Mathematical and philosophical thought about continuity has changed considerably over the ages, from Aristotle's insistence that a continuum is a unified whole, to the dominant account today, that a continuum is composed of infinitely many points. This book explores the key ideas and debates concerning continuity over more than 2500 years.

Stewart Shapiro received an M.A. in mathematics in 1975, and a Ph.D. in philosophy in 1978, both from the State University of New York at Buffalo. He is currently the O'Donnell Professor of Philosophy at The Ohio State University, and serves as Distinguished Visiting Professor at the University of Connecticut, and Presidential Fellow at the Hebrew University of Jerusalem. He has contributed to the philosophy of mathematics, philosophy of language, logic, and philosophy of logic.
Geoffrey Hellman received his BA and PhD from Harvard (PhD 1973). He has published widely in philosophy of quantum mechanics and philosophy of mathematics, developing a modal-structural interpretation of mathematics. He has also worked on predicative foundations of arithmetic (with Solomon Feferman) and pluralism in mathematics (with John L. Bell). In 2007 he was elected to the American Academy of Arts and Sciences. He and Stewart Shapiro co-authored Varieties of Continua: from Regions to Points and Back (Oxford, 2018).

• Introduction


• 1: Divisibility or indivisibility: the notion of continuity from the Presocratics to Aristotle, Barbara Sattler


• 2: Contiguity, continuity and continuous change: Alexander of Aphrodisias on Aristotle, Orna Harari


• 3: Infinity and continuity: Thomas Bradwardine and his contemporaries, Edith Dudley Sylla


• 4: Continuous extension and indivisibles in Galileo, Samuel Levey


• 5: The indivisibles of the continuum: seventeenth- century adventures in infinitesimal mathematics, Douglas. M Jesseph


• 6: The continuum, the infinitely small, and the law of conti- nuity in Leibniz, Samuel Levey


• 7: Continuity and intuition in 18th century analysis and in Kant, Daniel Sutherland


• 8: Bolzano on continuity, P. Rusnock


• 9: Cantor and continuity, Akihiro Kanamori


• 10: Dedekind on continuity, Emmylou Haner and Dirk Schlimm


• 11: What is a number?: continua, magnitudes, quantities, Charles McCarty


• 12: Continuity and intuitionism, Charles McCarty


• 13: The Peircean continuum, Francisco Vargas and Matthew E. Moore


• 14: Points as higher-order constructs: Whitehead's method of extensive abstraction, Achille C. Varzi


• 15: The predicative conception of the continuum, Peter Koellner


• 16: Point-free continuum, Giangiacomo Gerla


• 17: Intuitionistic/constructive accounts of the continuum today, John L. Bell


• 18: Contemporary innitesimalist theories of continua and their late 19th and early 20th century forerunners, Philip Ehrlich