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This monograph presents an interpretive essay on the foundations of Leibniz's calculus, accompanied by key texts in English translation. The essay examines Leibniz's evolving views on infinitesimals and infinite numbers, tracing their development from his early metaphysical ideas to his mature justifications of the calculus.
Leibniz first proposed treating infinitesimals as fictions in the 1670s, in line with the mathematical practices of his time, where abstract concepts could be used in calculations without implying their existence. By 1676, he rejected their status as quantities, yet continued to refine his arguments on this topic into the 1690s.
The essay concludes with an analysis of Leibniz's defense of his calculus in the early 18th century, showing how his later works naturally extended from earlier insights. This monograph will be a valuable resource for scholars and students of Leibniz and the history of science.
List of contents
- Part I Interpretive Essay.- Chapter 1. Introduction.- Chapter 2. On the Metaphysics of the Continuum (1669-1676).- Chapter 3. Mathematical Fictions.- Chapter 4. De Quadratura Arithmetica (DQA).- Chapter 5. Infinitesimals and Existence after 1676.- Chapter 6. Leibniz's Mature Justifications of the Calculus.- Chapter 7. Conclusion.- Part II A selection of translations of key texts.- Chapter 8: Texts for chapter 2, On the Metaphysics of the Continuum (1669-1676).- Chapter 9: Texts for chapter 3, Mathematical Fictions.- Chapter 10: Texts for chapter 4, De Quadratura Arithmetica (DQA).- Chapter 11: Texts for chapter 5, Infinitesimals and Existence after 1676.- Chapter 12: Texts for chapter 6, Leibniz's Mature Justifications of the Calculus.
About the author
Richard T. W. Arthur is Professor Emeritus at McMaster University, living in Toronto. Specializing in early modern natural philosophy, mathematics, and the foundations of physics, he focuses on time and the infinite. He holds degrees in physics (BA, Oxford, 1972) and philosophy (MA, McGill, 1976; PhD, Western Ontario, 1981). Arthur has taught applied mathematics and philosophy in Canada, Nigeria, the United States, and Italy. He has authored seven books and over 60 articles and chapters, primarily on seventeenth-century philosophy, including works on Leibniz, Newton, and Descartes. His notable books include "Leibniz" (Polity Press, 2014) and "Leibniz on Time, Space, and Relativity" (Oxford University Press, 2021). In philosophy of physics, he published "The Reality of Time Flow" (Springer, 2019). Currently, he is finishing four co-authored books, including studies on Leibniz’s calculus, unpublished writings on the infinite, translations of Leibniz’s scientific articles, and the influence of Leibniz on Bertrand Russell.
David Rabouin is a Senior Research Fellow (DR2) at the French National Centre for Scientific Research (CNRS), in the research group SPHERE (UMR 7219, CNRS – Université Paris Cité). His interest is in the history of philosophy and mathematics in the early modern period, with special focus on Descartes, Spinoza and Leibniz. He holds degrees in philosophy (PhD, Université Paris Sorbonne, 2002) and mathematics (MSc, Université Paris Diderot, 2014). He is currently the head of the ERC advanced grant PHILIUMM (Adg n°101020985): “The Philosophy of Leibniz in Light of His Unpublished Mathematical Manuscripts”. He is the author of more than 100 articles and several books including Mathesis universalis. L’idée de « mathématique universelle » d’Aristote à Descartes (Paris, PUF, 2009), Spinoza. Ethique locale (Paris, P.U.F., 2010), with Norma B. Goethe and Philip Beeley G.W. Leibniz, Interrelations between Mathematics and Philosophy (Dordrecht, Springer, 2015), with the “Mathesis group”, G. W. Leibniz. Mathesis universalis. Ecrits sur la mathématiques universelle (Paris, Vrin, 2018) and Mathématiques et Philosophie chez Leibniz. Au fil de l'analyse des notions et des vérités (Paris, Vrin, forthcoming).
