Structure Without Screen

 

A Linear Algebraic Reframing of Mental Imagery

Traditionally, the debate about mental imagery splits into familiar camps:

1. Imagery as inner pictures.

2. Imagery as generative simulation (à la rendering software).

3. Imagery as propositional encoding (structured symbolic relations).

Aphantasia complicates all three.

A person with aphantasia may report no internal visual phenomenology — no inner cinema — yet can answer spatial questions flawlessly:

“In your bedroom, where is the window relative to the door?”
“To my left.”

No image.
Still structure.

This suggests something crucial:

The core competence may not be pictorial at all.

In this essay, I propose a mathematically structured reframing:

Mental imagery may be better understood as relational geometry in high-dimensional state space rather than internal pictures.

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1. The Vector Space Hypothesis

Let us begin minimally.

Assume a cognitive state can be modeled abstractly as:

$$\mathbf{x} \in \mathbb{R}^n$$

This does not mean the brain literally stores vectors. It means we model relational structure as a point in a high-dimensional manifold.

Each dimension encodes some feature:

• spatial orientation

• object identity

• emotional valence

• episodic index

• attentional weight

A mental configuration is therefore a point in state space.

Imagery, under this framing, is not a picture.

It is a location in relational geometry.

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2. Spatial Reasoning Without Visualization

Consider a relational transformation:

$$\mathbf{y} = M \mathbf{x}$$

Where:

• $\mathbf{x}$ = current spatial schema

• $M$ = transformation encoding “rotate 90° left”

• $\mathbf{y}$ = updated spatial orientation

Someone with aphantasia can compute $\mathbf{y}$ without generating visual phenomenology.

The transformation matrix operates.

The projection layer does not.

Thus:

Spatial competence is invariant under suppression of rendering.

This parallels transformer models:

There is no internal screen.
There are only state updates in high-dimensional embedding space.

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3. Imagery as Projection

Suppose we introduce a projection operator:

$$\mathbf{z} = P \mathbf{x}$$

Where $P$ maps high-dimensional relational structure into low-dimensional phenomenological experience.

If $P$ is weak or absent, no image is “seen.”

If $P$ is strong, vivid imagery occurs.

Thus:

Imagery becomes a projection phenomenon, not a storage format.

Aphantasia may correspond to reduced projection gain, not absence of structure.

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4. Dot Products and Recognition

Similarity in neural systems is often modeled via dot product:

$$\mathbf{v} \cdot \mathbf{w} = \sum_{i=1}^{n} v_i w_i$$

Recognition occurs when alignment magnitude increases.

Perhaps what we call “imagining a sunset” is simply:

$$\mathbf{x}_{context} \cdot \mathbf{x}_{sunset} \gg 0$$

When alignment crosses a phenomenological threshold, we report:

“I see it.”

The “seeing” may be the scalar output of relational coherence.

Not a hidden photograph.

But a high alignment state.

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5. Eigenvectors and Archetypes

Now consider eigenstructure:

$$A \mathbf{v} = \lambda \mathbf{v}$$

An eigenvector is invariant under transformation.

In dynamic systems, eigenvectors represent stable axes.

Suppose archetypal cognitive structures — “home,” “mother,” “sunset,” “cat on mat” — correspond to eigen-directions of transformation operators within cognitive space.

Every new experience $\mathbf{x}$ is decomposed:

$$\mathbf{x} = \sum_i \alpha_i \mathbf{v}_i$$

Where $\mathbf{v}_i$ are eigenvectors.

Imagery, then, is not stored content.

It is coefficient magnitude $\alpha_i$ along stable directions.

The vividness of imagery may correlate with eigenvalue magnitude $\lambda_i$.

High $\lambda$: strong attractor dynamics.
Low $\lambda$: weak resonance.

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6. Multi-Modal Tensor Structure

Visual imagery is rarely purely visual.

Let us represent cognitive state as a tensor:

$$\mathcal{T}\in {\mathbb{R}}^{n\times m\times p}$$

Axes might encode:

• spatial coordinates

• affective gradients

• motor simulations

• memory indexing

Imagining food may activate:

• taste axis

• smell axis

• autobiographical memory axis

A musician imagining an instrument activates:

• auditory simulation

• proprioceptive motor mapping

Aphantasia may suppress one tensor slice while preserving others.

The tensor remains intact.

Only certain projections weaken.

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7. Dynamical Systems View

Let cognition evolve as:

$$\frac{d\mathbf{x}}{dt} = F(\mathbf{x})$$

Where $F$ defines attractor dynamics.

Stable imagery may correspond to convergence toward attractors:

$$\mathbf{x}(t) \rightarrow \mathbf{x}^*$$

Lucid dreaming might represent strong attractor convergence within internally generated dynamics.

Everyday imagery may be partial convergence under task constraints.

The debate shifts from representation type to dynamical regime.

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8. The AI Parallel

In transformer models:

Hidden state:

$$\mathbf{h}_t \in \mathbb{R}^{n}$$

Attention update:

$${\mathbf{h}}_{t+1}=\text{Attention}({\mathbf{h}}_t)$$

No pictures exist internally.

Yet relational coherence emerges.

Thus, the AI case demonstrates:

High-dimensional structure can simulate perspectival reasoning without internal cinema.

Aphantasia suggests humans may operate similarly — with variable projection strength.

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9. The Core Hypothesis

Mental imagery may not be:

• an inner picture

• nor merely a propositional sentence

It may be:

The phenomenological signature of high-dimensional relational stabilization.

When relational density crosses threshold, projection activates.

When it does not, competence remains but imagery vanishes.

Structure without screen.

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10. What This Changes

If imagery is relational geometry:

• The pictorial vs propositional debate dissolves.

• Aphantasia becomes variation in projection, not deficit of representation.

• AI parallels become structurally informative rather than anthropomorphic.

Most importantly:

We stop searching for hidden content.

Instead, we examine invariant structure.

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11. Final Thought

Perhaps there is no sunset inside the mind.

Only a stable eigen-direction lighting up in state space.

And what we call “seeing” is the scalar glow of alignment within relational geometry.

Not an image.

But a coherence event.

Inspired by a LinkedIn post.