 In 1912, the year of Poincaré’s death, Luitzen Egbertus Jan Brouwer in his inaugural lecture at Amsterdam characterized two opposing orientations in mathematics: intuitionism (in large part French) and formalism (in large part German). The two orientations were represented by the greatest mathematicians of the antagonistic Franco-German pairing: Henri Poincaré and David Hilbert (Brouwer 1912: 124–125). According to the historians, the modernists are thos eat the beginning of the 20th century who supported an algebraization of the contents up to and including the language of signs, while the conservative tendency finds the justification of mathematical knowledge before language, that is, in the intellect. The polemics of the period persist today, even though its political metaphor (conservatism/modernism) has been reversed. Since the death of Bourbaki, it is fashionable to emphasize the intuitive aspects of mathematics. Yet, it is remarkable that Brouwer himself criticizes Poincaré’s conflation of “the language of mathematics [and] the true mathematical construction” (Brouwer 1907: 176). Brouwer is quite right to emphasize that Poincaré does not separate language and mathematics. Poincaré ascribes to language—but not to logic—an essential role in mathematical reasoning. In this sense, we may say that Poincaré is, with respect to Brouwer, a semi-intuitionist who is located squarely between the two fronts of the dispute. For him, mathematical reasoning possesses and must conserve an intuitive content that defies the formal contents. In summary, the similarities and differences between Poincaré and Brouwer can be shown as follows: