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The history of continua. Philosophical and mathematical perspectives. (English) Zbl 1454.01002

Oxford: Oxford University Press (ISBN 978-0-19-880964-7/hbk). ix, 577 p. (2021).

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Publisher’s description: Mathematical and philosophical thought about continuity has changed considerably over the ages. Aristotle insisted that continuous substances are not composed of points, and that they can only be divided into parts potentially; a continuum is a unified whole. The most dominant account today, traced to Cantor and Dedekind, is in stark contrast with this, taking a continuum to be composed of infinitely many points. The opening chapters cover the ancient and medieval worlds: the pre-Socratics, Plato, Aristotle, Alexander, and a recently discovered manuscript by Bradwardine. In the early modern period, mathematicians developed the calculus the rise of infinitesimal techniques, thus transforming the notion of continuity. The main figures treated here include Galileo, Cavalieri, Leibniz, and Kant. In the early party of the nineteenth century, Bolzano was one of the first important mathematicians and philosophers to insist that continua are composed of points, and he made a heroic attempt to come to grips with the underlying issues concerning the infinite. The two figures most responsible for the contemporary hegemony concerning continuity are Cantor and Dedekind. Each is treated, along with precursors and influences in both mathematics and philosophy. The next chapters provide analyses of figures like du Bois-Reymond, Weyl, Brouwer, Peirce, and Whitehead. The final four chapters each focus on a more or less contemporary take on continuity that is outside the Dedekind-Cantor hegemony: a predicative approach, accounts that do not take continua to be composed of points, constructive approaches, and non-Archimedean accounts that make essential use of infinitesimals.
The articles of this volume will be reviewed individually.
Indexed articles:
Sattler, Barbara M., Divisibility or indivisibility, 6-26 [Zbl 1460.01004]
Harari, Orna, Contiguity, continuity, and continuous change, 27-48 [Zbl 1460.01003]
Sylla, Edith Dudley, Infinity and continuity, 49-81 [Zbl 1461.01006]
Levey, Samuel, Continuous extension and indivisibles in Galileo, 82-103 [Zbl 1462.01008]
Jesseph, Douglas M., The indivisibles of the continuum, 104-122 [Zbl 1464.01003]
Levey, Samuel, The continuum, the infinitely small, and the law of continuity in Leibniz, 123-157 [Zbl 1475.01011]
Sutherland, Daniel, Continuity and intuition in eighteenth-century analysis and in Kant, 158-186 [Zbl 1459.01014]
Rusnock, Paul, Bolzano on continuity, 188-218 [Zbl 1478.01009]
Kanamori, Akihiro, Cantor and continuity, 219-254 [Zbl 1472.01014]
Haffner, Emmylou; Schlimm, Dirk, Dedekind on continuity, 255-282 [Zbl 1464.01010]
McCarty, Charles, What is a number?, 283-298 [Zbl 1460.01009]
McCarty, Charles, Continuity in intuitionism, 299-327 [Zbl 1464.01013]
Vargas, Francisco; Moore, Matthew E., The Peircean continuum, 328-346 [Zbl 1471.01017]
Varzi, Achille C., Points as higher-order constructs, 347-378 [Zbl 1459.01019]
Koellner, Peter, The predicative conception of the continuum, 379-426 [Zbl 1471.01019]
Gerla, Giangiacomo, Point-free continuum, 427-475 [Zbl 1461.03010]
Bell, John L., Intuitionistic/constructive accounts of the continuum today, 476-501 [Zbl 1467.03032]
Ehrlich, Philip, Contemporary infinitesimalist theories of continua and their late nineteenth- and early twentieth-century forerunners, 502-570 [Zbl 1464.01008]

MSC:

01-06 Proceedings, conferences, collections, etc. pertaining to history and biography
03-06 Proceedings, conferences, collections, etc. pertaining to mathematical logic and foundations
03-03 History of mathematical logic and foundations
26-03 History of real functions
00A30 Philosophy of mathematics
00B15 Collections of articles of miscellaneous specific interest
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