# Toward a Functional Geometric Algebra for Natural Language Semantics

arXiv: https://arxiv.org/abs/2604.25902 · HTML: https://mlsanity.com/paper/2604.25902/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 78 references that are in the map.
- First posted on arXiv 28 Apr 2026.

## Builds on (most cited first, 5 of 5)

- [CliffordNet: All You Need is Geometric Algebra](https://mlsanity.com/paper/2601.06793/) (2026)
- [Clifford Algebraic Rotor Embeddings : Maybe embeddings should start to CARE](https://mlsanity.com/paper/2511.11665/) (2025)
- [Versor: A Geometric Sequence Architecture](https://mlsanity.com/paper/2602.10195/) (2026)
- [A Novel Spinor-Based Embedding Model for Transformers](https://mlsanity.com/paper/2410.00038/) (2024)
- [Composing or Not Composing? Towards Distributional Construction Grammars](https://mlsanity.com/paper/2412.07419/) (2024)

## Abstract (arXiv)

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 2n multivector algebra where base semantic concepts and their higher-order interactions are represented within a single, principled algebraic framework.