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Poincaré sets out a hierarchical view of the sciences inScience and Hypothesis(1902), although he does notexplicitly use this terminology. In his view the special sciencespresuppose physics, which presupposes geometry, which in turnpresupposes arithmetic. Poincaré treats topics in a serialorder—first arithmetic, then geometry, then physics, etc. In...

3.1 Logic and foundations: intuition and predicativity

Concerning logic and foundations of mathematics,Poincaré’s position is governed by two theses: 1. Logical inferences alone are epistemically inadequate to expressthe essential structure of a genuine mathematical reasoning in view ofits understandability (see Poincaré 1908: 159; 1913b:452). 2. As a consequence of the logical antinomies, one should avoid anyimpredicative concept formation. Historically, both theses are directed broadly against the founders ofmodern logic and set theory such as...

3.2 Geometry: conventions, intuition and aesthetics

Poincaré was strongly influenced by and attuned to the Frenchphilosophical scene, which refers to the Kantian tradition to show thelimits of methodological positivism. Members of the so-calledCritique de la sciencemovement (Benrubi 1928), whichincludes members of the “Boutroux Circle” (Nye 1979)advocate a mixture of positivism and Neo-Kantianism. They criticizeboth Comte’s determinism and Kant’s static view of themind’s structure. According to Poincaré, mathematicsrequires intuition not only...

Concerning the epistemological status of mechanics, Poincarépositions himself, as in his discussion of geometry, as holding aposition between empiricism and a priorism (Poincaré 1902: 111;2017: 71). The principles of mechanics certainly have, according toPoincaré, an empirical origin, but they nonetheless surpass thebounds of strict empiricism sinc...

There is no doubt that Poincaré’s work has been veryinfluential both in the sciences and in philosophy. It was alreadywidely discussed at the time it was first presented — not onlyin France but also in Germany (e.g. Ferdinand von Lindemann (1904) andEmil Meunier (1919))— and his geometric conventionalism alsogreatly influenced the logical empiricis...