State of the Art

The project aims to analyze several philosophical conceptions of mathematics and logic that were highly influential at the beginning of the twentieth century, but whose interdisciplinary and collaborative enterprises have often been underestimated in the analysis of their epistemological results.

These epistemologies can be traced back to the activities of different groups: 1) the members of the Peano School and in particular the collaborators to the Revue des Mathématiques and to the Formulaire des Mathématiques (Giuseppe Peano, Giovanni Vailati, Alessandro Padoa, Cesare Burali-Forti, Mario Pieri, Giovanni Vacca); 2) the editorial board of the Revue de Métaphysique et de Morale (RMM) (Xavier Léon, Leon Brunschvicg, Elie Halévy), in particular, the editors who encouraged the publication of epistemological articles (Louis Couturat, Maximilien Winter, Louis Weber, but also André Lalande, who was not part of the editorial board, but was an assiduous member of the meetings at Léon’s house); 3) the members and collaborators of the Vienna Circle (VC) engaged in work on the philosophy of logic and mathematics, in particular Hans Hahn, Karl Menger, Rudolf Carnap, and Kurt Gödel (among others).

The reasons for the misrepresentation of these three bundles of different epistemologies, and of their inner variants, as a coherent logicist understanding of mathematics are various. Firstly, there was an overestimation in the literature of the three major ‘isms’ in the foundations of mathematics, an overestimation that was partly a result of the activity of the Vienna Circle itself, as they devoted the 1930 Conference on The Epistemology of the Exact Sciences to logicism, formalism, and intuitionism. But there were also tendencies to assimilate several different positions to a unique, clearly stated point of view, as in the case of Russell’s remarks on Peano, or of Neurath’s effort to present a unitary scientific perspective of the Vienna Circle in the redaction of the 1929 manifest. Thirdly, these three groups have often been investigated in isolation, without a systematic investigation of their reciprocal connections. Finally, no truly interdisciplinary investigation has been conducted on these authors, i.e. an investigation that puts together experts in the history of mathematics (history of ideas and social history), in philosophy of science and in history of philosophy.

Although located in different regions (namely Italy, France, and the Austrian Empire) and active in different times (from late 19th century to mid 20th century), there were several interesting historical connections between scholars who belonged to these groups. For example, Louis Couturat, had lively interactions with several members of the Peano School (Peano, Padoa, Pieri, Vailati, and Vacca), as has been convincingly shown by Erika Luciano and Clara Silvia Roero; Ludovico Geymonat, a student of Peano, regularly attended the Vienna Circle meetings and was responsible for the diffusion of logical empiricism in Italy; Louis Rougier was in strict contact with Lalande (the supervisor of his dissertations), held contacts with several members of the Vienna Circle and co-organized international conferences which had the objective of promoting the ideas of the Vienna Circle in France and in Europe.