Objectivity in Mathematics: The Structuralist Roots of a Pragmatic Realism

Abstract

This paper proposes a reconsideration of mathematical structuralism, inaugurated by Bourbaki , by adopting the “practical turn” that owes much to Henri Poincare. By reconstructing his group theoretic approach of geometry, it seems possible to explain the main difficulty of modern philosophical eliminative and non-eliminative structuralism: the unclear ontological status of ‘structures’ and ‘places’. The formation of the group concept—a ‘universal’—is suggested by a specific system of stipulated sensations and, read as a relational set, the general group concept constitutes a model of the group axioms, which are exemplified (in the Goodmanian sense) by the sensation system. In other words, the shape created in the mind leads to a particular type of platonistic universals, which is a model (in the model theoretical sense) of the mathematical axiom system of the displacement group. The elements of the displacement group are independent and complete entities with respect to the axiom system of the group. But, by analyzing the subgroups of the displacement group (common to geometries with constant curvature) one transforms the variables of the axiom system in ‘places’ whose ‘objects’ lack any ontological commitment except with respect to the specified axioms. In general, a structure R is interpreted as a second order relation, which is exemplified by a system of axioms according to the pragmatic maxim of Peirce.

This paper draws on two talks given at the AIPS-conference at A Coruña and at the University of Toulouse. The work was carried out as part of the ANR/DFG project Mathematics: Objectivity by Representation.