David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Synthese 113 (3):381-421 (1997)
A continuum is here a primitive notion intended to correspond precisely to a path-connected subset of the usual euclidean space. In contrast, however, to the traditional treatment, we treat here continua not as pointsets, but as irreducible entities equipped only with a partial ordering ≤ interpreted as parthood. Our aim is to examine what basic topological and geometric properties of continua can be expressed in the language of ≤, and what principles we need in order to prove elementary facts about them. Surprisingly enough ≤ suffices to formulate the very heart of continuity (=jumpless and gapless transitions) in a general setting. Further, using a few principles about ≤ (together with the axioms of ZFC), we can define points, joins, meets and infinite closeness. Most important, we can develop a dimension theory based on notions like path, circle, line (=one-dimensional continuum), simple line and surface (=two-dimensional continuum), recovering thereby in a rigorous way Poincaré's well-known intuitive idea that dimension expresses the ways in which a continuum can be torn apart. We outline a classification of lines according to the number of circles and branching points they contain. The ordering (C,≤) is a topped and bottomed, atomic, almost dense and complete partial ordering, weaker than a lattice. Continuous transformations from C to C are also defined in a natural way and results about them are proved.
|Keywords||Philosophy Philosophy Epistemology Logic Metaphysics Philosophy of Language|
No categories specified
(categorize this paper)
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Begoña Carrascal (1994). Caracterización En Lenguajes Lógicos Infinitarios de Los Órdenes Parciales Diseminados Contables. Theoria 9 (1):173-184.
Paul Bankston (1999). A Hierarchy of Maps Between Compacta. Journal of Symbolic Logic 64 (4):1628-1644.
Scott Mann (2006). Space, Time and Natural Kinds. Journal of Critical Realism 5 (2):290-322.
Ernest Hartmann (2000). The Waking-to-Dreaming Continuum and the Effects of Emotion. Behavioral and Brain Sciences 23 (6):947-950.
Robert W. Latzer (1972). Nondirected Light Signals and the Structure of Time. Synthese 24 (1-2):236 - 280.
Ursula Martin & Elizabeth Scott (1997). The Order Types of Termination Orderings on Monadic Terms, Strings and Monadic Terms, Strings and Multisets. Journal of Symbolic Logic 62 (2):624-635.
Geoffrey Hellman (1994). Real Analysis Without Classes. Philosophia Mathematica 2 (3):228-250.
Philip Ehrlich (1986). The Absolute Arithmetic and Geometric Continua. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1986:237 - 246.
Martin Cooke (2005). To Continue With Continuity. Metaphysica 6 (2):91-109.
Michael J. White (1988). On Continuity: Aristotle Versus Topology? History and Philosophy of Logic 9 (1):1-12.
Added to index2009-01-28
Total downloads9 ( #245,720 of 1,724,745 )
Recent downloads (6 months)4 ( #167,193 of 1,724,745 )
How can I increase my downloads?