Jean Cavailles - On Logic and the Theory of Science

The preface and notice translated below read more like dares than invitations to study On Logic and the Theory of Science . Gaston Bachelard hints at the challenges to come in his description of Cavaillss sentences as often enigmatic in their concision. The works editors, Georges Canguilhem and Charles Ehresmann, were more frank when they presented the fruits of their original efforts in 1946. Apparently they considered authoring an introduction that might shed some light on the background context of philosophical and mathematical culture which Cavaills precisely wished to be taken for granted. But they feared the result would be mere commentary. Instead, apart from filling in references, they presented the work without apparatus and with the conjecture that Cavaills believed those who did not make the effort to understand did not deserve to be enlightened, a judgment all the more perplexing given that we are advised in the next paragraph that the problems raised in the pages to follow cannot honestly receive any definitive solution.

Readers should know that translating Cavaillss essay has conferred no special advantage on this score. Our strategy has been less to resolve the enigmas of Cavaillss writing than to preserve them in English. As for the context required even to understand them as enigmas, it is at least plausible that Cavaills could take some mathematical culture for granted on the part of his readership. But such is not the case today, given the historical distance and a new audience for whom Parisian philosophy in the interwar years has a hermetic, often parochial quality. Fortunately, a lot of ongoing work in intellectual history is giving scholars and students access to this moment.

The institutional setting for Cavaillss effort is in some sense the easy part. More problematic is his technical aptitude in scientific fields with notorious barriers to entry. To reconstruct the mental space in which Cavaills thought would require a complete study of the revolutions in logic and mathematics that shaped the first half of the twentieth century. Even so, obscurities would remain. There is something in On Logic and the Theory of Science to baffle everyone. Steeped in a philosophical formation at the crossroads of neo-Kantianism and phenomenology, Cavaills brings to mathematics questions that are beyond the mathematicians purview. And yet often he will make a mathematical allusion to illustrate a philosophical point, such as when he presents science as a Riemannian volume, closed and yet without any exterior. This concept from differential geometrymore typically known as a Riemannian manifoldis an illuminating metaphor for figuring science in its historical relationship to other regions of thought. It is even more illuminating for readers versed in differential geometry.

Jean Cavaillss doctoral work, published in two separate volumes in 1938, was in the history of mathematics, primarily in the formation of abstract set theory and the disputes that arose between formalists and intuitionists in this history. Here the key figures were Georg Cantor, David Hilbert, and L.E.J. Brouwer. Cavaills believed that the history of mathematics posed special problems for philosophy in that the truths articulated in this science were universal in their remit and yet the process of their discovery was temporal. The histories of empirical scienceschemistry, biology, physicsposed their own difficulties, ones that would be explored at length by Bachelard and Canguilhem, among many others. But to speak of discovery in the context of mathematics was already a polemical move. Platonist in his orientation, Cavaills inherited from his mentor Lon Brunschvicg a distrust of logicism, be it of an Oxford or Vienna variety. The truths discovered in mathematics were precisely that: mathematical truths, not tautologies reducible to logical syntax. And yet how is it that mathematics has a history, that is, is a science that is only disclosed to us in a historical way? Cavaills allowed that mathematicians might be lazy and fail to follow through on certain discoveries. But that is a contingent matter. Even if scientists never came along to discover dinosaur bones, it is still true that they would have been there. Is the transfinite of Cantors paradise really analogous in this regard?

In his defence of his doctoral work before the Socit franaise de philosophie in 1939, which he presented alongside his friend Albert Lautman, Cavaills justified his approach in terms of a mathematical experience. Typically we regard mathematics as distinctive from other sciences in not being experimental. But Cavaills maximises the semantic fullness of the French lexprience which connotes both experience as understood in English and experiment in the scientific senseto make a set of claims about mathematics as a becoming [ devenir ], that is, a historical process endowed with an essentially temporal quality. This becoming takes place in history but resists any collapse into historicism. It is both autonomous and unpredictable.

Emphasising the autonomy of mathematics was a means to suggest that it was not dependent upon, much less parasitic upon, any discursive framework or set of evidences external to itself. Note here the way in which Cavaillss view complicates Paul Ricoeurs later description of structuralism (in this case Claude Lvi-Strausss) as Kantianism without a transcendental subject. A terrific slogan, it captures the repeated gesture in French postwar thought to determine the conditionsbe they historical, semiotic, or psychicunder which an object appears. But Cavaills is not speaking of conditions; mathematics is truly absolute in his view.

Mathematics is a becoming. All we can do is try to understand its history; that is to say, to situate mathematics in relation to other intellectual activities, to discover certain characteristics of this becoming. [] I believe that it is possible, within the picturesque contingency of the concatenation of theories, to achieve this work. I have tried, for my part, to do so for set theory; I do not claim to have succeeded, but I do think that I was able to perceive, in the development of that theory which seems the very example of a theory of genius, and which was constructed through radically unforeseeable interventions, an internal necessity: certain problems in analysis gave rise to the essential notions, generated certain procedures already guessed at by Bolzano and Lejeune-Dirichlet, and which became fundamental procedures refined by Cantor. Autonomy, therefore necessity.

Cavaillss point is that the autonomy of mathematics entails as a necessary consequence that necessity itself belongs to mathematics. This is not a matter of a relation between the conditioning and the conditioned that might be reconfigured. Such a vision allows space for static, eruptions, all those ciphers for the event that would mark a history of thought inspired by Cavaillss work.