Learning at Port-Royal: Making Sense of Antoine Arnauld's "Nouveaux Éléments de Géométrie"

Abstract

To what end did Antoine Arnauld, a prominent theologian and philosopher of the seventeenth century, publish a work of elementary geometry? Why, in a century of such staggering intellectual achievements as the development of analytic geometry by Fermat and Descartes, and which promised the invention of the versions of the differential and integral calculus elaborated by Newton and Leibniz, did Arnauld believe it was necessary to rewrite the Elements of Euclid? Arnauld's Nouveaux éléments de géométrie, prefaced by his longtime collaborator Pierre Nicole, represents an intervention in the mathematical culture of the seventeenth century. In the early modern era the study of mathematics was often advocated for extra-mathematical ends and the Nouveaux éléments de géométrie, I argue, is a paradigmatic example of how mathematicians advocated its practice for propaedeutic reasons. More than the canonical figures Descartes, Pascal, and Leibniz, about whose projects in mathematics this has recently been argued, Arnauld's ambitions in the Nouveaux éléments de géométrie concern the cultivation of virtues among practitioners. To pursue the argument that Arnauld and Nicole advocate the practice of mathematics for the cultivation of virtue, I explore their broader philosophical and pedagogical commitments since it would be a mistake to consider the topic of mathematics in isolation. To this end, the early chapters of the dissertation explore the commitments that underwrite Arnauld and Nicole's recommendations for the pursuit of self-betterment. In their writings on the verbal and visual arts, and natural history and natural philosophy emerge values commensurate with their prescriptions to make use of mathematics for propaedeutic ends. What gives logic value but rhetoric none? What makes curiosity bad and natural philosophy useless? Why did Arnauld and Nicole advocate knowledge of logic but remain deeply ambivalent about an acquaintance with the visual arts? Answers to these questions are crucial to making sense of the values Arnauld brought to the project of rewriting Euclid’s Elements. Moreover, they help us to understand his and Nicole's prescriptions on topics including the passions and the senses, the explanations they offer about the nature of our souls and our capacity for reason, and the knowledge of method they hope to instil in practitioners. What virtues did Arnauld and Nicole believe accrued to those that studied geometry? In the dissertation I give an account of the moral, Christian, and epistemic virtues Arnauld and Nicole believed could be cultivated by a careful study of their edition of Euclid. I explore these questions in the final two chapters where I argue that Arnauld and Nicole's beliefs about what constitutes a well-lived Christian life motivates their prescription to study geometry. Since they believe we are bound by duty to be good and just, Arnauld and Nicole argue we should direct our education to this end. Not only does Arnauld's Nouveaux éléments de géométrie offer a subtle education in praiseworthy behaviour and deploys mathematical concepts to teach practitioners about their soul and God's nature, Arnauld and Nicole argue that we ought to cultivate epistemic virtues since being attentive, governed by reason, and truth-loving fulfils our moral and Christian obligations. In the final chapter of the dissertation, I examine their purely epistemological reasons for cultivating intellectual virtues, and argue that these constitute a surprising intervention from Port-Royalists in seventeenth century debates about the use and efficacy of reason. But for Euclid's Elements to serve these propaedeutic ends, Arnauld needed it to be a radically altered kind of mathematical practice. To conclude the dissertation, I examine Arnauld's arguments about the inadequacy of contemporary presentations of Euclid, and consider the corrections he offers in the Nouveaux éléments de géométrie. Few historians of philosophy have carefully examined Arnauld's work in mathematics and even fewer among their number have been anglophone scholars. Through a detailed contextualist study of the Nouveaux éléments de géométrie I show that in rewriting Euclid's Elements, Arnauld offered indispensable reflections on issues in moral philosophy, theology and epistemology, and detailed views commensurate with his broader philosophical project. The outcome of principled reflections on issues in contemporary philosophy and mathematics, Arnauld's rewriting of Euclid deserves our attention not least because of the insights it offers into French mathematical culture in the seventeenth century.