I propose a theory of space with infinitesimal regions called 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 Einstein algebras. (3) It solves the long-standing problem of interpreting smooth infinitesimal analysis (SIA) realistically, an alternative foundation of spacetime theories to real analysis (Lawvere Cahiers de Topologie et Géométrie Différentielle Catégoriques, 21(4), 277–392, 1980). SIA is formulated in intuitionistic logic and is thought to have no classical reformulations (Hellman Journal of Philosophical Logic, 35, 621–651, 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.

我提出了一种具有无穷小区域的空间理论，称为平滑无穷小几何(SIG)，它基于某些代数对象（即环），它规定了几何学家和物理学家启发式使用的推理模式（例如，圆由无限多条直线组成）。我认为 SIG 具有以下实用程序。(1) 它提供了向量场和切线空间的简单形而上学，否则会令人困惑。切线空间可以被认为是一个无穷小的空间区域。(2) 它概括了称为爱因斯坦代数的时空代数主义的标准实现（根据该实现，物理场基本上没有底层流形）。(3) 解决了长期存在的口译问题平滑无穷小分析(SIA) 是现实分析的时空理论替代基础 (Lawvere Cahiers de Topologie et Géométrie Différentielle Catégoriques , 21 (4), 277–392, 1980)。SIA 是用直觉逻辑制定的，被认为没有经典的重新表述（Hellman Journal of Philosophical Logic , 35 , 621–651, 2006）。与此相反，我认为 SIG 是这种重新制定的（部分）。但是 SIG 有一个非正统的理论，其中补充的原则失败了。