Introducing the ISE Methodology: A Powerful New Tool for Topological Redescription

This paper introduces a powerful new tool for topological redescription, the ISE Methodology. These tools allow us to remove and replace a theory's topological underpinnings just as easily as we can switch between different coordinate systems. Aspirationally, these novel topological redescription techniques can be used to provide new support for a roughly Kantian view of space and time; Rather than corresponding to any fundamental substances or relations, we can see the spacetime manifolds which appear in our theories as merely being an aspect of how we represent the world. This view of spacetime topology parallels the dynamic-first view of geometry as well as a Humean view of laws; The spacetime manifolds which feature in our best theories reflect nothing metaphysically substantial in the world beyond them it being one particularly nice way (among others) of codifying the dynamical behavior of matter. A parallel publication (namely, Grimmer (2023)) will explicitly characterize the power and scope of the topological redescription techniques offered to us by the ISE Methodology. The modest goal of this paper is simply to introduce the ISE Methodology by applying it to two example theories. Firstly, to familiarize ourselves with these techniques, I will show how they can be used to redescribe a spacetime theory via a Fourier transform. Secondly, I will show how the exact same techniques can be used to redescribe a lattice theory (i.e., a theory set on a discrete spacetime, M=RxZ) as existing on a continuous spacetime manifold,M=RxR.