How LLMs map language as mathematics—not definitions

Large language models are transforming how we understand word meaning through a mathematical approach that transcends traditional definitions. Unlike humans who categorize words in dictionaries, LLMs like GPT-4 place words in vast multidimensional spaces where meaning becomes fluid and context-dependent. This geometric approach to language represents a fundamental shift in how AI systems process and generate text, offering insights into both artificial and human cognition.

The big picture: LLMs don’t define words through categories but through location in high-dimensional vector spaces with thousands of dimensions.

• Each word exists as a mathematical point in this vast space, with its position constantly shifting based on surrounding context.
• The word “apple” might occupy one region when referring to fruit and completely different coordinates when referring to the technology company.

Behind the mathematics: When you type “apple” into an LLM, it transforms the word into a token mapped to a unique vector in 12,288-dimensional space.

• This initial vector represents a static first impression that then flows through neural network layers, being reweighted and reframed based on context.
• Words become geometric objects whose meaning is determined by their dynamic location rather than fixed definitions.

Why this matters: This approach represents a profound shift from the taxonomic, definition-based understanding of language to a fluid, contextual model.

• Traditional linguistics and AI systems organized words into taxonomies and categories, while vector-based systems allow for continuous meaning.
• The mathematical nature of these systems explains why LLMs can generate coherent language without truly “understanding” in the human sense.

Reading between the lines: LLMs reveal that language itself might be more mathematical and geometric than we previously realized.

• The success of these mathematical approaches suggests human language understanding might also rely on similar spatial-relational processes rather than strict definitions.
• This dimensional approach helps explain why human language is so adaptive and why words can instantly take on new meanings in different contexts.

The implications: Vector-based language processing opens new possibilities for AI systems to work with language in ways that mimic human flexibility.

• By representing meaning as geometry rather than definition, LLMs can handle nuance, ambiguity, and contextual shifts more effectively.
• This mathematical framework may ultimately provide insights into how our own brains process and understand language.