Chasles and the Projective Geometry: The Birth of a Global Foundational Programme for Mathematics, Mechanics and Philosophy.

This book is a contribution to the internalist history of mathematics. Contextual indications will be given, although these are conceived as a frame within which the main scene takes place, where “the main scene” is intended as the development of Chasles’ foundational programme for mathematics, mechanics and philosophy. A complete reading of the book requires a general knowledge of the mathematical techniques which were used in geometry and in mechanics in the nineteenth century, while no knowledge of Chasles’ work is necessary since all his discoveries will be explained in detail. To favour the reader’s understanding of Chasles’ mathematics, I have enriched the book with figures, commentaries and explanations. It follows that the scholars who deal with the technical history of mathematics are the ideal readers of this work. On the other hand, a partial reading of the book is also possible if the aim of the reader is to achieve a general idea of the thesis I support and of the argumentative technique I use, but she/he will possibly not be interested in all the mathematical details. In this perspective, I suggest reading the following sections: 1) the Introduction; 2) the initial part of every section and subsection where I present the general picture of a topic and my thesis before dealing with the mathematical details; 3) the conclusive considerations of every chapter; 4) the entire sixth chapter which concerns the philosophy of duality and presents few mathematical details; 5) the Conclusions. With this selection, the book can be of interest not only to the historians of mathematics, but also to the historians of science interested in under-standing the way in which the concept of foundation was conceived by a great mathematician who developed his thought on the subject mainly in the first half of the nineteenth century. The philosophers of mathematics who study the foundations of this discipline between the end of the nineteenth century and the beginning of the twentieth can also be interested in this book in order to compare the meaning of the locution “foundation of mathematics” in Chasles and in the period they study. Finally, the philosophers and the historians of philosophy can draw from this book some ideas regarding the philosophy of duality.