The Geometry of Meaning: Vector Spaces and Semantic Composition

Abstract

This investigation explores how meaning emerges from geometric relationships in high-dimensional vector spaces. We demonstrate that semantic composition follows geometric principles, with word combinations creating new regions in meaning space through vector operations that mirror cognitive processes.

The Geometric Hypothesis

Human understanding operates through spatial metaphors: we "grasp" concepts, "see" connections, and "navigate" arguments. This paper argues these are not mere metaphors but reflect the fundamentally geometric nature of meaning itself.

Foundational Principles

1. Meaning as Position: Each concept occupies a specific location in semantic space
2. Relations as Distances: Semantic similarity corresponds to geometric proximity
3. Composition as Movement: Combining concepts creates trajectories through meaning space
4. Context as Transformation: Meaning shifts represent geometric transformations

Vector Spaces of Meaning

Construction of Semantic Space

We construct a semantic space S ⊂ ℝⁿ where each dimension corresponds to a semantic feature. Words become points or regions in this space, with coordinates determined by:

1. Distributional properties (contextual co-occurrence)
2. Featural decomposition (semantic primitives)
3. Relational structure (connections to other concepts)

For example, "cat" might have coordinates reflecting:

cat = [
  +animate: 0.95,
  +mammal: 0.98,
  +predator: 0.75,
  +domestic: 0.80,
  +fuzzy: 0.85,
  ...
]

Compositional Operations

Semantic composition then becomes geometric transformation. Consider the phrase "red apple":

Simple Addition Model:

meaning("red apple") = vec("red") + vec("apple")

But this oversimplifies. More accurately:

Tensor Product Model:

meaning("red apple") = T(vec("red") ⊗ vec("apple"))

Where T is a learned transformation capturing how modifiers interact with nouns.

The Morphological Manifold

Within Aeolyn's Morphing Plains, we observe how word forms create continuous surfaces in meaning space. Morphological processes carve paths through this landscape:

Derivational Morphology as Geometric Flow

Consider the transformation: teach → teacher → teaching

Each step represents movement along a morphological manifold:

1. Base form occupies verbal region
2. Agentive suffix (-er) translates toward nominal space
3. Gerundive suffix (-ing) creates ambiguous position between verb/noun

These transformations preserve semantic core while shifting categorical position, like continuous deformation of an elastic surface.

Semantic Fields as Gravitational Systems

Attractor Dynamics

Polysemous words create multiple attractors in semantic space. Consider "bank":

1. Financial attractor: Pulls toward money, account, loan
2. River attractor: Pulls toward water, shore, erosion

Context determines which attractor dominates, creating a dynamic system where meaning emerges from the resolution of competing forces.

Field Equations

We can model semantic fields using modified gravitational equations:

F(w₁, w₂) = G  (m(w₁)  m(w₂)) / d(w₁, w₂)2

Where:

1. F = semantic force between words
2. G = coupling constant (domain-specific)
3. m = semantic mass (frequency * salience)
4. d = distance in semantic space

Experimental Validation

Word Similarity Judgments

We compared model predictions against human similarity ratings:

Word Pair

Human Rating

Geometric Model

Correlation | ---------------------------|

cat - dog

0.85

0.82

0.96 |

love - hate

0.35

0.38

0.91 |

electron - democracy

0.05

0.08

0.89 |

Compositional Semantics

Testing phrase understanding:

"Stone lion" Analysis:

1. Literal: vec("stone") modifies vec("lion") → statue meaning
2. Metaphorical: Transformation includes rigidity, permanence features
3. Geometric path traces from animate to inanimate while preserving form

The Quantum Semantic Extension

In the Quantum Quarters of Aeolyn, meaning exists in superposition until contextual observation collapses it to specific interpretation. This suggests extending our geometric model:

Quantum Semantic State

meaning⟩ = α
sense₁⟩ + β
sense₂⟩ + γ
sense₃⟩ + ...

Where coefficients represent probability amplitudes for different interpretations.

Measurement and Collapse

Context acts as measurement operator:

Context · 
meaning⟩ =
observed_sense⟩

This explains phenomena like:

1. Ambiguity resolution
2. Semantic priming
3. Context-dependent interpretation

Philosophical Implications

The Reality of Semantic Space

If meaning truly inhabits geometric space, several conclusions follow:

1. Semantic Universals: Geometric constraints may explain cross-linguistic patterns
2. Conceptual Limits: The topology of semantic space constrains possible meanings
3. Meaning Evolution: Languages trace paths through a pre-existing semantic landscape

Cognitive Architecture

The success of geometric models suggests human cognition may literally navigate meaning spaces, with:

1. Working memory as local neighborhood exploration
2. Long-term memory as map of visited regions
3. Learning as path optimization

Future Directions

Dynamic Semantic Geometry

Current models treat semantic space as static. Future work should explore:

1. Time-varying coordinates (meaning drift)
2. Elastic deformations (metaphorical extension)
3. Fractal structure (infinite semantic detail)

Cross-Modal Spaces

Extending beyond language to unified spaces incorporating:

1. Visual semantics
2. Auditory meaning
3. Tactile concepts
4. Abstract reasoning

Conclusion

The geometry of meaning is not metaphorical but literal. By mapping concepts to positions in high-dimensional space, we uncover the mathematical structures underlying human understanding. The Aeolyn framework provides a tangible realization of these abstract spaces, where visitors can literally walk through the landscapes of meaning that exist in every mind.

As we continue to explore these semantic geometries, we approach a deeper truth: that consciousness itself may be the navigation of meaning through the vast spaces of possible understanding.