On Identity

0.1 Introduction: On Identity

Traditionally, identity was seen as an intrinsic property, somewhat of a “self” defined by an immutable essence (e.g., Aristotle’s ousia), and while Hegel and Mead argued that identity arises dialectically through interaction with the “other”, I wonder if this holds true outside the human-specific context. I find quite a satisfaction “playing” with these notion of selves, identities and relationships mathematically, due to the implicit clarity it requires, while remaining delightfully challenging to be precise towards it. The dialect in question reflect a position when someone is at stage of comparison of someone else in a context. Which is congruent with A in relation to B. In mathematics, identity is generally relational as well, where objects (e.g., groups, spaces) are defined up to isomorphism [^cat:Category Theory], by their morphisms (structure-preserving maps), and said group \[ G \]’s identity lies in how it maps to other groups (e.g., via homomorphisms), not just its internal elements, or entities derive meaning from their place in a hierarchy [^type:Type Theory] (e.g., natural numbers in Peano arithmetic).
But is it sound to frame identity as requiring relation? Accepting it for now for the sake of exercise, and carrying it into a bio-physical realm, a cell’s identity, for instance, is potentially given by what allows for metabolic interfaces with its environment, a membrane. Without it, a cell would not be a cell, but loose environmental material. Perhaps identity is better put as identity requiring clear bound of self, allowing for relation means. In question is the dual nature of identity, where a clear relation to itself must be persistence, but from this persistence something else happens, persistence seems to be then connected to whatever is not part ot that identity almost as a constructive-coupling. Moreover, both Mead and Hagel posed that this relationship with a “generalised-other” allows for the development of self-awareness [^human: social-dynamics and self-awareness].

The dialectical tension between being and relating, poses predicates as:

0.1.1 Axiom 1: Boundary Necessity

Identity requires a boundary to mediate interaction as a system’s coherence depends on a demarcation separating “self” from “other.” In category theory, this is formalized as:
\[ \forall S \in \text{Ob}(\mathcal{C}), \, \exists B_S : \text{Hom}(S, E) \to \text{Hom}(S, S) \]
where \[B_S\] filters permissible morphisms (e.g., a cell membrane regulating metabolic exchange).

0.1.2 Axiom 2: Persistence via Invariance

Identity persists if relations preserve structural invariants, where type-theoretically:
\[ \text{Id}(S) \coloneqq \{ R \mid \forall f: S \to T, \, f \circ R = R \circ f \} \]
This ensures a group \[G\]’s identity element \[e_G\] maps to \[e_H\] under homomorphisms, just as a cell maintains homeostasis despite environmental flux.

0.1.3 Axiom 3: Duality of Cohesion and Interaction

Boundaries and relations are adjoint processes, where formally:
\[ B_S \dashv R_S : \mathcal{C} \to \mathcal{D} \]
The boundary functor \[B_S\] (e.g., psychological self-concept) is inseparable from the relation functor \[R_S\] (e.g., social recognition), mirroring Mead’s “generalized other.”

These axioms in turn face somewhat further tension when applied cross-domain, as boundary ambiguity requires having an answer to how “sharp” must a boundary be? And how stable? Do ranges the ranges of variation in identity alike the established “tree-ness” throughout the seasons impose a clear contrain on the idenitarian boundaries of a tree as an object, that according to a substance-ontology change is part of persistence. This seems incomplete as we have established in the Axiom 2 that persistence occurs via invariance, and in Axiom 3 that boundaries and relations are adjoint processes, persisting not despite change, but through it like a river maintaining form despite molecular turnover via the terrain where if flows. If we juxtapose this river analogy and Heraclitus’ aphorism “No man ever steps in the same river twice, …” [^rive: No man ever steps in the same river twice, for it’s not the same river and he’s not the same man.] then persistence is both enabled and destabilized by interaction, and identity persistence is both bounded and allow by the asymmetry of being and relating.

This sounds rather familiar with Hoffmeyer’s and Emmeche’s notion of Code-duality and the semiotics of nature, where something encondes through its morphological composition its identity, while because it it, then can be translated as a sign, pointer to it through abstraction. Now that I thing about it, when they (Hoffmeyer and Emmeche) state that “Biological information is not a substance”, when referring to morphology, from the Latin morph, which puts a great smile on my face with such an unforeseen and elegant overload (potentially a coincidence that for it’s beauty requires rigorous scrutiny as I’m not a subscriber of an elegant universe), [^informare: to bring something into form. And this again is the root of the now fashionable word information.] when combining Hegel and Heraclitus, identity becomes a two-way morphism: \[ \text{River} \xleftrightarrow[\text{terrain + observer}]{\text{flow + perception}} \text{Man} \] where: 1. The river’s identity persists
2. Each interaction (stepping) creates a new pullback diagram:
\[ \begin{CD} \text{Man}_t @>>> \text{River}_t \\ @VVV @VVV \\ \text{Man}_{t+1} @>>> \text{River}_{t+1} \end{CD} \]

Here, both man and river are transformed through their interface, the ability to interact, and what it interacts with. Hegel’s becoming identity is a fixed point in the dialectic of boundary maintenance and relational adaptation, and via reflexivity of higher-order adjunctions, boundaries become objects of interaction, suggesting that to exist is to be a morphism in the category of becoming.

Thus, to frame identity as relational may be sound, as long as we accept that:
1. Boundaries are necessary for interaction, acting as filters (Axiom 1).
2. Persistence and relation are adjoint (Axiom 3), a duality formalized in category theory’s \[B_S \dashv R_S\].

From this perspective, identity is neither static essence nor chaotic flux but a structure through recurrence. Given a recursive loop where boundaries enable interaction, which in turn reinforces boundaries.

Heraclitus’ aphorism adds two critical twists:
A. The river is never the same (molecular turnover, identity as “rive” is a pointer to the object).
B. The man is never the same (the man changes through the act of stepping into the “river”, is also the author of the pointer).

The river’s identity depends on both its internal flow and its interaction with the man, and pointed to as a reference by the man. The river’s identity is co-constitutive. The man’s identity is also altered by the river (e.g., wet feet, shifted perspective), suggesting a relational asymmetry.

0.1.4 Conclusion: To Step Again is to Redefine

Hegel and Heraclitus together reveal that identity is neither static nor chaotic but a directed, relational process. The river and the man are different at each step, yet recognizable as themselves because their interactions follow structured rules (terrain, social norms, biological constraints). This mirrors the mathematical insight that isomorphism requires not just structural similarity but directional compatibility (e.g., homomorphisms).

Viewing identity as a limit constrained property such as \[ \text{Id}(X) = \lim_{\to \text{interactions}} X_t \] where \[ X_t \] are the entity’s temporal states. The “self” is the limit that coheres these states, even as interactions (Heraclitus) push the system toward a new limit.

To “step into the river” is to engage in a non-commutative operation: the man affects the river differently than the river affects the man.
The river’s terrain (boundary) is eroded and reshaped by the river’s flow, just as the man’s psychological boundaries evolve through experience, and here boundaries are permeable and reflexive.

Hegel’s river maintains its identity because it changes—its flow is stabilized by the terrain (boundary) that guides it.
The river \[ R \] is a functor mapping time slices \[ t_i \] to states \[ R(t_i) \], with the terrain \[ T \] as a boundary natural transformation ensuring commutativity: \[ T \circ R(t_i) = R(t_{i+1}) \circ T \] because the terrain acts as a “relational invariant,” allowing the river to persist as a process, and a cell’s identity persists via metabolic turnover, allowed its membrane.

The man observing the river is also observing his own changing reflection—a recursive feedback loop akin to Gödel’s incompleteness (systems cannot fully formalize their own consistency).

Thus, identity is a commuting diagram of transformation, where persistence and flux are reconciled through the very act of interaction. To “step into the river twice” is to participate in this dance, a dialectic where boundaries and relations co-define what it means to be.

1 On Identity: A Mathematical Theory of Relational Being

Traditionally, identity was seen as an intrinsic property, somewhat of a “self” defined by an immutable essence (e.g., Aristotle’s ousia), and while Hegel and Mead argued that identity arises dialectically through interaction with the “other”, I wonder if this holds true outside the human-specific context[1]. I find quite a satisfaction “playing” with these notion of selves, identities and relationships mathematically, due to the implicit clarity it requires, while remaining delightfully challenging to be precise towards it.

In mathematics, identity is generally relational as well, where objects (e.g., groups, spaces) are defined up to isomorphism[5], by their morphisms (structure-preserving maps), and said group \[ G \]’s identity lies in how it maps to other groups (e.g., via homomorphisms), not just its internal elements. Following the Law of Identity in formal logic[3]: \[ \forall x(x = x) \], we extend this to relational contexts where identity becomes a relative predicate[8]: \[ x =_F y \] (x is the same F as y).

But is it sound to frame identity as requiring relation? Accepting it for now for the sake of exercise, and carrying it into a bio-physical realm, a cell’s identity, for instance, is potentially given by what allows for metabolic interfaces with its environment—a membrane. Without it, a cell would not be a cell, but loose environmental material.

The dialectical tension between being and relating poses predicates as:

1.0.1 Axiom 1: Boundary Necessity (Categorical Separation)

Identity requires a boundary to mediate interaction as a system’s coherence depends on a demarcation separating “self” from “other.” In category theory, this is formalized as[5]:
\[ \forall S \in \text{Ob}(\mathcal{C}), \, \exists B_S : \text{Hom}(S, E) \to \text{Hom}(S, S) \]
where \[B_S\] is a boundary functor that filters permissible morphisms, creating what we call the identity closure of S: \[ \overline{S} = \{ f \in \text{Hom}(S, S) \mid B_S(f) = \text{id}_S \} \]

1.0.2 Axiom 2: Persistence via Invariance (Structural Stability)

Identity persists iff relations preserve structural invariants[6]. Following Einstein’s insight that physical laws remain invariant under coordinate transformations, we define identity as the maximal invariant substructure: \[ \text{Id}(S) \coloneqq \{ R \mid \forall f: S \to T, \, f \circ R = R \circ f \} \]
This ensures that like the speed of light in relativity, certain properties remain constant across all valid transformations—a group \[G\]’s identity element \[e_G\] satisfies \[ \forall \phi \in \text{Hom}(G,H): \phi(e_G) = e_H \].

1.0.3 Axiom 3: Duality of Cohesion and Interaction (Adjoint Correspondence)

Boundaries and relations are adjoint functors[5], expressing a fundamental duality:
\[ B_S \dashv R_S : \mathcal{C} \leftrightarrows \mathcal{D} \]
with natural isomorphism: \[ \text{Hom}_{\mathcal{D}}(B_S(X), Y) \cong \text{Hom}_{\mathcal{C}}(X, R_S(Y)) \]

This mirrors Einstein’s mass-energy equivalence—boundary (cohesion) and relation (interaction) are different manifestations of the same underlying structure.

1.1 Semiotic Mathematics: Information as Morphological Encoding

When Hoffmeyer and Emmeche state that “biological information is not a substance”[1], they point toward a profound mathematical truth. Information, from Latin informare (“to bring into form”), is precisely a morphism—a structure-preserving map that encodes identity through relational patterns.

Let us formalize this Code-Duality Principle: \[ \text{Info}: \text{Morph}(\mathcal{C}) \to \text{Sign}(\mathcal{S}) \] where each morphism \[ f: A \to B \] in category \[\mathcal{C}\] corresponds to a sign \[ \sigma_f \] in semiotic space \[\mathcal{S}\], such that:

1. Encoding: \[ \text{Info}(f \circ g) = \text{Info}(f) \star \text{Info}(g) \] (composition preserves meaning)
2. Abstraction: \[ \forall f \in \text{Iso}(\mathcal{C}): \text{Info}(f) = \text{Info}(\text{id}) \] (isomorphic structures carry identical information)
3. Translation: \[ \text{Info}(f^{-1}) = \overline{\text{Info}(f)} \] (inverse morphisms yield complementary signs)

This explains why a DNA sequence (morphological composition) can encode organismal identity while simultaneously serving as a pointer (sign) to that organism through abstraction—a beautiful mathematical overload indeed!

1.2 The Heraclitean Transform: Dynamic Identity as Temporal Functor

Juxtaposing Hegel’s river analogy with Heraclitus’ aphorism reveals identity as a temporal functor \[\mathcal{F}: \text{Time} \to \text{Cat}\], where each moment maps to a category of states.

The River-Man System can be expressed as a pullback square in the category of interactions: \[ \begin{CD} \text{Man}_t \times \text{River}_t @>>> \text{Experience}_t \\ @V{\pi_1}VV @V{f_t}VV \\ \text{Man}_t @>{g_t}>> \text{Context}_t \\ @V{h}VV @V{k}VV \\ \text{Man}_{t+1} @>{g_{t+1}}>> \text{Context}_{t+1} \end{CD} \]

The Heraclitean Transform \[\mathcal{H}: S_t \mapsto S_{t+1}\] satisfies: 1. Non-Commutativity: \[\mathcal{H}_{\text{man}} \circ \mathcal{H}_{\text{river}} \neq \mathcal{H}_{\text{river}} \circ \mathcal{H}_{\text{man}}\] 2. Mutual Determination: \[\mathcal{H}_{\text{man}}(t+1) = f(\mathcal{H}_{\text{river}}(t))\] 3. Information Conservation: \[|\text{Info}(\mathcal{H}(S))| = |\text{Info}(S)|\] (total information preserved across transformation)

1.3 Identity as Limit: The Convergence Theorem

Following Einstein’s method of finding invariants, we define identity as the categorical limit: \[ \text{Id}(X) = \lim_{\to \text{interactions}} X_t \]

Theorem (Identity Convergence): For any system \[S\] with bounded interaction space, the sequence of states \[\{S_t\}\] converges to a unique fixed point \[S^*\] such that: \[ \mathcal{H}(S^*) = S^* \text{ and } \forall t: d(S_t, S^*) \leq \epsilon \cdot \lambda^t \] where \[\lambda < 1\] is the identity contraction factor.

Proof Sketch: The boundary functor \[B_S\] acts as a contraction mapping on the space of possible states. By the Banach Fixed-Point Theorem, iteration of \[\mathcal{H}\] must converge to a unique fixed point—the system’s essential identity.

1.4 Self-Reference as Gödel Morphism

The recursive feedback loop where “the man observing the river observes his own changing reflection” can be formalized as a Gödel morphism: \[ \varphi: \text{Obs}(S) \to S \] such that \[\varphi\] embeds the observation of \[S\] into \[S\] itself.

This creates a diagonal construction similar to Gödel’s incompleteness proof: \[ \text{SelfRef}(S) = \{ s \in S \mid \varphi(\text{Obs}(s)) = s \} \]

Incompleteness Principle for Identity: No system \[S\] can completely formalize its own identity relations. There always exists a morphism \[\psi: S \to S\] such that: \[ S \nvdash \text{"}\psi \text{ preserves identity"} \text{ and } S \nvdash \neg\text{"}\psi \text{ preserves identity"} \]

1.5 Conclusion: Identity as Commuting Diagram of Becoming

Following Einstein’s geometric approach to physics, identity emerges as the curvature of relational space—a measure of how interactions bend the trajectory of becoming.

The fundamental equation of relational identity: \[ \boxed{\frac{D\text{Id}}{Dt} = \kappa(\text{Boundary}, \text{Relation}) \cdot \text{Id}} \]

where \[\kappa\] is the identity curvature tensor encoding how boundaries and relations co-determine persistence.

Final Insight: Just as Einstein revealed that matter tells spacetime how to curve and curved spacetime tells matter how to move, boundaries tell relations how to flow, and flowing relations tell boundaries how to persist. Identity is neither substance nor process, but the invariant geometry of this interaction—the fixed point where being and becoming coincide in perfect mathematical harmony.

To “step into the river twice” is to trace a geodesic in the manifold of becoming, where each step reveals new curvatures while preserving the essential metric of what it means to be.

Note: This suggests that identity, like physical law, may exhibit both local symmetries (individual persistence) and global invariances (relational consistency) pointing towards a theory of relational ontology.