Thesis ID: CBB001560801

Theories of Continuity and Infinitesimals: Four Philosophers of the Nineteenth Century (2008)


Keele, Lisa (Author)

Indiana University
McCarty, David C.

Publication Date: 2008
Edition Details: Advisor: McCarty, David C.
Physical Details: 349 pp.
Language: English

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.


Description 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

Similar Citations

Article Błaszczyk, Piotr; Katz, Mikhail G.; Sherry, David; (2013)
Ten Misconceptions from the History of Analysis and Their Debunking (/isis/citation/CBB001252715/) unapi

Article Moore, Matthew E.; (2002)
A Cantorian Argument Against Infinitesimals (/isis/citation/CBB000300363/) unapi

Book Cantor, Georg; Ferreirós, José; (2006)
Fundamentos para una teoría general de conjuntos: Escritos y correspondencia selecta (/isis/citation/CBB000930382/) unapi

Article Mechthild Koreuber; (2015)
Zur Einführung einer begrifflichen Perspektive in die Mathematik: Dedekind, Noether, van der Waerden (/isis/citation/CBB619659341/) unapi

Book Gray, Jeremy; (2008)
Plato's Ghost: The Modernist Transformation of Mathematics (/isis/citation/CBB000950298/) unapi

Article Emmylou Haffner; (2019)
From Modules to Lattices: Insight into the Genesis of Dedekind's Dualgruppen (/isis/citation/CBB728342102/) unapi

Article Moore, Gregory H.; (2008)
The Emergence of Open Sets, Closed Sets, and Limit Points in Analysis and Topology (/isis/citation/CBB000950203/) unapi

Article Gardies, Jean-Louis; (1989)
La conception néo-platonicienne de l'abstraction chez Dedekind, Cantor, Frege et Peano (/isis/citation/CBB000033632/) unapi

Article Belna, Jean-Pierre; (1997)
Les nombres réels: Frege critique de Cantor et de Dedekind (/isis/citation/CBB000072831/) unapi

Book Belna, Jean-Pierre; (1996)
La notion de nombre chez Dedekind, Cantor, Frege: Théories, conceptions et philosophie (/isis/citation/CBB000074591/) unapi

Article Medvedev, F. A.; (1984)
Über die Abstrakten Mengenlehre von Cantor und Dedekind (/isis/citation/CBB000015927/) unapi

Article Campos, Daniel G.; (2007)
Peirce on the Role of Poietic Creation in Mathematical Reasoning (/isis/citation/CBB001023426/) unapi

Thesis David E. Dunning; (2020)
Writing the Rules of Reason: Notations in Mathematical Logic, 1847–1937 (/isis/citation/CBB517005733/) unapi

Article Gana, Francesco; (1985)
Peirce e Dedekind: La definizione di insieme finito (/isis/citation/CBB000052979/) unapi

Authors & Contributors
Belna, Jean-Pierre
Grattan-Guinness, Ivor
Medvedev, F. A.
Gardies, Jean-Louis
Gana, Francesco
Moore, Matthew E.
Berichte zur Wissenschaftsgeschichte
Historia Mathematica
Transactions of the Charles S. Peirce Society
Revue Philosophique de la France et de l' Étranger
Revue d'Histoire des Sciences
Princeton University Press
Princeton University
Set theory
Number theory; number concept
Philosophy of mathematics
Cantor, Georg Ferdinand Ludwig
Dedekind, Richard
Frege, Gottlob
Peirce, Charles Sanders
Peano, Giuseppe
Russell, Bertrand Arthur William
Time Periods
19th century
20th century, early
Central Europe: Germany, Austria, Switzerland
North America

Be the first to comment!

{{ comment.created_by.username }} on {{ comment.created_on | date:'medium' }}

Log in or register to comment