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THE CLASSICAL CONTINUUM WITHOUT POINTS

  • GEOFFREY HELLMAN (a1) and STEWART SHAPIRO (a2)

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.

Copyright

Corresponding author

*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

References

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Aristotle, . (1984). The Complete Works of Aristotle: The Revised Oxford Translation. In Barnes, J., editor. Bollingen Series 61. Princeton, NJ: Princeton University Press.
Bell, J. L. (1998). A Primer of Infinitesimal Analysis. Cambridge, UK: Cambridge University Press.
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. 1924.
Bell, J. L. (2009). Cohesiveness. Intellectica, 51, 145168.
Bishop, E. (1967). Foundations of Constructive Analysis. New York, NY: McGraw.
Cartwright, R. (1975). Scattered objects. In Lehrer, K., editor. Analysis and Metaphysics. Dordrecht, the Netherlands: Reidel, pp. 153171.
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.
Gierz, G. et al. . (1980). A Compendium of Continuous Lattices. Berlin:Springer.
Gruszczyński, R. & Pietruszczak, A. (2009). Space, points, and mereology: On foundations of point-free Euclidean geometry. Logic and Logical Philosophy, 18, 145188.
Hellman, G. (1989). Mathematics without Numbers: Towards a Modal-Structural Interpretation. Oxford:Oxford University Press.
Hellman, G. (1996). Structuralism without Structures. Philosophia Mathematica, 4, 100123.
Hellman, G. (2006). Mathematical pluralism: The case of smooth infinitesimal analysis. Journal of Philosophical Logic, 35, 621651.
Hellman, G., & Shapiro, S. (2012). Towards a point-free account of the continuous. Iyyun: The Jerusalem Philosophical Quarterly, 61, 263287.
Johnstone, P. T. (1983). The point of pointless topology. Bulletin (New Series) of the American Mathematical Society, 8, 4153.
Lewis, D. (1991). Parts of Classes. Oxford:Blackwell.
Menger, K. (1940). Topology without points. Rice Institute Pamphlets, 27, 80107.
Roeper, P. (1997). Region-based topology. Journal of Philosophical Logic, 26, 251309.
Roeper, P. (2006). The Aristotelian continuum: A formal characterization. Notre Dame Journal of Formal Logic, 47, 211231.
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.
White, M. J. (1992). The Continuous and the Discrete: Ancient Physical Theories from a Contemporary Perspective. Oxford: Oxford University Press.
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The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
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