# Some reasonable requirements

Regardless of which mathematical representation we choose, certain properties of time are regarded as fundamental and common to all mathematical images of time. These are listed below.

## Primary observers

Mathematicians routinely make truth statements relative to sets of axioms and postulates stated in contextually incomplete form, no primary observer being required in pure mathematics. However, in Chapter 2 we argued that in physics, all relevant statements should be contextually complete, meaning that we should always ask *for whom is this or that assertion meaningful*? Therefore, whenever mathematicians apply their mathematics to the physical universe, they should identify the primary observers involved.

Two notable examples come to mind where the contextual incompleteness of mathematics created difficulties for physicists, one involving the structure of space and the other the structure of time:

- 1. For over two thousand years, geometers assumed that the postulates and axioms of Euclidean geometry were absolute, until Gauss, Bolyai, and Lobachevsky independently showed that Euclid’s Fifth Postulate is independent of the other axioms and postulates. Assuming homogeneity and isotropy,
^{[1]}a primary observer embedded in three-dimensional physical space could in principle measure angles and distances and hence determine empirically which one of three mutually exclusive spaces they were in. These correspond to spherical, Euclidean, and hyperbolic geometries respectively. - 2. Newton’s Absolute Time [Newton, 1687] makes sense if we assume that he was describing the universe as seen by some exophysical primary observer (God) standing outside Absolute Space and Time with total information about everything in it. Classical physicists became as conditioned to think in terms of Absolute Time as mathematicians had become conditioned to believe that Euclidean geometry was absolute. The advent of relativity changed this conditioning.

- [1] A homogeneous medium has the same physical properties everywhere, whilst there is no preferreddirection at any point in an isotropic medium.