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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
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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
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

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