Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-23T06:18:53.553Z Has data issue: false hasContentIssue false
  • English
  • Français

THE CLASSICAL CONTINUUM WITHOUT POINTS

Published online by Cambridge University Press:  25 March 2013

GEOFFREY HELLMAN  and
STEWART SHAPIRO
GEOFFREY HELLMAN*
Affiliation:
Department of Philosophy, University of Minnesota
STEWART SHAPIRO*
Affiliation:
Department of Philosophy, The Ohio State University
*
*DEPARTMENT OF PHILOSOPHY, UNIVERSITY OF MINNESOTA, 831 HELLER HALL, 271-19TH AVENUE SOUTH, MINNEAPOLIS, MN 55455
†DEPARTMENT OF PHILOSOPHY, THE OHIO STATE UNIVERSITY, 350 UNIVERSITY HALL, 230 NORTH OVAL MALL, COLUMBUS, OH 43210
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a point-free construction of the classical one-dimensional continuum, with an interval structure based on mereology and either a weak set theory or a logic of plural quantification. In some respects, this realizes ideas going back to Aristotle, although, unlike Aristotle, we make free use of contemporary “actual infinity”. Also, in contrast to intuitionistic analysis, smooth infinitesimal analysis, and Eret Bishop’s (1967) constructivism, we follow classical analysis in allowing partitioning of our “gunky line” into mutually exclusive and exhaustive disjoint parts, thereby demonstrating the independence of “indecomposability” from a nonpunctiform conception. It is surprising that such simple axioms as ours already imply the Archimedean property and the interval analogue of Dedekind completeness (least-upper-bound principle), and that they determine an isomorphism with the Dedekind–Cantor structure of ℝ as a complete, separable, ordered field. We also present some simple topological models of our system, establishing consistency relative to classical analysis. Finally, after describing how to nominalize our theory, we close with comparisons with earlier efforts related to our own.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 6 , Issue 3 , September 2013 , pp. 488 - 512
Copyright
Copyright © Association for Symbolic Logic 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Aristotle, . (1984). The Complete Works of Aristotle: The Revised Oxford Translation. In Barnes, J., editor. Bollingen Series 61. Princeton, NJ: Princeton University Press.Google Scholar
Bell, J. L. (1998). A Primer of Infinitesimal Analysis. Cambridge, UK: Cambridge University Press.Google Scholar
Bell, J. L. (2001). The continuum in smooth infinitesimal analysis. In Shuster, P., Berger, U., and Osswald, H., editors. Reuniting the Antipodes: Constructive and Nonstandard Views of the Continuum. Dordrecht, the Netherlands: Kluwer, pp. 19–24.CrossRefGoogle Scholar
Bell, J. L. (2009). Cohesiveness. Intellectica, 51, 145–168.Google Scholar
Bishop, E. (1967). Foundations of Constructive Analysis. New York, NY: McGraw.Google Scholar
Cartwright, R. (1975). Scattered objects. In Lehrer, K., editor. Analysis and Metaphysics. Dordrecht, the Netherlands: Reidel, pp. 153–171.CrossRefGoogle Scholar
Dedekind, R. (1963). Continuity and Irrational Numbers. Translation of Stetigkeit und irrationale Zahlen (1901). In Beman, W. W., editor. Essays on the Theory of Numbers. New York, NY: Dover.Google Scholar
Gierz, G. et al. . (1980). A Compendium of Continuous Lattices. Berlin:Springer.CrossRefGoogle Scholar
Gruszczyński, R. & Pietruszczak, A. (2009). Space, points, and mereology: On foundations of point-free Euclidean geometry. Logic and Logical Philosophy, 18, 145–188.CrossRefGoogle Scholar
Hellman, G. (1989). Mathematics without Numbers: Towards a Modal-Structural Interpretation. Oxford:Oxford University Press.Google Scholar
Hellman, G. (1996). Structuralism without Structures. Philosophia Mathematica, 4, 100–123.Google Scholar
Hellman, G. (2006). Mathematical pluralism: The case of smooth infinitesimal analysis. Journal of Philosophical Logic, 35, 621–651.Google Scholar
Hellman, G., & Shapiro, S. (2012). Towards a point-free account of the continuous. Iyyun: The Jerusalem Philosophical Quarterly, 61, 263–287.Google Scholar
Johnstone, P. T. (1983). The point of pointless topology. Bulletin (New Series) of the American Mathematical Society, 8, 41–53.Google Scholar
Lewis, D. (1991). Parts of Classes. Oxford:Blackwell.Google Scholar
Menger, K. (1940). Topology without points. Rice Institute Pamphlets, 27, 80–107.Google Scholar
Roeper, P. (1997). Region-based topology. Journal of Philosophical Logic, 26, 251–309.Google Scholar
Roeper, P. (2006). The Aristotelian continuum: A formal characterization. Notre Dame Journal of Formal Logic, 47, 211–231.Google Scholar
Tarski, A. (1956). Foundations of the geometry of solids. In Logic, Semantics and Metamathematics. Corcoran, J., editor (second edition). Oxford: Clarendon Press. Indianapolis: Hackett Publishing Company, 1983.Google Scholar
White, M. J. (1992). The Continuous and the Discrete: Ancient Physical Theories from a Contemporary Perspective. Oxford: Oxford University Press.Google Scholar