zbMATH — the first resource for mathematics

Convention: Poincaré and some of his critics. (English) Zbl 1044.00003
The paper aims at clarifying H. Poincaré’s notion of conventionalism by a close interpretation of chapters III–V of [Science and Hypothesis, (New York, Dover) (1952; Zbl 0049.29106)]; reprint of the edition Walter Scott Publishing Company (1905), further English translations (1907; JFM 38.0093.12), (1913; JFM 44.0086.16), (1929; JFM 55.0634.10)], for German translations cf. (1904; JFM 35.0080.01), (1906; JFM 37.0057.01), (1914; JFM 45.0121.16), (1928; JFM 54.0049.03), translation from the French original “La science et l’hypothèse”, Paris, Flammarion (1902; JFM 34.0080.12), (1925; JFM 51.0053.05)].
These chapters expose different arguments for the conventionality of geometry. Poincaré holds that geometric axioms are conventions similar to definitions or systems of measurements. He argues that different geometries are equivalent in the sense that they form equally valid alternatives. This equivalence arises from the possibility of translating one geometry into the other. It is puzzling that Poincaré gives two arguments for this inter-translatability, a strong argument in Ch. III and a weaker one in Ch. V establishing the empirical equivalence of inter-translatable theories as theories of physical space. A central issue of the paper is a clarification of this strange fact that a strong argument is supported by a weaker one. “This is embarrassing: if the strong argument is correct, there seems to be no need for an independent argument supporting the weaker claim of empirical equivalence. If, on the other hand, an independent argument is required for the weaker thesis, the role of the stronger inter-translatability argument becomes perplexing” (p. 473).
The examination of the relevant chapters of “Science and hypothesis” leads to the following results (cf. p. 492): although starting from a Kantian setting, Poincaré considers the theorems of geometry as neither necessary nor synthetic a priori truths. He furthermore challenges Kant’s conception of a pure a priori intuition of space. Nevertheless geometry is based upon a priori concepts, like the concept of group. Spatial relations as such are unobservable, applied geometry is a synthesis of geometry and physics. Experimental tests of geometry are inconclusive, “equivalent descriptions of any result can be constructed on the basis of geometric inter-translatability relations.” Geometry is conventional insofar as we have the freedom to adopt particular geometries, but the choice is not arbitrary. Conventions can also be found in physics, but they differ from geometrical conventions.
The paper closes with responses to Poincaré’s views, especially by Einstein (pp. 494-504) and more recent understandings of inter-translatability.

00A30 Philosophy of mathematics
01A60 History of mathematics in the 20th century
Full Text: DOI