The Temporal Continuum

TL;DR

This paper explores the philosophical and mathematical distinctions between spatial and temporal continua, proposing a constructive model of the temporal continuum emphasizing orientation and continuity.

Contribution

It introduces a novel constructive model of the temporal continuum using Dedekind cuts and defines two topologies, highlighting the role of orientation.

Findings

Every total function from the oriented to the ordinary topology is continuous.

The model distinguishes the temporal continuum from the spatial one based on orientation.

Constructive Dedekind cuts are used to model the temporal continuum.

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.