Measurement, invariance and representation in Cavaillès, Desanti and Vuillemin
Kingston University, London
This thesis will investigate the problem of measurement in 20th-century French epistemology of mathematics, as developed by Jean Cavaillès, Jean-Toussaint Desanti and Jules Vuillemin, in order to clarify certain fundamental problems within structuralism and contemporary French philosophy. Executed for his role in the French resistance, Cavaillès' work is incomplete and wrought with hermeneutical difficulties. Nevertheless, his project exerted a tremendous influence on 20th-century French philosophy, extending far beyond a narrowly 'mathematical' sphere. A central tenet was to make visible an ‘internal necessity’ driving the development of mathematical problems, distinct from the empirical history of theories. A crucial and unresolved question for this project was how to account for the invariance of mathematical structures as they were transformed in novel theories, such as in the grouptheoretic representation of the theorems of elementary geometry in the 19th century. In problematizing the identity of mathematical concepts as represented in different formal apparatuses, Cavaillès instigated a line of investigation into the historicity of scientific concepts that resonated through figures such as Georges Canguilhem, Claude Lévi-Strauss, Michel Foucault and Alain Badiou. My contention is that within Cavaillès' own terrain thisproblem of identifying invariance under transformation is, paradigmatically, a problem for the theory of measurement, wherein the concepts of 'invariance'—preservation across different representations—and 'artifacts'—features idiosyncratic to a particular representation—are formally articulated. Differing approaches to this problematic orientate the divergences of Cavaillès' students Desanti and Vuillemin, who also played crucial roles in 20th-century French philosophy, yet have received scant attention in the anglophone literature. Focusing onengagement with formal theories of measurement will delineate a shared and contested conceptual problematic around invariance, representation and the history of science. This will contribute to central debates in contemporary European philosophy concerning the philosophical import of formalism and the unity of the sciences.