• 1 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

• 3 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.

• 4 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.

• 5 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.

• 9 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).

• 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 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.

• 16 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.

• 17 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:

• 19 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:

• 22 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:

• 24 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:

• 28 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.

• 29 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

• 35 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:

• 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:

• 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:

• 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”.

• 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.

• 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:

• 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:

• 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.