Keele, Lisa (Author)

Indiana University

McCarty, David C.

Publication Date: 2008

Edition Details: Advisor: McCarty, David C.

Physical Details: 349 pp.

Language: English

Edition Details: Advisor: McCarty, David C.

Physical Details: 349 pp.

Language: English

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The concept of continuity recurs in many different philosophical contexts. Aristotle and Kant believed it to be an essential feature of space and time. Medieval scholars believed it to be the key to unlock the mysteries of motion and change. Bertrand Russell believed that, while everyone talked about continuity, no one quite knew what it was they were talking about. The subject of this dissertation is mathematical continuity in particular. By mathematical continuity, I mean continuity as it applies to or is found in mathematical systems such as sets of numbers. Mathematical continuity is a relatively recent concern. The need to address whether numerical systems are continuous came about with the creation of calculus, specifically, of limit theory. The dissertation focuses on four mathematicians/philosophers from the late nineteenth and early twentieth centuries who were concerned with mathematical continuity. Richard Dedekind and Georg Cantor, in the 1870s and 1880s, developed the concept of a 'point-continuum;' i.e. a continuum composed of discrete entities, such as a collection of numbers arranged on a straight line. Paul du Bois-Reymond, in 1882, and Charles S. Peirce, especially in his post-1906 essays, criticized this compositional point-continuum. Du Bois-Reymond believed infinitesimals were necessary for continuity; Peirce believed no compositional continuum could ever satisfy our intuitions. My ultimate conclusions are that (1) the concept of the mathematical point-continuum does suffer from philosophical difficulties, (2) the concept of the infinitesimal is neither as philosophically problematic nor as mathematically useless as is often charged, but that (3) infinitesimals by themselves cannot solve the problems raised by a compositional view of continuity.

...MoreDescription Looks at the work of Richard Dedekind, Georg Cantor, Paul du Bois-Reymond, and Charles S. Peirce. Cited in *Diss. Abstr. Int. A* 69/08 (2009). Pub. no. AAT 3319910.

Citation URI

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Błaszczyk, Piotr;
Katz, Mikhail G.;
Sherry, David;
(2013)

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(/isis/citation/CBB001252715/)

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Li, Di;
(2005)

Transmission of the mathematical theories of German mathematicians in China during the first half of the 20th century
(/isis/citation/CBB000503127/)

Article
Moore, Matthew E.;
(2002)

A Cantorian Argument Against Infinitesimals
(/isis/citation/CBB000300363/)

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Cantor, Georg;
Ferreirós, José;
(2006)

Fundamentos para una teoría general de conjuntos: Escritos y correspondencia selecta
(/isis/citation/CBB000930382/)

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Ehrlich, Philip;
(2006)

The Rise of Non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of Non-Archimedean Systems of Magnitudes
(/isis/citation/CBB000670056/)

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Schlimm, Dirk;
(2011)

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(/isis/citation/CBB001211472/)

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Mechthild Koreuber;
(2015)

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(/isis/citation/CBB619659341/)

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(2008)

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(/isis/citation/CBB000950298/)

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Emmylou Haffner;
(2019)

From Modules to Lattices: Insight into the Genesis of Dedekind's Dualgruppen
(/isis/citation/CBB728342102/)

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Moore, Gregory H.;
(2008)

The Emergence of Open Sets, Closed Sets, and Limit Points in Analysis and Topology
(/isis/citation/CBB000950203/)

Article
Gardies, Jean-Louis;
(1989)

La conception néo-platonicienne de l'abstraction chez Dedekind, Cantor, Frege et Peano
(/isis/citation/CBB000033632/)

Book
Grattan-Guinness, I.;
(2000)

The Search for Mathematical Roots, 1870-1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel
(/isis/citation/CBB000111675/)

Book
Grattan-Guiness, I.;
(2000)

Search for Mathematical Roots, 1870-1940: Logics, Set Theories and the Foundations of Mathematics from Cantor Through Russell to Gödel
(/isis/citation/CBB000102346/)

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Belna, Jean-Pierre;
(1997)

Les nombres réels: Frege critique de Cantor et de Dedekind
(/isis/citation/CBB000072831/)

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Belna, Jean-Pierre;
(1996)

La notion de nombre chez Dedekind, Cantor, Frege: Théories, conceptions et philosophie
(/isis/citation/CBB000074591/)

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Medvedev, F. A.;
(1984)

Über die Abstrakten Mengenlehre von Cantor und Dedekind
(/isis/citation/CBB000015927/)

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Peirce on the Role of Poietic Creation in Mathematical Reasoning
(/isis/citation/CBB001023426/)

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(/isis/citation/CBB517005733/)

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(/isis/citation/CBB001023424/)

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Gana, Francesco;
(1985)

Peirce e Dedekind: La definizione di insieme finito
(/isis/citation/CBB000052979/)

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