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The Logico-philosophical Foundations of Geometry: Historical and Systematic Perspectives, (ANCPyT, Argentina)

Research project funded by the Agencia Nacional de Promoción Científica y Tecnológica (Project number PICT 2021-0221) and located at the Instituto de Humanidades y Ciencias del Litoral, Universidad Nacional del Litoral. 

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The aim of this project is to investigate a series of philosophical, historical, and mathematical problems concerning the logical foundations of geometry. From a historical perspective, the project aims to elaborate the first systematic study on the origins, evolution, and varieties of logicism in the philosophy of geometry. On the one hand, the prehistory of geometric logicism in Leibniz's philosophy of mathematics will be studied. On the other hand, a historical reconstruction and a detailed analysis of Russell's and Carnap's proposals for a logicist foundation of geometry in the period between 1900 and 1940, that is, during the rise of logicism in the philosophy of mathematics, will be provided. The overarching goal is to contribute to a more global understanding of logicism in the philosophy of mathematics, by investigating and elucidating relatively neglected aspects of this position in relation to the foundations of geometry. This historical investigation is articulated subsequently with a systematic exploration of the philosophical significance of the logical investigations on the foundations of geometry developed by Tarski and his collaborators (Schwabhäuser, Szczerba, and Szmielew, among others), in the second half of the twentieth century. Specifically, the relevance of a series of metatheoretical results on the formalization and axiomatization of classical geometric theories for recent discussions on the formal criteria of theoretical equivalence and the notion of formal content of a mathematical statement will be examined. The project expects to make an original contribution by exploring the significance of the conceptual framework developed by Tarski and his collaborators in their (meta)geometrical research, in particular their formal notion of interpretability of theories, for the logical explanation of the notions of theoretical equivalence and formal content. This approach will  allow us to integrate in an original way two important lines of research in the field of formal approaches in philosophy of science and contemporary philosophy of mathematics.

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The project runs from April  2023 to March 2025.

Non-classical Logicism: Geometry, Ramsification, and Interpretability from Russell to Carnap (CONICET, Argentina)

Research project funded by the National Scientific and Technological Research Council (CONICET), and located at IHUCSO LITORAL, CONICET/Universidad Nacional del Litoral.  

The aim of the present research is to carry out a systematic study of the non-classical variants of logicism in the philosophy of mathematics, i.e., the philosophical position according to which pure mathematics is a part of (or reducible to) higher-order logic. More specifically, the project will undertake a systematic inquiry and comparative analysis of a number of important attempts at a logical foundation of geometry. First, a detailed study of the logicist conception of geometry developed by Bertrand Russell in his Principles of Mathematics (1903) will be offered. In particular, the implications of his theory of space for the proposal of a purely logical construction of geometry will be examined. Secondly, a systematic reconstruction and critical analysis of Rudolf Carnap's attempts at a logicist reduction of geometry in the period between 1920 and 1940 will be provided. The focus of the research will be on several variants of geometrical logicism based on the technique of ramsification and the notion of interpretability of theories. Thirdly, the above historical analysis of the logical foundations of geometry will be related to recent discussions of the method of abstraction in the contemporary philosophy of mathematics. Taking the synthetic tradition of modern axiomatic geometry as a case study, the philosophical significance of this mathematical method for the foundations of geometry will be examined. 

The project runs from January 2023 to December 2024.

Purity and Abstraction in Modern Geometry: Historical and Philosophical Perspectives (FWF, Austria)

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Research project funded by the Austrian Research Fund  (Project Number M 2803-G) and located at the Institute of Philosophy, University of Vienna. 

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The aim of the project is to investigate a cluster of historical, philosophical, and mathematical problems in relation to a central program in modern axiomatic geometry, which aimed at the elimination of numbers from the foundations of geometry. Historically, the most influential instance of this research program is David Hilbert’s axiomatization of Euclidean geometry presented in Grundlagen der Geometrie (1899). Philosophically, the program aimed at providing a novel answer to an ancient problem in the philosophy of geometry, i.e., to define the role that numbers must play in the foundation of geometry.

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The project ran from February 2020 to January 2022. For details, see the project website.  

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Impossibility Results in Geometry: Historical and Systematic Perspectives (ANPCyT, Argentina)

Research project funded by the Agencia Nacional de Promoción Científica y Tecnológica (Project number PICT 2017-0443) and located at the Instituto de Humanidades y Ciencias del Litoral, Universidad Nacional del Litoral.   

In this project, we explore a series of historical, mathematical, and philosophical problems related to producing impossibility results in geometry. On the one hand, we undertake a historical and systematic study of the emergence and development of different methods of proving impossibility results. On the other hand, we carry out a detailed examination of the historical evolution of the understanding of the meaning and nature of impossibility results. The investigation focuses on the period spanning from the introduction of algebraic methods in geometry in the seventeenth century, to the emergence of the first formal axiomatizations towards the end of the 19th century. The inquiry proceeds mainly by means of case studies, among which the following stand out: the impossibility of carrying out classical constructions (quadrature of the circle, duplicity of the circle, duplicity of the circle and the of the circle, duplication of the cube, trisection of the angle) with elementary instruments, the introduction of ideal elements in geometry, the introduction of ideal elements in projective geometry, the proofs of (relative) consistency of non-Euclidean geometries, and novel independence results in formal axiomatic geometry. 

The project ran from March 2019 to September 2019.

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