Subjective Numbers: A Category-Theoretic Construction

Abstract

This paper introduces subjective numbers, a novel mathematical framework constructed within category theory that formalizes intrinsic perspective as a fundamental property of mathematical objects. Traditional mathematics assumes objective universality, with mathematical truths independent of observer. In contrast, subjective numbers possess inherent "viewpoints" dictating their relationships with other mathematical entities. These perspectives create a mathematics where relations are perspective-dependent, challenging fundamental principles of equality and equivalence relations. We present a comprehensive category-theoretic construction, demonstrating emergent properties such as path-dependent truth and perspective incompleteness (where certain equivalences are discoverable only through cross-perspective inference). Rigorous proofs establish consistency relative to ZFC and non-triviality. This paradigm shift offers a mathematically robust approach to modeling phenomena where viewpoint and context are intrinsic determinants of truth and relation.

1. Introduction

1.1 Motivation and Central Idea

Mathematics, in its conventional form, operates under an implicit assumption of objective universality: mathematical truths are considered independent of any observer or context. This foundational principle of objectivity, particularly powerful in fields prioritizing symmetry and invariance, introduces significant limitations in mathematics' ability to formally represent perspective-dependent phenomena. The elegance and efficacy of traditional mathematics are deeply rooted in symmetry, from the symmetric nature of equality itself to the commutative properties of fundamental operations and the symmetries underpinning geometric understanding. However, this focus on symmetry presents challenges when formalizing inherently asymmetrical, directional, or viewpoint-dependent patterns.

Consider a basic symmetry principle: in conventional mathematics, if a = b, then necessarily b = a. Yet, in many contexts, evaluations fundamentally defy this symmetry. Evaluations of quality, preference, or relevance are often directional and perspective-dependent. In social dynamics, one agent may perceive another as equal, while the reverse perception may not hold. In cognitive systems, information relevance is viewpoint-specific. This fundamental asymmetry of relations manifests across diverse domains, yet lacks precise mathematical formalization within standard frameworks.

This paper introduces subjective numbers as a rigorous mathematical solution to this representational gap. A subjective number is constructed within category theory to inherently embody not just a value, but an intrinsic perspective that fundamentally governs its relationships with other mathematical entities. Unlike approaches that treat perspective as an external parameter, we construct subjective numbers as mathematical objects where perspective is an intrinsic property, defined through category-theoretic morphisms that formalize how perspectives relate and interact. In our notation for the relation function R_p(s, t, a, b), the first perspective p is always the vantage perspective evaluating the relationship between subjective numbers with perspectives s and t and values a and b.

For instance, a subjective number 5_(p) might consider itself equal to 5_(q) from its perspective (denoted 5_(p) =_p 5_(q)), while 5_(q) might not reciprocate this evaluation from its perspective (denoted ¬(5_(q) =_q 5_(p))). This inherent asymmetry represents a foundational departure from traditional mathematical assumptions, yet provides a consistent framework for modeling perspective-dependent phenomena. Subjective numbers offer a novel mathematical lens for examining systems where viewpoint, directionality, and intrinsic perspective are not merely contextual factors but are mathematically constitutive of the objects and relations themselves. This framework allows for the formalization of path-dependent truths, perspective incompleteness, and non-symmetric relations within a rigorous mathematical structure.

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1.2 Key Distinctions and Innovations

Subjective numbers are distinguished from related concepts by several key innovations:

• Category-Theoretic Construction: Subjective numbers are rigorously constructed within existing mathematical frameworks—specifically category theory—analogous to the construction of complex numbers from real numbers. This constructive approach centers on morphisms that intrinsically embody perspective through explicit relation functions and composition rules that preserve perspective information. Unlike systems that simply parameterize existing mathematical objects by perspective labels, our construction builds perspective into the fundamental identity of the mathematical objects themselves.
• Intrinsic Perspective: Perspective is not treated as an external parameter but as an intrinsic property of each subjective number, deeply embedded within its mathematical identity through the categorical representation of subjective numbers as families of morphisms. These morphisms formalize how perspectives interact, transform, and evaluate each other, creating a unified framework where perspective is a fundamental mathematical concept rather than just contextual metadata.
• Local Consistency, Global Asymmetry: The framework rigorously maintains logical consistency within each perspective, ensuring mathematical coherence at the local level. Simultaneously, it allows for controlled asymmetry and perspective-dependence across perspectives, enabling the modeling of phenomena characterized by viewpoint-dependent relations and potential global inconsistencies without systemic collapse.
• Beyond Contextual Logic: While modal and paraconsistent logics address context-dependent truth values, subjective numbers extend context-dependence to the very nature of mathematical objects themselves. This represents a more foundational shift than simply modifying logical operators to accommodate context.

2. Preliminaries and Metatheoretical Framework

2.1. Category Theory Background

We briefly recall essential category theory concepts used in our construction. For readers less familiar with category theory, we present these concepts intuitively before providing formal definitions.

Intuitively, a category consists of objects (which can be thought of as mathematical structures) and morphisms (which represent structure-preserving maps between objects). The key feature of category theory is that it focuses on relationships between objects rather than their internal structure. This relationship-centric approach makes category theory particularly suitable for our construction of subjective numbers, where perspective-dependent relationships are fundamental.

Definition 2.1 (Category): A category 𝒞 consists of:

1. A collection of objects, denoted by Ob(𝒞).
2. For each pair of objects A, B ∈ Ob(𝒞), a set Mor_𝒞(A, B) of morphisms from A to B. If f ∈ Mor_𝒞(A, B), we write f: A → B.
3. For each triple of objects A, B, C ∈ Ob(𝒞), a composition law that assigns to each pair of morphisms f: A → B and g: B → C a morphism g ∘ f: A → C, called the composition of g and f.
4. For each object A ∈ Ob(𝒞), an identity morphism id_A: A → A.

These data are subject to the following axioms:

• Associativity of Composition: For morphisms f: A → B, g: B → C, and h: C → D, we have h ∘ (g ∘ f) = (h ∘ g) ∘ f.
• Identity Morphisms: For each perspective p ∈ P, the identity morphism is defined by id_p = f_{(idV, p → p)}, where id_V ∈ V denotes the designated identity element of the value space. For the standard case where V = ℝ, this identity element is 0, making id_p = f_{(0, p → p)}. For the case where V is a Boolean algebra, the identity element is true₀, making id_p = f_{(true0, p → p)}. This choice of identity element ensures that when we later define operations in Section 6, the identity morphism composes appropriately with other morphisms while preserving the algebraic value semantics: if f_{(a, p → q)} represents a subjective number with value a, then f_{(a, p → q)} ∘ id_p = f_{(a, p → q)} both as a morphism and in terms of the represented value a.

Definition 2.2 (Quotient Category): A quotient category is formed when we define an equivalence relation on the morphisms of a category that is compatible with composition. Two morphisms are considered equivalent if they belong to the same equivalence class. The objects of the quotient category remain the same as the original category, but the morphisms become equivalence classes of the original morphisms. For a quotient category to be well-defined, the equivalence relation must satisfy the following compatibility conditions:

1. If f ≡ g, then dom(f) = dom(g) and cod(f) = cod(g) (i.e., equivalent morphisms must have the same source and target objects).
2. If f₁ ≡ f₂ and g₁ ≡ g₂, and if f₁ and g₁ are composable, then f₁ ∘ g₁ ≡ f₂ ∘ g₂ (i.e., composition must be well-defined on equivalence classes).
3. For each object A, id_A ≡ id_A (i.e., identity morphisms must be equivalent only to themselves).

These conditions ensure that the quotient category preserves the essential structural properties of the original category while grouping morphisms according to the desired equivalence relation.

Functors are structure-preserving maps between categories, capturing how entire categorical structures relate to each other:

Definition 2.3 (Functor): A functor F: 𝒞 → 𝒟 between categories 𝒞 and 𝒟 consists of:

1. A mapping F: Ob(𝒞) → Ob(𝒟) that assigns to each object A in 𝒞 an object F(A) in 𝒟.
2. For each pair of objects A, B in 𝒞, a mapping F_A,B: Mor_𝒞(A, B) → Mor_𝒟(F(A), F(B)) that assigns to each morphism f: A → B in 𝒞 a morphism F(f): F(A) → F(B) in 𝒟.

These mappings must preserve composition and identity morphisms:

• F(g ∘ f) = F(g) ∘ F(f) for all composable morphisms f and g.
• F(id_A) = id_F(A) for every object A in 𝒞.

2.2. Metatheoretical Framework for Subjective Mathematics

Before proceeding to the formal construction of subjective numbers, we must establish the metatheoretical framework within which we reason about perspective-dependent mathematics. This addresses a fundamental question: How do we, as mathematicians, study a system where mathematical truth itself depends on perspective?

2.2.1 Metatheoretical Stance: In developing and analyzing subjective numbers, we adopt the standard metatheoretical stance of mathematics. This stance allows us to reason about the entire system of perspectives while remaining outside that system. Just as a set theorist reasons about sets without requiring an axiom positing the existence of a "meta-set theorist," we reason about perspective-dependent mathematics without requiring an axiomatized "meta-perspective" within the system.

This approach recognizes the distinction between:

1. Object-level mathematics: The subjective numbers themselves, with their intrinsic perspectives and perspective-dependent relations
2. Meta-level mathematics: Our reasoning about the system of subjective numbers, which follows standard mathematical methodology

When we make statements about "all perspectives" or system-wide properties, these are metatheoretical observations about the subjective numbers framework, not statements made from within any particular perspective in the system.

2.2.2 Philosophical Foundation of Perspective: Within our framework, a perspective is not merely a formal index or label but represents a mathematical vantage point with its own internally consistent logic and evaluation criteria. Each perspective represents:

1. A coherent system of evaluation with its own standards of equality and relation
2. An origination point for mathematical assertions that may differ from other perspectives
3. A mathematical entity that can itself be studied, related to other perspectives, and combined through various operations

This philosophical foundation grounds our mathematical formalization while clarifying that perspectives are not arbitrary but represent meaningful mathematical viewpoints with their own internal structure.

2.2.3. Metatheoretical Principle of Static Continuity

Formal expression: ∀p, q, r ∈ P, ∀a, b, c ∈ V, ∀φ ∈ Operations: (a_(p) φ_p b_(q) = c_(r) at time t₁) → (a_(p) φ_p b_(q) = c_(r) at time t₂).

Motivation: This principle does not forbid the introduction of time. Rather, it specifies that if time is to be part of the framework, it must be introduced explicitly. It establishes a baseline of stability for operations in the subjective numbers framework, ensuring predictability and mathematical coherence in the absence of explicitly time-dependent elements. This principle allows us to initially focus on the core innovation of perspective-dependence, setting a foundation for controlled extensions into dynamic systems in future work.

2.3. Formal Definitions

2.3.1 Relation Function Formalization

The relation function R_p for each perspective p is a core component of our framework. Formally, we define:

For any perspectives s, t ∈ P and values a, b ∈ V, the expression R_p(s, t, a, b) indicates whether perspective p considers the subjective number a_(s) to be equal to b_(t).

Note on “Undefined” vs. “False”: Throughout examples, we may informally say the relation R_p(s, t, a, b) is undefined when perspective p has no information about a_(s) and b_(t). Formally, however, R_p always maps into {true, false}. Whenever R_p(s, t, a, b) is not explicitly declared true, it defaults to false. Hence, undefined is just an informal way of saying not recognized as equal in perspective p.

When a relation involves perspective p itself, we often write R_p(p, q, a, b) to denote how p relates its own value a to perspective q's value b.

This characterization ensures that while perspective p may differ from others in how it evaluates equality, it remains internally coherent and respects basic numeric identities.

2.3.2. Cross-Perspective Adoption Function

This function represents a form of "trust" or "acceptance" between perspectives. When CPA_p(q, r) = true, perspective p is willing to incorporate q's judgments about r into its own reasoning. The Cross-Perspective Inference axiom (Axiom 4, see Section 4) references CPA_p directly: if p trusts q's evaluation about r, then p extends transitivity across perspectives accordingly. Formally,

The Cross-Perspective Inference axiom (Axiom 4) references CPA_p directly: if p trusts q’s evaluation about r, then p extends transitivity across perspectives accordingly. Formally,

We treat CPA_p(q,r) as a primitive function in the system, so there is no circular definition: the axiom system introduces CPA_p as a parameter, and Axiom 4 describes precisely how cross-perspective acceptance connects to multi-perspective transitivity. This ensures that each perspective retains control over which other viewpoints it “trusts,” preventing unrestricted chains of inference that might lead to contradictions.

2.3.3 Perspective Transition Notation

For any two perspectives p, q ∈ P, we introduce the notation a_{(p → q)} to represent a subjective number with value a that temporarily adopts perspective q while originating from perspective p. This perspective transition construct is essential for defining how perspectives can shift and interact with the cross-perspective adoption function. Formally:

a_{(p → q)} is a subjective number with relation function R_{p → q} such that for all r ∈ P and all x, y ∈ V:

This means that a_{(p → q)} evaluates relations in exactly the same way that perspective q would, despite having originated from perspective p. This notation provides a formal bridge between our categorical construction and the axiom system, especially for Axiom 6 (Perspective Adoption).

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3. Formal Framework and Definitions

3.1. Definition of the Category 𝓢𝓝

We define the category 𝓢𝓝 (the category of subjective numbers) as follows:

1. Objects: The objects of 𝓢𝓝 are the perspectives, which form a non-empty set P = {p, q, r, ...}.
2. Morphisms: For any two perspectives p, q ∈ P and for any value a ∈ V (where V is a value space such as ℝ or ℂ), we define a morphism f_{(a, p → q)}: p → q. The set of morphisms from p to q is given by:
   Mor_𝓢𝓝(p, q) = {f_{(a, p → q)} | a ∈ V}
3. Composition: For morphisms f_{(a, p → q)}: p → q and f_{(b, q → r)}: q → r, we define composition as:
   f_{(b, q → r)} ∘ f_{(a, p → q)} = f_{((a,b), p → r)}
   This enhanced composition rule records both values in an ordered pair, preserving information about the intermediate perspective. This approach:
   • Records both the original value and intermediate value, creating a richer structure that captures the path of perspective transitions
   • Makes morphisms "path-aware," which is crucial for modeling domains where evaluation history matters
   • Forms a quotient category structure through explicit equivalence relations to satisfy categorical axioms

   Morphism Composition in Practice

   Let's illustrate the composition of morphisms with a concrete example. Consider three perspectives p, q, and r, with values in ℝ. Suppose we have:

   • A morphism f_{(3, p → q)}: p → q representing value 3 transitioning from perspective p to q
   • A morphism f_{(5, q → r)}: q → r representing value 5 transitioning from perspective q to r

   To compose these morphisms, we follow our composition rule:

   f_{(5, q → r)} ∘ f_{(3, p → q)} = f_{((3,5), p → r)}

   The resulting morphism f_{((3,5), p → r)}: p → r records the history of values (first 3, then 5) as we transition from perspective p to r via q.

   To understand why this composition stores the value history, consider a case where values undergo transformations between perspectives. For example, if perspective q doubles incoming values and perspective r adds 1 to incoming values, then:

   • The value 3 from perspective p becomes 6 in perspective q
   • The value 6 from perspective q becomes 7 in perspective r

   By recording (3,6) in our morphism, we capture this transformation history, which would be lost if we only stored the final value 7. This is crucial for scenarios where the sequence of perspectives matters and not just the final result.

   To simplify notation in cases where the complete value history isn't needed, we define a projection function π₁((a,b)) = a that extracts the first component.
4. Identity Morphisms: For each perspective p ∈ P, the identity morphism is defined by id_p = f_{(idV, p → p)}, where id_V ∈ V denotes the designated identity element of the value space. For the standard case where V = ℝ, this identity element is 0, making id_p = f_{(0, p → p)}. For the case where V is a Boolean algebra, the identity element is true₀, making id_p = f_{(true0, p → p)}. This choice of identity element ensures that when we later define operations in Section 6, the identity morphism composes appropriately with other morphisms while preserving the algebraic value semantics: if f_{(a, p → q)} represents a subjective number with value a, then f_{(a, p → q)} ∘ id_p = f_{(a, p → q)} both as a morphism and in terms of the represented value a.

3.2. Relationship Between Categorical Construction and Relation Functions

The connection between the categorical morphisms and the relation functions is established through the following equivalence:

For perspectives p, q ∈ P and values a, b ∈ V:

This means that subjective equality from perspective q between subjective numbers a_(p) and b_(t) corresponds precisely to the relation function R_q evaluating to true when comparing these subjective numbers. This equivalence provides the crucial link between our categorical construction and the axiom system presented in Section 4, allowing us to move seamlessly between the two formulations.

3.2.1. Quotient Category Construction

In order to define the category 𝓢𝓝 properly, we introduce an equivalence relation ≡ on the morphisms to identify certain morphisms as "the same" in the quotient. Specifically, for values a, b in the underlying space V, we let f_{(a, p→q)} and f_{(b, p→q)} be two morphisms with the same source p and target q. We say f_{(a, p→q)} ≡ f_{(b, p→q)} if and only if their respective "value histories" belong to the same class under our value-history equivalence (denoted ∼).

Definition 3.2.1 (Value-History Equivalence Relation): We define the equivalence relation ∼ on value histories as the smallest equivalence relation satisfying the following conditions:

1. Identity Properties:
   • For any value a ∈ V, (id_V, a) ∼ a and (a, id_V) ∼ a, where id_V is the identity element in V.
2. Associativity: For any values a, b, c ∈ V, ((a, b), c) ∼ (a, (b, c)).
3. Closure Properties: The relation ∼ is closed under:
   • Reflexivity: For all value histories h, h ∼ h.
   • Symmetry: If h₁ ∼ h₂, then h₂ ∼ h₁.
   • Transitivity: If h₁ ∼ h₂ and h₂ ∼ h₃, then h₁ ∼ h₃.

Definition 3.2.2 (Morphism Equivalence Relation): Two morphisms

are declared equivalent (written f_{(h1, p → q)} ≡ f_{(h2, p → q)}) if and only if:

1. They have the same source and target objects (i.e., the same p and q), and
2. Their value histories h₁ and h₂ are equivalent under the value-history equivalence relation ∼.

Proposition 3.2.1 (Compatibility with Composition): The morphism equivalence relation ≡ is compatible with morphism composition. That is, if f₁ ≡ f₂ and g₁ ≡ g₂, and if the compositions g₁ ∘ f₁ and g₂ ∘ f₂ are defined, then g₁ ∘ f₁ ≡ g₂ ∘ f₂.

This quotient construction yields the category 𝓢𝓝 where morphisms are equivalence classes under ≡, ensuring that the category structure is well-defined. In particular, this construction guarantees that:

• f_{(a, p→q)} ∘ id_p ≡ f_{(a, p→q)} (right identity)
• id_q ∘ f_{(a, p→q)} ≡ f_{(a, p→q)} (left identity)
• (f_{(c, r→s)} ∘ f_{(b, q→r)}) ∘ f_{(a, p→q)} ≡ f_{(c, r→s)} ∘ (f_{(b, q→r)} ∘ f_{(a, p→q)}) (associativity)

When we work with the category 𝓢𝓝, we are actually working with these equivalence classes rather than with the individual morphisms themselves. This abstraction enables us to focus on the essential structure of perspective transitions without being distracted by irrelevant distinctions in representation.

3.3. Verification of Category Axioms for 𝓢𝓝

3.4. Perspective Distance Function

We define the perspective distance d(p, q) = ∫_{V × V} |R_p(p, q, a, b) − R_q(q, p, b, a)| dμ(a,b). For this integral to be well-defined, we fix a σ-finite measure μ on V×V (typically a product of Lebesgue measures on a bounded domain, or any measure for which the indicator functions arising from R_p remain integrable).

In particular, we require:

• Local Boundedness: If V is unbounded (like ℝ), then R_p and R_q must differ on at most a region of measure zero beyond a bounded subset, ensuring the integral converges.
• Diagonal Sets: When we encode true as 1 and false as 0 (effectively using characteristic functions), we treat the “diagonal” {(a,b) : a = b} in ℝ² as measure-zero under standard Lebesgue measure, so μ({(a,b) : a = b}) = 0. This ensures d(p, q) can be computed by integrating over all pairs (a,b) where a ≠ b without extra terms.

Under these assumptions, Theorem 3.2 holds: d(p, q) is finite for standard examples like comparison relations or ≤-based definitions. If V is unbounded, we either (1) impose that R_p differs from R_q only on a subset of bounded measure, or (2) define d_K(p, q) restricted to a compact region K and then consider a limit as K expands, whenever the limit exists. Thus the definition can be adapted to preserve well-definedness in various practical settings.

3.4.1 Measure-Theoretic Specification for Perspective Distance

We formalize our definition of the perspective distance function to ensure its well-definedness on both bounded and unbounded domains under rigorous measure-theoretic principles.

Definition 3.3 (Measure for Perspective Distance): Let V be equipped with a σ-algebra Σ_V (e.g., the Borel σ-algebra if V = ℝ). We define the product measure space (V × V, Σ_V ⊗ Σ_V, μ) where:

• Σ_V ⊗ Σ_V is the product σ-algebra generated by all measurable rectangles in V × V.
• μ is a σ-finite product measure (typically the product of Lebesgue measures when V = ℝ).

Measurability Requirement: For the perspective distance function d(p, q) = ∫_V×V |R_p(p, q, a, b) − R_q(q, p, b, a)| dμ(a,b) to be well-defined, we require that for all perspectives p, q in P, the function (a,b) ↦ |R_p(p,q,a,b) − R_q(q,p,b,a)| is measurable with respect to Σ_V ⊗ Σ_V.

Numerical Encoding for Integration: For integration purposes in the perspective distance function, we numerically encode true as 1 and false as 0, creating characteristic functions of the form:

• χ_{Rp(p,q,·,·)}(a,b) = 1 when R_p(p,q,a,b) = true
• χ_{Rp(p,q,·,·)}(a,b) = 0 when R_p(p,q,a,b) = false

This encoding ensures that the sets {(a,b) ∈ V × V : R_p(p,q,a,b) = true} are measurable subsets of V×V, allowing the perspective distance function to be well-defined. Under standard assumptions (e.g., when the relation functions are defined using Borel-measurable operations like comparisons, arithmetic functions, or continuous transformations of values), the measurability condition is naturally satisfied.

Localized Perspective Distance for Unbounded Domains: For practical computation and to ensure convergence for unbounded domains, we restrict the integral to a compact set K ⊂ V × V:

As K increases to cover V×V, we can define d(p, q) = lim_K→V×V d_K(p,q) when this limit exists. This approach ensures d(p,q) remains finite when R_p and R_q satisfy appropriate boundedness conditions.

Convergence Conditions: The perspective distance function is guaranteed to converge in the following cases:

1. When V is a bounded domain (e.g., a closed interval [a,b] ⊂ ℝ).
2. When the difference between relation functions vanishes outside a bounded subset of V×V.
3. When the relation functions exhibit "locality properties" where perspectives primarily differ in their evaluations of nearby values.

This distance function provides a concrete mathematical tool for quantifying differences between perspectives, which is valuable for both theoretical analysis and practical applications of subjective numbers.

4. Axioms for Subjective Mathematics

We present a formal axiom system for subjective numbers. These axioms provide the rigorous foundation for subjective mathematics by specifying the essential properties and operations of subjective numbers. This axiomatic system is directly connected to the categorical construction presented in Section 3, with each axiom corresponding to specific properties of the category 𝓢𝓝.

Axiom 1 (Subjective Reflexivity): For any subjective number a_(p), a_(p) =_p a_(p).

Formal expression: ∀p ∈ P, ∀a ∈ V: R_p(p, p, a, a) = true.

Motivation: This axiom guarantees that every subjective number is self-identical from its own perspective, reflecting the identity property in 𝓢𝓝. Categorically, this corresponds to the identity morphism property in the category 𝓢𝓝.

Axiom 2 (Non-Symmetric Equality): If a_(p) =_p b_(q), this does not necessarily imply that b_(q) =_q a_(p).

Formal expression: ∃p,q ∈ P, ∃a,b ∈ V: R_p(p, q, a, b) ≠ R_q(q, p, b, a).

Motivation: This axiom introduces perspective-dependence by allowing the evaluation of equality to differ between perspectives. In the categorical framework, this corresponds to the potential asymmetry in morphisms between perspectives. Example 5.1 demonstrates this axiom.

Axiom 3 (Subjective Transitivity): For any perspectives ℓ, p, q, r ∈ P and any values a, b, c ∈ V, if R_ℓ(p, q, a, b) and R_ℓ(q, r, b, c) are true, then R_ℓ(p, r, a, c) must be true.

Formal expression: ∀ℓ, p, q, r ∈ P, ∀a, b, c ∈ V: (R_ℓ(p, q, a, b) ∧ R_ℓ(q, r, b, c)) → R_ℓ(p, r, a, c).

Motivation: This axiom ensures that the evaluation within a single perspective is consistent and transitive. In category theory terms, this corresponds to the associativity of composition in the category 𝓢𝓝.

Axiom 4 (Cross-Perspective Inference): If a_(p) =_p b_(q) and b_(q) =_q c_(r), then a_(p) =_p c_(r) if and only if CPA_p(q, r) = true.

Formal expression: ∀p, q, r ∈ P, ∀a, b, c ∈ V: (R_p(p, q, a, b) ∧ R_q(q, r, b, c) ∧ CPA_p(q, r)) ↔ R_p(p, r, a, c).

The bidirectional implication (↔) in Axiom 4 is necessary for two key reasons:

1. The forward direction (→) ensures that when perspective p trusts q's evaluation of r, transitivity can extend across perspectives.
2. The reverse direction (←) ensures that transitivity across perspectives must be explicitly authorized through the CPA function, preventing uncontrolled inference chains.

Without this bidirectional constraint, the framework could either allow unauthorized transitive inferences or fail to properly utilize authorized trust relationships. In the category 𝓢𝓝, this axiom governs how morphisms can be composed across different perspectives while preserving the proper evaluation semantics.

Motivation: This axiom governs the conditions under which transitivity extends across different perspectives.

Axiom 5 (Value Consistency): If a_(p) =_p b_(q) and a = c (in V), then c_(p) =_p b_(q).

Formal expression: ∀p, q ∈ P, ∀a, b, c ∈ V: (R_p(p, q, a, b) ∧ (a = c)) → R_p(p, q, c, b).

Motivation: This axiom ensures that the inherent numerical equality is preserved within any subjective evaluation, reflecting the category-theoretic property that morphisms with the same numerical value are equivalent when their source and target perspectives are the same.

Axiom 6 (Perspective Adoption): A subjective number a_(p) may temporarily adopt another perspective q, denoted by a_{(p → q)}, to evaluate relations as if observed from q.

Formal expression: ∀p, q ∈ P, ∀a ∈ V: Define a_{(p → q)} such that ∀r ∈ P, ∀b ∈ V: R_{p → q}(p → q, r, a, b) = R_q(q, r, a, b).

Motivation: This axiom provides a mechanism for shifting perspectives, which is crucial for dynamic interactions. In the category 𝓢𝓝, this corresponds to the perspective transition notation introduced in Section 2.3.3 and used in defining morphisms.

Axiom 7 (Perspective Distinctness): For any two distinct perspectives p and q in P, there must exist at least one pair of subjective numbers a_(p) and b_(q) such that R_p(p, q, a, b) ≠ R_q(q, p, b, a).

Formal expression: ∀p, q ∈ P: (p ≠ q) → ∃a, b ∈ V: R_p(p, q, a, b) ≠ R_q(q, p, b, a).

Motivation: This axiom ensures that distinct perspectives yield non-trivial differences in evaluations, which corresponds to the category-theoretic property that different objects in 𝓢𝓝 exhibit distinct morphism behaviors.

4.1 Local Consistency

4.2 Independence of Axiom 7

4.3 Minimal Axiomatization

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5. Properties of Subjective Numbers

From our category-theoretic construction and axiomatization of subjective numbers, several important mathematical properties emerge naturally. This section examines these properties to develop a deeper understanding of the framework.

Property 5.1 (Subjective Reflexivity): For any subjective number a_(p), it holds that a_(p) =_p a_(p). Subjective equality is reflexive from within its own perspective. This property is derived from Axiom 1.

This property derives directly from Axiom 1 and is a fundamental characteristic of the framework. It establishes that every subjective number equals itself from its own perspective, providing a consistent starting point for evaluations.

Property 5.2 (Non-Symmetric Equality): If a_(p) =_p b_(q), it is not necessarily the case that b_(q) =_q a_(p). Subjective equality is inherently non-symmetric. This property is derived from Axiom 2.

This property, which follows from Axiom 2, represents a foundational departure from classical mathematics. It formalizes the notion that evaluations can be direction-dependent and perspective-specific, capturing asymmetric relations that occur in many complex systems.

Property 5.3 (Subjective Transitivity): If a_(p) =_r b_(q) and b_(q) =_r c_(s), then a_(p) =_r c_(s). Subjective equality is transitive from the perspective of the evaluator (r in this case). This property is derived from Axiom 3.

This property, derived from Axiom 3, ensures that reasoning within a single perspective remains consistent. While equality across perspectives may be asymmetric, the framework maintains logical coherence within each individual perspective.

Property 5.4 (Controlled Cross-Perspective Inference): If a_(p) =_p b_(q) and b_(q) =_q c_(r), then a_(p) =_p c_(r) holds if and only if CPA_p(q, r) = true. Transitivity across perspectives is controlled and conditional, not automatic. This property is derived from Axiom 4.

This property, which follows directly from Axiom 4, regulates how information and evaluations propagate across different perspectives. It formalizes the idea that accepting another perspective's evaluation is an explicit choice governed by the CPA function.

Property 5.5 (Value Consistency): If a_(p) =_p b_(q) and a = c as numerical values (in the value space V), then c_(p) =_p b_(q). If two subjective numbers share the same numerical value when viewed from a given perspective, their subjective equality relations from that perspective will be consistent with respect to value substitution. This property is derived from Axiom 5.

Derived from Axiom 5, this property ensures that subjective evaluations respect the underlying numerical equality in the value space V. This maintains mathematical consistency while allowing for the novel features of subjective evaluation.

6. Operations and Algebraic Structure

To make subjective numbers a functional mathematical framework, we define operational rules for combining and manipulating them, with particular attention to perspective propagation through operations. This section establishes the algebraic structure of subjective numbers and proves key properties that distinguish it from classical number systems.

6.1 Perspective Propagation in Operations

When subjective numbers are combined through operations, determining the perspective of the resulting subjective number is crucial. We define three fundamental perspective propagation rules with formal definitions and illustrated examples.

Comparison of Perspective Propagation Rules

Definition 6.1 (Dominant Perspective Rule): In this rule, the perspective of the first operand in an operation becomes the dominant perspective, determining the perspective of the result.

Formal Definition: For any binary operation φ: V × V → V on values in V and subjective numbers a_(s) and b_(t), the dominant perspective rule yields:

where the subscript d in +_d indicates the dominant perspective. The resulting subjective number inherits perspective s from the first operand.

Definition 6.2 (Perspective Fusion Rule): In this rule, the perspectives of both operands are combined to create a new, composite perspective for the result.

Formal Definition: For a binary operation φ: V × V → V and perspective fusion, the rule is:

where s ⊕ t represents a new, composite perspective with a relation function defined as:

for any perspective r and values x, y ∈ V. Here, f: {true, false} × {true, false} → {true, false} is a fusion function that combines evaluations from perspectives s and t. Common choices for f include:

• Conjunctive fusion: f(u, v) = u ∧ v
• Disjunctive fusion: f(u, v) = u ∨ v
• Weighted fusion: f(u, v) = 𝟙_{w1u + w2v > θ} for weights w₁, w₂ ∈ [0,1] with w₁ + w₂ = 1 and threshold θ ∈ [0,1]

The choice of fusion function depends on the specific application and determines how the composite perspective evaluates relations with other perspectives.

Definition 6.3 (Novel Perspective Rule): In this rule, the operation generates a completely new perspective for the result, distinct from the perspectives of either operand.

Formal Definition: For operation φ: V × V → V and novel perspective generation:

where u is a new perspective object distinct from s and t. The relation function R_u for this novel perspective is defined as:

where g: {true, false} × {true, false} × [0,1] → {true, false} is a generation function that creates new perspective relations, and η ∈ [0,1] is a source of randomness or creativity in the generation process.

The parameter η can be systematically defined in various ways:

• As a fixed real number in [0,1] representing a bias toward one perspective
• As a random variable with a defined distribution
• As a function of the values and perspectives involved: η = h(a, b, s, t) for some function h: V × V × P × P → [0,1]

This allows for both deterministic and non-deterministic novel perspective generation.

6.2 Formal Definitions of Basic Operations

Using the perspective propagation rules established in 6.1, we formally define the basic arithmetic operations for subjective numbers.

Definition 6.4 (Subjective Addition): For subjective numbers a_(s) and b_(t), their sum under perspective propagation rule r ∈ {dominant, fusion, novel} is defined as:

a_(s) +_r b_(t) = (a + b)_(u)

where the perspective u of the resulting subjective number is determined by rule r applied to perspectives s and t.

Definition 6.5 (Subjective Multiplication): For subjective numbers a_(s) and b_(t), their product under perspective propagation rule r is:

a_(s) ×_r b_(t) = (a × b)_(u)

where again, the perspective u is determined by rule r.

Definition 6.6 (Subjective Subtraction): For subjective numbers a_(s) and b_(t), their difference under perspective propagation rule r is:

a_(s) -_r b_(t) = (a - b)_(u)

Definition 6.7 (Subjective Division): For subjective numbers a_(s) and b_(t) with b ≠ 0, their quotient under rule r is:

a_(s) ÷_r b_(t) = (a ÷ b)_(u)

Definition 6.8 (Subjective Power): For subjective numbers a_(s) and b_(t), the power operation under rule r is:

a_(s) ^_r b_(t) = (a^b)_(u)

where a^b represents exponentiation in the value space V, and perspective u is determined by rule r.

6.3 Algebraic Properties

The introduction of perspective into mathematical operations creates algebraic structures with properties that differ significantly from classical number systems.

6.3.1 Non-Commutativity of Subjective Operations

6.3.2 Perspective-Dependent Associativity

6.3.3 Local Transitivity and Preorder Structure

As established in Theorem 5.1, for any fixed perspective p ∈ P, the subjective equality relation =_p is reflexive (Axiom 1) and transitive (Axiom 3), thus forming a preorder. It generally fails antisymmetry, so it does not constitute a partial order.

This preorder structure provides a foundation for algebraic operations within the subjective numbers framework by ensuring that equality comparisons remain consistent from a single perspective's viewpoint, even as we allow for perspective-dependent evaluations across different perspectives.

6.3.4 Distributivity Properties

In Theorem 6.3, we showed that subjective multiplication ×_r distributes over subjective addition +_r under perspective rules that preserve the underlying algebraic properties of V. Specifically, when perspective fusion is defined by an associative and commutative operator (such as conjunctive fusion f(u,v) = u ∧ v), the usual distributive laws hold locally within the quotient category.

However, if the fusion operator ⊕ is neither associative nor commutative, or if p ⊕ q is defined by a more complex weighting or non-idempotent rule, then distributivity may fail. In such cases, we only guarantee local distributive laws for operations within a single perspective or under fusion rules that satisfy the relevant algebraic identities. Consequently, the scope of Theorem 6.3 is limited to fusion functions f that respect associativity and identity behavior. If f does not meet these properties, further analysis is required to determine which distributive or associative laws continue to hold.

6.4 Subjective Algebraic Structures

The algebraic properties of subjective numbers lead to novel mathematical structures that extend and generalize classical algebraic structures like groups, rings, and fields.

Definition 6.9 (Subjective Group): A set of subjective numbers G equipped with a binary operation ∘_r using perspective propagation rule r forms a subjective group if the following conditions are met:

1. Closure: For all a_(p), b_(q) ∈ G, the result a_(p) ∘_r b_(q) ∈ G.
2. Subjective Associativity: For all a_(p), b_(q), c_(r) ∈ G, the operation ∘_r is associative when evaluated from a fixed perspective.
3. Subjective Identity: There exists a subjective identity element e_(i) ∈ G such that for all a_(p) ∈ G, either a_(p) ∘_r e_(i) =_p a_(p) or e_(i) ∘_r a_(p) =_p a_(p) or both, depending on the perspective propagation rule.
4. Subjective Inverse: For each a_(p) ∈ G, there exists a subjective inverse a^{-1}_(q) ∈ G such that either a_(p) ∘_r a^{-1}_(q) =_p e_(i) or a^{-1}_(q) ∘_r a_(p) =_p e_(i) or both, depending on the perspective propagation rule.

Unlike a classical group, a subjective group allows for non-commutative operations and perspective-dependent identities and inverses. The specific characteristics of a subjective group depend on the chosen perspective propagation rule.

Definition 6.10 (Subjective Ring): A set of subjective numbers R equipped with two binary operations +_r and ×_r using perspective propagation rule r forms a subjective ring if:

1. (R, +_r) forms a subjective abelian group (with respect to a fixed evaluation perspective).
2. (R, ×_r) forms a subjective monoid (a subjective group without the inverse property).
3. ×_r distributes over +_r from the left and right, when evaluated from a fixed perspective.

Definition 6.11 (Subjective Field): A subjective ring (F, +_r, ×_r) forms a subjective field if:

1. (F \ {0_(p)}, ×_r) forms a subjective abelian group (with respect to a fixed evaluation perspective).
2. The distributive property holds for perspective-consistent operations.

1. For fixed perspective p and dominant perspective rule:
• (V_p, +_d) is isomorphic to (V, +) as an algebraic structure.
• The isomorphism is given by the mapping a_(p) ↦ a.

2. For mixed perspectives and dominant perspective rule:
• The structure is a disjoint union of classical algebraic structures, each corresponding to a single perspective.
• The global structure forms a categorically enriched algebraic system where operations reflect the category-theoretic construction.

3. For perspective fusion rule with conjunctive fusion:
• The resulting structure creates a hierarchy of increasingly restrictive evaluation criteria.
• This forms a lattice structure on the set of perspectives, where the meet operation corresponds to perspective fusion.

4. For novel perspective rule:
• The resulting structure forms a free algebra over the generating set of perspectives.
• Each operation creates new perspectives that are not reducible to the original perspectives.

This classification contextualizes subjective numbers within the broader landscape of algebraic structures, showing how they both extend and transform classical algebraic systems. □

6.5 Consistency Relative to ZFC

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7. Perspective Relations and Equivalence Classes

7.1 Subjective Equivalence Relations

A fundamental concept in subjective mathematics is equivalence from a specific perspective. We define subjective equivalence relations as follows:

Definition 7.1 (Subjective Equivalence Relation): A subjective equivalence relation =_s from perspective s is characterized by:

1. Reflexivity: a_(s) =_s a_(s) for all a_(s) (by Axiom 1)
2. s-Transitivity: If a_(p) =_s b_(q) and b_(q) =_s c_(r), then a_(p) =_s c_(r) (by Axiom 3)
3. Potential Non-Symmetry: a_(p) =_s b_(q) does not necessarily imply b_(q) =_s a_(p) (by Axiom 2)

This differs from traditional equivalence relations by not requiring symmetry, making it a reflexive and transitive relation (a preorder) but not necessarily an equivalence relation in the standard sense.

7.2 Subjective Equivalence Classes

From perspective s, we define a subjective equivalence class [a_(p)]_s as the set of all subjective numbers that are considered equal to a_(p) from perspective s:

Definition 7.2 (Subjective Equivalence Class): [a_(p)]_s = {b_(q) | a_(p) =_s b_(q)}

These subjective equivalence classes possess properties distinct from classical equivalence classes:

This non-partition property is a fundamental difference between subjective mathematics and classical mathematics, reflecting how perspective-dependent evaluation creates overlapping rather than cleanly separated categories.

8. Connections to Established Mathematics

8.1 Relation to Modal and Multi-valued Logics

Subjective mathematics exhibits profound connections to modal logic, multi-valued logic, and non-classical logical systems designed to handle context-dependent truth, modality, and non-standard truth values. These connections provide valuable insights into the nature of subjective mathematics while highlighting its unique contributions.

Modal Logic Correspondence

Modal logic extends classical logic by introducing operators that qualify statements according to different "modes" of truth—notably possibility (◇) and necessity (□). This aligns with subjective mathematics' treatment of perspective-dependent truth.

8.2 Beyond Modal Logic: Unique Aspects of Subjective Mathematics

While the modal logic translation demonstrates a deep connection, subjective mathematics extends beyond what modal logic typically addresses in several key ways:

1. Intrinsic Perspective: In subjective mathematics, perspective is an intrinsic property of mathematical objects themselves, not just an operator applied to propositions. This differs from modal logic where modalities are typically external operators.
2. Non-Symmetric Relations: Unlike the accessibility relations in Kripke frames, which can be symmetric or asymmetric depending on axioms chosen, subjective mathematics inherently supports non-symmetric equality as a foundational property.
3. Object-Level vs. Proposition-Level: Modal logic operates primarily at the level of propositions, whereas subjective mathematics builds perspective into the very nature of mathematical objects.
4. Algebraic Structure: Subjective mathematics inherits and extends algebraic properties through operations like addition and multiplication, which don't have direct parallels in standard modal logic.

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9. Selective Fusion of Perspectives

In earlier sections, we introduced the notion of perspective fusion, denoted p ⊕ q, which formally merges the evaluations of two perspectives into a single, composite perspective. While fusion is a powerful way to combine knowledge or viewpoints, it can also lead to paradox if performed indiscriminately—especially when self-referential statements are involved. In this section, we show how restricting fusion to “compatible” perspectives (i.e., those that do not generate contradictions) preserves consistency, thereby avoiding liar-like paradoxes, yet still allows beneficial “wisdom of crowds” effects when contradictions do not arise.

9.1 Contradiction-Free (Selective) Fusion

To formalize the restriction, we define a compatibility condition that detects whether two (or more) perspectives can be safely fused without producing inconsistency or self-referential paradox.

9.1.1 Category-Theoretic Foundation of Selective Fusion

To provide a rigorous foundation for selective fusion within our category-theoretic framework, we now formalize the compatibility test as a functorial construction in 𝓢𝓝.

Definition 9.1.1 (Compatibility Functor): Let Comp: 𝓢𝓝 × 𝓢𝓝 → Set be a functor that maps each pair of perspectives (p, q) to the set:

This functor identifies all pairs of values (x, y) for which perspectives p and q agree on their mutual evaluations.

Definition 9.1.2 (Categorical Compatibility Test): Two perspectives p, q ∈ P are compatible, denoted Compatible(p, q) = true, if and only if:

In other words, perspectives are compatible when they agree on all mutual evaluations.

Definition 9.1.3 (Categorical Selective Fusion): When Compatible(p, q) = true, the fused perspective p ⊕ q is constructed as a pullback in 𝓢𝓝, with canonical projections:

The relation function for this fused perspective is defined as:

where f: {true, false} × {true, false} → {true, false} is a fusion function that combines the evaluations from the constituent perspectives.

Theorem 9.1.1 (Functorial Properties of Selective Fusion): Selective fusion, when defined on compatible perspectives, gives rise to a partial functor ⊕: 𝓢𝓝_c × 𝓢𝓝_c → 𝓢𝓝, where 𝓢𝓝_c is the subcategory of 𝓢𝓝 restricted to compatible perspectives.

Corollary 9.1.1 (Paradox Avoidance): The category-theoretic construction of selective fusion ensures that no contradiction or paradox can arise from fusing compatible perspectives.

This category-theoretic formalization provides a rigorous foundation for selective fusion, showing how it integrates with the broader framework of subjective numbers and establishing its consistency-preserving properties. The partial functorial structure captures precisely the essential property that fusion should only be allowed when it preserves mathematical coherence.

10. Conclusion and Future Directions

We have presented subjective numbers as a systematic way to embed perspectives into the very structure of mathematical objects. By formalizing perspective as an intrinsic property of each number, rather than an external parameter, we capture asymmetries and directional evaluations that classic frameworks overlook. This approach clarifies how truth and equality can depend on the specific vantage point from which they are assessed—extending or modifying standard algebraic structures without sacrificing local consistency or coherence.

Through a precise category-theoretic construction, we demonstrated that subjective numbers satisfy an axiomatic system (Axioms 1–7) ensuring internal consistency, path-dependent evaluations, and a rich algebraic structure. We showed how these constructions remain compatible with familiar Boolean operations and preserve classical logic within each individual perspective, with non-classical effects emerging only when multiple perspectives interact. Our measure-theoretic extension further quantifies disagreement between perspectives, illustrating how subtle perspective shifts can be captured in a rigorous manner.

10.1 Summary of Key Contributions

• Intrinsic Perspective: We placed perspective within the identity of each subjective number, unifying context-sensitive phenomena under standard mathematical tools.
• Category-Theoretic Rigor: By defining objects as perspectives and morphisms as perspective-indexed values, we ensured that composition and identities remain well-defined through a quotient construction.
• Extended Algebraic Operations: We explored how addition, multiplication, and other operations become non-commutative or non-associative under different perspective propagation rules, introducing novel “fusion” or “dominant” perspective behaviors.
• Consistency and Applicability: The axioms and proofs illustrate how subjective numbers sit consistently within ZFC, revealing new ways to analyze preference, directionality, and context dependence in mathematics.

10.2 Philosophical and Mathematical Outlook

The framework invites new ways to model settings where knowledge, assessment, or meaning depends crucially on the observer’s standpoint—aligning well with domains like cognitive science, social choice theory, and logic of agency. Rather than treating perspective as a nuisance, our construction shows it can be a first-class concept in mathematics. By connecting category theory and perspective-laden valuations, we gain a flexible means to study asymmetry and context-dependence in myriad applications.

10.3 Future Directions

There remain significant frontiers for subjective numbers:

• Enriched Operations: Investigating perspective-dependent exponentiation, integrals, or other higher-level constructs.
• Computational Implementations: Realizing the theory in proof assistants or programming languages, where perspective-laden data types might naturally manage conflicting viewpoints.
• Extensions Beyond Bool: Replacing the Boolean space with fuzzy, probabilistic, or intuitionistic structures, thus expanding the expressiveness for real-world uncertain or partial information scenarios.
• Dynamic Perspective Shifts: Developing time-evolving or event-driven perspectives that capture real-time updates in multi-agent systems.

Overall, this foundation lays out a coherent vision in which mathematics of perspective evolves past strict symmetry, enabling further research to bring subjective numbers into broader logical, computational, and philosophical discussions.

Appendix: Mathematical Examples for Subjective Numbers

This appendix provides detailed examples that illustrate key concepts of the subjective numbers framework, helping to build intuition for this novel mathematical structure.

Example A1: Illustrative Model of Subjective Numbers with Explicit Relation Functions

Let us construct a concrete model of subjective numbers with three perspectives P = {p, q, r} and values in ℝ. We explicitly define relation functions that demonstrate key properties of the framework.

Perspectives:

• p, q, and r are distinct perspectives

Relation Functions: For perspective p, define R_p as follows:

• R_p(p, p, a, a) = true for all a ∈ ℝ (reflexivity)
• R_p(p, q, a, b) = true if and only if a = b (value equality with q)
• R_p(p, r, a, b) = true if and only if a ≤ b (value ordering with r)

For perspective q, define R_q as follows:

• R_q(q, q, a, a) = true for all a ∈ ℝ (reflexivity)
• R_q(q, p, a, b) = true if and only if a ≥ b (reverse ordering with p)
• R_q(q, r, a, b) = true if and only if a = 2b (scaling relation with r)

For perspective r, define R_r as follows:

• R_r(r, r, a, a) = true for all a ∈ ℝ (reflexivity)
• R_r(r, p, a, b) = true if and only if a = b + 1 (offset relation with p)
• R_r(r, q, a, b) = true if and only if a = b/2 (inverse scaling with q)

Verification of Axioms: We can verify that this model satisfies all axioms:

1. Subjective Reflexivity: Each perspective satisfies R_s(s, s, a, a) = true, demonstrating Axiom 1.
2. Non-Symmetric Equality: Consider R_p(p, r, 3, 5) = true since 3 ≤ 5, but R_r(r, p, 5, 3) = false since 5 ≠ 3 + 1, demonstrating Axiom 2.
3. Subjective Transitivity: For each perspective, if a_(s) =_p b_(t) and b_(t) =_p c_(u), then a_(s) =_p c_(u), demonstrating Axiom 3.

Specific Evaluations: From perspective p:

• 5_(p) =_p 5_(q) (by value equality)
• 3_(p) =_p 7_(r) (since 3 ≤ 7)
• ¬(7_(p) =_p 3_(r)) (since 7 ≰ 3)

From perspective q:

• 8_(q) =_q 4_(p) (since 8 ≥ 4)
• 10_(q) =_q 5_(r) (since 10 = 2 × 5)
• ¬(3_(q) =_q 7_(p)) (since 3 ≱ 7)

From perspective r:

• 6_(r) =_r 5_(p) (since 6 = 5 + 1)
• 4_(r) =_r 8_(q) (since 4 = 8/2)
• ¬(3_(r) =_r 9_(p)) (since 3 ≠ 9 + 1)

This model demonstrates perspective-dependent evaluation, non-symmetric relations, and distinctive relation patterns for each perspective, validating the subjective numbers framework.

Example A2: Perspective Propagation in Operations

Let us examine perspective propagation through operations in detail using the three fundamental rules.

Consider the perspectives P = {a, b, c} and subjective numbers 3_(a), 4_(b), and 5_(c). We define relation functions such that:

• R_a(a, b, 3, 4) = true but R_b(b, a, 4, 3) = false
• R_a(a, c, 3, 5) = false but R_c(c, a, 5, 3) = true
• R_b(b, c, 4, 5) = true and R_c(c, b, 5, 4) = true

Dominant Perspective Rule: When performing addition with the dominant perspective rule:

1. 3_(a) +_d 4_(b) = 7_(a)
2. 4_(b) +_d 3_(a) = 7_(b)
3. 5_(c) +_d 3_(a) = 8_(c)

For subjective numbers 7_(a) and 7_(b), consider their relationships:

• From perspective a: 7_(a) =_a 7_(b) iff R_a(a, b, 7, 7) = true (which would be true if we extend our relation to value equality for same values)
• From perspective b: 7_(b) =_b 7_(a) iff R_b(b, a, 7, 7) = true (which would be true if we extend our relation to value equality for same values)

Despite having the same numerical value, 7_(a) and 7_(b) are distinct subjective numbers with different perspectives.

Perspective Fusion Rule: For perspective fusion, we define the composite perspective relation function R_ab such that:

• R_ab(ab, a, x, y) = R_a(a, a, x, y) (inherit a's relation to itself)
• R_ab(ab, b, x, y) = R_b(b, b, x, y) (inherit b's relation to itself)
• R_ab(ab, c, x, y) = R_a(a, c, x, y) ∨ R_b(b, c, x, y) (recognize c if either component does)

Now, when performing addition with perspective fusion:

1. 3_(a) +_a∨b 4_(b) = 7_(a⊕b)
2. 4_(b) +_b∨c 5_(c) = 9_(b⊕c)

For evaluating relations:

• Since R_a(a, c, 3, 5) = false but R_b(b, c, 4, 5) = true, the disjunctive fusion gives R_a⊕b(a⊕b, c, 7, 5) = true
• This means 7_(a⊕b) =_a⊕b 5_(c), showing how the fused perspective evaluates relations differently from either constituent perspective.

Novel Perspective Rule: For the novel perspective rule, we generate entirely new perspectives with distinct relation functions:

1. 3_(a) +_n 4_(b) = 7_(z) where z is a new perspective
2. 4_(b) +_n 5_(c) = 9_(w) where w is a new perspective

For the novel perspective z created from a and b, we define the relation function R_z to include elements from both source perspectives:

• R_z(z, a, 7, 3) = true (the result recognizes its components)
• R_z(z, b, 7, 4) = true (the result recognizes its components)
• R_z(z, c, 7, y) = g(R_a(a, c, 3, y), R_b(b, c, 4, y), 0.5) (new relationships with other perspectives defined through the generation function g)

This example illustrates how each perspective propagation rule creates distinct patterns of relations in the resulting subjective numbers, demonstrating the flexibility of the framework for modeling diverse perspective interactions.

Example A3: Path-Dependent Truth

This example demonstrates how truth in subjective mathematics can be path-dependent, where different evaluation paths yield different valid conclusions.

Consider perspectives P = {α, β, γ} with the following specific relation functions:

1. For perspective α:
   • Reflexivity: R_α(α, α, a, a) = true for all a ∈ ℝ (reflexivity)
   • Cross-Perspective: R_α(α, β, 5, 5) = true (perspective α considers its 5 equal to perspective β's 5)
   • R_α(α, α, 5, 3) = false (perspective α considers 5 and 3 different in direct evaluation)
   • R_α(α, γ, a, b) is undefined for all a, b (no direct relation with γ)
2. For perspective β:
   • Reflexivity: R_β(β, β, a, a) = true for all a ∈ ℝ (reflexivity)
   • Cross-Perspective: R_β(β, γ, 5, 5) = true (perspective β considers its 5 equal to perspective γ's 5)
   • R_β(β, α, a, b) is undefined for all a, b (no direct relation with α)
3. For perspective γ:
   • Reflexivity: R_γ(γ, γ, a, a) = true for all a ∈ ℝ (reflexivity)
   • Cross-Perspective: R_γ(γ, α, 5, 3) = true (perspective γ considers its 5 equal to perspective α's 3)
   • R_γ(γ, β, a, b) is undefined for all a, b (no direct relation with β)

We also define the cross-perspective adoption relation:

• CPA_α(β, γ) = true (perspective α accepts β's evaluation of γ)
• CPA_α(γ, α) = true (perspective α accepts γ's evaluation of α)

Now, let's analyze the subjective equality relation between 5_(α) and 3_(α) from perspective α using different evaluation paths:

Direct Evaluation from Perspective α: By direct evaluation from perspective α's relation function, we have: R_α(α, α, 5, 3) = false Therefore, from perspective α's direct viewpoint, 5_(α) ≠_α 3_(α).

Evaluation via Perspective Chain α → β → γ → α:

1. Start with 5_(α)
2. From perspective α: 5_(α) =_α 5_(β) (since R_α(α, β, 5, 5) = true)
3. Perspective α adopts perspective β's evaluation of γ: 5_(β) =_α 5_(γ) (since R_β(β, γ, 5, 5) = true and CPA_α(β, γ) = true)
4. Perspective α adopts perspective γ's evaluation of α: 5_(γ) =_α 3_(α) (since R_γ(γ, α, 5, 3) = true and CPA_α(γ, α) = true)
5. By transitivity (within perspective α's adopted viewpoints): 5_(α) =_α 3_(α)

Thus, we have derived 5_(α) =_α 3_(α) by following the perspective chain α → β → γ → α, even though direct evaluation gives 5_(α) ≠_α 3_(α). This example demonstrates path-dependent truth.

Example A4: Perspective Incompleteness

This example demonstrates the Perspective Incompleteness phenomenon with explicit relation functions showing that certain mathematical truths are inaccessible from any single perspective but become available through perspective fusion.

Consider a system with three perspectives P = {α, β, γ} focused on the subjective numbers 7_(α), 4_(β), and 10_(γ).

Define relation functions with the following specific knowledge distribution:

1. For perspective α:
   • R_α(α, α, 7, 7) = true (reflexivity)
   • R_α(α, β, 7, 4) = true (perspective α recognizes equivalence with β's 4)
   • R_α(α, γ, 7, 10) is undefined (perspective α has no knowledge about relation to γ's 10)
2. For perspective β:
   • R_β(β, β, 4, 4) = true (reflexivity)
   • R_β(β, γ, 4, 10) = true (perspective β recognizes equivalence with γ's 10)
   • R_β(β, α, 4, 7) is undefined (perspective β has no knowledge about relation to α's 7)
3. For perspective γ:
   • R_γ(γ, γ, 10, 10) = true (reflexivity)
   • R_γ(γ, α, 10, 7) = false (perspective γ explicitly rejects equivalence with α's 7)
   • R_γ(γ, β, 10, 4) is undefined (perspective γ has no knowledge about relation to β's 4)

Now, we analyze what mathematical truths are accessible from each individual perspective:

From Perspective α:

• 7_(α) =_α 4_(β) (accessible truth)
• Relationship between 7_(α) and 10_(γ) is inaccessible (undefined in R_α)

From Perspective β:

• 4_(β) =_β 10_(γ) (accessible truth)
• Relationship between 4_(β) and 7_(α) is inaccessible (undefined in R_β)

From Perspective γ:

• 10_(γ) ≠_γ 7_(α) (accessible truth - false)
• Relationship between 10_(γ) and 4_(β) is inaccessible (undefined in R_γ)

Now, create fused perspective α⊕β with relation function R_α⊕β defined as:

• R_α⊕β(α⊕β, p, x, y) = R_α(α, p, x, y) ∨ R_β(β, p, x, y) for all p ∈ P and x, y ∈ ℝ

From this fused perspective α⊕β:

• 7_(α) =_α⊕β 4_(β) (from α's knowledge)
• 4_(β) =_α⊕β 10_(γ) (from β's knowledge)
• By transitivity: 7_(α) =_α⊕β 10_(γ)

This demonstrates perspective incompleteness: the relation between 7_(α) and 10_(γ) is inaccessible from any single perspective (α, β, or γ), yet becomes accessible through perspective fusion (α⊕β). In fact, perspective γ explicitly contradicts this derived relation (10_(γ) ≠_γ 7_(α)), showing how the fusion of perspectives can reveal mathematical truths that are not just unknown but potentially contradicted by individual perspectives. This example demonstrates Perspective Incompleteness.

This realization 7_(α) =_α⊕β 10_(γ) represents an emergent mathematical truth that exists only at the collective level, not derivable from any individual perspective in isolation.

Example A5: Constructing a Penrose Stairs Analogue with Subjective Numbers

To further solidify the intuition behind subjective numbers and their capacity to model paradoxical, viewpoint-dependent structures, we present a concrete construction that mirrors the famous Penrose stairs illusion. This example demonstrates how subjective numbers can create a mathematical analogue of an impossible object, exhibiting local consistency while forming a globally paradoxical loop due to perspective-dependent relations.

Construction of the Penrose Stairs Analogue:

1. Perspectives as Steps: We define a set of four perspectives, P = {p₁, p₂, p₃, p₄}, each representing a "step" in our subjective staircase.
2. Subjective Numbers as Levels: We consider subjective numbers representing different "levels" of ascent, using integer values for simplicity:
   • level_0(p1): Subjective number with value 0 and perspective p₁.
   • level_1(p2): Subjective number with value 1 and perspective p₂.
   • level_2(p3): Subjective number with value 2 and perspective p₃.
   • level_3(p4): Subjective number with value 3 and perspective p₄.
3. Asymmetric Subjective Equality Relations for "Ascent": We define the relation functions R_p1, R_p2, R_p3, and R_p4 to establish an asymmetric subjective equality that mimics an ascending staircase:
   • From Perspective p₁: "Level 0" is subjectively equal to "Level 1" (from perspective p₂): R_p1(p₁, p₂, 0, 1) = true. In other words, level_0(p1) =_p1 level_1(p2).
   • From Perspective p₂: "Level 1" is subjectively equal to "Level 2" (from perspective p₃): R_p2(p₂, p₃, 1, 2) = true. That is, level_1(p2) =_p2 level_2(p3).
   • From Perspective p₃: "Level 2" is subjectively equal to "Level 3" (from perspective p₄): R_p3(p₃, p₄, 2, 3) = true. Meaning, level_2(p3) =_p3 level_3(p4).
   • Closing the Loop: From Perspective p₄: "Level 3" is subjectively equal to "Level 0" (from perspective p₁): R_p4(p₄, p₁, 3, 0) = true. Concretely, level_3(p4) =_p4 level_0(p1).
   • Non-Equality for Direct Comparison: To maintain the illusion of distinct levels from each perspective and to ensure asymmetry, we stipulate that direct self-comparisons of different levels within the same perspective are not subjectively equal. Formally: R_pi(p_i, p_i, n, m) = false if n ≠ m, and R_pi(p_i, p_j, n, m) ≠ R_pj(p_j, p_i, m, n) for i ≠ j, n ≠ m. Reflexivity still holds as R_pi(p_i, p_i, n, n) = true.
4. The Penrose Stairs Effect: Path-Dependent "Ascent" and Paradoxical Loop: Starting from level_0(p1), let us trace the subjective equality relations, simulating a "walk" on our subjective Penrose stairs:
   • Step 1 (Perspective p₁): We have level_0(p1) =_p1 level_1(p2) . We effectively "ascend" to level_1(p2) .
   • Step 2 (Perspective p₂): We have level_1(p2) =_p2 level_2(p3) . We further "ascend" to level_2(p3) .
   • Step 3 (Perspective p₃): We have level_2(p3) =_p3 level_3(p4) . We continue to "ascend" to level_3(p4) .
   • Step 4 (Perspective p₄ - Closing the Loop): We have level_3(p4) =_p4 level_0(p1) . Paradoxically, we "ascend" back to something subjectively equal to our starting point, level_0(p1) , completing a loop of continuous ascent.
   This cyclical chain of subjective equalities, level_0(p1) → level_1(p2) → level_2(p3) → level_3(p4) → level_0(p1) , mirrors the impossible loop of the Penrose stairs. Each step appears to ascend locally (from one perspective to the next), yet globally, the chain returns to its starting level, creating a paradoxical closed loop of continuous ascent. This example demonstrates path-dependent truth.

This construction can be visualized as a directed graph where each node represents a subjective number (level_0(p1), level_1(p2), etc.), and a directed edge from node A to node B exists if A =_p B from the perspective p associated with node A.