The correct distance
1Jean Cavaillès, the foremost representative of the French tradition of philosophy of mathematics, can be described as the thinker within this tradition who gave logic, both philosophical and mathematical, the most prominent place—as reflected in the title of his works, by the number of pages he devoted to set theory, and by his constant discussion of both Hilbert and Husserl.
2In another sense, however, logic proves to be the poor relation in Cavaillès’s research. It is approached essentially through the metamathematical formalization of mathematical theories (of their operations as well as their proofs), that is to say through the formalization of axiomatic method and subsequently [1] that of proof theory as represented by Hilbert and Gentzen. The venerable tradition which, since Aristotle, has linked logic to grammar, is disregarded. Moreover, Cavaillès’s examination of logicism—and the discussion of the nature of logic that such an examination implies—goes no further than a critique of the Principia Mathematica. Apart from that, the entire logicist corpus only seems to merit a few brief references. Finally, the logicism of Frege and Russell is frequently conflated with the program developed by Carnap in The Logical Syntax of Language, as well as with Tarski’s early works. Logicism as a whole is erroneously equated with the summary thesis that “there is nothing more in mathematics than certain formal systems.” [2]
3The problem is not one of historical accuracy alone: the role assigned to logic determines the type of philosophy of mathematics one will go on to develop. For the problem faced by any philosophy of mathematics is that of situating philosophy at the correct distance from mathematics: too close and philosophy will end up being nothing more than an illustrative paraphrase of the concepts and results of mathematics; too far away and philosophy of mathematics merges into a general theory of abstraction—of both abstract objects and the conditions under which they can be known—with the risk of losing the specificity of mathematics, particularly mathematics qua symbolic activity. In short, philosophy of mathematics risks being confused either with a “philosophy of mathematical practice” or with a general theory of knowledge. It is not easy to find the correct distance, and in order to do so some kind of mediation is indispensable. As shown by Sébastien Gandon, [3] this mediation has taken various forms throughout history: Descartes’s theory of method, Kant’s theory of the faculties, Russell’s theory of relations. But although the figures have changed, in each case it is basically a question of logic in the broadest sense of the term. Although logic is not a branch of mathematics, it has historically designated, not a discipline in its own right, but rather the very place where exchanges between philosophy and mathematics take place. It is logic that can provide the philosopher with the mediation that allows mathematics to be seen in terms of a theory of the sign and meaning, of syntactic forms, and of generality.
4Although these issues are amply and explicitly discussed in Cavaillès’s texts, the theoretical position that unites them, and which the Russell of the Principles of Mathematics calls “philosophical logic,” never appears in its own right. There is undoubtedly an explicit reason for this: namely, that in Cavaillès’s writings, when logic is not assumed to be synonymous with proof theory, the term refers primarily to the transcendental logic of Kant and Husserl. However, advocating the advances that he considers are made possible by Hilbertian formalism, Cavaillès writes:
The difference with Kant is that there is no such thing as pure logical thought, logic is only a constituent, which cannot be isolated from any truly functioning thought. Therefore, the problem of the connection between abstract thought and intuition no longer arises, at least not in the same place. If logic no longer exists as an autonomous discipline, its role can only be defined negatively by eliminating the role of concrete intuitions, which guarantee both the fruitfulness and the validity of reasoning. [4]
6Philosophy of mathematics thus revokes the understanding of logic as something concerned with nonintuitive knowledge or, at least, the purely intellectual components of knowledge: since mathematical signs undo any distinction between the intellectual and the intuitive, they invalidate any such concern from the outset. What is more, insofar as “the very essence of mathematics is the rulegoverned play of symbols,” [5] and the formalization of reasoning by means of “symbolic logic” is constitutive of its very elaboration, Cavaillès is able to conclude, again following Hilbert, that
[l]ogic and mathematics, undergoing a common fate, are distributed equally between the domains of formalism and intuition, separation no longer being made between them but bearing upon mixtures of the two, mixtures within which it is futile to try and isolate one from the other. [6]
8In short, philosophy of mathematics cannot take the form of a logic, because logic cannot be separated from mathematics. As GillesGaston Granger notes, [7] “the notion of logical form indispensable to a proof theory,” [8] which Cavaillès seeks to isolate, in reality encompasses the constitutive forms of mathematical objectivity.
9Hourya Benis Sinaceur has done remarkable work in underlining and illuminating Cavaillès’s thesis of an autonomous dialectic of mathematical concepts, and its Hegelian inspiration. [9] However, Cavaillès does not maintain that mathematics is in some sense is its own philosophy, or that the only task for the philosopher of mathematics is to explain the internal philosophy of mathematics. In fact, if mathematical concepts are constantly enriched from within, it is because they presuppose a whole sphere that no mathematical concept itself can capture—the sphere of pure thought. What is proper to mathematics is to nurture and even instigate the sphere of pure thought—that regime in which thought gives its objects to itself, and through its very operations constitutes a field of objectivity in its own right. However, mathematics, despite the autonomy of its concepts, still has an irreducible need for a general theory of “rational or dialectical sequences” that only philosophical logic can provide. And indeed, we find that, under the auspices of that general theory, the thread of philosophical logic runs through the whole of Cavaillès’s work.
10Nevertheless, it is never a matter of a logic that would place mathematical discursivity and conceptuality within the ranks of the fundamental forms of language and thought. Cavaillès’s conviction can be summed up briefly as follows: if in mathematics the in concreto construction of the concept in intuition is “apprehension in the proof of the act of the very conditions that make it possible,” and if “the synthesis is coextensive with the production of what is synthesized,” [10] it is because the experience of mathematical activity differs from experience in the ordinary sense—just as the autonomous dialectic of mathematical concepts distinguishes them from any other type of concept. The emphasis here is placed upon the specificity of mathematics, [11] and in particular upon the specificity of mathematical experience, [12] without addressing the question of the place of mathematical language within language as a whole. On the contrary, this question is rejected. [13] Essentially, every mathematical theory is its own grammar, which means that there is no point in looking for links between mathematical concepts and the most general grammatical forms.
A Wittgensteinian before Wittgenstein
11The obscured presence of philosophical logic in Cavaillès’s work, along with his (to say the least) limited engagement with Frege’s and Russell’s logicism, have had an additional consequence: the absence of any comparative reading of Cavaillès not only with the Wittgenstein of the Philosophical Grammar and the Remarks on the Foundations of Mathematics, which Cavaillès was not able to read, but also with the Wittgenstein that Cavaillès had read, he of the Tractatus LogicoPhilosophicus.
12Analogies with the mathematical philosophy of Wittgenstein’s later period are not lacking. [14] In fact, they are too numerous to be regarded as indicating only a superficial resemblance. In what follows, two sections of the Philosophical Grammar will interest us in particular, [15] and with them, two examples of mathematical recurrence to which Wittgenstein refers many times. The first concerns the periodicity of the division of 1 by 3, as expressed by the notation
131 / 3 = 0.3
141.
15By highlighting the two instances of the number “1” here, Wittgenstein’s notation suggests what is to be understood, namely that the remainder is the same as the dividend, and that therefore the division will continue in the same way indefinitely.
16The second example given by Wittgenstein is the recursive proof of the associativity of addition on integers, a proof based on the following complex of equations, which Wittgenstein notates as follows (B):
17a + (b + 1) = (a + b) + 1
18a + (b + (c + 1)) = a + ((b + c) + 1) = (a + (b + c)) + 1
19(a + b) + (c + 1) = ((a + b) + c) + 1.
20Here it is to be understood that the respective first terms of the second and third lines, a + (b + (c + 1)) and (a + b) + (c + 1), are identical since their last terms, (a + (b + c)) + 1 and ((a + b) + c) + 1, are identical, which is precisely the claim of the recurrence hypothesis. But then to emphasize this point Wittgenstein adds “connection lines” to this complex, for example between the (c + 1) of a + (b + (c + 1)) and the c of (a + (b + c)) + 1, on the one hand, and between the (c + 1) of a + (b + (c + 1)) and the c of ((a + b) + c) + 1 on the other. This addition yields a new schema (R), which shows the recurrence rule as such, that is, the generality that was to be discovered. This generality is found only in the passage from (B) to (R). On the contrary, in Wittgenstein’s view, the assertion of the conclusion of recurrence in the form “a + (b + n) = (a + b) + n is valid for all n” is philosophically misleading, because it makes the reality of the proof disappear beneath an apparently magical calling forth of the totality of all integers.
21The proof of the periodicity of division and the proof of the associativity of addition both proceed via the displaying of a particular aspect of the symbolic complex in question, which exhibits the internal relation upon which the proof bears. Proof, according to Wittgenstein, consists of nothing other than the paradigmatization that displays the relevant aspect by means of correlations. [16]
22Cavaillès insists upon the unpredictability of the becoming of mathematics, [17] tracing this to the “unpredictability of the synthesis” [18] of the sensible and intellectual parts that are mixed together in the sign, and specifies that this unpredictability of the synthesis “is the definition of its existence.” However, just as the sign, although governed by an intellectual rule, “is a condition of creation owing to its capacity to move within the sensible,” [19] so Wittgenstein identifies the discovery of the periodicity of the division 1/3 with the recognition of a certain internal relation of a logical nature, as exemplified by a certain schematic configuration of signs. This recognition brings about a new theory, a new framework of symbolic calculus: “I should like to say: the proof shews me a new connexion, and hence it also gives me a new concept.” [20] Moreover, the aspect that is recognized is all the more unpredictable because “We don’t see that something can be looked at in a certain way until it is so looked at. We don’t see that an aspect is possible until it is there.” [21] What better expression could there be of the unpredictability of the novel aspect than Cavaillès’s formula, that it “is the definition of its existence”? Here we find the principle of the two thinkers’ shared advocacy of a form of a priori synthesis, [22] in spite of their common opposition to Kant.
23Cavaillès presents the becoming of mathematics as something that is necessary without being necessitated, insofar as the reasons for the superceding of a given theory are to be found within it, but cannot be derived from a deduction within that theory. Similarly, Wittgenstein remarks: “We might have gone on dividing without ever becoming aware of recurring decimals [as in 1/3]. When we have seen them, we have seen something new,”—that is to say “constructed a new calculus.” [23] According to Cavaillès, the becoming of a mathematical theory is articulated according to “dialectical moments” that are irreducible to the mathematical procedures internal to that theory. These moments correspond to different types of reorganization that produce a movement from one theory to a new theory. They are intractable to the kind of systematic analysis that logic might undertake. The driving force of the process, in fact, “seems to escape all investigation: it is here that the full meaning of experience, the dialogue between conscious activity as a power to undertake experiments subject to conditions, and these conditions themselves.” [24] Similarly, for Wittgenstein the passage from one calculus to another constitutes a leap that is irreducible to any logical derivation: “When I said that the new sign with the marks of emphasis must have been derived from the old one without the marks, that was meaningless, because of course I can consider the sign with the marks without regard to its origin.” [25] The emergence of a novelty in mathematics is fundamentally the product of an experiment with signs. While this is explicit in Cavaillès, Wittgenstein compares calculus to an experiment understood as the fixing of a paradigm, [26] and emphasizes that a new sign arises each time from a certain way of drawing attention to an aspect schematized by a coordination between paradigms: “It expresses a new way of looking at the calculation and therefore a way of looking at a new calculation.” [27]
24Such a description of the dynamic of mathematics is based primarily on a clear distinction between inert signs and properly mathematical symbols. A symbol is a sign that cannot be separated from the way it is employed—that is, from its actual use, from the work that gives it its full meaning. This distinction between sign and symbol, between a sensible mark and the expression of a grammatical connection, is a constant in Wittgenstein’s analysis from the time of the Tractatus. [28] For his part, Cavaillès also emphasizes the extent to which the symbols manipulated by the mathematician are always already caught up in mathematical activity. According to Cavaillès, the error of logicism consists in trying to grasp the sign in abstraction from the operation that gives it its meaning. [29] Wittgenstein makes what may be regarded as a symmetrical critique of logicism: he charges it not with confusing a sign with a symbol, believing that the former by itself can constitute the latter, but with confusing a symbol with a sign, by wanting to incorporate into a sign the way in which it should be understood. [30]
25The proximity of the two authors does not end there. Indeed, for Cavaillès, any mathematical calculus deploys its space and sets out its dimensions according to “certain degrees of freedom” in the combination of symbols, just as Wittgenstein compares the passage to a new calculus to the emergence of a new dimension and more generally to the institution of a new space, thereby entirely redefining the meaning of symbols. [31] Just as, for Cavaillès, mathematical understanding consists in “catching the gesture,” for Wittgenstein it consists in the grasping of a certain aspect, which is each time a new aspect. [32] In the same way that, according to Cavaillès, no overarching point of view or theory can grasp the whole of mathematics, so Wittgenstein insists on the “motley of techniques of proof” [33] which mathematics resolves into. And finally, just as Cavaillès is sensitive to the fact that every calculation is situated and that the enunciation of a mathematical situation is itself a mathematical situation, Wittgenstein also recognizes that mathematical activity is rooted in a situation, and compares the search for the solution to a mathematical problem to a state of disorientation. [34]
26The common focus of these numerous elements of similarity between the two authors is the notion of form. Cavaillès attributes to logicism and formalism “the illusion of an irreducible formality.” [35] There is no sense—that is, no form—without an act; that is to say that no form—still less an ultimate form—is there, already present, just waiting to be discovered: it is at best only the retrospective form of a determinate sequence of acts. The form lies in the very gesture that brings it into view. Similarly, Wittgenstein insists on the fact that there is no form—that is, no formal property or internal relation—except in the features of the symbols that bring it out. Furthermore, a form is always the form of a transition: it is that which paradigmatizes the supplementary addition that changes the way of seeing things by making a certain schema appear, itself the paradigm of a certain situation of calculation, as the paradigm of a new situation of calculation, and consequently—since the transition has modified the situation without itself constituting an additional step in the calculation—as the paradigm of a situation relative to a new calculation. The form lies in the way in which attention is drawn to a new way of seeing. We find here the same proposition as the “mathematical moves” that Cavaillès talks about in relation to the paradigm, “where the connecting act, as soon as it is realized, becomes the connecting type.” [36]
27However, if for Cavaillès form is a gesture, the feature of a certain act operating on symbols, for Wittgenstein it is the feature of certain symbols operated by an act. As Wittgenstein says in the Tractatus, “the expression for a formal property is a feature of certain symbols” (4.126). What he calls “a feature of certain symbols” is what he will later call an “aspect,” as exemplified by a certain accentuation of a symbolic paradigm. A paradigm is thus the expression of a form, but this common formula masks a gulf between Cavaillès and Wittgenstein: for Cavaillès, a paradigm is the actualization of the meaning of a sequence, while for Wittgenstein it is the formalization of a symbolic reconfiguration. For Cavaillès, the actual experience of the possibility of an act; for Wittgenstein, the bringing to light of the possibility of a structure. We know, moreover, that it is in these terms—the possibility of the structure of an image—that Wittgenstein, in the Tractatus (at 2.15), characterizes the “representational form” of this image. A form can only ever be shown, within the internal property of certain symbols. And again, it is only a variation of forms or an identity of forms that can be shown. And indeed Wittgenstein (at 5.24–5.241) characterizes an operation as “the mark […] of a difference between [propositional] forms.”
28The difference in perspective between the two authors can be further clarified as follows: while Cavaillès understands form to mean the actualization by which an act assures its own sense and generality, he does not, however, specify the symbolic structure that underlies such an experience. And in fact, the passage in LTS devoted to the moments of paradigm and thematization does not specifically mention the medium of a symbolic calculation. Granger thus shows, taking the example of conics, that the operation that emerges from a paradigm is not necessarily a strictly symbolic operation. [37] The example used in LTS is that of addition: the form of addition is produced by paradigmatization—that is, it is considered independently of the nature of the quantities to which it is applied, and then thematized according to the possible formal properties of commutativity or associativity. In Cavaillès, a paradigm extracts a form that is the form of an operation. On the contrary, in Wittgenstein, a paradigm schematizes an operation, and thereby a connection between forms: the operation does not have a form but reveals a form, while showing that the latter can never be extracted as such. What remains is to clarify the origin of this discrepancy between the two authors.
Logicism regained
29The passages of the Philosophical Grammar with which we have been able to compare many of Cavaillès’s remarks naturally lead back to their source: the discussion of the notion of form in the Tractatus. But this source itself refers back to a primary source: the analysis of “denoting complexes” conducted by Russell in the wake of the Principles of Mathematics.
30Russell is surprisingly neglected by Cavaillès, whose critique of Russellian logicism focuses on the twofold postulate of extralogical axioms (in particular the axiom of reducibility) and of logical notions that the signs used in symbolic logic are supposed to express transparently. The absence of any reference to the Russell of the Principles of Mathematics in Cavaillès’s texts is nevertheless somewhat strange. For with his theory of the sign (developed within the framework of his “modified formalism”) and in his discussion of Husserl, Cavaillès, like Russell, attempted to deepen what could be called the articulation of philosophical logic and mathematical logic. Above all, the relative neglect of Russell is all the more strange given that, from the time of the Principles of Mathematics, Russell continually sought to identify the logical concepts that articulate what Cavaillès calls “the moment of the variable”: the notions of propositional function, formal implication, denoting complex, etc. Like Cavaillès, Russell carried out an analysis of mathematical generality—that is, of mathematical processes of generalization. And, for Russell as for Cavaillès, mathematical generality, inseparable from its symbolization, essentially constitutes a formalization. This formalization, in Cavaillès’s case—and also, as we shall see, in Russell’s—is supported by an idea of a combinatorial nature, the omnipresence of which is attested by its mention in all the key passages where Cavaillès tries to think the essence of mathematics:
The very essence of mathematics is a regulated play of symbols, in which these symbols are not an adjunct to memory, but define a kind of abstract space with as many dimensions as there are degrees of freedom in the concrete and unpredictable operation of the combination. […] If abstract thought implies necessity, if mathematical becoming is the appearance of a true novelty, then the creation involved must be situated within that sensible realm that combinatorial space represents. [38]
32The role of the notion of “combinatorial space” in Cavaillès’s analysis cannot be overestimated. [39] Every mathematical construction, through projection of the possible manipulations upon which it is based, effectively deploys or redeploys a “combinatorial space” in which it takes place. This space is not a real space, any more than the manipulated signs resemble simple empirical marks. It is the field of operation “in which the application of the rules of a system constitutes the signs into ideal units and inserts them into an abstract structure.” [40] Nowhere is the theme of substitution more present than in the oftquoted passage where Cavaillès introduces the concept of paradigm:
And so the synthesis is coextensive with the production of what is synthesized. The later developments will furnish examples of this. What is important here is the disengagement brought about with each suppression of singularity. This is what is represented in logical calculus by the rule of substitution, i.e., the possibility of replacing in the new element that from which it actually proceeds with something else equivalent to it from the new point of view. [41]
34To which he adds:
This is the moment of the variable. By replacing the determinations of acts by the place vacated for a substitution, we progressively raise ourselves to a degree of abstraction which gives the illusion of an irreducible formality. This is what Leibniz tried in passing to the absolute through the mirage of an infinite whose simplicity makes conditions and conditioned simultaneous. […] Here the spatial image of juxtaposition, the utilization of an elementary combinatory, which preserves the simple characters of the finite as it reveals itself to itself (resigned to justifying every hiatus in an opaque way) within the infinite, are at once tied to the origin of the enterprise and a cause of its failure. [42]
36This combinatorics, which Cavaillès raises into a veritable metaparadigm of mathematical experience [43] is not to be confused with the combinatorial system suggested by a narrow interpretation of the Hilbert program, [44] nor with permutation theory—as if one were trying to explain mathematics by means of a particular mathematical theory. Cavaillès’s combinatorics has essentially to do with the elementary idea of substitution within a symbolic complex. Indeed, what is “the moment of the variable” if not the experience that, when replacing a with b, one could replace b in turn with something else, and consequently that one can replace a by any term x? In this way the possibility of an operation becomes an object of consideration in its own right. Thematization is therefore already at work within any paradigm. Conversely, thematization basically amounts, in accordance with the axiomatic method, to making the sign of the operation itself a variable: thematization is in this sense a form of paradigmatization. And indeed Cavaillès is well aware of the interplay between the two procedures. [45]
37In any case, a form, for Cavaillès, is always the form of a symbolic complex. In his approach to the two procedures of paradigmatization and thematization, as we have said above, he considers principally the example of the operation of addition and its formal properties. Starting from the complex 2 + 3, we can extract the abstract form of the addition operation, conceived independently of the terms to which it applies; then we can consider its formal properties, such as commutativity. These two steps do not take us away from the consideration of complexes: the paradigm of addition is in a sense nothing other than the formal complex x + y (a complex containing symbols of variables), and the thematic commutativity of addition is nothing other than a relation of equality between two schematic complexes, x O y = y O x, where “O” designates any operation.
38Now, the Tractatus (2.15) also deals with symbolic complexes. And these complexes come into Wittgenstein’s work via Russell. Indeed, in Russell’s manuscripts of 1904–1905, we find a long reflection on complexes, which Russell calls “denoting complexes,” and their forms, as fundamental vectors of generality, and especially of mathematical generality. Let us present the main lines of this reflection, which extends over hundreds of pages and several distinct manuscripts. Russell’s guiding idea here [46] is that a complex, insofar as it is endowed with an internal structure, is unified according to a certain mode of combination, but that this internal structure is not a constituent of the complex of which it is the mode of combination. For example, the mode of combination of the propositional complex A is greater than B can be designated by “x̂R̂ŷ.” The constituents of the complex are A, the relation greater than, and B. A mode of combination is also an entity in its own right, and can be a constituent in other complexes. For Cavaillès, this is illustrated by the transversal abstraction of thematization—the aim of an operation transformed into an object, “a gesture on a gesture” [47]—which requires that a form can in turn be considered as an entity. Nevertheless, Φ‘x̂, which in Russell’s text refers to the mode of combination of the complex Φ‘x, is not a constituent of Φ‘x; therefore Φ‘x is not a function of Φ. In particular, x ∊ u is not a function of u, which prohibits the introduction of the complex x ∊ x. Russell believes that here he has found a solution to his paradox. [48]
39The consequence of this guiding idea is that it is necessary to distinguish two types of variations and consequently two types of variables, which Russell calls “entityvariables” and “modeofcombinationvariables.” [49] Thus Russell notes as “Φ‘x̂” the variable way in which an entity can appear in a complex: the values of this variable are all “the particular ways in which an entity is combined with other entities” to form a complex. Any complex that has a particular way of combining f‘x̂ will be noted as “F‘(f‘x̂).” One can then introduce any complex , which Russell calls a “function of functions,” to designate any combination of a mode of combination (that is, a “function”) with other constituents (entities or functions). We can see how a theory of complex types can be constructed in this way. But we can also see the difficulty Russell faces: the mode of combination of a complex is not accessible outside this complex. Russell therefore ends up using both “Φ‘ẑ” and “ẑ(Φ‘z)” to denote [50] a variable of the mode of combination of a complex, and “Φ‘x” to denote any complex containing a certain entity x. The ambiguity is subtle: in the first case, the symbol “Φ” is a variable; in the second, it is a schematic letter. Mixing up the two roles, Russell equates the mode of combination Φ‘x̂ of a complex Φ‘x, that is, its form, with the associated function x ↦ Φ‘x, which amounts to thinking of instantiation as a special case of substitution. Russell thus considers positing: . And, indeed, how can we take aim at the form of a complex as such, and quantify over it, without going through this identification? But it is an impossible identification. Because to speak of the value of ẑ(Φ‘z) for the argument x is to make the targeted mode of combination the constituent of a new complex Co(x, ẑ(Φ‘z)). The two complexes Φ‘x and Co(x, ẑ(Φ‘z)) = Φ‘z[x/z] have the same denotation, but do not have the same meaning, since they do not share the same internal constitution. Russell takes note of this: Φ‘x ≠ {ẑ(Φ‘z)}‘x. He comments on this matter in another manuscript: “We are here in the region of the fundamental nature of complexity.” [51]
40A last problem, a correlative of the difficulties just indicated, is highlighted by the original notation that Russell proposes for complexes: the notation “(C≬x̂),” which combines the respective parentheses of the C kernel and of the x̂ argument of a complex, in order to indicate the inseparability of these two components within the complex. The problem, notes Russell, is that in , (C≬x̂) appears as a meaning and therefore cannot vary. In fact, a simple letter such as “C” cannot be used to designate a term which, being in “meaningposition,” has a structure that is essential to it. [52] Russell summarizes the diagnosis of all the problems encountered as follows:
We avoid contradictions by writing (C≬x̂) for a mode of combination, and refusing to vary C. In this case, (C≬x̂) is immediately recognized as an instance, and is not to be regarded as what is denoted by . I believe this is right.
If “the xinstance of (C≬x̂)” is legitimate, it still only denotes (C≬x̂) and does not mean it. In use, when we assert (C≬Soc) [for example, Socrates is mortal], we don’t go through the process . […] in , if legitimate, (C≬x̂) is in a meaningposition. [53]
42The solution will be to abandon the focus on the modes of combination themselves, and to try to identify only types of complexes; to clearly distinguish between substitution on the one hand, and the assignment of a particular value to a variable symbol on the other; to do without any “meaningvariation”; and, in fact, to do without the very notion of meaning, that is, the notion of denoting concept: this is what will happen in “On Denoting.” The first steps in this reform are set out in “On Fundamentals”:
We may use (C≬x) for a general complex containing x. Here the C is inseparable, and is nothing by itself; for it is part of the meaning of (C≬x), and is therefore not necessarily one entity at all. But we can form types of complexes, by keeping an invariable structure, and merely permitting one constituent to vary. The type of which (C≬x) is an instance will be denoted by (C≬x̂). Here (C≬x̂) does not stand for an instance, but for the general type of which (C≬x) is an instance. The recognition of (C≬x) as an instance of (C≬x̂), both in general and for particular types, is one of the prerequisites of all reasoning from the general to the particular. We cannot further symbolize this […]. [54]
44A mode of combination is nothing by itself: it cannot be symbolically objectified. This conclusion is the ancestor of the remarks in the Tractatus concerning the form of representation. The clearest solution to which Russell’s reflections led him can be found in the substitution theory he elaborated at the end of 1905, the main published form of which is “The Paradoxes of Logic.” [55] In this theory, the substitution operation is adopted as a primitive: the replacement, by another entity, of a constituent appearing as an entity in a proposition that is itself taken as a complex entity. For a proposition p and a constituent a of p, the expression “p/a;b!q” means that q results from the replacement of a by b in p. The expression “p/a;b” is then only a definite description: “the result of the replacement of a by b in p.” An additional step of abstraction, and the symbol “p/a,” which Russell calls a “matrix,” represents the attribute whose extension is the class of all x such that p/a;x is a true statement. The definition of x∊p/a is indeed: (Ǝq)((p/a;x!q)&q). Similarly, u/x ∊ q/(p,a) is a shortened form of (Ǝr)((q/p,a;u,x!r)&r). The p/a matrix represents, as it were, the schematic form of the classes; p/a,b, the schematic form of the binary relations, etc. The theory leaves no room for predicate or relation symbols, since the only simple terms are schematic letters of entities (“p,” “a”) and of entity variables (“x”). However, the use of schematic matrices such as p/a, together with quantification over entities, makes it possible, even in the absence of any predicate variable, to express the equivalent of a quantification over firstorder predicates. Similarly, quantification over secondorder predicates is reconstructed using q/p,a matrices, and so on. In this way, Russell reconstructs a hierarchy of types integrated into substitutional grammar—that is, expressed by the syntactic constraints—of number and places of arguments—imposed on the possible substitutions within a matrix. [56]
45Let us now return to Cavaillès. When discussing the concept of paradigm, commentators often speak of a structure that it would be responsible for “extracting” as such. Granger writes: “It is, of course, about the process of the separation of a form.” And then, speaking more specifically of the concept of paradigm: “it lays the operation bare, so to speak, by emptying it of its variable contents.“ [57] Similarly, Benis Sinaceur sums it up as follows: “The paradigm extracts the form of the operation that is invariant across the multiplicity of its applications […].Thematization arises when we consider the form itself as a variable […]. [58] CassouNoguès’s commentary is along similar lines: “Throughout the process, thought maintains the same orientation, continually facing toward the level of objects, where complexes are formed from which it extracts the structure. […] Paradigmatization is oriented toward the level of objects, from which it extracts structures.” [59]
46But let us speak of complexes, precisely. As we have seen, Cavaillès’s example lends itself to a presentation in terms of complexes. However, is it really possible to “separate” or “extract” the form or structure of the complex that schematizes an operation? “[T]he signification of an operation as ‘operated’,” [60] in order to be a signification, must correspond to a complex as a “meaning,” not as an “entity,” to use Russell’s terms. The form of that complex cannot, therefore, be “extracted.” Cavaillès here leaves unexploited the rich logicist reflection on the symbolization of form. But isn’t a theory of operations and sequences devoid of any logical theory of their symbolization condemned to remain on the side of a philosophy of the act, of experience, of consciousness? If the dialectic specific to mathematics is inseparable from the fact that mathematics is always written, then what is a “philosophy of the concept” without an analysis of the logical forms that articulate the writing of mathematics?
47In this respect, the first lesson to be drawn from logicism is that a form can only be symbolically exhibited, not grasped for itself. Cavaillès would not disagree, but then what status will he give to the form that a paradigm “actualizes”? Rather than capturing forms, one can schematize logical types expressed by means of substitution matrices: this is what Russell does, and in so doing he moves from the impasse of a variation of forms to a theory of forms of variation. And what is a “logical prototype,” as mentioned in the Tractatus at 5.522, if not a schematic complex—that is, a complex with one or more variables to indicate possible substitutions? Forms are not just “immanent,” in some vague sense, to mathematical sequences. The formal can and must lend itself to a theory, even if it cannot be a science of forms; and this theory is none other than philosophical logic as conceived in the tradition of logicism.
48The second lesson of logicism is that one cannot invoke forms without knowing how the variables that “give prominence” [61] to them vary, how they are assigned their possible values. The question is no longer the emergence of a form of variation, “but the quomodo of it,” to paraphrase Cavaillès. [62] Form is a meaningless word if the generality it covers is left unexplained, if the determination of the values of a variable, that is to say both their specification and their production, is deferred by being referred to a quasiexperimental process. Russell’s answer to the question of such determination is that the variables vary without restriction: it is precisely the task of logic to ensure that ultimately the range of variation of each variable covers absolutely all entities, whatever they may be. Wittgenstein’s answer in the Tractatus (4.1252) is that in each case it is a “series of forms” that determines the values of the variable as being all the propositions having the formal property exhibited by the series. (In this respect, Wittgenstein’s fundamental criticism of Russell is that he assumed the operation of substitution—that is, the highlighting of an argument—to be available as if it did not already presuppose a certain form of generality.) [63] In both cases, the determination of formal generality is indeed a matter of philosophical logic as inspired by logicism. And one can thus reinterpret in a logicist perspective what Cavaillès calls the “positing sense” of an operation: [64] if the posited sense of an operation is indeed, as Cavaillès says, the moment of the introduction of a variable, the paradigm as a complex that exhibits a certain mode of combination, then one can also—diverging from Cavaillès—understand its positing sense as being the moment of the generation of the possible values of this variable.
49Cavaillès’s examination of the essence of mathematical thinking emphasizes the concept of form: it places particular importance on showing “how a systematic study of forms is indispensable.” [65] But let us take Cavaillès at his word: what is a form? What is a paradigm? And what kind of theory can really be expected to answer such questions?
50Of course, it is clear what Cavaillès means by “forms”: they are both the inference forms characteristic of the “rational sequences” specific to mathematics, and the forms that crystallize the formalizations that make it possible to extract mathematical “structures,” in a sense already mobilized in set theory. However, if the philosopher does not undertake to analyze the concept of form for itself, they run the risk of leaving it either to mathematics or to a philosophy of consciousness that is precisely what Cavaillès is trying to escape. The theory of science capable of carrying out this work of analysis is Cavaillès’s ἐπιστήμη ζητοῦμένη, his “soughtafter science” in Aristotle’s words: it is to be found neither in Kant nor in Bolzano; neither in proof theory nor in Husserlian mathesis, since both of these (in the latter case because it is based on a formal apophantics as much as on a formal ontology) already assume the availability of the notion of form, and prove incapable of accessing the structures proper to mathematics.
51Our thesis is quite simply that Cavaillès could quite well have drawn nearer to this soughtafter science—logic as the locus of analysis of the concept of form and mathematical forms of generality—by way of the philosophical logic of the logicists. In this respect, the lesson of Russell’s writings from 1905 is that there is no theory of forms outside of a substitution theory: forms are logical types; while the lesson of the Tractatus is that there is no formal generality without the elaboration of a series of forms, and that a form is always the form of a transition between forms.
52Beyond the absent reference to logicism, what is also at issue here is, as mentioned at the outset, the need for the philosopher to take up the correct distance from mathematics. In this respect, Cavaillès robbed himself of a certain type of relationship between philosophy and mathematics by giving up the anchoring offered by philosophical logic, and thus depriving himself of an escape route from the dilemma of a philosophy of mathematics condemned to be either external to mathematics—in other words, in the role of a spectator—or “immanent” to it—in other words, annexed by it.
Notes

[1]
Jean Cavaillès, Méthode axiomatique et formalisme (Paris: Hermann, 1981), 88–89. Henceforth referred to as MAF.

[2]
Jean Cavaillès, ‘On Logic and Theory of Science’, trans. Theodore Kisiel, in Phenomenology and the Natural Sciences, ed. Joseph J. Kockelmans and Theodore Kisiel (Evanston: Northwestern University Press, 1970 [1947]), 347–409: 379. Henceforth referred to as LTS.

[3]
Sébastien Gandon, “Quelle philosophie pour quelle mathématique?,” Archives de philosophie 76, no. 2 (2013).

[4]
MAF, 92. See also LTS, 376.

[5]
MAF, 93. On this point, see Pierre CassouNoguès, De l’expérience mathématique. Essai sur la philosophie des sciences de Jean Cavaillès (Paris: Vrin, 2001), 140.

[6]
MAF, 99.

[7]
GillesGaston Granger, Pour la connaissance philosophique (Paris: Odile Jacob, 1988), 70.

[8]
LTS, 375 (translation modified).

[9]
Hourya Benis Sinaceur, Cavaillès (Paris: Les Belles Lettres, 2013), 153–178.

[10]
See MAF (27) and LTS (376) respectively.

[11]
“Mathematical activity is an object of analysis and has an essence: but like a smell or a sound, it is itself” (“Mathématique et formalisme,” in Jean Cavaillès, Œuvres complètes de philosophie des sciences (Paris: Hermann, 1994), 664). This statement of Cavailles’s about mathematical activity is strikingly reminiscent of Russell’s statement about (and here lies the crucial difference) logical constants, at the beginning of the preface to the Principles of Mathematics: “The discussion of indefinables—which forms the chief part of philosophical logic—is the endeavour to see clearly, and to make others see clearly, the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of pineapple.” Bertrand Russell, Principles of Mathematics (London: Routledge Classics, 2009), xliii.

[12]
On this point, see Benis Sinaceur, Cavaillès, 82–83 and 152.

[13]
MAF, 34.

[14]
The literature on Cavaillès, however, makes little mention of these analogies. A notable exception is the comparison, albeit heavily qualified, that Granger proposes between Cavaillès’s idea that an object is its mode of construction, and the Wittgensteinian idea that the meaning of a theorem is in its proof (see Granger, Pour la connaissance philosophique, 74).

[15]
Ludwig Wittgenstein, Philosophical Grammar, edited by Rush Rhees, translated by Anthony Kenny (Oxford: Blackwell, 1974), 430–445 (Part Two, §36–37).

[16]
On the role that Wittgenstein makes paradigms play in mathematics, and in particular on the fact that any paradigm is valid only through the correlation established between two paradigms, see JeanPhilippe Narboux, “L’exemplarité de la preuve mathématique selon Wittgenstein,” Revue de Métaphysique et de Morale 46, no. 2 (2005): 301. This point echoes Wittgenstein’s essential observation: “For it only makes sense to speak of understanding in cases where we understand one thing as opposed to another. And it is this contrast that signs express” (Wittgenstein, Philosophical Grammar, 445).

[17]
Cavaillès, Œuvres complètes, 601.

[18]
MAF, 178.

[19]
MAF, 94.

[20]
Ludwig Wittgenstein, Remarks on the Foundations of Mathematics, edited by G.H. von Wright, R. Rhees, and G.E.M. Anscombe, translated by G.E.M. Anscombe (Oxford: Blackwell, 1998), 297.

[21]
Wittgenstein, Philosophical Grammar, 444. On the Wittgensteinian idea of surprise in mathematics, insofar as it is linked to the discovery of a new aspect schematized by a proof, see Juliet Floyd, “On Being Surprised. Wittgenstein on AspectPerception, Logic, and Mathematics,” in Seeing Wittgenstein Anew: New Essays on Aspect Seeing, ed. William Day and Victor J. Krebs (New York and Cambridge: Cambridge University Press, 2014), 332–34.

[22]
On this point, two texts in particular should be considered: MAF (93) and Wittgenstein, Philosophical Grammar (410).

[23]
Wittgenstein, Philosophical Grammar, 441, 404; see MAF, 178, and Cavaillès, Œuvres complètes, 600–601.

[24]
MAF, 178.

[25]
Wittgenstein, Philosophical Grammar, 440.

[26]
Wittgenstein, The Blue and Brown Books: Preliminary Studies for the “Philosophical Investigations” (Oxford: Blackwell, 1958), II, §67–76.

[27]
Wittgenstein, Philosophical Grammar, 438.

[28]
MAF (174) and LTS (376); Ludwig Wittgenstein, Tractatus LogicoPhilosophicus, trans. C.K. Ogden (Mineola, NY: Dover, 1999), 3.326–3.328.

[29]
MAF (174) and LTS (382–383). See also “Réflexions sur le fondement des mathématiques,” in Cavaillès, Œuvres complètes, 577.

[30]
Wittgenstein, Tractatus LogicoPhilosophicus, 3.33–3.331.

[31]
MAF, 93; Wittgenstein, The Blue and Brown Books, I, §44.

[32]
MAF, 178; Wittgenstein, Philosophical Grammar, 438; Wittgenstein, The Blue and Brown Books, III, §46; IV, §30; IV, §33.

[33]
Wittgenstein, The Blue and Brown Books, III, §46.

[34]
MAF (174) and Cavaillès, Œuvres complètes (602); Wittgenstein, The Blue and Brown Books, III, §§80–81; IV, §31.

[35]
LTS, 377.

[36]
LTS, 375.

[37]
Granger, Pour la connaissance philosophique, 71–72 and 76–77. The same applies to Cavaillès’s references to Lebesgue’s exponential function and integration.

[38]
MAF, 93.

[39]
As Pierre CassouNoguès rightly insisted (CassouNoguès, De l’expérience mathématique, 174–180).

[40]
CassouNoguès, De l’expérience mathématique, 174.

[41]
LTS, 376. This text directly echoes MAF, 102.

[42]
LTS, 376–377.

[43]
See Hourya Benis Sinaceur, “Formes et concepts,” in La connaissance philosophique. Essais sur l’œuvre de G.G. Granger (Paris: Presses Universitaires de France, 1995), 102 and 107–108.

[44]
See Cavaillès, Œuvres complètes, 664.

[45]
See LTS, 378.

[46]
Bertrand Russell, “On Functions,” in Collected Papers (London: Routledge, 36 vols., 1983–), vol. 4, 98; Bertrand Russell, “Fundamental Notions,” in Ibid., 138 and 154.

[47]
MAF, 178.

[48]
See Russell, “On Functions,” 96 (Editors’ Introduction); see also Russell, “Fundamental Notions,” 154.

[49]
Russell, “On Functions,” 98–99; Bertrand Russell, “On Fundamentals,” in Collected Papers, vol. 5, 360.

[50]
See, for example, Russell, “Fundamental Notions,” 114, 123, and 254.

[51]
Bertrand Russell, “On the Nature of Functions,” in Collected Papers, 265.

[52]
One of Russell’s examples is the complex “the present Prime Minister of England” (“x” corresponding to the constituent “England”), a complex which expresses a meaning by virtue of the unified combination of the words that make it up.

[53]
Russell, “On Fundamentals,” 360–361.

[54]
Ibid., 366.

[55]
Bertrand Russell, “The Paradoxes of Logic,” in Collected Papers, vol. 5, 278–296.

[56]
On this point, see Gregory Landini, Russell’s Hidden Substitutional Theory (New York: Oxford University Press, 1998), 140–145.

[57]
Granger, Pour la connaissance philosophique, 70 and 71.

[58]
Benis Sinaceur, “Formes et concepts,” 108.

[59]
CassouNoguès, De l’expérience mathématique, 272.

[60]
LTS, 378.

[61]
To borrow an expression from the Tractatus LogicoPhilosophicus, 5.522.

[62]
LTS, 378.

[63]
While Wittgenstein acknowledges, at 5.501, that other types of generality exist (other ways of determining a class of propositions), the generality founded on the underlying iterability and exhibited by a series of forms constitutes the main type of generality. On this point, as on Wittgenstein’s critique of Russell, see Jérôme Sackur, Formes et faits. Analyse et théorie de la connaissance dans l’atomisme logique (Paris: Vrin, 2005) (208–213) and Brice Halimi, “The Concept of ‘Essential’ General Validity in Wittgenstein’s Tractatus,” in Early Analytic Philosophy. New Perspectives on the Tradition, vol. 80 of Western Ontario Series in Philosophy of Science, ed. Sorin Costreie (Cham: Springer, 2016) (288–290 and 297–298).

[64]
LTS, 378.

[65]
LTS, 379.