Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-16T19:21:20.861Z Has data issue: false hasContentIssue false
  • English
  • Français

THE TEMPORAL CONTINUUM

Part of: General and miscellaneous specific topics Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  12 April 2024

MOHAMMAD ARDESHIR  and
RASOUL RAMEZANIAN
MOHAMMAD ARDESHIR
Affiliation:
DEPARTMENT OF MATHEMATICS SHARIF UNIVERSITY OF TECHNOLOGY 11365-9415 TEHRAN, IRAN E-mail: mardeshir@sharif.edu
RASOUL RAMEZANIAN*
Affiliation:
DEPARTMENT OF ECONOMICS, UNIVERSITY OF LAUSANNE INTERNEF 1015 LAUSANNE, SWITZERLAND
*
E-mail: rasoul.ramezanian@unil.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

The continuum has been one of the most controversial topics in mathematics since the time of the Greeks. Some mathematicians, such as Euclid and Cantor, held the position that a line is composed of points, while others, like Aristotle, Weyl, and Brouwer, argued that a line is not composed of points but rather a matrix of a continued insertion of points. In spite of this disagreement on the structure of the continuum, they did distinguish the temporal line from the spatial line. In this paper, we argue that there is indeed a difference between the intuition of the spatial continuum and the intuition of the temporal continuum. The main primary aspect of the temporal continuum, in contrast with the spatial continuum, is the notion of orientation.

The continuum has usually been mathematically modeled by Cauchy sequences and the Dedekind cuts. While in the first model, each point can be approximated by rational numbers, in the second one, that is not possible constructively. We argue that points on the temporal continuum cannot be approximated by rationals as a temporal point is a flow that sinks to the past. In our model, the continuum is a collection of constructive Dedekind cuts, and we define two topologies for temporal continuum: 1. oriented topology and 2. the ordinary topology. We prove that every total function from the oriented topological space to the ordinary one is continuous.

Keywords

intuitionismcontinuityconstructive mathematics

MSC classification

Primary: 00A30: Philosophy of mathematics 03A05: Philosophical and critical 03A10: Logic in the philosophy of science
Type
Research Article
Information
The Review of Symbolic Logic , First View , pp. 1 - 19
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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

Ardeshir, M., & Ramezanian, R. (2009). Decidability and Specker sequences in intuitionistic mathematics. Mathematical Logic Quarterly, 55(6), 637–648.CrossRefGoogle Scholar
Aristotle. (1999). Physics. Oxford: Oxford University Press. Translation by R. Waterfield.Google Scholar
van Atten, M. (2006). Brouwer Meets Husserl. On the Phenomenology of Choice Sequences. Berlin: Springer.Google Scholar
van Atten, M., van Dalen, D., & Tieszen, R. (2002). Brouwer and Weyl: The phenomenology and mathematics of the intuitive continuumm. Philosophia Mathematica (3), 10, 203–226.CrossRefGoogle Scholar
Brouwer, L. E. J. (1907). On the Foundation of Mathematics. Ph.D. Thesis, University of Amsterdam, in [6].Google Scholar
Brouwer, L. E. J. (1975). Collected Works, Vol. 1, ed. by Heyting, A. Amsterdam: North-Holland.Google Scholar
Cantor, G. (1883). Foundations of a General Theory of Manifolds: A Mathematico-Philosophical Investigation into the Theory of the Infinite, in [11].Google Scholar
Connes, A., & Consani, C. (2011). The hyperring of adele classes, Journal of Number Theory, 131(2), 159–194.CrossRefGoogle Scholar
Connes, A., & Consani, C. (2010). From monoids to hyperstructures: In search of an absolute arithmetic. Casimir Force, Casimir Operators and the Riemann Hypothesis: Mathematics for Innovation in Industry and Science, edited by Gerrit van Dijk and Masato Wakayama, Berlin, New York: De Gruyter, 2010, pp. 147–198. https://doi.org/10.1515/9783110226133.147.CrossRefGoogle Scholar
Dummett, M. (1964). Bringing about the past. Philosophical Review, 73(3), 338–359.CrossRefGoogle Scholar
Ewald, W. (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics , Vol. 2. Oxford: Oxford Science Publications.Google Scholar
Heath, T. L. (1956). The Thirteen Books of Euclids Elements , Vol. 1. New York: Dover. Unabridged republication of the second edition from 1925.Google Scholar
Krasner, M. (1983). A class of hyperrings and hyperfields. International Journal of Mathematics and Mathematical Sciences, 6(2), 307311.CrossRefGoogle Scholar
Kuiper, J. J. C. (2004). IDEAS AND EXPLORATIONS, Brouwers Road to Intuitionism. Ph.D. Thesis, Utrecht University.Google Scholar
Martel, I. (2000). Probabilistic Empiricism: in Defence of a Reichenbachian Theory of Causality and the Direction of Time. Ph.D. Thesis, University of Colorado at Boulder.Google Scholar
Reichenbach, H. (1956). The Direction of Time . Berkeley: University of California Press.CrossRefGoogle Scholar
Troelstra, A. S., & van Dalen, D. (1988). Constructivism in Mathematics: An Introduction , Vol. 1. Amsterdam: North-Holland.Google Scholar
Veldman, W. (1995). Some intuitionisitic variations on the notion of a finite set of natural numbers. In deSwart, H. C. M. and Bergmans, L. J. M., editors. Perspectives on Negation, Essays in honour of Johan J. de Iongh on his 80th birthday. Tilburg: Tilburg University Press, pp. 177–202.Google Scholar
Weyl, H. (1994). The Continuum: A Critical Examination of the Foundation of Analysis. Translated by Stephen Pollard and Thomas Bole. New York: Dover Publications. Originally published in 1918.Google Scholar