Geometrization of Mental States

• Geometrization of Mental States is defined as the rigorous mathematical representation of internal cognitive states as points, trajectories, or flows in geometric spaces.
• The framework applies Banach spaces and differential equations to ensure continuity, existence, and uniqueness of cognitive trajectories through dynamical systems principles.
• It integrates intuitionistic representations with classical analysis to capture evolving thought processes while addressing measure-theoretic and semantic limitations.

The geometrization of mental states refers to the rigorous mathematical representation of internal cognitive states and their dynamics as structures within geometric, topological, or functional analytic spaces. This approach synthesizes ideas from dynamical systems, topology, measure theory, logic, and intuitionistic mathematics to formalize the continuous, evolving nature of mental processes—such as thoughts, emotions, and memories—without recourse to categorical abstractions. Models in this field encode mental states as points, trajectories, or flows within a possibly infinite-dimensional state space, allowing one to paper existence, uniqueness, continuity, and limitations concerning measurability and formal semantic expressibility.

1. Abstract State Space and Topological Structure

The primary mathematical foundation is an abstract state space $M$, equipped with a topology $\tau$ that defines continuity of mental state evolution. In this context, a continuous trajectory $x: \mathbb{R} \to M$ models the progression of a mental state over time: for any open set $U \in \tau$, the preimage $x^{-1}(U)$ must be open in $\mathbb{R}$. This condition enforces that small changes in time manifest as small changes in state, faithfully reflecting the gradual flow of cognitive processes. Frequently, $M$ is chosen as a Banach space—a complete normed vector space—which serves as an analytic substrate for formulating differential equations governing mental evolution.

Measure-theoretic structure (i.e., a measure $\mu$ or a σ-algebra $\mathcal{F}$ on subsets of $M$) is considered to quantify probabilistic or statistical properties of mental states, although non-measurability of certain subsets (Vitali sets, Banach–Tarski paradox) limits the scope of classical volume assignment within the full continuum.

2. Trajectories, Flows, and Dynamical Evolution

Mental state evolution is modeled as a trajectory $x: \mathbb{R} \to M$, encapsulating the notion of a "flowing" or "proceeding" thought sequence. The analytic theory employs ordinary differential equations (ODEs), e.g.,

$\dot{x}(t) = F(x(t)),\quad x(0) = s_0$

where $F: M \to M$ defines the cognitive dynamics. The Banach fixed-point theorem (contraction mapping principle) guarantees that under a Lipschitz condition on $F$, there exists a locally unique, continuously differentiable solution on an interval $[–T, T]$. This formalism constructs cognitive trajectories, occasionally extended to flows $\{\Phi^t: M \to M\}_{t \in \mathbb{R}}$ satisfying $\Phi^{t+s} = \Phi^t \circ \Phi^s$, encoding the semigroup property fundamental to dynamical systems.

Infinite-dimensional Banach spaces allow modeling the high complexity and nonlinearity intrinsic to abstract mental constructs, with the possibility of defining cognitive recurrence phenomena analogously to Poincaré recurrence in physical systems.

3. Intuitionistic Representation: Choice Sequences and Creating Subject

The framework blends classical analytic techniques with intuitionistic logic to reflect the phenomenology of "free" mental evolution. Rather than treating a trajectory as a completed function, the intuitionistic approach conceptualizes evolution as a choice sequence $(s_0, s_1, s_2, \ldots)$, where each state is generated stage-by-stage without a predetermined law—a direct invocation of Brouwer's "creating subject." This perspective posits the continuum not as a fixed, completed totality, but as an open-ended process, closely mirroring the subjective experience of thought formation.

Notably, intuitionistic principles (e.g., continuity principle, fan theorem) ensure that all observables on these choice sequences are uniformly continuous, avoiding nonconstructive pathologies and offering robust regularity in modeling mental phenomena.

4. Existence, Uniqueness, and Analytic Foundations

Analytic results include rigorous existence and uniqueness criteria for mental state trajectories. If $F$ is Lipschitz continuous, the contraction mapping argument establishes the unique solvability of the ODE: $$x(t) = s_0 + \int_0^t F(x(\tau)) \, d\tau, \quad |F(u) - F(v)| \leq L |u - v|$$ in the space $C([–T, T], M)$ with uniform norm. Iterative application of $\Phi$ (the evolution operator) constructs successive approximations to the trajectory, confirming not only existence but also guide for numerical simulation or analytic studies of mental processes.

Under strengthened global conditions, solutions extend for all $t \in \mathbb{R}$, offering a robust model for indefinitely proceeding cognitive evolution.

5. Non-Measurability and Logical Semantic Limits

The richness and continuity of the state space entail foundational restrictions. Classical set theory and the axiom of choice imply the existence of nonmeasurable sets (e.g., via the Vitali construction), which preclude a countably additive measure from assigning a volume or probability to every subset of $\mathbb{R}$ or its image in $M$. This result directly impacts attempts to quantify "how much time" a system spends in a region or probability assignments for mental phenomena—certain fragments of the cognitive continuum remain outside quantitative reach.

Logical semantics are formalized via a two-sorted language with atomic predicates like $X(t, s)$ ("at time $t$, the system is in state $s$"). Tarski's inductive definition of truth is applied in the structure $\mathcal{M}$ with time domain $\mathbb{R}$ and state domain $M$: $$X(t, s) \text{ is true in } \mathcal{M} \iff s = x(t)$$ Despite this, the set of all true sentences (the exhaustive theory of a mental flow) is not definable within $\mathcal{M}$ due to the Tarski undefinability theorem and diagonal lemma, asserting a fundamental limit to self-representation and introspective completeness.

6. Theoretical Influences and Methodological Contrasts

The development draws on several mathematical traditions:

• Intuitionism (Brouwer): Models the continuum as an ongoing act of creation via choice sequences, capturing the flexibility and generativity of mental evolution.
• Functional Analysis (Banach): Provides existence and uniqueness proofs for the analytic dynamics of state flows.
• Logical Semantics (Tarski): Supplies the inductive framework for interpreting truth in formal languages concerning mental state trajectories.
• Dynamical Systems (Poincaré, Hadamard): Motivates recurrence phenomena and underlines the complexity of high-dimensional flows.
• Measure Theory: Illuminates the limitations imposed by nonmeasurable sets and paradoxes (e.g., Banach–Tarski) in representing the full structure of continuously evolving mental states.

The paper deliberately avoids category-theoretic approaches (sheaves, geodesic flows, higher categorical unifications), emphasizing classical topological, analytic, and intuitionistic constructs as a more direct means to model continuous cognition.

7. Synthesis and Implications

The geometrization scheme situates mental states as points and flows within topological or functional spaces, models their time evolution as analytic trajectories (with rigorous existence and uniqueness), and represents their progression as unfolding, nonpredetermined choice sequences. The analytic–intuitionistic blend allows expressing both the mathematical regularity of continuous change and the phenomenological openness of thought.

The inability to fully measure or semantically encapsulate all aspects of the evolving continuum reveals intrinsic limitations for both quantitative approaches and formal representational completeness in cognitive modeling. This suggests a plausible implication: any full theory of cognition must accommodate not only analytically tractable laws but also foundational constraints arising from the nature of the continuum and the logic underlying its generation.

Overall, this integrated model provides a mathematically rigorous, geometrically-structured, and phenomenologically faithful account of mental state evolution in cognitive spaces, suitable for further research into formal cognitive science, mathematical psychology, and dynamical systems theory of thought (Alpay et al., 31 Aug 2025).