Distributional and neural approaches to natural language semantics have been built almost exclusively on conventional linear algebra: vectors, matrices, tensors, and the operations that accompany them. These methods have achieved remarkable empirical success, yet they face persistent structural limitations in compositional semantics, type sensitivity, and interpretability. I argue in this paper that geometric algebra (GA) -- specifically, Clifford algebras -- provides a mathematically superior foundation for semantic representation, and that a Functional Geometric Algebra (FGA) framework extends GA toward a typed, compositional semantics capable of supporting inference, transformation, and interpretability while retaining full compatibility with distributional learning and modern neural architectures. I develop the formal foundations, identify three core capabilities that GA provides and linear algebra does not, present a detailed worked example illustrating operator-level semantic contrasts, and show how GA-based operations already implicit in current transformer architectures can be made explicit and extended. The central claim is not merely increased dimensionality but increased structural organization: GA expands an $n$-dimensional embedding space into a $2^n$ multivector algebra where base semantic concepts and their higher-order interactions are represented within a single, principled algebraic framework.

翻译：分布语义和神经语义方法几乎完全建立在传统线性代数之上：向量、矩阵、张量及其伴随运算。这些方法取得了显著的实证成功，但在组合语义、类型敏感性和可解释性方面仍面临结构性局限。本文论证几何代数（GA）——特别是克利福德代数——为语义表示提供了数学上更优越的基础，而功能性几何代数（FGA）框架则将几何代数扩展为一种类型化、组合化的语义，能够在保持与分布学习和现代神经架构完全兼容的同时，支持推理、变换和可解释性。本文阐述了形式基础，指明了几何代数提供而线性代数所不具备的三种核心能力，呈现了一个详细的工作示例以说明算子级语义对比，并展示了当前Transformer架构中已隐含的基于几何代数的运算如何被显式化并加以扩展。核心论点并非仅仅是维度的增加，而是结构化组织的增强：几何代数将n维嵌入空间扩展为$2^n$维多向量代数，使基础语义概念及其高阶交互能够在一个统一的、有原则的代数框架内得以表示。