Inventing, Demonstrating: Poincaré Reading Kant On Mathematics Fabien Chareix Philosophy Department Université Paris-Sorbonne (Paris IV) Paris, France fabien.chareix@paris-sorbonne.fr Fabien Chareix CERPHI: Centre d'Etudes en Rhétorique, Philosophie et Histoire des Idées UMR-CNRS 5037 Lyon, France Abstract—How far did Henri Poincaré go into Kant's understanding of Mathematics? The interactions between Mathematics and Philosophy were vivid on the eve of the 20 th Century and this paper is intended to clarify one aspect of the impact philosophy can have on scientific thought. The concept of a priori synthetical judgment, central in Kant's critique reappraisal of Metaphysics, is used by Poincaré in a very peculiar way that has little to do with Kant and kantism. Keywords-component; Mathematics; Poincaré; Kant; Intuition; Hilbert; Russell; Logic I. INTRODUCTION: IS KANT'S PHILOSOPHY OF MATHEMATICS DEAD? In a paragraph quoted from the original paper, deleted afterwards in the final printed version of Science et Méthode, Henri Poincaré refers to Louis Couturat's evaluation of what remains in Kant's legacy: "this is what Mr. Couturat has presented in the papers I just mentioned; this is what he states in an even clearer way in the speech he delivered on the occasion of Kant's jubilee, so that I heard the person next to me saying in low voice: 'we obviously see that this is the centennial of Kant's death'" [1]. In the wake of public controversies that followed the publication of Couturat's book, Les mathématiques et la logique, Poincaré took the opportunity to clarify the meaning and the role of invention and imagination in the making of mathematical reasoning. Far from agreeing on his definition of Mathematics, Poincaré nonetheless maintains some of the distinctions that Kant used in order to support his own view on the matter. "Those philosophers, he adds, are right in some way; in order to practice Arithmetic, or Geometry, or any other Science, we need something more than pure Logic. That other thing, we have no other way to name it than using the word Intuition. But how many different ideas are hidden under the same words?" Poincaré then goes as follows: "Let's compare the four axioms: 1. Two quantities equal to a third one, are equal 2. If a theorem is true for 1, and if we demonstrate that it is true for n+1, provided it is for n, then it will be true for all integer numbers. 3. If on a line point c is between a and b, and point d between a and c, then d will be located between a and b. 4. By one point, only one parallel exists for any given line. All four of them must be referred to Intuition, yet the first one is the wording of a rule belonging to formal Logic; the second is a real a priori synthetical judgment, the foundation of rigorous mathematical induction; the third is a call to imagination; the fourth is an definition under disguise" [2]. The a priori synthetical judgment, a true kantian legacy, is seized as the ground of intuitions that are mandatorily used in the making of Mathematics. Poincaré makes benevolent use of this peculiar concept of Intuition against Couturat and the other "lazy philosophers", as he calls them, who, just like Le Roy [3], are keen on assuming that Mathematics are of pure deductive and formal essence. On the contrary, Intuition is, according to Poincaré, this non-analytical part of mathematical reasoning that evades any logical deduction. Hilbert himself should be willing to admit it, if only, Poincaré adds, he could realize that analytic rigor only relies on the reasoning made through recurrence, that is to say: on an a priori synthesis: "Therefore not only Hilbert's reasoning implies the induction principle, but it also implies that this principle is given to us, not as a definition, but as an a priori synthetical judgment. To put it shortly: any demonstration is necessary. The only possible demonstration is done by complete recurrence. It is valid only if we admit the induction principle, and if we consider it not as a definition, but as a synthetical judgment" [4]. In an effort to generalize the statement, very far from Kant's strict separation between Geometry and Arithmetic, Poincaré extends the central role of Intuition and a priori synthetical judgments to Geometry where, though it only rarely applies as such, Induction forms the very ground of any demonstrative processes [5]. Poincaré adds in the same paper: "Make no mistake. What is indeed the fundamental theorem of Geometry? It is that the axioms of Geometry do not imply contradiction, and this, we can't demonstrate it without the principle of induction". Many things could be said about this extension, starting with what Gerhard Heinzmann [6] rightly reminds us: there aren't two distinct theories of Mathematics in Poincaré's works, one, rationalist, that would cover discrete quantities found in Arithmetic, and another one, conventionalist with regards to