  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.
