We are using the same link as last spring. If you don’t have it, email us at princetonbucharestseminar@gmail.com.

DATE: Tuesday, December 7

TIME: 1 PM Princeton time (ET) / 8 PM Bucharest time (UTC+2)

PANEL: Logic, Diagrams and Geometry in Early Modern Thought

SPEAKERS: Vincenzo De Risi (CNRS & Max-Planck-Gesellschaft) & David Rabouin (CNRS & ERC Philiumm)

ABSTRACT:

**Geometry without Diagrams from Patrizi to Leibniz**

Vincenzo De Risi

In the talk I discuss the philosophy and practices concerning diagrams in early modern geometry. I argue that in the course of the seventeenth century an important transformation in mathematical epistemology took place, leading more and more philosophers to distrust the explanatory power of diagrams, and more and more mathematicians to do without them in their demonstrations. I link this new approach to diagrammatic reasoning to the most important revolution that took place in geometry in the modern age: the transformation of this discipline from a science of figures into a science of space. I also discuss the development of logic during the seventeenth century in relation to the abandonment of diagrammatic practices in geometry, and argue that the emergence of a new logic of relations is in part due to this transformation of mathematical epistemology. I consider especially the thought of philosophers and mathematicians such as Oronce Fine, Francesco Patrizi, Claude Richard, Giovanni Alfonso Borelli, and Gottfried Wilhelm Leibniz.

**On the dream of a ‘purely intellectual’ mathematics in Descartes**

David Rabouin

This talk is a follow up to several studies I did on the role of imagination in Descartes’ mathematics, in connection to the relation between geometry and algebra. I would like to approach this question in a more philosophical and less technical context. More specifically, I would like to focus on the famous passage in the *Méditations Métaphysiques* where Descartes tackles the issue of the relationship between “pure intellect” and “imagination” by mentioning mathematical examples. I am struck by the persistence of a reading of this passage which, to my mind, goes in a deeply wrongheaded direction. According to this interpretation, Descartes criticizes the role of imagination in mathematics and, by the same token, defends an *alternative* way of doing mathematics; a kind of “purely intellectual” mathematics. Based on this background, one then assumes that this is what Descartes did when algebraizing geometry: ie., replacing the role of imagination by that of the intellect. Contrary to this view, I will argue that Descartes is not referring to a *new* mathematics in the *Méditations*. Moreover, Descartes is not opposing a mathematics based on the pure intellect to a mathematics based on imagination. He is opposing two usages of imagination in mathematics (in order to make clear the epistemological difference between two faculties of the mind). Finally, I will claim that what Descartes says in the *Méditations* is fully compatible with what he claims elsewhere, that is to say that in mathematics imagination can function as an “aid” to the intellect. I will show that this stance is fully compatible with his own practice as a mathematician.