the attempt to delimit “concrete” mathematics in terms of an informal notion of effective definability was unsuccessful on the terms in which the French analysts posed it, insofar as, despite the expressed scruples of Borel, Lebesgue et al., the theories that they had developed in order to advance mathematical analysis were shown to depend on the very aspects of set theory which they contested.13 In the introduction to *TAE*, Cavaillès frames the “paradox” posed by set theory in terms of “the ever more widespread recourse to a contested theory within incontestable mathematics”, and expands on this by stating that “[t]he theoretical crisis which commenced in 1905 has only embarrassed restrictions and scruples concerning set theoretical modes of reasoning without arresting their internal power of expansion.”14 We can understand this “paradox” to be that the practical efficacy of set theory had proved it to be an essential matter for mathematical research even in domains and amongst mathematicians who had wished to delimit its content in purely instrumental terms. This is a paradigmatic example for what Cavaillès will call in the introduction to *TAE* the “problem of explaining the unequal fecundity of methods.”15 On Cavaillès’ reading, the French analysts had posed this problem for themselves without having the adequate formal means to resolve it. He saw Hilbertian proof theory, and in particular the broad history of the expanded notion of recursive or effective definitions as it had been developed following Richard Dedekind’s work, as being essentially continuous with this unresolved problematic in Borel and Lebesgue. As such, interrogation of these two concepts of *the effective *and of *mathematical work* should be seen to inform each other *mutatis mutandis* throughout Cavaillès’ oeuvre.

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^{ }The relevant distinction between “empiricism” and “idealism” at stake here has its roots in Emil (...)

11Two innovations are important in Cavaillès’ reading of this epistemological conjuncture. In the first place, Cavaillès progressively imbues the concept of the effective with a* temporal* connotation, viz. as a problematic of the relation between *already completed* (or effectuated) mathematics and future mathematical production. This means that Cavaillès’ wish to do justice to the open-ended *effectiveness *of mathematical work is inseparable from his ultimate rejection of any *a priori* theory of mathematics, and his partial sympathy with an “empiricist” project (in the sense of the French analysts).16 Second, the dialectic of effectiveness and effectuation at play here must be indexed to Cavaillès’ changing views on the idea of an intuitive basis for mathematical construction and hence to a shifting relation both to French neo-Kantianism and to the Kantian underpinnings of finitist theories in mathematics.

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^{ }J. Tannery, “De l’infini mathématique”,*Revue générale des sciences pures et appliquées*, vol. 8, (...)

12Turning to the “Five Letters”, the debate there, which sets the terms for the notions of the effective subsequently developed by Borel and Lebesgue, concerned what constitutes an admissible definition in mathematics. In this respect, a shared reference for the French analysts is a prior distinction drawn by Jules Tannery in his 1897 review of Louis Couturat’s 1896 *De l’infini mathématique*, in which Tannery affirmed the validity of mathematical correspondences that could be *determined* but which could not be *described*.17 Tannery’s position on such correspondences is framed as an explicitly Cartesian commitment to the “idea of determination”, i.e. that we can intuit mathematical determinations (or laws determining correspondences) independently of the possible realisation (or description) of such correspondences:

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^{ }Ibid., p. 134.[I]t is important to observe that the idea of determination is independent of the possibility of formulating what this determination consists of. It is not necessary, for a set to be determined, that one can effectively identify with respect to [

effectivement reconnaître sur] a given number whether it [effectively] belongs or does not belong to that set, it is sufficient that one knows that it [either] does or does not belong to it, and any reasoning based on the fact that one has the choice between these two suppositions will be valid.18

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^{ }For an analysis of the same debate as an exemplary case for what Jean-Toussaint Desanti calls an (...)

13We can thus gloss this distinction as that between* abstract –*or *postulated– correspondences *(which can only be “determined” but not “described”), and *effective correspondences* (correspondences that can be concretely “described”). In the “Five Letters”, Hadamard, Borel, Baire and Lebesgue each invoke this distinction, but in opposed terms. Indeed, the shifting lexical terrain around this referent in the discussion can be read as the sign of a contested concept.19 For Hadamard (following Tannery), abstract or postulated* determination* is sufficient for mathematical *definition*; for Borel and Lebesgue *definition *or *determination* must equally entail (*effective*)* description*. In this sense, the later moves by Borel and Lebesgue to transform the “effective” into a quasi-formal* *notion can be understood as differing attempts to give a *descriptive definition of definition* in mathematics (it is in this sense that we are discussing the genesis of descriptive set theory). The different proposals made by Borel and Lebesgue respectively push in *constructive *and purely *descriptive* directions. The two definitions they advance are as follows:

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^{ }E. Borel,Leçons…,op. cit., p. 65. Reprint of E. Borel, “Les ‘Paradoxes’ de la théorie des ensem (...)Borel’s definition of the

effectivement énumérable(1908): A set is “effectively enumerable”, if it is possible to “state, by means of a finite number of words, a definite procedure for unambiguously attributing a determined rank to each of its elements.”20

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^{ }H. Lebesgue, “Sur les fonctions représentables analytiquement”,Journal de Mathématiques Pures et(...)Lebesgue’s definition of the

nommableorl’effectif(1905): “An object is defined or given when a finite number of words applying to this object and to this object alone have been uttered; that is, when a characteristic property of the object has beennamed.”21

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^{ }Appendix to E. Borel,*Leçons*…,*op. cit.*, p. 219, which is a modified version of E. Borel, “Le cal (...)

14In Borel’s case, it is clear that what he has in mind is the individuation of mathematical objects. He will often speak of certain individual “incommensurable numbers” such as π for which we can nevertheless determine the decimal representation up to any desired degree of approximation (these will be “effectively enumerable” in a way that, e.g., “the set of all the incommensurable numbers” is not). Borel intends this stricture to formally characterise the epistemological notion of a mathematical object being agreed upon in an unambiguous manner by working mathematicians. He thus wishes to characterise mathematical objectivity by reference to intersubjectivity –i.e. that any given mathematician will be able to determine the “same” object to the same degree– and in turn to specify this notion of the intersubjective availability of the object by reference to calculability. This is clear when Borel later develops an even more restrictive notion of “calculable numbers” or “calculable functions”: “We state that number α is calculable when, given an arbitrary whole number *n*, we can obtain a rational number which differs from α by less that 1/*n*.”22 Again, the privileged referent for Borel in specifying his notion of effectiveness or calculability is the possibility of decimal expansion. In a footnote to this definition of calculable numbers, Borel then develops what I will call Borel’s Thesis, i.e. that effectively calculable mathematics coincides with (or exhausts) applicable mathematics:

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^{ }E. Borel,Leçons…,op. cit., p. 219, emphasis mine.Borel’s Thesis: “I am intentionally leaving aside questions concerning the greater or lesser practical length of the operations; what is essential is that each of these operations is executable in a finite time, by a certain and unambiguous method. For example, a decimal number

β, such that itsn^{th }decimal is equal to the decimal of π of rankn! must be considered as definite, even though its calculation, with the use of ten exact characters, may require, in the state of Analysis as it stands [l’état actuel], a time far exceeding the length of human life.From the practical point of view, it may be said that the numbers effectively needed[dont on aeffectivement besoin]can, in general, be effectively calculated[effectivement calculés]to the desired degree of approximation[…].”23

15I call this Borel’s Thesis because there is an identification at stake, one which crosses our two modalities of the effective, as the actual and as the calculable. The thesis is that all of the mathematical objects that practising mathematicians are going to need for what Borel will think of as *actual* mathematical work (i.e. which are of relevance to “concrete” or “applicable” mathematics) are those which can be *effectively* *calculated* (i.e. given in a computational manner). The identification at stake is thus between two different senses of the same term, as neatly condensed in the above citation: Borel is attempting to give a univocal interpretation to the word *effectivement *by using a formal notion (*effectivement calculés*) to specify an informal or conceptual one (those mathematics *dont on a effectivement besoin* “from a practical point of view”).

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^{ }Lebesgue makes this clear when, in the 2nd edition of H. Lebesgue,*Leçons sur l’intégration et la*(...) - 25
^{ }*OC*23-24. Cavaillès is selective in his treatment of Lebesgue here. Lebesgue uses both descriptiv (...)

16It is this identification that Cavaillès will oppose by giving a temporal reading of the restrictions at stake: for Cavaillès, Borel’s definitions are constructive *–*and the Kroneckerian legacy is clear here, in that we are discussing *constraints* placed on mathematical reasoning– insofar as they are *prescriptive* with respect to *future* mathematical work. By contrast, Cavaillès takes Lebesgue’s definition of effective mathematics (above) to be more *open* (or less restrictive), in that definitions are to be introduced according to the immanent demands of specific theories, rather than as a propodeautic to their development.24 As Cavaillès summarises Lebesgue’s position, “it is the notion of definition that has changed here: far from demanding [qua Borel] a possible construction solely based on a type of object posed once and for all, the effective [*l’effectif*, i.e. Lebesgue’s notion] on the contrary demands that only that which is important for the present reasoning [*le raissonnement présent*] appears in the characterisation of the object.”25 Thus for Lebesgue, all we need to do to define an object that we wish to describe is to *have named* its characteristic properties (and the past tense is crucial here), but there is no *particular* *type* of (practically effectuable) object which is being prescribed (nor proscribed) by this definition of the effective, and thus there is no prescription with respect to future mathematical work. The characteristic example here is Lebesgue’s definition of measure, viz. that to every set of points *E *we can associate a non-negative number *m *(*E*) that satisfies the following three conditions:

(I) Any two equal sets have the same measure;

(II) A set that is the sum of a finite or of a denumerable infinity of sets, without any points in common between them, has as its measure the sum of the measures of these sets;

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^{ }H. Lebesgue,Leçons…,op. cit., p. 103.(III) The measure of the set of all the points (0,1) is 1.26

17This definition simply defines the properties that we wish the concept of measure to have, but does not prescribe anything in particular about “calculation”: the stated stricture tells us exactly the conditions that the concept of measure is to fulfil, and *nothing more*, i.e. it does not extend to delimiting the modes of effectuation involved. Cavaillès thus expresses his sympathy for Lebesgue’s project (at least at this point) in terms of the attempt to honour the *immanence *of modes of definition in mathematics to what Cavaillès will call *effective work*. Having cited the above definition of measure and then referenced the general advances made as a result in the theory of integration, Cavaillès writes:

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^{ }OC25-26.They [these advances] stem from the simplifications achieved thanks to the descriptive character of the definition: all that matters is what is named there, whereas a construction that has nothing to do with the problem posed [again, i.e. Borel’s strictures] could only be a burden or an arbitrary restriction. In a way, what is affirmed here is the homogeneity of the materials of an enterprise, the simultaneity of mathematics with the work before it: this remains empiricism, since it is only a question of describing effective work [

le travail effectif], but it is an empiricism of actual thought [la pensée en acte], with no other reference than to the unpredictable becoming of mathematics. The modes of definition are left up to the variations and demands of its movement: with each new acquisition, new possibilities appear. The enrichment of the nameable coincides with the enrichment of science itself.27

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^{ }In a letter to Paul Labérenne (written at the end of 1938), Cavaillès claims explicitly that it i (...)

18Once again, in praising this more capacious definition of effective work (or actual thought), Cavaillès has progressively imbued the notion of effectiveness with a temporal connotation. This informs Cavaillès’ qualified aligning of himself with an “empiricist” project for the description of “mathematical experience”, and against the possibility of an *a priori* theory of science.28 Indeed, this conviction constitutes the core of the Cavaillèsian programme: the effectiveness of mathematics is defined in terms of the *unpredictability *of its development; the rejection of the *a priori* is a rejection of *prediction*.

# II. Effective Process and Effectuated Process in *Transfini et Continu*

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^{ }The essay was intended for*Revue philosophique*and submitted in September 1941, but the journal w (...) - 30
^{ }A note on Cavaillès’ sources: with respect to Church and Kleene, the papers cited by Cavaillès in (...)

19Written in 1940-1941 and published posthumously in 1947, *Transfini et continu *marks a new moment, both in Cavaillès’ thinking on the concept of the effective and in the broader mathematical history of the concept.29 The essay is predominantly concerned with Cavaillès’ technical exposition of two new mathematical developments: first, the formalisation of the Church-Kleene notion of effective calculability (or computability) in the papers they authored from 1935 to 1938, i.e. the formation of the so-called Church-Turing thesis; second, Gödel’s 1938/1940 results on the consistency of the continuum hypothesis.30 Cavaillès then concludes with a dense few pages in which he develops a new critique of Kantian projects in the foundations of mathematics. The point I wish to focus on is in what way Cavaillès’ use of the concept of the effective with respect to the earlier debate between the French analysts is transformed by his engagement with the Herbrand-Gödel notion of recursive function and the Church-Kleene formalisation of effective calculability in this new moment. For our purposes here, it is significant that Cavaillès directly relates Borel and Lebesgue’s notions of the effective to Church’s recasting of the concept of “effective procedure (or process)” by introducing a split between what Cavaillès will call “effective calculation” and “effectuated calculation”, which then informs his critique of the “Kantian theory of mathematics” on the grounds that it conflates “effective process” and “effectuated process”.

- 31
^{ }Exceptions include J. Avigad and V. Brattka, “Computability and analysis: The legacy of Alan Turi (...) - 32
^{ }The major points of this “pre-history” are all surveyed by Cavaillès at an earlier moment in chap (...) - 33
^{ }Cf. Church’s review of Turing’s, “On computable numbers, with an application to the Entscheidungs (...) - 34
^{ }I thank Brice Halimi and Walter Dean for clarifying conversations on this point.

20The fact that Cavaillès understands Church and Kleene to be formalising the same notion as Borel and Lebesgue merits comment, given that neither Church’s own papers nor prominent commentaries on the Church-Turing hypothesis tend to mention this connection.31 Church, Kleene and Turing’s formalisations are instead usually indexed to the general pre-history of the notion of recursive definition and the search for a resolution to the “decision problem”, whose major points include the introduction of the chain rule in Dedekind’s *Was sind und was sollen die Zahlen? *(1888), Hilbert’s use of transfinite recursion and the epsilon calculus in the attempted proof of the continuum hypothesis in the “Über das Unendliche” lecture of 1925, work by Hilbert, Ackermann and Bernays on the *Entscheidungsproblem*, and the Herbrand-Gödel notion of quasi-recursive function.32 These are doubtlessly the major references explicitly made by Church. It is important to note, however, that Church repeatedly refers to his and Turing’s definitions as providing a formalisation of an already widespread “ordinary (not explicitly defined)” or “vague” notion of effective calculability (indeed, the status of the Church-Turing thesis as a “thesis” is that it suggests such an identification between a formal and an informal notion, and thus is not “provable” in formal terms).33 I take Cavaillès to have seen this informal notion to be precisely what was at stake in the debates over the notion of effective definability among the French analysts. In having seen this connection, Cavaillès in *TC* is to an extent anticipating the history of effective descriptive set theory.34

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^{ }*OC*462.

21When Cavaillès introduces the notion of effective calculation in *TC* –and the immediate reference here is to Gödel, but the middle section of *TC *will be concerned with the Church-Kleene formalisation in a way that shows that Cavaillès is connecting the two developments– he does so by immediate reference to the same point which framed the introduction to *TAE*, i.e. the discussion between Borel and Lebesgue around 1905 concerning “the notions of the nameable and of the effective.”35 He then cites Lebesgue’s definition of this notion (see above, p. 221), before remarking that:

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^{ }OC461-462. The (non-standard) terminology ofcalcul régulier, as well as that of “quasi-recursiv (...)The vagueness of this requirement –which was intended to limit the domain of authentic mathematics– is due to the indeterminacy of [the notion of]

words. If these include “all objects [such] that… there is an object that…”, where is the finitude? The development of formalisms, in particular thanks to Gödel’s methods, makes it possible to avoid this pitfall. By obliging a calculation to be effective [être effectif], Lebesgue was led to transform it into an effectuated calculation [calcul effectué], [at least] if he wanted to chase away the indeterminacy of “a calculation”. But this would have made all progress impossible. The correlative notions of the regulated calculus [calcul régulier] and of quasi-recursive functions introduced thanks to Gödel’s work escape this dilemma.36

- 37
^{ }E.g. Lebesgue frames his commitment to descriptive definitions as being distinct from Borel’s con (...)

22What is going on here? The point concerning the “indeterminacy of words” is clear: if we say that all we need to do in order to have defined a mathematical object is to name a characteristic property as the *definiens* of the *definiendum*, without having placed any constraints on the acceptable modes of definition involved, then nothing prevents us from naming precisely those “paradoxical” properties that the notion of effectiveness was intended to prohibit, i.e. the problems of impredicativity return in full force. The only practical means advanced for avoiding this problem at Lebesgue’s time was to (implicitly or explicitly) refer back to the kind of constraints on calculability that characterised Borel’s notion of the effectively enumerable. This explains the ambiguity of Lebesgue’s position, wherein he both seems to have a wider notion of the effective than Borel and states that he is attempting to express the same constraint.37 From Cavaillès’ perspective this means that *in practice* Lebesgue’s openness to the demands of effective work collapses into constraining mathematics solely to the domain of *effectuated* calculations, i.e. that Lebesgue’s quasi-formal notion of effectiveness is only clarified by reference to a particular class of (already) effectuated objects. Lebesgue’s position thus collapses back into Borel’s in the course of its elaboration. Cavaillès’ claim is that first Gödel and then Church-Kleene, in formalising the modern notion of effective computability, allow us to avoid this pitfall:

- 38
^{ }OC462. Again, Cavaillès is drawing on Bernays’ exposition ofregelrecht auswertbarenfunctions ( (...)By means of this numbering and these recursive relations, there is a permanent control of the effectiveness of the calculation [

l’effectivité du calcul]: each of these steps falls under our grasp since we are sure to reach it by means of a canonical sequence of steps. The determination lies in the process of recursion. The truth is that this is not fixed once and for all, which is precisely what prevents [future] mathematics from being limited to [already completed] contemporary mathematics [A la vérité celui-ci n’est pas fixé une fois pour toutes, ce qui évite justement de borner les mathématiques aux mathématiques actuelles]. But a function, for example, will be said to becalculable in a regulated manner[régulièrement calculable] if it is possible to construct a special formalism for it that meets the conditions laid down.38

- 39
^{ }*OC*463. The editors of*Philosophie Mathématique*add a footnote at this point to A. Church, “The C (...) - 40
^{ }*OC*472.

23The development here is dialectically subtle. As Cavaillès notes, it is Gödel’s work that inspired Kleene and Church to develop what Church calls an *absolute* definition of constructibility or effectiveness, where “absolute” is understood to signify that the notion is “independent of reference to a given formal system.”39 This absoluteness or independence of the new formal definition of effectiveness is what makes it adequate as a proposed formalisation of the informal notion. Yet the inverse of this movement of absolutisation is that the notion of an effectuable object is now purely *relative* to what Cavaillès will call “a given conceptual system”^{ }(i.e. to a given formalism).40^{ }In terms of the temporalised reading that we have been discussing, Cavaillès’ is asserting that it is precisely *because* the notion of effectiveness in the Church setting is solely a notion of *procedure *or *process *that it doesn’t attempt to limit mathematics to any particular class of effectively or intuitively graspable objects, and thus is more supple with respect to the demands of mathematical theories that have *not yet* been developed.

- 41
^{ }*OC*464.

24Thus the novel point here –all of which comes to depend “on the definition of effective process [*processus effectif*]”41– is that the same movement in the history of the concept of the effective can at once be read as an *absolutisation* of the notion of constructibility and a *relativisation* of the notion of intuition. The notion of intuition thus no longer refers to any given class of “evident” objects upon which construction could be founded, but rather, in Cavaillès’ terms, intuition is internal to concatenation. It is this inversion which grounds Cavaillès’ new critique of Kant, or, more precisely, of the Kantian underpinnings of competing projects of mathematical finitism:

- 42
^{ }OC469-470.To posit an irreducible intuition is only to come to a thoughtless halt [

Une intuition irréductible n’est qu’un arrêt sans pensée]. The Kantian theory of mathematics […] conflates the dialectical moment of the positing of the concept and the transcendental moment of its schematisation. […] Moreover, the misunderstanding appears which takes the construction itself to be representable; this is the confusion already noted between effective process and effectuated process [procés effectif et procés effectué]. […] The fundamental ambiguity of transcendental philosophy is that it poses the conditions of knowledge as relative to a synthetic activity which is defined and directly apprehended in a knowledge that itself escapes from critique: the condition of representation is conflated with the representations of this condition.42

25To summarise, the problem with Borel’s position was that his proto- constructivist notion of the effectively enumerable sought to condition mathematics by something extrinsic to mathematics, i.e. the notion of a finite number of words or characters. This is connected to the notion of an intuitive basis on which (future) mathematical work could be grounded. In turn, Lebesgue’s more liberal “descriptive” notion of the effective was tied to the same strictures. We can say that Borel and Lebesgue share a certain “Kantianism” that we also find in Kronecker, Hilbert, Brouwer and other finitist projects (albeit in very different ways) insofar as they make a double attempt, at once descriptive and restrictive: the first is to describe the constructive (or intuitive) activity of mathematical understanding as it actually (or effectively) proceeds; the second is that they aim to ground this constructive activity in certain well-defined or intuitively graspable class of effectuated objects (or intuitions), that is to say well-defined (or individuated) with respect to* our consciousness *and its limitations.

- 43
^{ }D. Hilbert, “Die Grundlagen de Mathematik”,*Abhendlungen aus dem mathematischen Seminar der Hambu*(...) - 44
^{ }*Ibid*., p. 464, emphasis mine.

26Hilbert’s methodological comments in the 1927 Hamburg address on “The Foundations of Mathematics” are exemplary in this regard. On the one hand, he positions the project of proof theory in descriptive terms: “The fundamental idea of my proof theory is none other than to describe the activity [*Tätigkeit*] of our understanding, to make a protocol of the rules according to which our thinking actually [*tatsätlich*] proceeds.”43 On the other, this ambition to describe construction is prefaced by the assertion that “as a condition for the use of logical inferences and the performance of logical operations, something must already be given to us in our faculty of representation [*in der Vorstellung*], *certain extralogical concrete objects that are intuitively* [*anschaulich*]* present as immediate experience prior to all thought*.”44 What Cavaillès argues, to the contrary, is that by delinking effectiveness and effectuation, Gödel and Church allow mathematical philosophy to maintain its descriptive ambition without its restrictive supplement. This is founded in the recognition that the meanings given to any type of object that we might wish to serve as a foundation or as an intuitive basis for mathematical reasoning (such as points, marks, whole numbers, “the continuum”) are continually altered and rewritten by the formalisms (or concatenations) that specify them. There is nothing prior to thought: this is the consequence of Cavaillès’ characteristic conjunction “mathematical experience”.

27What is at stake in the critique of the Kantian theory of mathematics in the conclusion* *to *TC* is Cavaillès’ confrontation with various projects of finitism and constructivism spanning the preceding 50 years, one that is remarkable because it displaces certain technical problematics from the history of mathematical finitism onto Cavaillès’ own philosophy of mathematical history. The new theory of intuition that is outlined here is thus Cavaillès’ contribution to a relativisation or historicisation of the transcendental:

- 45
^{ }OC471-472.What marks history is the submission of the transcendental to its stages […]. Necessity appears after the fact [

aprés coup]. No analysis of the consciousness of acts permits prediction, any more than it can procure an arbitrary permanence. Intuitive is a synonym for effective (or effectuating) consciousness, transcendental for constitutive (or constituting) consciousness, relative to a given conceptual system. No more than there is a closed system is there an absolute transcendental: formal unity is the continuity of a concatenation of acts, the concrete is the novelty of unifying a multiplicity that is still conceptually recognised as such.45

- 46
^{ }Cf. J. Petitot, “Jean Cavaillès et le Continu”, address delivered at a commemoration for Cavaillè (...)

28Mapping this passage onto the one cited just above (p. 230), the effective or effectuating pole is placed on the side of “effective process”, while the transcendental or constitutive pole is placed on the side of “effectuated process”. For Cavaillès, the (temporal) disjunction here between the effective (or effectuating) and the effectuated poles is not something that is contingent or *extrinsic* to the mathematical concept, but is something that is *constitutive *of the concept itself. The new concept of concept presented here is continuous with Cavaillès’ redefinition of the notion of the effective, which now names the whole array of symbolic and conceptual material (objectively) available to a working mathematician at a given moment. In other words, the effective becomes the site of what Petitot has called Cavaillès’ theory of the “*continuous actualisation *of intuition [actualisation continue *de l’intuition*]”, an actualisation which is rather a perpetual reactualisation that is each time mediated by the specificities of mathematical work, and not through any reference to the putatively *a priori* structure of consciousness.46 The conclusion to *TC *should then be read as the transition to the object of Cavaillès’ mature work in *LTS*, namely that of providing a theory of the logic of the passages between the positing of objects and their transformations* *in new formal settings, which is to say a theory of *logical time *that is no longer to be indexed to the time of consciousness, as with the Kantian and Husserlian theories.

# Conclusion: *On Logic and the Theory of Science *as a theory of logical time

- 47
^{ }The conjunction*temps logique*is found in both J. Lacan, “Le temps logique et l’assertion de cert (...) - 48
^{ }*OC*504, translation from*On Logic…*,*op. cit.*, p. 65.

29In designating *LTS* as a theory of logical time, and positioning *TC* as the transitional text for the development of that theory, I am importing a concept from outside of Cavaillès’ own discourse.47 This framing is intended to capture what Cavaillès himself states in more aporetic terms in the course of his analysis of Bolzano in the first section of *LTS*: “[…] science moves outside of time –if by time we mean reference to the lived experience [*vécu*] of a consciousness.”48 This formula is paradoxical: if science moves outside of time, what is the medium of this movement? I suggest that *LTS* is to be understood as providing a theory of this time of construction, understood not as the time *internal* to the construction of a particular mathematical theory, but rather as the time of the passage *between *incommensurable mathematical theories, or, as Cavaillès will sometimes say (whence his confrontation with Husserl), between *consciousnesses*. As I have indicated, Cavaillès’ transition to consideration of this logical time passes in an essential manner through reflection on the mathematical history of the effective or (it comes to the same thing) the problem of recursive definitions. In this respect, it is important to note that the kernel of this new theoretical horizon was already present in *MAF *when Cavaillès treats Jacques Herbrand’s definition of intuitionistic or finitary reasoning from “Sur la non-contradiction de l’Arithmétique” (1931), i.e. a crucial point in the genesis of Gödel’s formalisation of general recursive functions:

- 49
^{ }Footnote 3 in J. Herbrand, “Sur la non-contradiction de l’Arithmétique”,Journal für die reine un(...)By an intuitionistic argument we understand an argument satisfying the following conditions: in it we never consider anything but a given finite number of objects and of functions; these functions are well-defined, their definition allowing the computation of their value in a

univocalway; we never state that an object exists without giving the means of constructing it; we never consider the totality of all the objectsxof an infinite collection; and when we say that an argument (or a theorem) is true for all thesex, we mean that, for eachxtaken by itself, it is possible to repeat the general argument in question, which should be considered to be merely the prototype of these particular arguments.49

- 50
^{ }J. Herbrand, “Sur la non-contradiction…”, art. cit., p. 5, translation from*Logical Writings*,*op.*(...)

30Herbrand then goes on to introduce the idea of recursively defined functions with the demand that –considered “intuitionistically” in the sense just cited– such functions must “must make the effective computation [*permettent de faire effectivement le calcul*]” of values possible “for every particular set of numbers.”50 With respect to these definitions, Cavaillès states:

- 51
^{ }OC161.Indeterminacy now only remains with respect to the construction: it can be understood as a progression of stages, each of which is finite in the sense mentioned above. There is no longer a way to encompass all of these steps in a single glance [

Il n’y a plus moyen d’embrasser du regard toutes les démarches]: if consciousness ceases to be coextensive with their succession, it is at least present in each of them.51

- 52
^{ }*OC*560, translation from*On Logic…*,*op. cit.*, p. 136, emphasis mine.

31This remarkable formulation anticipates Cavaillès’ definition of formal concatenation in *LTS* with which I began: the characteristic of the intelligible is that everything does not be established all at once [*d’un seul coup*], i.e. that no single consciousness (for which we can read: no single formalism, no single theory, no single transcendental) can simultaneously survey the elements of mathematical work. This insight that modern mathematics dissolves the ambition to enumerate a unitary structure of consciousness then comes to found Cavaillès’ final break with Husserl in the conclusion to *LTS*, on the grounds that in the passage between an old and a new mathematical theory “[t]here is more consciousness in it –*and* *not the same consciousness as before*. The term ‘consciousness’ does not entail *univocity *of application any more than ‘thing’ does an isolable unity.”52 Between the composition of these two texts in 1938 and 1942, there has been a crucial development: we have moved from the assertion that consciousness is *present at each step* to the assertion that *at each step* *it cannot be the same consciousness that is present.* My claim is that this modification in Cavaillès’ view coincides with his treatment of the mathematical developments around the concept of the effective that we have traced in his treatment of Gödel, Church and Kleene in *TC*, and is continuous with his profound reorientation with respect to the concept of intuition: if the new formalisation of effective procedure destroys the idea that mathematics is to be grounded in any *particular *intuitive basis, it equally shows that there can only be a multiplicity of consciousnesses (or transcendentals), which are each immanent to the progress of mathematical work. This discovery, which implies that there can only ever be a progression of incommensurable formalisations, is understood by Cavaillès not as a consequence of the limitations of our (finite) faculties, but rather as a characterisation of reason itself.

## Notes

1 ^{ }This paper is a development of a presentation given at FPMW 13, hosted at the Université Côte d’Azur, 7-9 October 2021. I thank the organisers and respondents, whose engagement has improved this work immeasurably.

2 ^{ }Cavaillès responding to Albert Lautman in “La pensée mathematique”, a transcription of their joint presentation to the Société française de Philosophie on 4^{th} February 1939. Cited from Cavaillès, *Œuvres complètes de philosophie des sciences*, Paris, Hermann, 1994, p. 628. Hereafter references to Cavaillès’ main works will be cited from the *Œuvres complètes* as *OC*. Translations are my own unless otherwise noted.

3 ^{ }*OC* 226.

4 ^{ }Cf. *OC *85,* OC *100,* OC *106,* OC* 180, *OC* 187, *OC* 540,* OC *628.

5 ^{ }Cf. H. B. Sinaceur, *Cavaillès*, Paris, Les Belles Lettres, 2013, p. 86: “L’effectivité hégélienne […] constitue un fil conducteur essentiel pour lire Cavaillès […].”

6 ^{ }To resume the polemical term that Cavaillès adopts from Léon Brunschvicg.

7 ^{ }*OC* 517-518, translation from R. Mackay and K. Peden, *On Logic and the Theory of Science*, Falmouth, Urbanomic, 2021, p. 82-83, translation modified. Mackay and Peden translate *effectif* throughout by the English “actual” (with corresponding choices for all noun and verb forms), as is the contemporary norm in translations from French to English. I tend to agree with the conceptual move here; however, for reasons mentioned above, I have chosen to suspend the lexical equivalence for the purposes of this exposition.

8 ^{ }*OC *14.

9 ^{ }For an overview of some of the vagaries of this nascent “concrete” set theory, cf. T. Murata, “French Empiricism I –One of the studies of the foundations of Mathematics”, *Commentarii mathematici Universitatis Sancti Pauli*,* *vol. 6, n^{o} 2, 1958, p. 93-114, “French Empiricism II”, *Commentarii…*, vol. 7, n^{o} 1, 1959, p. 57-63, “French Empiricism III”, *Commentarii…*,* *vol. 7, n^{o} 2, 1959, p. 99-116, “French Empiricism IV”, *Commentarii…*, vol. 8, n^{o} 2, 1960, p. 81-96, H. Gispert, “La théorie des ensembles en France avant la crise de 1905: Baire, Borel, Lebesgue… et tout les autres”, *Revue d’histoire des mathématiques*, n^{o} 1, 1995, p. 39-81, and P. Bertin, *Jeu d’espace, espace de jeu: Philosophie et genèse hausdorffienne des espaces topologiques*, PhD Thesis, Université Paris Diderot-Paris 7, 2019, p. 87-91.

10 ^{ }The “Five Letters” are initially published as R. Baire, E. Borel, J. Hadamard and H. Lebesgue, “Cinq lettres sur la théorie des ensembles”, *Bulletin de la Société Mathématique de France*, n^{o} 33, 1904, p. 261-273, and republished as an appendix to the second and subsequent editions of Borel’s *Leçons sur la théorie des fonctions*, Paris, Gauthier-Villars, 1914, p. 150-160. Translation in G. H. Moore, “Zermelo’s Axiom of Choice: It’s Origins, Developments, & Influence”, in *Studies in the History of Mathematics and Physical Sciences*, 8, New York, Springer-Verlag, 1982, p. 311-320.

11 ^{ }Cf. *OC *362 and *OC *29.

12 ^{ }I take this as one reason why Cavaillès published *MAF* as the sequel to *TAE*. While *MAF* is Cavaillès’ *thèse principale* and *TAE* is the *thèse complémentaire*, and *MAF* was completed prior to *TAE*, *TAE *appeared as volumes 606 and 607 of the collection *Actualités scientifiques et techniques* published by Hermann, and *MAF* as volumes 608-610. Cavaillès states that *MAF *is the “continuation” of *TAE *(cf. *OC *15). Cf. P. Cassou-Noguès, *De l’expérience mathématique: Essai sur la philosophie des sciences de J. Cavaillès*, Paris,* *Librarie Philosophique J. Vrin, 2001, p. 19.

13 ^{ }For an analysis of the history of the relevant results here, cf. G. H. Moore, “Lebesgue’s Measure Problem and Zermelo’s Axiom of Choice: The Mathematical Effects of a Philosophical Dispute”, in J. W. Dauben and V.S. Sexton (eds.), *History and philosophy of science: selected papers*,* *New York, New York Academy of Sciences, 1983, p. 129-154.

14 ^{ }*OC *224.

15 ^{ }*OC *227.

16 ^{ }The relevant distinction between “empiricism” and “idealism” at stake here has its roots in Emil du Bois-Reymond. See the commentary in L. Brunschvicg, *Les étapes de la philosophie mathématique*, Paris, Alcan, 1912, p. 527-533.

17 ^{ }J. Tannery, “De l’infini mathématique”, *Revue générale des sciences pures et appliquées*, vol. 8, n^{o} 4, Paris, Octav Doin, 1897, p. 129-140, p. 132, reviewing L. Couturat, *De l’infini mathématique*, Paris, Felix Alcan, 1896.

18 ^{ }*Ibid.*, p. 134.

19 ^{ }For an analysis of the same debate as an exemplary case for what Jean-Toussaint Desanti calls an “epistemological problem” in mathematics, cf. J.-T. Desanti, “Qu-est-ce qu’un problème épistémologique?”, *Porisme*, n^{os} 3, 4-5, 1965, reprinted in *La philosophie silenceuse*, Paris, Éditions du Seuil, 1975, p. 110-132.

20 ^{ }E. Borel, *Leçons*…, *op. cit.*, p. 65. Reprint of E. Borel, “Les ‘Paradoxes’ de la théorie des ensembles”, *Annales scientifiques de l’École Normale Supérieure*, vol. 3, n^{o} 25, 1908, p. 443-448, wherein the constraint is initially articulated negatively in response to Richard’s paradox. Cited by Cavaillès at *OC *17.

21 ^{ }H. Lebesgue, “Sur les fonctions représentables analytiquement”, *Journal de Mathématiques Pures et Appliquées*, vol. 6, n^{o} 1, 1905, p. 139-216. Cited by Cavaillès at *OC *23.

22 ^{ }Appendix to E. Borel, *Leçons*…, *op. cit.*, p. 219, which is a modified version of E. Borel, “Le calcul des intégrales definies”, *Journal de Mathématiques Pures et Appliquées*, vol. 6, n^{o} 8, 1912, p. 159-210. The definition of “calculable function” given immediately afterwards follows the same schema.

23 ^{ }E. Borel, *Leçons*…, *op. cit.*, p. 219, emphasis mine.

24 ^{ }Lebesgue makes this clear when, in the 2nd edition of H. Lebesgue, *Leçons sur l’intégration et la recherche des fonctions primitives professées au Collège de France*, Paris, Gauthier-Villars, 1904, published in 1928, he adds a footnote to the effect that Hilbert’s approach in the *Grundlagen* *der Geometrie *exemplifies the axiomatic method insofar as it presents an axiomatisation as a “completed whole” prior to the elaboration of the theory, whereas “descriptive definitions laid down in the course of the development of a theory, such as the [i.e. Lebegue’s] definition of the integral, do not claim to enumerate all the axioms on which they are based; they do not form a complete whole and cannot be isolated from the rest of the theory.” H. Lebesgue, *Leçons…*, *op. cit*., p. 207.

25 ^{ }*OC *23-24. Cavaillès is selective in his treatment of Lebesgue here. Lebesgue uses both descriptive and constructive definitions, and sometimes positions his work on measure as orientated towards producing constructive definitions of the central concepts of the theory of integration that would be *equivalent* to previously given descriptive ones. Cf. H. Lebesgue, *Leçons…*,* op. cit.*, p. 100, and the commentary in A. Michel,* Constitution de la théorie moderne de l’intégration*, Paris, Vrin, 1992, p. 85.

26 ^{ }H. Lebesgue, *Leçons…*,* op. cit.*, p. 103.

27 ^{ }*OC *25-26.

28 ^{ }In a letter to Paul Labérenne (written at the end of 1938), Cavaillès claims explicitly that it is the “same empiricism” of the French analysts which, in manner more radical than he had expressed in *MAF*, allows Cavaillès to see that “all recourse to any *a priori *whatsoever is dishonest: thus my complete break with idealism, even of the Brunschvicgian variety”, given that “there is no kind of way to pre-emptively dominate or decide the effective work of the mind –that is, of human technique– in experience.” Cited from H. Mougin, “Jean Cavaillès”, *La Pensée*, 4, Paris, Centre d’études et de recherches marxistes, 1945, p. 70-83, p. 79.

29 ^{ }The essay was intended for *Revue philosophique *and submitted in September 1941, but the journal was abolished by the Nazi occupying forces before it could be published. A typescript, annotated by Cavaillès, was preserved by Suzanne Lautman (the widow of Albert Lautman), and corrected by Henri Cartan and Claude Chevalley for publication as the fourth and final volume of the *Essais philosophiques* collection that Cavaillès (along with Raymond Aaron) had edited for Hermann before the war. See the (anonymous) introductory note by Georges Canguilhem, *OC *452.

30 ^{ }A note on Cavaillès’ sources: with respect to Church and Kleene, the papers cited by Cavaillès in the original publication of *TC* are S.C. Kleene, “A Theory of Positive Integers in Formal logic. Part I”, *American Journal of Mathematics*, vol. 57, n^{o} 1, 1935, p. 153-173, S.C. Kleene, “A Theory of Positive Integers in Formal logic. Part II”, *American Journal of Mathematics*, vol. 57, n^{o} 2, 1935, p. 219-244, and A. Church, “An unsolvable problem of elementary number theory”, *American Journal of Mathematics*, vol. 58, 1936, p. 345-363 (cf. J. Cavaillès, *Transfini et continu*, Paris, Hermann, coll. “Essais philosophiques” vol. 4, 1947, p. 15). The editors of J. Cavaillès, *Philosophie Math*é*matique*, Paris, Hermann, 1962 (i.e. the version reproduced in *Oeuvres complètes*) insert references to S. C. Kleene, “On Notation for Ordinal Numbers”, *The Journal for Symbolic Logic*, n^{o} 3, 1938, p. 150-155, A. Church and S.C. Kleene, “Formal definitions in the theory of ordinal numbers”, *Fundamenta Mathematicae*, vol. 28, n^{o} 1, 1937, p. 11-21, and A. Church, “The Constructive Second Number Class”, *Bulletin of the American Mathematical Society*, vol. 44, n^{o} 4, 1938, p. 224-232, without signalling that these were not necessarily read by Cavaillès (cf. *OC* 463)*. *With respect to Gödel’s results, the only paper cited directly by Cavaillès in *TC *is K. Gödel, “Consistency Proof for the Generalized Continuum Hypothesis”, *Proceedings of the National Academy of the Sciences of the United States of America*, vol. 25, n^{o} 4, 1939, p. 220-224, which does not directly mention the new definition of general recursive functions (again, the editors of the 1962 *Philosophie Mathématique *collection insert additional citations; compare the 1947 edition of *Transfini et Continu*, p. 17 with *OC *467). Cavaillès does however appear to directly quote Gödel’s 1938 abstract announcing the results (i.e. K. Gödel, “The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis”, *Proceedings of the National Academy of the Sciences of the United States of America*, vol. 24, n^{o} 12, 1938, p. 556-557), albeit in paraphrased form (compare Gödel’s 1938 abstract, p. 556 with *OC *467); again, Gödel makes no direct mention of general recursion there. Cavaillès does not seem to have had access to the canonical formulation of the notion of general recursive functions in Gödel’s Princeton lectures as transcribed by Kleene and J. B. Rosser and published as K. Gödel, “On undecidable propositions of formal mathematical systems” (reprinted with revisions in M. Davis (ed.), *The undecidable: basic papers on undecidable propositions, unsolvable problems, and computable functions*, New York, Raven Press, 1965, p. 39-74 and K. Gödel, *Collected Works: Volume* *1*, Oxford, Oxford University Press, 1986, p. 346-371). However, Cavaillès’ exposition in *TC *draws on Paul Bernays’ Supplement II to the second volume of Hilbert and Bernays’ *Grundlagen der Mathematik *(D. Hilbert and P. Bernays, *Grundlagen der Mathematik* *II*, Berlin, Springer, 1939, p. 392-421, miscited by Cavaillès as being published in 1940), which recapitulates Gödel’s innovations (as does A. Church, “An unsolvable problem…”, art. cit., 1936, which, as I have noted, was read by Cavaillès). This Supplement –entitled “A precise explication of the concept of calculable function and Church’s Theorem on the decision problem [*Eine Pr*ä*zisierung des Begriffs der berechenbaren Funktion und der Satz von CHURCH über das Entscheidungsproblem*]”– likely influenced the specific manner in which Cavaillès interrelates the new results by Church, Gödel and Kleene.

31 ^{ }Exceptions include J. Avigad and V. Brattka, “Computability and analysis: The legacy of Alan Turing”, in Downey R. (ed.), *Turing’s Legacy: Developments from Turing’s Ideas in Logic* (Lecture Notes in Logic), Cambridge, Cambridge University Press, 2014, p. 1-47 and W. Dean and S. Walsh, “The Prehistory of the Subsystems of Second-Order Arithmetic”, *Review of Symbolic Logic*, vol. 10, n^{o} 2, 2017, p. 357-396, as well as some Russian histories, probably owing to the influence of Nikolai Luzin (e.g. A. L. Semenov and V.A. Uspensky, “What are the gains of the theory of algorithms?: Basic Developments Connected with the Concept of Algorithm and with Its Applications in Mathematics”, in A. P. Ershov and D. E. Knuth (eds.), *Algorithms in Modern Mathematics and Computer Science*, New York, Springer-Verlag, 1981, p. 100-234, in particular p. 105-106 and p. 176-181). Whilst Church doesn’t directly index the notion of effective procedure to Borel et al., he does cite Borel’s notion of the effectively enumerable in A. Church, “A Bibliography of Symbolic Logic”, *Journal of Symbolic Logic*, Cambridge, Cambridge University Press, vol. 1, n^{o} 4, p. 121-216, p. 157, along with other works by the French analysts.

32 ^{ }The major points of this “pre-history” are all surveyed by Cavaillès at an earlier moment in chapter III of *MAF*.

33 ^{ }Cf. Church’s review of Turing’s, “On computable numbers, with an application to the Entscheidungsproblem”, *Journal of Symbolic Logic*, vol. 2, n^{o} 1, 1937, p. 42-43, and Church, “The Constructive Second Number Class”, art. cit., 1938, p. 226-227.

34 ^{ }I thank Brice Halimi and Walter Dean for clarifying conversations on this point.

35 ^{ }*OC *462.

36 ^{ }*OC *461-462. The (non-standard) terminology of *calcul régulier*, as well as that of “quasi-recursive functions” are both drawn from Bernays’ Supplement II to the second edition of the *Grundlagen*, i.e. I understand *calcul régulier* to be Cavaillès’ translation of the calculus of *regelrecht auswertbaren *functions introduced by Hilbert and Bernays as a generalisation of Church, Gödel and Kleene’s definitions (cf. Hilbert and Bernays,* Grundlagen der Mathematik* *II*, *op. cit.*, 1939, p. 394-395). Wilfrid Sieg suggests “reckonable functions” as a translation for this notion (cf. W. Sieg, “Step by Recursive Step: Church’s Analysis of Effective Calculability”, *The Bulletin of Symbolic Logic*, vol. 3, n^{o} 2, 1997, p. 154-180, p. 166); the French translation by F. Gaillard, E. Guillaume and M. Guillaume renders it by *fonctions à évaluation réglée* (cf. D. Hilbert and P. Bernays, *Fondements des mathématiques* *2*, Paris, L’Harmattan, 2001, p. 466-467).

37 ^{ }E.g. Lebesgue frames his commitment to descriptive definitions as being distinct from Borel’s constructive definitions; on the other hand, immediately preceding his definition of the effective Lebesgue writes: “I shall try never to speak of a function without effectively defining it; I shall thus adopt a point of view very similar to that taken by M. Borel in his *Leçons sur la théorie des fonctions.*” H. Lebesgue, “Sur les fonctions…”, art. cit., 1905, p. 205.

38 ^{ }*OC *462. Again, Cavaillès is drawing on Bernays’ exposition of *regelrecht auswertbaren *functions (see footnote 36, above).

39 ^{ }*OC *463. The editors of *Philosophie Mathématique *add a footnote at this point to A. Church, “The Constructive…”, art. cit., 1938, which Cavaillès does appears to be paraphrasing: “Much of the interest of the proposed definition lies […] in its absoluteness, and would be lost if it could be shown that it was in any essential sense relative to a particular scheme of notation or a particular formal system of logic.” (p. 224). As noted above, Cavaillès does not cite this paper in the original version of *TC* (see footnote 30). The question of whether Cavaillès read Church’s 1938 paper is significant, given that it would have been a source available to Cavaillès which mentions Alan Turing’s contribution to the so-called “Church-Turing Thesis”, and Turing is never mentioned by Cavaillès.

40 ^{ }*OC *472.

41 ^{ }*OC *464.

42 ^{ }*OC *469-470.

43 ^{ }D. Hilbert, “Die Grundlagen de Mathematik”, *Abhendlungen aus dem mathematischen Seminar der Hamburgischen Universität*, n^{o} 6, 1928, translation from J. van Heijenoort (ed.), *From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931*, Cambridge, Harvard University Press, 1967, p. 464-479, p. 475.

44 ^{ }*Ibid*., p. 464, emphasis mine.

45 ^{ }*OC *471-472.

46 ^{ }Cf. J. Petitot, “Jean Cavaillès et le Continu”, address delivered at a commemoration for Cavaillès held at the ENS, 17^{th }February 2014, transcript available at: http://jeanpetitot.com/ArticlesPDF/Petitot_Cavailles.pdf [accessed 22/12/2021]. On Petitot’s reading, Cavaillès’ gesture in *TC* is in fact to simultaneously historicise the two Kantian regimes of intuition and the concept: on the one hand, “intuition (i.e. the transcendental aesthetic) is historicised through reactualised intuitive superposition, [that is] intuition is connected to effective practices [*pratiques effectives*] and no longer to an original donation as with Husserl”; on the other, “categoricity (i.e. the transcendental analytic) is historicised through the dialectic of the concept.” (*ibid*.) In other words, what interests Petitot in *TC* is that in the same movement by which Cavaillès abolishes the idea of an external donation which is to act as the basis for mathematical synthesis, he reworks the doctrine of the schematism as internal to mathematical work.

47 ^{ }The conjunction *temps logique* is found in both J. Lacan, “Le temps logique et l’assertion de certitude anticipée. Un nouveau sophisme”, *Cahiers d’art, 1940-1944*, 1944, p. 32-42 and G. Bachelard, *Le rationalisme appliqu*é, Paris, PUF, 1949, p. 60, p. 96.

48 ^{ }*OC* 504, translation from *On Logic…*,* op. cit.*, p. 65.

49 ^{ }Footnote 3 in J. Herbrand, “Sur la non-contradiction de l’Arithmétique”, *Journal für die reine und angewandte Mathematik*, vol. 66, n^{o} 1, 1931, p. 1-8, p. 3, translation from J. Herbrand, *Logical Writings*, ed. W. D. Goldfarb, Cambridge, Harvard University Press, 1971, p. 288-289, emphasis mine. Herbrand’s term “intuitionistic argument [*raisonnement intuitionniste*]” should be understood as referring to the demands of *finitary *reasoning in the sense of the Hilbert programme, rather than to *intuitionistic arguments* in the sense of Brouwer, hence why Cavaillès introduces the cited passage at *OC *161 as a way to “enlarge the notion of finitary reasoning [*raisonnement fini*]” (on this point, see van Heijenoort’s introduction to Herbrand’s 1931 paper in J. Herbrand, *Écrits Logiques*, Paris, PUF, 1968, translated in *Logical Writings*, *op. cit.*, 1971, p. 282-284). In a letter to Lautman written 13^{th }June 1936 whilst making revisions to *MAF *Cavaillès writes that “the great merit of Herbrand –which I haven’t emphasised enough– is to strip his approach of all intuitive signification (i.e. interpretation of quantitative signs).” H. B. Sinaceur, “Lettres inédites de Jean Cavaillès à Albert Lautman”, *Revue d’histoire des sciences*, vol. 40, n^{o} 1, 1987, p. 117-128, p. 120.

50 ^{ }J. Herbrand, “Sur la non-contradiction…”, art. cit., p. 5, translation from *Logical Writings*, *op. cit.*, p. 291, translation modified to replace Goldfarb’s “actual” with “effective”. See W. Sieg, “Mechanical procedures and mathematical experiences”, in George A. (ed.), *Mathematics and Mind*, Oxford, Oxford University Press, 1994, p. 71-117 and “Only Two Letters: The Correspondence between Herbrand and Gödel”, *The Bulletin of Symbolic Logic*, vol. 11, n^{o} 2, p. 172-184 for commentary on the profound conceptual developments between Herbrand’s definitions and Gödel’s.

51 ^{ }*OC* 161.

52 ^{ }*OC* 560, translation from *On Logic…*,* op. cit.*, p. 136, emphasis mine.

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**Matt Hare**, « The Effective as the Actual and as the Calculable in Jean Cavaillès », *Noesis*, 38 | 2022, 213-235.

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