Jean Cavaillès: On the Logic and Theory of Science

Massimiliano Simons

On Logic and the Theory of Science Book Cover On Logic and the Theory of Science
Jean Cavaillès. Introductory notice by Georges Canguilhem and Charles Ehresmann. Introduction by Knox Peden. Translated by Translated by Knox Peden and Robin Mackay
Urbanomic/Sequence Press
Paperback $18.95

Reviewed by: Massimiliano Simons (Ghent University)

Why read an obscure, enigmatic and technical treatise on philosophy of science by a half-forgotten philosopher of mathematics, let alone translate it? It seems that there are plenty of reasons to do so. The work of Jean Cavaillès (1930-1944), and especially his final work Sur la logique et la théorie de la science, is typically seen as the starting point of a ‘philosophy of the concept’. Together with Gaston Bachelard, Georges Canguilhem and Michel Foucault, Cavaillès is labeled as one of the main representatives of the French tradition of historical epistemology.

The project of a philosophy of the concept mainly gained fame due to Foucault, who popularized it in his foreword to the English translation of Canguilhem’s The Normal and The Pathological in 1978. Here he claims that to understand Canguilhem, one needs to fall back on the work of Cavaillès. Taking Husserl’s Paris 1929 lecture series as their starting point, phenomenology was developed in France in two radically different ways. On the one hand there is the well-known existential phenomenology, starting with Sartre’s ‘Transcendence de l’Ego’ (1935). But on the other hand there was Cavaillès, who put Husserl’s philosophy of science and mathematics at the center. According to Foucault, these two opposing phenomenological projects shaped 20th-century French philosophy:

It is the line that separates a philosophy of experience, of sense and of subject and a philosophy of knowledge, of rationality and of concept. On the one hand, one network is that of Sartre and Merleau-Ponty; and then another is that of Cavaillès, Bachelard and Canguilhem. In other words, we are dealing with two modalities according to which phenomenology was taken up in France, when quite late – around 1930 – it finally began to be, if not known, at least recognized. Contemporary philosophy in France began in those years. (Foucault 1978, 8)

The source of this ‘philosophy of the concept’ lies in the last few, rather enigmatic pages of Cavaillès ultimate work, here delivered in a new translation by Robin Mackay and Knox Peden for Urbanomic Press. The notion itself is in fact only introduced in the penultimate line of the book: “It is not a philosophy of consciousness but a philosophy of the concept that can yield a doctrine of science. The generative necessity is not that of an activity, but of a dialectic.” (p. 136) The book itself, which is both dense and short, is mainly a critique of the existing philosophies of science in early 20th-century philosophy with a focus on Kant, Carnap and Husserl. Only in the final pages Cavaillès makes some suggestions on what his alternative would be. For sure, Cavaillès intended to develop this alternative elsewhere, perhaps in the intended introduction he wanted to write. These plans did not materialize due to his early death.

Nevertheless, his promise of an alternative to the ‘philosophy of consciousness’ was taken up in French philosophy, in the form of a mythology around the figure of Cavaillès. When I say ‘myth’ I do not so much mean ‘wrong’ or ‘false’, but first of all that it functions as a myth. This means that the truth value of the myth is secondary to its effects – which have been plenty in the self-understanding of a number of French philosophers. The myth consists of three parts: (1) Cavaillès’ essay contains a set of devastating arguments against Husserlian phenomenology, making any phenomenological philosophy untenable; (2) Cavaillès replaced phenomenology with his own alternative, a philosophy of the concept; and (3) this alternative boils down to a form of Spinozism. Let us have a closer look at these three components of this mythology.

The first part is that Cavaillès’ work contains a number of knock-out arguments against Husserlian phenomenology. The final part of the book indeed deals with Husserl’s philosophy of science, as developed by the latter in Formale und transzendentale Logik (1929). Though Cavaillès formulates numerous criticisms, two main objections stand out: the issue of incompleteness and the dilemma of Cavaillès.

The incompleteness issue concerns Husserl believe that it is both possible and desirable to develop a fully-formalized mathematics in the line of David Hilbert’s formalism. That entails that it is capable to articulate and prove its own completeness: any true statement that can be articulated within the formal axiomatic system concerning its object of research can also be proven within that system. And though a common belief when Husserl’s book was published, a few years later it was radically shattered when in 1931 Kurt Gödel published his incompleteness theorems, showing that Hilbert’s program was impossible. Cavaillès was one of the first who saw the consequences of Gödel’s theorems for the Husserlian project, since it too relied on Hilbert’s program, endorsing its ideal of completeness as a central aspect of the phenomenological logic:

For we are familiar with Gödel’s result: any theory containing the arithmetic of whole numbers – which is to say basically any mathematical theory – is necessarily non-saturated. […] For the Husserlian conception of logic and mathematics this affair is particularly serious. In the first place, the very notion of a theory that can be fully dominated and isolated can no longer be maintained. (pp. 128-129)

A second issue consists in a fundamental dilemma that Cavaillès presents to Husserl. The latter aims not only for mathematical completeness, but also for a foundation of science that is both transcendental and absolute. Husserl had the ambition to ground all logical norms in the transcendental subject and its subjective acts. But simultaneously, Husserl hoped for an absoluteness of these norms: they cannot be questioned or relativized in any way. For Cavaillès the trouble arises if one combines both claims: if logical norms are constituted through subjective acts, these acts themselves call for a set of norms to ground them. But these underlying norms would in turn demand a foundation in subjective acts. To avoid an infinite regress, Husserl has to choose between the transcendental and the absolute character of his project. Hence the dilemma of Cavaillès: “If transcendental logic truly grounds logic then there is no absolute logic (i.e., a logic governing absolute subjective activity). If there is an absolute logic it can draw its authority only from itself; it is not transcendental.” (p. 120)

Similar to Husserl and Hilbert, and many other early 20th-century philosophers and scientists, Cavaillès was concerned with the new scientific developments, provoking a foundational crisis of mathematics: if radical historical transformations of mathematics were possible, what then was the foundation of rationality, mathematics, science? In a typical French manner, Cavaillès searched for the answer not in the atemporal, but in the historical: mathematics was rational not despite, but because of its history. The normative force, required to elevate mathematics above contingency, was to be found in its historical developments, where each new result bolstered the past results, but also provoked the next steps. In his primary thesis Méthode axiomatique et formalisme, Cavaillès would define understanding in a similarly dynamic way: “to understand is to capture the gesture, and to be able to continue it” (Cavaillès 1994, 186). Mathematics, and science in general, is understood, not so much as a set of thoughts, but rather in terms of gestures, which Cavaillès often referred to, somewhat misleadingly, under the banner of ‘experience’ (see Cortois 1996).

While terms such as gesture and experience suggests a philosophy of consciousness, this is countered through the third part of the mythology: Spinozism. Cavaillès was the face of a revival of interest in Spinozism, again seen as the alternative to phenomenology. In the words of Gilles-Gaston Granger, another proponent of this myth, it came down to “Jean Cavaillès or the climb to Spinoza” (Granger 1947). Cavaillès, sive Spinoza. The Spinozist reading it provided must have indeed sounded as music in the ears of the anti-humanist turn in the 1960s. Cavaillès, according to this reading, “set out to develop a philosophy without a subject” (Canguilhem 1994, 686).

We thus have a philosophy of science seen from the perspective of gestures. But these gestures are not grounded in a transcendental subject, but, following Spinoza, in the effective development of the successive theoretical concepts. The legitimacy of the concepts is found in these concepts themselves, in the normative force by which they call forth, by a form of necessity, one another. It is in this sense that another famous passage of the book is typically read:

Yet one of the essential problems for the doctrine of science is precisely that progress cannot be a mere increase in volume by juxtaposition, the prior subsisting with the new, but must be a perpetual revision of contents by way of deepening and erasure [rature]. What is after is more than what was before, not because it contains it or even because it extends it, but because it exists from it necessarily and bears within its content the singular mark, each time, of its superiority. There is more consciousness in it – and not the same consciousness as before. (pp. 135-136)

As a result, and especially from the 1960s onwards, Cavaillès’ work was used in French philosophy as an argument for the obsolescence of phenomenology. For instance, when Sartre attacked Foucault’s Les mots et les choses as a reactionary work, Canguilhem responded by invoking the authority of Cavaillès:

Cavaillès assigned the phenomenological enterprise its limits even before that enterprise had exhibited its unlimited ambitions – even in France itself, which is to say, with a certain lag – and he assigned, twenty years in advance, the task that philosophy is in the process of accepting today – the task of substituting for the primacy of experienced or reflexive consciousness the primacy of concepts, systems, or structures. (Canguilhem 1994, 92)

This mythological authority similarly found its way in to the work of many prominent French scholars, ranging from Jean-Toussaint Desanti, Gilles-Gaston Granger, Étienne Balibar to Alain Badiou. Despite the lack of any substantial elaboration by Cavaillès’ of what this philosophy of the concept entailed, his work was nevertheless used as a shovel to bury the philosophies of the past, most notably phenomenology and existentialism. Cavaillès’ alternative was indeed often quite easily equated with the structuralism and anti-humanism of the 1960s. These ideas not only inspired Canguilhem or Foucault, but were central to the Marxism of Louis Althusser as well, who mobilized Spinoza to purge Marxism from any form of Humanism. When Althusser, in his later self-criticisms, turned his gaze to his previous work, he could proudly proclaim that he was no structuralist: “We were guilty of an equally powerful and compromising passion: we were Spinozists.” (Althusser 1976, 132)

But Cavaillès’ biography gave this Spinozism an additional, political dimension. Similar to his friend Albert Lautman and Canguilhem, Cavaillès joined the resistance. While Canguilhem survived, both Cavaillès and Lautman tragically died. In the case of Cavaillès, he was first arrested in Narbonne in 1942, by the French police and ended up in a prison camp in Montpellier. It was during this imprisonment that he wrote this book, which partly explains the dense character of the text and the often liberal quotations, since Cavaillès was restricted in time and resources (though Lautman brought him several books). But with the manuscript in hand, he soon escaped from the prison camp though, due to betrayal, he was arrested again in August 1943 and eventually shot. The book On Logic and the Theory of Science was thus only published posthumously in 1947, by Georges Canguilhem and the mathematician Charles Ehresmann, who also picked the generic title for the book. In a letter of 1941 to Brunschvicg, Cavaillès announced that he was writing a work called L’expérience mathématique, while to Canguilhem and Ehresmann he spoke of a Traité de la logique. The truth seemed to be somewhere in between.

This political dimension of Cavaillès’ biography has been mobilized extensively in this mythology. Raymond Aron, who was a good friend of Cavaillès, later reported how Cavaillès entrusted to him in 1943 that “I am a Spinozist. I believe that we are seizing the necessary everywhere. The necessary in the sequences of mathematics, the necessary even in the stages of mathematical science, the necessary also in this struggle that we lead.” (quoted in Canguilhem 1976, 31) Or, in a variation that Aron cites in his own preface to Cavaillès’ Philosophie mathématique: “I am a Spinozist, we must resist, fight, face death. This is what truth and reason demand.” (Aron 1962, 14) Philosophers such as Canguilhem or Foucault did not hesitate to use this political activism against their opponents. In the same reply of Canguilhem to Sartre, he immediately added:

Shot by the Nazis for his Resistance activity, Cavaillès, who called himself a Spinozist and did not believe in history in the existential sense, refuted in advance – by the action he felt himself impelled to undertake, by his participation in the history that he lived out tragically until his death – the argument of those who seek to discredit what they call structuralism by condemning it to generate, among other misdeeds, passivity in the face of reality. (Canguilhem 1994, 92)

Foucault, in a similar vein, invokes this image in an interview in the 1980s, when he was focusing on Greek philosophy and techniques of the self:

The key to the personal poetic attitude of a philosopher is not to be sought in his ideas, as if it could be deduced from them, but rather in his philosophy-as-life, in his philosophical, life, his ethos. Among the French philosophers who participated in the Resistance during the war, one was Cavaillès, a historian of mathematics who was interested in the development of its internal structures. None of the philosophers of engagement – Sartre, Simone de Beauvoir, Merleau-Ponty – none of them did a thing. (Foucault 1984, 374)

It is thus this triple mythology – Cavaillès burying phenomenology, replacing it with a philosophy of the concept, inspired by Spinoza – that still motivates many philosophers to take up this otherwise challenging and technical treatise on the philosophy of science. The fact that this book is now translated in English (again) testifies to a similar growing interest in the Anglo-American world. There has indeed been a booming cottage industry of Anglophone papers dealing with Cavaillès’ philosophy of the concept, and its criticisms of Husserlian phenomenology (e.g. Hyder 2003; Thompson 2008; Peña-Guzmán 2020). Most of this literature tends to take over this mythology, though often stressing more continuity with phenomenology – an element already found in Foucault. The most exemplary case, perhaps, is an earlier book by Knox Peden, who helped realize this novel translation of Cavaillès and provided it with a helpful introduction. In his Spinoza Contra Phenomenology (2014) Peden repeated this mythology of how Cavaillès is the source of an anti-phenomenological tendency in France, continued by Bachelard, Desanti, Althusser and Gilles Deleuze.

Ironically, while this productive mythology is spreading in the Anglophone world, it is more and more problematized in France. The last decades of scholarship on Cavaillès have further substantiated a number of doubts concerning this triple narrative. First of all, the central role of Spinoza have been questioned, or at least completed with an acknowledgment that Cavaillès’ philosophy of the concept was inspired by other authors as well, such as Hegel (e.g. Sinaceur 2013).

The second main question is whether Cavaillès’ criticisms of Husserl are indeed as devastating as often portrayed. For instance, the young Jacques Derrida already tried to defend Husserl’s project against Cavaillès’ use of Gödel. In his master thesis on Le problème de la genèse dans la philosophie de Husserl (only published in 1990), he suggested that completeness is merely a kind of regulative ideal for Husserl, one that does not need to be realizable. Another option is offered by the daughter of Gaston, Suzanne Bachelard (1957) who suggests that there are multiple notions of completeness at work in Husserl’s philosophy, and that only some of them were refuted by Gödel. Hence, Husserl’s project can remain meaningful, even if one acknowledges the incompleteness theorems.

As part of the secondary literature has noticed, Cavaillès is at many points not that far off from phenomenology itself. It could thus be read, not so much as the destruction, but as a transformation of phenomenology. This is also announced in the lines immediately following Cavaillès’ dilemma: “Perhaps subsequent phenomenological investigations will allow us at least to contest such a brutally posited dilemma.” (p. 120) And to his credit, Peden seems to follow this suggestion in his introduction, playing down his earlier claims in Spinoza contra Phenomenology, and acknowledges that “it is both anachronistic and an overstatement to suggest that there was anything anti-phenomenological about Cavaillès’s work in its original conception. Moreover, criticizing Husserl for his errant steps is par for the course in the phenomenological tradition and Cavaillès is no exception to this tendency.” (p. 11)

And indeed the work of Cavaillès is a reminder of an alternative way phenomenology could have been developed in France, one not focused so much on anthropology and daily experience, but one that continued the Husserlian ambition to explore science in a phenomenological manner. In fact, this tradition exist, but not so much associated with a philosophy of the concept, but rather with authors such as Suzanne Bachelard, Trần Đức Thảo and Jean-Toussaint Desanti. Each in their own way attempted to use phenomenological methods to analyze the methods and products of science.

But even such a reappraisal of Cavaillès as a phenomenologist still risk to fall for another form of anachronism, as if phenomenology was the dominant and only philosophy around in France in the 1930s. This is indirectly suggested by Peden’s introduction (as well as most other Anglophone literature), ignoring Cavaillès’ interactions with other forms of philosophy of science. In the secondary literature, the motivation to focus on phenomenology seems to be one of relevance: given that Cavaillès is nowadays anachronistically classified as continental philosophy, the audience expects a focus on Husserl, not David Hilbert or Rudolf Carnap. This is somewhat surprising, since most of Cavaillès’ essay is focused on other theories, ranging from Kantianism to Logical Positivism. What is therefore perhaps missing in the introduction of Peden, is a sketch of these debates as well.

Moreover, in its current form, the book risks to fall prey not only to a form of anachronism, but also to a typical Anglophone tendency to canonize. The danger is not only to forget minor philosophers (a fate that Cavaillès shared for a long time), but also to immediately canonize newly discovered philosophers in an abstract, contextless hall of fame. This leads to abstract discussions where Cavaillès is compared to Spinoza, Kant, Husserl and Heidegger in a ahistorical fashion, as if they were all part of an eternal and continuous philosophical debate. This tends to forget that the history of philosophy is often more fragmentary.

First of all, it is necessary to ask how the work of the ‘great’ philosophers reached Cavaillès: who introduced him to Spinoza or Kant? The main interpreter of Kant and Spinoza in the early 20th-century was Léon Brunschvicg (1869-1944), supervisor of, among others, (Gaston) Bachelard, Cavaillès, Lautman and Aron. Brunschvicg was the most influential French philosopher at that time, with the possible exception of Henri Bergson (see Terzi 2022). But both Brunschvicg and Bergson were two very divergent products of the French tradition of spiritualism, the specific way how Kant was taken up in France, with a stress on the faculty of judgment and the notion of reflexivity. Though originally opposed to scientific philosophy, it transformed itself at the end of the 19th century, under the name of reflexive analysis (analyse réflexive), represented by authors such as Jules Lagneau and Jules Lachelier, in a Kantian philosophy that developed an interest in science as well, exemplified by the work of Émile Boutroux and Brunschvicg. It was this French Kantianism that fundamentally shaped the thought of Bachelard, Cavaillès and Canguilhem.

Being a clear form of subject-centered philosophy, it is most likely that Cavaillès’ attacks on a ‘philosophy of consciousness’ are not merely aimed at the new, and at that time in France still mostly ignored movement of phenomenology, but at French spiritualism as well. From this angle, the fact that he starts his book with Kantianism appears in a new light, and Cavaillès does extensively engage with the work of his supervisor, Brunschvicg (59-61). Therefore the claim that the philosophy of the concept came into being through a confrontation with phenomenology might be misleading. Quite telling, when Foucault later published a French version of his introduction, he left phenomenology out of the picture and broadened his claims: “Without doubt, this cleavage comes from a long way and we could trace it back through the 19th century: Bergson and Poincaré, Lachelier and Couturat, Maine de Biran and Comte.” (Foucault 1985, 4)

Such a broader history of the philosophy of the concept raises a number of additional questions, well-articulated by Cassou-Noguès and Pascale Gillot (2009): is the philosophy of concept a normative project, to be realized in a future philosophy, or a descriptive claim, referring to an eternal and recurrent constant in the history of philosophy? And against what is a philosophy of the concept precisely opposed? Too often terms such as consciousness, subjectivity, experience and meaning are simply used interchangeably in these often rash oppositions. It becomes especially difficult once one acknowledges that several of the early protagonists of the philosophy of the concept, such as Gaston Bachelard, Alexandre Koyré or the early Canguilhem, defended philosophies that had a clear place for the subject, exemplified perhaps best by Bachelard’s ‘new scientific spirit’ (Bachelard 1934). Only after the Second World War did the anti-humanism and anti-psychologism, that are typically associated with the philosophy of the concept, become dominant.

The originality of Cavaillès resides in the fact that his philosophy was indeed one of the first French philosophies who did – but never as radical as later generations would have it – take a distance from a philosophy of science centered around the subject. Cavaillès’s motivations to do so most likely were linked to the fact that he was one of the first in France to really engage with the new German developments in logic and philosophy of science. But again, it would be misleading to simply assume that this was bound to happen and that France was simply one of the last resisting strongholds doomed to fall for the new logic. The real history of philosophy of science is more contingent. That these developments would become so central for philosophy of science’s self-identity was never a necessity. Cavaillès’ fate could equally have been one of complete oblivion.

This leads us to our second correction of the mythology: there was also an atmosphere of complete ignorance and even dismissal of what was going on in philosophy outside of France. Though it was Brunschvicg who co-invited Husserl to Paris in 1929, he was not that interested. He did not even attend his lectures, but only met Husserl a few days later, when the latter went to the doctoral defense of Alexandre Koyré (who had studied with Husserl). Gaston Bachelard was similarly absent. It highlights how far apart philosophical worlds at that time were.

Cavaillès’ interest in and familiarity with German philosophy was rather exceptional at that time. In 1930-1931 he would stay in Germany on a Rockefeller scholarship to study German Protestantism, during which he met Husserl. He also attended the infamous Davos debate between Cassirer and Heidegger (see Cavaillès 1929) and the Wiener Kreis Vorkonferenz in Prague in 1934, where the Vienna Circle presented itself to the international philosophical scene for the first time (see Cavaillès 1935). Though Brunschvicg and Bachelard were invited, they declined. This highlights another perhaps underappreciated dimension of Cavaillès book, namely that the main perceived threat for Cavaillès was perhaps not so much phenomenology or French spiritualism, but the new movement of logical positivism. More than phenomenology, logical positivism challenged French philosophy, even at home. In 1935, logical positivism held its first international conference in Paris. And despite attempts by French representatives of logical positivism, such as Louis Rougier, to invite Brunschvicg, Bachelard and Cavaillès, most French philosophers responded with indifference and hostility (see Dewulf and Simons 2021). French philosophers of science, exemplified by Brunschvicg, had a distaste for logic, rejecting the idea that science and mathematics could be reduced to them.

Again Cavaillès was the exception in taking this movement seriously. But again this engagement should not be simply understood as the moment when the great lessons of German and British philosophy – Cavaillès discusses Russell and Wittgenstein as well – finally seeped into French philosophy. Cavaillès absorptions of these developments was one that immediately also digested them in order to produce something new. Cavaillès was not the only to do so. In fact, in 1938 Rougier reported to Hans Reichenbach how, in response to the 1935 conference, French philosophers were closing ranks and aimed for a counteroffensive. One first step towards this project was a gathering in September 1938 at Amersfoort in the Netherlands. Ferdinand Gonseth, the central organizer, would write about this conference: “We were quite a large group, from which I still recall Bachelard and his daughter Suzanne, Barzin, Bayer, Dupréel, Destouches, Paulette Février, Ebbinghaus, Tarski, Tatarkiewicz, and especially Jean Cavaillès.” (quoted in Emery 2000, 177-178)

Whether and in what form such a French counter-alliance would have taken shape could never be tested, since it was interrupted, literally during the conference, by incoming news: the first mobilization. Germany and France were at the brink of war. Cavaillès and some others, such Bachelard, decided to leave the conference and return to France. Bachelard returned to Paris and mainly start writing on poetics and imagination; Gonseth went back to Switzerland; both Cavaillès and Lautman joined the war effort and tragically died. No genuine French counter-alliance was formed, but for both Cavaillès and Lautman it was clear that the Vienna Circle was the adversary, and hence the reason why it plays a prominent part in the second part of this book. Especially Carnap was the opponent. While in captivity, Cavaillès would write to Lautman about Carnaps as “their old enemy of the Logische Syntax der Sprache” (quoted in Ferrières, 1950: 164).

Cavaillès’ On Logic and the Theory of Science is thus misleadingly technical in its content, as if what is at stake is nothing but highly specialized arguments in philosophy of science and mathematics. In truth, the stakes were higher and the book is a witness to a genuine struggle for identity: what is the task of philosophy of science? What is the correct way to approach scientific practices and their history? This little book is the battleground of all the major candidates at the beginning of the 20th century: spiritualism, logicism, phenomenology, logical positivism, intuitionism, formalism, and so on. Cavaillès did not pick sides, but attempted to develop his own alternative. In that sense, he practiced what he preached: to understand is to capture the gesture, and to be able to continue it. It is this aborted continuation that, though often shed of its contextual skin, has been the fertilizer for the philosophy of the concept in France and historical epistemology in general.


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