Poincaré and Intuitionism

Poincaré’s position is often presented as semi-intuitionist, especially in relation to his position in arithmetic. This article examines Poincaré’s relationship to an intuitionist position. In the first part, we examine the relationship between the Kantian tradition and intuitionism, and the reception of Poincaré’s approach by Brouwer and Heyting. In the second part, we present a detailed analysis of Poincaré’s conception of induction as an a priori synthetic judgment based on pure intuition and in relation to the foundations of geometry. For the latter, Poincaré presupposes a power of the mind to form groups and continua, of which we possess an intuitive knowledge.We conclude that although we are accustomed to some mathematical conceptions of Poincaré and Brouwer are rather closely related, Poincaré and Brouwer differ fundametally in their approaches from a conceptual point of view. Contrary to the neo-intuitionists of the Brouwerian tradition, for Poincaré the use of language is not only an efficient tool for memorizing or communicating, but also, and above all, an essential part of the genesis of mathematical conceptualization.