Smooth Infinitesimals in the Metaphysical Foundation of Spacetime Theories

I propose a theory of space with infinitesimal regions called $\mathit{\text{smooth}}\mathit{\text{infinitesimal}}\mathit{\text{geometry}}$ (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 $\mathit{\text{Einstein}}\mathit{\text{algebras}}$. (3) It solves the long-standing problem of interpreting $\mathit{\text{smooth}}\mathit{\text{infinitesimal}}\mathit{\text{analysis}}$ (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.