On Bolzano and Greek Concepts of Continuity

Piotr Błaszczyk () and Marlena Fila ()
Additional contact information
Piotr Błaszczyk: Pedagogical University of Cracow, Institute of Mathematics
Marlena Fila: Pedagogical University of Cracow, Institute of Mathematics

A chapter in Handbook of the History and Philosophy of Mathematical Practice, 2024, pp 1563-1594 from Springer

Abstract: Abstract In ancient Greek, continuity referred to lines, figures, and solids, or time, place, and motion. All these objects were considered continuous in the same sense: they were divisible into infinitely divisible parts. Bolzano 1817 Rein analytischer Beweis (Bolzano, Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein ent-gegengesetzes Resultat gewäihren, wenigstens eine reelle Wurzel der Gleichung liege. Gottlieb Hasse, Prague. In: [53], 253–277, 1817a) gave rise to two brand new ideas: εδ concept of continuous function and continuity of total order in terms of the least upper bound principle. The first was related to motion, the second to a straight line. The chapter aims to reveal the mathematical motives that lead Bolzano to these ideas. In the first part, we discuss Aristotle’s definition and examples of continuity, Euclid’s concept of magnitude and number, and Pappus’ account of mechanical curves. Then we show how the idea of the punctiform curve emerged in Descartes’ (Descartes, La Géométrie. Jan Maire, Leiden, 1637). The second part includes a detailed analysis of (Bolzano, Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein ent-gegengesetzes Resultat gewäihren, wenigstens eine reelle Wurzel der Gleichung liege. Gottlieb Hasse, Prague. In: [53], 253–277, 1817a).

Keywords: Continuity; Continuous function; Least upper bound; Magnitudes; Real numbers; Topological continuum (search for similar items in EconPapers)
Date: 2024
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-40846-5_18

Ordering information: This item can be ordered from
http://www.springer.com/9783031408465

DOI: 10.1007/978-3-031-40846-5_18

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().