Séminaire PhilMath

 

Résumé : Peano’s remarks on the principle of permanence
The paper investigates the principle of permanence in the Peano School, suggesting a reading that differs from the one usually found in contemporary literature, where 1) it is usually associated to logical consistency as a criterion for the creation (rather than presentation) of new theories, and in
particular for the extension of theories ; 2) It is investigated as a metatheoretical criterion related to questions of conservativity and is often couched in proof-theoretic jargon, emphasizing deduction.
The paper will analyse the principle of permanence with respect to linguistic issues, to specific mathematical examples it can be applied to and in relation to structuralism. The principle of permanence, as analysed in the works by Peano, is a principle of practice and a principle of economy. It is a principle of practice because what one preserves and the notation that one adopts to highlight it, might depend on the context, just as the choice of definitions cannot be
made based on purely logical virtues, but involves also practical issues (Peano discusses the principle of permanence in the context of the analysis of definitions). The principle of permanence is a principle of economy, because it exhibits the structural features of a theory: an algebraic presentation of the theory of groups allows one to present the main features of several systems that
are groups and whose difference can be easily perceived by specifying some further properties that pertain to a given group and not to groups in general.