Leibniz, Newton, and beyond: History and Philosophy of Mathematics from the 17th to the 19th Century

Members:

Ettore Casari (Scuola Normale Superiore)
Vincenzo De Risi (Max Planck Institut für Wissenschaftsgeschichte - Berlin)
Mariano Giaquinta (Scuola Normale Superiore)
Niccolò Guicciardini (Università di Bergamo)
Gabriele Lolli (Scuola Normale Superiore)
Massimo Mugnai (Scuola Normale Superiore)

Research goals:

The Research Group aims to investigate the relationships between history of mathematics, history of logic and history of philosophy in the modern age, stressing the role of mathematical practice in the development and shaping of a plurality of epistemological approaches to the exact sciences. The period covered is roughly the one between Leibniz and Bolzano. In fact, as soon as the creation of the Calculus gave birth to modern mathematics, it also opened a large debate on the method, meaning and aims of the exact sciences themselves. On the one hand, the new mathematical discoveries seem to press towards a more abstract and symbolic approach in geometry and arithmetic, and a closest link between mathematics and logic. The foundational studies in elementary geometry of the late Renaissance, and the rapid development of algebra, find their way in the wide Leibnizian project of a new mathematical logic and a complete formalization of geometry. On the other hand, some mathematicians (mainly working in Newton's entourage) stress the necessity of intuition and imagination in mathematical reasoning, and explicitly theorize the superiority of the synthetic and constructive method of the Ancients over the algebraic modern approach in mathematics - even looking for a mechanical foundation of geometry itself. The divide between these two parties, however, was in fact much more blurred and vague than it is normally recognized, and the followers of both Leibniz and Newton in the 18th century carried on a very complex debate on the nature of mathematics, mixing together logical considerations, metaphysical stances, mathematical discoveries, psychological and perceptual theories, physical and philosophical arguments. Eventually, the intertwining threads of the two traditions culminate at the dawn of the new century in the opposite research projects of Kant's transcendental logic and Bolzano's Wissensschaftslehre. In a very strong sense, these two different philosophical approaches represent the synthesis of the previous foundational attempts in each direction, and a new start for the contemporary epistemological debate on logic and mathematics.

Main Research Topics:

Leibniz's philosophy of mathematics; Newton mathematical methodology; history of logic in the 17th-19th century; history of geometry in the 18th century; philosophy of space in the modern age; Kant's philosophy of mathematics; Bolzano's logical theory.