Smooth Infinitesimals in the Metaphysical Foundation of Spacetime Theories

TL;DR

This paper introduces smooth infinitesimal geometry (SIG), a novel algebraic framework for understanding spacetime that clarifies vector fields, generalizes Einstein algebras, and offers a realistic interpretation of smooth infinitesimal analysis.

Contribution

It develops SIG as a new algebraic approach to spacetime, addressing foundational issues and providing a realistic interpretation of SIA within a classical framework.

Findings

SIG clarifies the metaphysics of tangent spaces and vector fields.

It generalizes Einstein algebras to include infinitesimal regions.

SIG offers a reformulation of smooth infinitesimal analysis compatible with classical logic.

Abstract

I propose a theory of space with infinitesimal regions called \textit{smooth infinitesimal geometry} (SIG) based on certain algebraic objects (i.e., rings), which regiments a mode of reasoning heuristically used by geometricists and physicists (e.g., circle is composed of infinitely many straight lines). I argue that SIG has the following utilities. (1) It provides a simple metaphysics of vector fields and tangent space that are otherwise perplexing. A tangent space can be considered an infinitesimal region of space. (2) It generalizes a standard implementation of spacetime algebraicism (according to which physical fields exist fundamentally without an underlying manifold) called \textit{Einstein algebras}. (3) It solves the long-standing problem of interpreting \textit{smooth infinitesimal analysis} (SIA) realistically, an alternative foundation of spacetime theories to real analysis (Lawvere 1980). SIA is formulated in intuitionistic logic and is thought to have no classical reformulations (Hellman 2006). Against this, I argue that SIG is (part of) such a reformulation. But SIG has an unorthodox mereology, in which the principle of supplementation fails.