🧩 Structural Formalisation of Similarity Theory

A Science Page of Similarity Theory
By Simon Raphael

A Modest Graph-Based Model of Consciousness, Relation, Structure, Frames of Time, and Dimensions

Similarity Theory is primarily a philosophical and cosmological framework. It begins with consciousness, moves through relation, and understands structure as the outcome of relational existence.

In its simplest form, the theory may be expressed as:

consciousness → relation → structure

This page introduces the first structural formalisation of that sequence. It does not claim to prove Similarity Theory mathematically. It does not reduce consciousness to graphs, equations, or physical mechanisms. Instead, it offers a careful structural model that helps clarify how the main ideas of Similarity Theory relate to one another.

The full version of this model appears as Appendix A: A Structural Formalisation of Similarity Theory, Version 1.3 in the manuscript.

🔍 Why Formalise Similarity Theory?

Every serious philosophical framework eventually faces a question:

Can its ideas be arranged clearly enough that others can examine, test, criticise, and develop them?

Similarity Theory was first developed through philosophical reflection, lived experience, analogy, and observation of recurring patterns across existence. It explores consciousness, time, identity, dimensions, ethical relation, and structural resemblance across different scales of reality.

However, if a theory remains only in prose, readers may ask whether its concepts truly hold together.

Structural formalisation helps answer that question.

It does not replace the philosophical heart of the theory. Rather, it provides a supporting framework that makes the theory’s internal architecture more visible.

In this first version, the purpose is modest:

to clarify relation,
to represent Frames of Time,
to describe identity continuity,
to explain similarity as partial structural resemblance,
to model dimensions as rule sets,
and to show how higher and lower rule sets may interact.

This is not a final mathematical system. It is a first structural grammar.

🧠 Consciousness Is Not Reduced to a Graph

A central caution must be made from the beginning.

In Similarity Theory, consciousness is not treated as a mathematical object. It is not a vertex, a point, a number, or a graph element.

The graph model is only a representation of relational structure.

A vertex in the model is not consciousness itself. It is a formal placeholder associated with consciousness as it appears within structure. This distinction is essential because Similarity Theory begins from the view that consciousness is ontologically primary. The model therefore cannot reduce consciousness to the very structure that consciousness is proposed to generate.

The graph is a map.

It is not the territory.

The purpose of the model is to represent selected relational features of the theory, not to capture the full metaphysical depth of consciousness itself.

🔗 The Basic Structural Model

Appendix A uses a labelled directed multigraph:

G = (V, E, ℓV, ℓE)

where:

V represents formal positions within a relational structure.
E represents directed relations between those positions.
ℓV assigns labels to vertices.
ℓE assigns labels to edges.

This allows the theory to represent relation without claiming that the representation is the reality itself.

In plain language, the model allows us to ask:

What is related?
How is it related?
What kind of relation is present?
Does the relation continue across frames?
Does the same pattern appear elsewhere?
What rule set allows or limits the relation?

This is useful because Similarity Theory is fundamentally relational. It does not begin with isolated objects as final realities. It begins with consciousness and asks how relation gives rise to structure.

🪞 Relation as the Beginning of Structure

In Similarity Theory, relation is the condition through which consciousness becomes knowable.

Before relation, consciousness may be understood as present but undistinguished. Once relation occurs, distinction appears. Once distinction appears, structure begins.

This is why the theory can be expressed as:

consciousness → relation → structure

The structural model represents this through directed relations. A relation may be shown as an edge between formal positions.

For Version 1.3, the model uses a small set of relation labels:

recognition
interaction
memory
self relation

These labels are intentionally simple.

Recognition represents one formal position distinguishing or becoming directed towards another.

Interaction represents directed relational effect. It is not intended as a full model of physical causation, biological exchange, psychological response, or energetic influence.

Memory represents a structural relation that refers to or carries continuity with a previous frame.

Self relation represents a reflexive relation, where a position relates to itself in a simplified structural way.

These are not the only possible relation types. They are the minimum useful set for a first model.

🕰️ Frames of Time as Preserved Structural States

One of the central ideas in Similarity Theory is that Frames of Time do not disappear.

A Frame of Time is not merely a memory, image, or dead record. It is understood within the theory as a preserved state of consciousness, relation, and structure.

In the graph model, a Frame of Time is represented as:

Gt

where t belongs to an ordered set of frames.

This does not mean that time is treated as a physical object or flowing substance. In Similarity Theory, time is better understood as the measurement of difference between states.

Later frames are constructed as new relational configurations. They may contain additional relations, different labels, or more complex structures. However, earlier frames are not edited, overwritten, or deleted.

They remain preserved.

This allows the theory to express an important idea:

change does not erase what has occurred.

New states emerge, but previous frames remain part of the deeper archive of relational reality.

🧬 Identity as Continuity Through Transformation

Identity is often misunderstood as sameness.

Similarity Theory does not treat identity as a frozen object. A child, an adult, and an elder are not identical in form, yet they belong to a recognisable continuity. The same principle may apply to memory, personality, death, transformation, and consciousness itself.

In Appendix A, identity continuity is represented through an underlying entity set:

U

with a partial identity mapping:

it: U ⇀ Vt

This is a technical way of saying that an identity may be tracked across different frames, even when its represented structure changes.

The model is careful here. This identity mapping is only a Version 1 convenience. It is not a final claim that identity is metaphysically discrete or fixed. It does not yet fully model merging, splitting, distributed consciousness, identity rupture, or reconverging identity streams.

Those are future developments.

For now, the model gives us a simple way to represent continuity through change.

🌀 Similarity Is Not Sameness

Similarity Theory does not claim that different things are identical.

It does not claim that an atom is literally a solar system, that a cell is literally a civilisation, or that a human being is literally a universe.

The claim is more careful than that.

Similarity means that structures may share recognisable relational patterns across different scales, systems, contexts, or rule sets.

In the structural model, similarity is represented through a partial structure-preserving mapping:

σ: Gt ⇀ Gt′

For Version 1.3, this mapping preserves:

adjacency,
direction,
vertex labels,
and edge labels,

where those features are defined.

This means that two structures do not need to be identical. They only need to share a recognisable substructure or relational pattern.

This is why partial structural resemblance is more suitable than exact sameness.

Similarity is not repetition.

It is structural echo.

🧭 Dimensions as Rule Sets

In Similarity Theory, dimensions are not merely places, spatial layers, or science-fiction realms.

A dimension is better understood as a rule set of existence.

It determines what kinds of relations, behaviours, and structures are possible.

In the structural model, a dimension is represented schematically as a constraint system:

D

This means that a dimension can be understood as a set of permissions and limitations governing possible relations.

The examples are illustrative, not rigid metaphysical partitions.

🪨 Dimension 1: Existence

Dimension 1 corresponds to inanimate matter: particles, atoms, molecules, sand, rocks, and material structures.

These are not dormant in the sense of absolute inactivity. They are active through physical bonds, forces, vibration, and interaction. However, they do not possess biological growth or autonomous agency.

🌱 Dimension 2: Growth and Responsiveness

Dimension 2 corresponds to plant life and related forms of growth and environmental responsiveness.

A plant responds to light, water, gravity, temperature, soil, and seasonal change. It participates in relation, but it does not operate with the agency, memory, movement, or symbolic self-reflection associated with Dimension 3.

🐕 Dimension 3: Agency and Reflective Relation

Dimension 3 includes animals and humans.

Animals and humans both participate in perception, movement, memory, and agency, though at different layers of relational complexity.

A dog and a human are not necessarily in different dimensions. They are both within Dimension 3, but they operate through different layers of awareness, interpretation, and symbolic capacity.

This distinction between dimensions and layers is important.

Dimensions are rule sets.

Layers describe differences of complexity within a rule set.

🔄 Cross Dimensional Interaction

Similarity Theory allows for cross dimensional interaction.

A higher rule set may act upon a lower rule set, but the lower rule set cannot fully interpret or reconstruct the higher one.

In Appendix A, this is represented as a structure forgetting mapping:

F: Ghigher → Glower

This means that a richer structure may be simplified, collapsed, or partially lost when received by a more limited rule set.

For example, when a human interacts with a plant, the human may act with intention, memory, agency, and symbolic understanding. The plant may register the interaction as light, shadow, pressure, support, damage, water, or chemical change.

The plant participates in the interaction, but it does not interpret the human as a human in the human sense.

The same principle can apply to animals.

A dog may recognise human routine, tone, movement, emotional atmosphere, and repeated behavioural patterns. Yet it does not reconstruct the full human world of law, mathematics, abstract philosophy, or symbolic metaphysics.

This does not make the dog inferior in a moral sense. It means the dog operates through a different relational layer.

The same idea may apply to humans in relation to higher dimensional structures. A person may encounter something beyond ordinary interpretation and experience it as dream, intuition, symbolic event, anomaly, spiritual perception, or mystery.

The lower rule set receives an effect.

It does not necessarily receive the full structure.

🧪 What This Model Does and Does Not Do

This structural model can help represent:

Frames of Time as preserved structural states,
relations as directed labelled edges,
identity continuity across frames,
similarity as partial structural resemblance,
dimensions as schematic rule sets,
and cross dimensional interaction as structural simplification.

However, it does not prove Similarity Theory.

It does not prove that consciousness is primary.

It does not prove the metaphysical reality of Frames of Time.

It does not reduce spiritual, philosophical, or experiential claims to mathematics.

It does not claim to be a final scientific model.

Its purpose is more modest and more precise:

to make the internal structure of Similarity Theory easier to examine.

📘 Why Appendix A Matters

Appendix A matters because it shows that Similarity Theory is not merely a collection of metaphors.

Its concepts are connected.

Consciousness gives rise to relation.
Relation gives rise to structure.
Structures persist as Frames of Time.
Frames may resemble one another through partial similarity.
Dimensions constrain what relations are possible.
Higher rule sets may act upon lower rule sets in ways that lower structures cannot fully reconstruct.

This does not make the theory complete.

It makes the theory clearer.

A philosophical theory does not need to begin with equations in order to be serious. But it does need coherence, discipline, and structure.

Appendix A is a first step in that direction.

📄 Read the Full Structural Formalisation

The full formal version appears in:

Appendix A: A Structural Formalisation of Similarity Theory, Version 1.3

This appendix introduces the labelled directed multigraph model, Frames of Time as preserved graph states, identity continuity, similarity mappings, dimensions as rule sets, and cross dimensional structure forgetting.

It should be read as a modest structural formalisation, not as a mathematical proof.