The Poincaré Paradox and Non-Classical Logics

Abstract

At various occasions H. Poincaré emphasized that the physical continuum is not transitive. In his popular books on Fondement de la Géometrie, La Science et I’Hypothèse and La Valeur de la Science (cf. [Poincaré, 1902; Poincaré, 1904]) he symbolized this situation by the following formulae

$$A = B,B = C,A \ne C$$

where he interprets the equality sign as indistinguishability—i.e. A is indistinguishable from B, B is indistinguishable from C, but A might be very well discernible from C. Poincaré viewed this formula as a philosophical principle which was not accepted by all of his contemporaries (see e.g. E. Borel’s criticism). In the meantime there exist a large variety of important examples indicating the validity of formulae (1)—e.g. K. Menger’s work on Positivistic Geometry (cf. [Menger, 19791) or the whole field of Cluster Analysis. As a simple example we recall the non-transitive, symmetric and reflexive relation ≈ _∈ describing the magnitude of real numbers

$$a{ \approx _ \in }b \Leftrightarrow |a - b| < \in $$

(2)

where ∈ is a positive real number depending on the given system. In particular we say that the real numbers a and b have the same magnitude if and only if a ≈ _∈ b holds.