Neo-Kantian Perspectives on Mathematical Knowledge

The project will investigate some of the interactions between the epistemologies of Marburg neo-Kantians and the structural turn that marked the history of mathematics in 1870-1910. Several studies have emphasised the mediating role of Marburg neo-Kantianism in the transition from the philosophical discussion of the Kantian theory of space to the debate on the epistemological status of mathematics after modern axiomatics (see e.g., Coa 1991, Richardson 1998, Friedman 2001). More recently, there has been interesting scholarly work on how Cassirer contributed to elucidate conceptual mathematical patterns such as idealisation and abstraction (Mormann 2008, Heis 2011, Yap 2017). However, a detailed study of the mathematical and conceptual background to this debate is still missing. In general, much remains to be done to elucidate the connections between mathematics and philosophical conceptions of logic in the formation period of modern axiomatics (cf. however Peckhaus 1997, 2018). This project aims to contribute to filling this gap by investigating the epistemological themes at work in Felix Kleins Erlangen Programme and related mathematical developments, such as the theory of transformation groups and the projective models of non-Euclidean geometries. The study of these themes will provide the relevant context for a novel assessment of the neo-Kantian strategies to re-establish the link between higher mathematics and human experience. This project proposal is highly innovative in investigating how some of these strategies reached beyond the debate on the Kantian philosophy of mathematics to address issues that are now again in the focus of philosophical approaches to mathematics informed by mathematical and scientific practices (e.g., Ferreirs 2016, Giardino 2017). This includes the coexistence of dierent research programmes in mathematics, the epistemological status of mathematics, the relation of mathematical knowledge with other scientific disciplines and cultural phenomena. In order to address these issues, the project relies on a methodology that is commonly employed in the philosophy of mathematical practice (Mancosu et al. 2008). This consists in integrating more standard logical and foundational approaches to mathematical theories with the study of special examples from the practices of mathematics. In order to better appreciate the connections between mathematical practices and philosophical ideas, the project relies, furthermore, on the historico-critical approach that is characteristic of the history of philosophy of science (e.g., Heidelberger & Stadler 2002) and of more recent historiography of analytic philosophy (Reck et al. 2013). The applicant will conduct research on the topics described in the present project proposal in collaboration with the research team of the ERC funded project, The Roots of Mathematical Structuralism, directed by Georg Schiemer at the Department of Philosophy at the University of Vienna.