Purity and Abstraction in Modern Geometry

The aim of this project is to investigate a series 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 Hilberts 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. This project has three principal objectives. The first is to contribute to filling a gap in the history of modern geometry, by providing the first systematic study of the program of the geometrization of geometry, as I shall call the research movement which aimed at the elimination of numerical considerations from the foundations of geometry. Specifically, the focus will be set on a series of important and innovative geometrical developments as well as in their philosophical consequences and underpinnings. The second principal objective of the project is to articulate the previous investigation on this important tradition in modern geometry with a more general and systematic discussion of the problem of purity of method in the context of geometrical reasoning. Roughly, purity is here connected with the search of specific arguments or proofs for mathematical propositions or theorems, where the means of proofs are considered as appropriate (or inappropriate) in relation to the conditions explicitly stated in such statements. The project will provide a detailed analysis of the application of this central methodological principle within the modern program which pursued the elimination of numbers from geometry. Finally, the third principal objective of the project is to explore further the historical and philosophical significance of this foundational program in modern geometry, by examining its relevance and potential for current investigations into abstraction and abstraction principles in contemporary philosophy of mathematics. Schematically, By abstraction principles one understands schemata, by which "concrete" mathematical entities are combined with the help of an equivalence relation and thus new "abstract" entities are defined. The present project will contribute to current investigations into the method of abstraction in modern geometry, by providing a detailed account of the application of this mathematical technique in the geometrical program under scrutiny here. On the one hand, I will investigate the foundational and methodological role played by the method of abstraction within this specific geometrical framework. On the other hand, I will conduct a more systematic analysis of a series of epistemological, ontological and semantical issues posed by the use of abstraction principles in the foundations of geometry.

The aim of this project is to investigate a series 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. This project has three principal objectives. The first is to contribute to filling a gap in the history of modern geometry, by providing the first systematic study of the program of the "geometrization of geometry", as I shall call the research movement which aimed at the elimination of numerical considerations from the foundations of geometry. Specifically, the focus will be set on a series of important and innovative geometrical developments as well as in their philosophical consequences and underpinnings. The second principal objective of the project is to articulate the previous investigation on this important tradition in modern geometry with a more general and systematic discussion of the problem of 'purity of method' in the context of geometrical reasoning. Roughly, 'purity' is here connected with the search of specific arguments or proofs for mathematical propositions or theorems, where the means of proofs are considered as appropriate (or inappropriate) in relation to the conditions explicitly stated in such statements. The project will provide a detailed analysis of the application of this central methodological principle within the modern program which pursued the elimination of numbers from geometry. Finally, the third principal objective of the project is to explore further the historical and philosophical significance of this foundational program in modern geometry, by examining its relevance and potential for current investigations into abstraction and abstraction principles in contemporary philosophy of mathematics. Schematically, By abstraction principles one understands schemata, by which "concrete" mathematical entities are combined with the help of an equivalence relation and thus new "abstract" entities are defined. The present project will contribute to current investigations into the method of abstraction in modern geometry, by providing a detailed account of the application of this mathematical technique in the geometrical program under scrutiny here. On the one hand, I will investigate the foundational and methodological role played by the method of abstraction within this specific geometrical framework. On the other hand, I will conduct a more systematic analysis of a series of epistemological, ontological and semantical issues posed by the use of abstraction principles in the foundations of geometry.