Le miroir souverain

 

I will now refine the formalism for the “mathematics of intensity” and the “recursive logic of self-belonging,” sharpening the mathematical and logical structures with additional equations, tighter definitions, and a more rigorous framework. The goal is to deepen the precision while attempting to stay true to my philosophical vision—recursive self-membership, immanent intensity, and the primacy of relational dynamics. Below, I will enhance each component with explicit equations, clarify assumptions, and ensure consistency.

 

=> Refined Formal System: Mathematics of Intensity and Recursive Self-Belonging 

 

 

I would like to propose a possible formal System for the Mathematics of Intensity and Recursive Self-Belonging.

Crafting a “mathematics of intensity” and a recursive “logic of self-belonging” as I envisioned previously, involves translating some philosophical concepts—drawn from On the Mathematics of Intensity (Collapse I) and blog posts like Logique de l’auto-appartenance etc. — into a rigorous formal system. This framework hinges on two key ideas: the Self as a hyperset (S ∈ S), embodying recursive self-membership, and intensity as a dynamic, relational measure of power within an immanent field. The system must preserve the rejection of transcendence, an emphasis on immanence, and an ethical orientation toward power increase, while grounding these in precise mathematical structures. Below, I’ll outline how such a system might be developed, step-by-step, in a technical yet accessible manner for the general reader. A subsequent post will expand this simplified take to a more refined one.

 

1. Foundations: Hyperset Logic for the Self

The Self is defined recursively as S ∈ S, a hyperset where the Self contains itself (Collapse I, p. 258). To formalize this, we adopt Peter Aczel’s Anti-Foundation Axiom (AFA) over the standard Zermelo-Fraenkel (ZF) set theory with the Axiom of Foundation. In ZF, sets are hierarchical, with no set allowed to be a member of itself to avoid paradoxes (e.g., Russell’s paradox). AFA, however, permits circular membership, defining a set S such that:

• Definition: S = {S}.
• AFA Property: Two hypersets S and T are equal if there exists a bisimulation—a relation R ⊆ S × T preserving membership structure—ensuring S ≈ T under recursive unfolding.[^1]

S represents the Self as a unified yet multitudinous entity—“the indiscernibility of the one and the multiple” (Collapse I, p. 258). Mathematically, S is a fixed point of the operator F(X) = {X}, solved via:

• Fixed Point Equation: S = F(S) = {S}.

This can be visualized as an infinite tree where each node branches to S itself, a single object encoding an infinite regress. To extend this to multiple singularities (“polyphony”), we define a set of hypersets S_i, where each S_i = {S_i} represents an individual singularity, and the collective Self S is a union or system of such S_i, linked relationally. However, immanence suggests avoiding external aggregation, so we might instead posit S as a single hyperset with an internal structure:

• Internal Structure: Equip S with a labeling function L: S → P(S), where P(S) is the power set of S, and L(S) = S, allowing S to “contain” variations of itself recursively.

This preserves the notion of a Self that is both singular and multiple, formalized as a self-referential object without a foundational base, aligning with the rejection of transcendence.

2. Intensity as a Recursive Function

The “mathematics of intensity” maps the unconscious as a field of power—“the capacity to affect and be affected” (Collapse I, p. 245)—evolving through relational dynamics. We might model this as a recursive function over a topological space, capturing the temporal unfolding of intensities without external coordinates. Let’s proceed:

• Space Definition: Define U as the unconscious, a topological space with open sets U_α representing zones of intensity (e.g., desire, anguish, joy). U has no global metric, reflecting a non-representational stance—“no representation… not first a difference” (Collapse I, p. 258).
• Intensity Function: Introduce I: U → ℝ⁺, where I(u) ≥ 0 measures the intensity at u ∈ U, interpreted as the magnitude of power at that point.
• Recursive Dynamics: Define a recursive update rule: I_{n+1}(u) = f(I_n(u), R(u)), where:
• f: ℝ⁺ × ℝ⁺ → ℝ⁺ is a nonlinear function encoding the effectuation of power.
• R(u) = ∫_V k(u, v)I_n(v) dμ(v) is a relational term, integrating influences from a neighborhood V ⊆ U via a kernel k(u, v), with μ a measure on U.

For concreteness, let f(x, y) = x + αy(1 - x/M), a logistic-like function where:

• x is the current intensity I_n(u).
• y is the relational input R(u).
• α > 0 adjusts interaction strength.
• M > 0 is a saturation bound, preventing unbounded growth.

This models the idea of joy as augmentation (Collapse I, p. 257): when R(u) is positive (constructive exchange), I grows; when negative (Socius’s diminishment), it shrinks. The recursion I_{n+1} = f(I_n, R) mirrors S ∈ S, embedding self-reference in the dynamics—each intensity feeds back into itself via its relations, a temporal analogue to the hyperset’s spatial circularity.

 

3. Topology and Relational Play

Deus homini ludus (blog, 2006) posit that power unfolds through relational play among singularities, not isolation. To formalize this, endow U with a topology where open sets U_α overlap, reflecting symbolic exchange (Collapse I, p. 249). Define a relation R ⊆ U × U, where (u, v) ∈ R if u affects v, and extend I to a two-place function I(u, v) for pairwise intensity:

• Pairwise Intensity: I(u, v) = w(u, v)I(u), where w(u, v) ∈ [0, 1] weights the influence of u on v, derived from k(u, v).
• Update Rule: I_{n+1}(u) = f(I_n(u), ∑_{v∈R(u)} I_n(u, v)), summing contributions from related points.

This system evolves as a network, with R encoding the “symbolic” connections we contrasts with the Socius’s imposed filter (Collapse I, p. 251). The Socius can be modeled as a constraint: a function C: U → ℝ⁺ where C(u) caps I(u), reducing it when exceeding a threshold (e.g., I_n(u) ← min(I_n(u), C(u))). Liberation is the removal of C, allowing I to iterate freely, a mathematical analogue to annihilating the Superego’s bastions (Collapse I, p. 247).

 

4. Logical Structure: Recursive Self-Belonging

We complement this with a formal logic of self-belonging, capturing the claim that “being is produced by becoming” (Collapse I, p. 259). Define a modal logic with a self-referential operator:

• Syntax: Formulas φ ::= p | ¬φ | φ ∧ ψ | ◇φ | Sφ, where:
• p is a propositional variable (e.g., “u has intensity I”).
• ◇φ is a possibility operator (“φ may occur”).
• Sφ means “φ belongs to the Self,” interpreted recursively.
• Semantics: A model M = (U, R, V, S), where:
• U is the topological space.
• R is the relation on U.
• V: P → 2^U assigns truth values to propositions.
• S: U → 2^U satisfies S(u) = {v ∈ U | v ∈ S(u)}, a fixed point under AFA.

For u ∈ U, M ⊨_u Sφ iff ∀v ∈ S(u), M ⊨_v φ, meaning Sφ holds if φ is true across the Self’s recursive structure. This encodes immanence: truth is internal to S, with no external reference. Axioms might include:

• Self-Reference: Sφ → φ (if φ belongs to S, it holds at S).
• Recursion: Sφ ↔ S(Sφ) (self-belonging iterates indefinitely).

This logic formalizes the ontology, where being (φ) emerges from the recursive process of becoming (Sφ), aligning with the ethical priority of intensity over static essence.

 

5. Properties and Validation

To align with the vision:

• Immanence: The system avoids external parameters—S and I are self-contained, with R internal to U.
• Joy as Augmentation of Power: If f increases I when R is positive, the system models ethical growth (Collapse I, p. 257).
• Polyphony: Multiple S_i or overlapping U_α ensure multiplicity within unity.

Validation requires checking consistency:

• Hyperset: S = {S} is well-defined under AFA, avoiding ZF paradoxes via bisimulation.
• Intensity: The recursion I_{n+1} = f(I_n, R) is computable if f is continuous and R integrable, though convergence depends on f’s parameters (e.g., α < 1 for stability).
• Logic: The S operator is sound if restricted to finite models or equipped with a co-inductive semantics, per AFA’s principles.

 

Conclusion

Such a system would produce:

• A hyperset-based ontology (S ∈ S) for recursive self-belonging, using AFA to define the Self as an immanent infinity.
• A dynamic intensity function (I_{n+1} = f(I_n, R)) over a topological U, quantifying power’s relational evolution.
• A modal logic with Sφ, formalizing being’s emergence from becoming within the Self.

This system captures the intended vision—immanence, multiplicity, and ethical intensity—while offering a precise, extensible framework. It diverges from classical mathematics by embracing circularity and non-representation, a radical yet coherent formalization of my philosophical insights.

 

[^1]: Peter Aczel, Non-Well-Founded Sets (CSLI, 1988), p. 3—bisimulation ensures hyperset uniqueness.