Information Theory of Individuality

• Information Theory of Individuality is a framework that defines, distinguishes, and quantifies entities using algorithmic complexity, quantum statistics, and entropy measures.
• It employs constructs such as Kolmogorov complexity, KL divergence, and mutual information to measure uniqueness and predictability within complex systems.
• Applications span biological systems, artificial intelligence, and social networks, enabling analysis of emergent behaviors and dynamic individual boundaries.

The information theory of individuality addresses how the informational content of systems, entities, or agents can define, distinguish, and quantify their status as "individuals" in complex environments—spanning the domains of computation, physics, biology, social systems, and artificial intelligence. It seeks to generalize the concept of individuality from classical notions rooted in identity and autonomy toward mathematically rigorous frameworks specifying boundaries, uniqueness, and the propagation of information.

1. Algorithmic Information and Individual Objects

Algorithmic Information Theory (AIT) provides an objective and robust quantification of the information content of individual objects, in contrast to classical information theory which describes the statistical properties of ensembles [0703024]. In AIT, the Kolmogorov complexity $K(x)$ of a binary string $x$ is the length of the shortest program (for a fixed universal Turing machine $U$) that generates $x$:

$K(x) = \min \{\, |p| : U(p) = x \,\}$

This quantifies how “random” or “structured” an object is; if $K(x)$ is close to $|x|$, $x$ is deemed algorithmically random and possesses maximal individuality in the sense of incompressibility. Chaitin’s $\Omega$ number,

$\Omega = \sum_{p \text{ halts}} 2^{-|p|}$

is a concrete instance of a maximally unknowable real number—exemplifying a form of absolute individuality unattainable by effective description.

Philosophically, AIT diverges from Shannon’s entropy-based classical information theory, rejecting statistical ensembles in favor of intrinsic computational description. Individuality is thus reduced to minimal algorithmic representation, independently of external context.

2. Quantum Individuality and Non-Individuality

Contemporary quantum theory reveals drastic deviations from classical individuality (Ronde et al., 2012, Pylkkänen et al., 2014, Kastner, 2023). Quantum entities (e.g., particles) are fundamentally indistinguishable, and any attempt at labeling leads to nonphysical surplus structures. Quantum statistics enforce non-classical counting—Pauli exclusion for fermions, symmetrization for bosons—such that the number of possible configurations cannot be expressed by classical identifiers.

Formal operationalizations in quasi-set theory allow non-individuals (“m-atoms”) that do not obey the idempotence law $x = x$ in the usual sense. The Kochen-Specker theorem shows the impossibility of assigning definite, non-contextual classical properties to quantum observables, reinforcing the necessity to abandon classical individuality.

Superposition further prevents any assignment of exclusive properties; a state $|\psi \rangle = a|up\rangle + b|down\rangle$ encodes amplitudes, not clear-cut individual attributes. Informationally, quantum systems store correlations, probabilities, and contextual relations rather than fixed individual properties—individuality is replaced by structure defined in the space of observables and outcomes.

In Bohmian mechanics (Pylkkänen et al., 2014), individuality can be partially restored as definite trajectories subject to non-local and context-dependent quantum potentials:

$Q = -\frac{\hbar^2}{2m} \frac{\nabla^2 R}{R}$

However, this individuality is not absolute—it is constrained by global processes (implicate order) and the geometry of the underlying system (symplectic/metaplectic group symmetries).

3. Information-Theoretic Partitioning and Autonomy

In biological, cognitive, or engineered systems, information theory formalizes individuality through algorithmic decomposition and measures of informational autonomy (Krakauer et al., 2014):

Given time-series observations $\{S_n, E_n\}$ (system, environment), information-theoretic boundaries that support individuality are determined by maximizing past-to-future information propagation:

$$I(S_n, E_n; S_{n+1}) = H(S_{n+1}) - H(S_{n+1} | S_n, E_n)$$

Decomposing this,

$$I(S_{n+1}; S_n) \quad \text{(autonomy;\ system informs its own future)},$$

$$I(S_{n+1}; E_n | S_n) \quad \text{(non-closure;\ environment's contribution)}$$

Partitions are incrementally grown by absorbing environmental variables that increase autonomy. Genuine individuals correspond to partitions with maximal “informational closure”—where the autonomous propagation of past-to-future information is strongest and outside influence is minimized. Individuality becomes a question of dynamical predictive power, not replication or static boundaries.

Notably, this approach allows for classification of individuality in cultural evolution, viruses, multicellular assemblies, and collective phenomena where conventional replication-based definitions fail.

4. Information Bottlenecks and Hierarchical Individuality

Transitions in evolutionary or developmental organization—such as the emergence of multicellularity—are described by informational phase transitions, with bottlenecks acting as filters for individuality (Smith et al., 2015). In multilevel selection models, individuals at one level (e.g., cells) may cluster and encode functions across higher levels (e.g., biofilms, multicellular individuals).

The size of the propagule (sporesize $N_S$) in reproduction acts as an information bottleneck:

$$V = 1 \ \text{if} \quad \bigcup_{i=1}^{N_S} G_i = \{1, ..., 1\},\quad \text{else}\ V=0$$

where $G_i$ are bitstrings representing essential functions. As $N_S$ decreases, distributed functions compress into multifunctional individuals—modeling a phase transition in individuality encoding. Horizontal gene transfer smooths fitness transitions, facilitating higher-level individuality. This mechanism, akin to the Eigen error-threshold, exemplifies hierarchical individuality emergence through informational constraints.

5. Entropy, Composability, and Individuality in Complex Systems

Generalized entropy frameworks, including group-theoretical approaches (Sicuro et al., 2015), clarify how individuality can be ascribed to systems through composable information measures. For independent systems $\mathcal{A}, \mathcal{B}$,

$$S(\mathcal{A} \cup \mathcal{B}) = \Phi(S(\mathcal{A}), S(\mathcal{B})), \quad \Phi(x, y) = G(G^{-1}(x) + G^{-1}(y))$$

The associated information measure,

$I_G(\mathcal{A}) = G^{-1}(S_G(\mathcal{A}))$

guarantees additivity:

$$I_G(\mathcal{A} \cup \mathcal{B}) = I_G(\mathcal{A}) + I_G(\mathcal{B})$$

This structure provides a universal formalism for encoding individuality in physical, quantum, and social systems by ensuring that each subsystem’s information content is precisely defined and combined according to group laws.

Einstein’s likelihood principle emerges as a measure of combinatorial individuality:

$$\mathcal{W}_G(\mathcal{A}) = \exp(I_G(\mathcal{A})) = e_G^{S_G(\mathcal{A})}$$

This principle bridges the statistical weight of individual configurations with their entropy-based information content.

6. Quantitative Measures of Individuality: Divergence and Entropy

Shannon entropy, KL divergence, and mutual information quantify individuality in data, biological systems, and artificial agents (Chodrow, 2017, Lairez, 2022). KL divergence,

$$D_{KL}(P || Q) = \sum_i P(i) \log \frac{P(i)}{Q(i)}$$

measures the “surprise” or distinctiveness of an individual’s traits compared to a background, establishing nonconformity as mathematical individuality. Entropy,

$H(p) = -\sum_x p(x) \log p(x)$

provides a measure of uncertainty; lower entropy indicates higher specialization and uniqueness. Mutual information,

$I(X;Y) = H(X) - H(X|Y)$

tracks interdependencies within a system, serving as a signature of internally coherent individuality.

Subjectivity is inherent—entropy depends on the observer’s resolution, and individuality emerges as an informational parameter representing the discriminability of entities.

7. Applications Across Domains: Biological, Artificial, Social, Quantum

The conceptual machinery developed for the information theory of individuality finds substantive application in diverse domains:

• Biological systems: Algorithmic decomposition, autonomy measures, and information bottlenecks formalize individuality in genetic, cellular, multicellular, and even viral and sociocultural contexts (Krakauer et al., 2014, Smith et al., 2015).
• Artificial agents: Individuality is operationalized as the mutual information between agent identity and observation; intrinsic rewards can induce emergent specialization and division of labor in multi-agent reinforcement learning (Jiang et al., 2020).
• Social networks and collective intelligence: Spontaneous agent differentiation via social interaction (e.g., LLM-based agents) demonstrates individuality as an emergent phenomenon rooted in memory, experience, and social clustering (Takata et al., 5 Nov 2024).
• Quantum and computational metaphysics: Quantum haecceity and algorithmic idealism generalize individuality to the field of indistinguishable objects and informational constructs, challenging classical identity (Kastner, 2023, Sienicki, 16 Dec 2024).

In all cases, individuality is no longer viewed as static or intrinsic, but as a function of information content, semantic autonomy, and structural relations—dynamically realized through algorithmic, physical, and statistical processes.

Conclusion

The information theory of individuality rigorously advances from classical identity and replication-based notions toward nuanced, mathematically grounded definitions. It incorporates algorithmic complexity, entropy, mutual information, and compositional frameworks, spanning quantum non-individuality, biological autonomy, artificial differentiation, and the emergent coherence of informational agents. Individuality is encoded, quantified, and propagated according to the minimal descriptive, predictive, and structural properties of the system in context—establishing a unified paradigm for analyzing the nature and boundaries of individuals in complex, adaptive, and interactive environments.