Time and Space

The aim of this paper is to introduce the basic theoretical (both conceptual and technical) tenets of Mallios’s Abstract Differential Geometry (ADG), as well as to present and to summarise the main results from its applications in the last quarter of a century towards formulating an entirely homological-algebraic, purely gauge-theoretic, finitistic, quantal and manifestly background geometrical smooth spacetime-manifold-independent vacuum Einstein gravity to a wider readership of mathematicians, mathematical physicists, and philosophers of mathematics and physics alike. This is an abstract and generalised version of the usual pseudo-Riemannian spacetime manifold-based vacuum Einstein gravity of General Relativity (GR), here called ADG-gravity. ADG-gravity, like the mathematical theory of ADG on which it is based, is shown here to rely crucially on and to derive from a sheaf $\mathcal{A}$ of commutative algebras representing the structure sheaf of generalised arithmetics or the module of sheaf cohomological coefficients in ADG, being physically interpreted as a sheaf of abelian algebras of generalised local coordinates in ADG-gravity. As a result, ADG-gravity is seen to support and be supported by abstract and generalised ADG-theoretic versions of both the Principle of General Covariance (PGC) of GR, here coined the Principle of Algebraic Relativity of Differentiability (PARD), and Einstein’s General Principle of Relativity, here coined the Principle of $\mathcal{A}$-Relativity ($\mathcal{A}\text{-}R$). The PARD and the $\mathcal{A}\text{-}R$ are seen to be effectively equivalent principles, both being categorical variants of what Mallios originally coined the Principle of $\mathcal{A}$-Invariance ($\mathcal{A}\text{-}I$). This is because both can be formulated functorially in terms of natural transformation type of morphisms between the relevant ADG-theoretic sheaf categories involved. In the last section, we discuss the physical importance and significance of ADG-gravity—as well as of the PARD, the $\mathcal{A}\text{-}R$ and the $\mathcal{A}\text{-}I$ that it supports and is supported by—in various important technical (structural-mathematical) and conceptual (physico-philosophical) issues in current and, in potentially future, Classical (GR) and Quantum Gravity (QG) research. The paper closes with a recurring theme of ours, already touched and elaborated upon throughout the last two and a half decades of applying ADG to QG, namely the importance of developing new mathematics and associated theoretical concepts in QG research. This paper is an extended, more mathematically slanted, sequel to our latest physico-philosophical review of applications of ADG in QG in the light of further recent categorical developments and results on the functorial character and nature of ADG-gravity and $\mathcal{A}$-Relativity. Full article