Infinitesimal Differences : Controversies Between Leibniz and His Contemporaries

The essays offer a unified and comprehensive view of 17th century mathematical and metaphysical disputes overstatus of infinitesimals, particularly the question whether they were real or mere fictions. Leibniz's development of the calculus and his understanding of its metaphysical foundation are taken as both a point of departure and a frame of reference for the 17th century discussions of infinitesimals, that involved Hobbes, Wallis, Newton, Bernoulli, Hermann, and Nieuwentijt. Although the calculus was undoubtedly successful in mathematical practice, it remained controversial because its procedures seemed to lack an adequate metaphysical or methodological justification. The topic is also of philosophical interest, because Leibniz freely employed the language of infinitesimal quantities in the foundations of his dynamics and theory of forces. Thus, philosophical disputes over the Leibnizian science of bodies naturally involve questions about the nature of infinitesimals. The volume also includes newly discovered Leibnizian marginalia in the mathematical writings of Hobbes. Read more... Abstract: The development of the calculus during the 17th century was successful in mathematical practice, but raised questions about the nature of infinitesimals: were they real or rather fictitious? Read more... Frontmatter......Page 1 Table of Contents......Page 5 Introduction......Page 7 Leery Bedfellows: Newton and Leibniz on the Status of Infinitesimals......Page 13 Infinity, Infinitesimals, and the Reform of Cavalieri: John Wallis and his Critics......Page 37 Indivisibilia Vera – How Leibniz Came to Love Mathematics......Page 59 Indivisibles and Infinitesimals in Early Mathematical Texts of Leibniz......Page 101 Archimedes, Infinitesimals and the Law of Continuity: On Leibniz’s Fictionalism......Page 113 An Enticing (Im)Possibility: Infinitesimals, Differentials, and the Leibnizian Calculus......Page 141 Productive Ambiguity in Leibniz’s Representation of Infinitesimals......Page 159 Generality and Infinitely Small Quantities in Leibniz’s Mathematics - The Case of his Arithmetical Quadrature of Conic Sections and Related Curves......Page 177 Leibniz’s Calculation with Compendia......Page 191 Nieuwentijt, Leibniz, and Jacob Hermann on Infinitesimals......Page 205 Truth in Fiction: Origins and Consequences of Leibniz’s Doctrine of Infinitesimal Magnitudes......Page 221 Rule of Continuity and Infinitesimals in Leibniz’s Physics......Page 241 Leibniz on Infinitesimals and the Reality of Force......Page 261 Dead Force, Infinitesimals, and the Mathematicization of Nature......Page 287 Backmatter......Page 313

The essays offer a unified and comprehensive view of 17th century mathematical and metaphysical disputes overstatus of infinitesimals, particularly the question whether they were real or mere fictions. Leibniz's development of the calculus and his understanding of its metaphysical foundation are taken as both a point of departure and a frame of reference for the 17th century discussions of infinitesimals, that involved Hobbes, Wallis, Newton, Bernoulli, Hermann, and Nieuwentijt. Although the calculus was undoubtedly successful in mathematical practice, it remained controversial because its procedures seemed to lack an adequate metaphysical or methodological justification. The topic is also of philosophical interest, because Leibniz freely employed the language of infinitesimal quantities in the foundations of his dynamics and theory of forces. Thus, philosophical disputes over the Leibnizian science of bodies naturally involve questions about the nature of infinitesimals. The volume also includes newly discovered Leibnizian marginalia in the mathematical writings of Hobbes.

The essays offer a unified and comprehensive view of 17th century mathematical and metaphysical disputes over status of infinitesimals, particularly the question whether they were real or mere fictions. Leibnizs development of the calculus and his understanding of its metaphysical foundation are taken as both a point of departure and a frame of reference for the 17th century discussions of infinitesimals, that involved Hobbes, Wallis, Newton, Bernoulli, Hermann, and Nieuwentijt. Although the calculus was undoubtedly successful in mathematical practice, it remained controversial because its procedures seemed to lack an adequate metaphysical or methodological justification. The topic is also of philosophical interest, because Leibniz freely employed the language of infinitesimal quantities in the foundations of his dynamics and theory of forces. Thus, philosophical disputes over the Leibnizian science of bodies naturally involve questions about the nature of infinitesimals. The volume also includes newly discovered Leibnizian marginalia in the mathematical writings of Hobbes. "The development of the calculus during the 17th century was successful in mathematical practice, but raised questions about the nature of infinitesimals: were they real or rather fictitious? This collection of essays, by scholars from Canada, the US, Germany, United Kingdom and Switzerland, gives a comprehensive study of the controversies over the nature and status of the infinitesimal. Aside from Leibniz, the scholars considered are Hobbes, Wallis, Newton, Bernoulli, Hermann, and Nieuwentijt. The collection also contains newly discovered marginalia of Leibniz to the writings of Hobbes."--Jacket. Review text: "As a philosopher, Leibniz thought deeply about the nature of the infinite and the infinitesimal. ... Infinitesimal Differences makes compulsory reading for anyone interested in how Leibniz defined and manipulated infinitesimals as geometrical, algebraic, and physical magnitudes, and how Leibniz's ideas on infinitesimals were received-and often misunderstood-by his contemporaries, including acolytes of his, such as Johann Bernoulli."Niccolò Guicciardini in: NTMZeitschrift für Geschichte der Wissenschaften, Technik und Medizin, Band18/ 2010