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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.02613 |
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| _version_ | 1866918134486663168 |
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| author | Alpay, Faruk Alpay, Taylan |
| author_facet | Alpay, Faruk Alpay, Taylan |
| contents | We construct a rigorous mathematical framework for an abstract continuous evolution of an internal state, inspired by the intuitive notion of a flowing thought sequence. Using tools from topology, functional analysis, measure theory, and logic, we formalize an indefinitely proceeding sequence of states as a non-linear continuum with rich structure. In our development, a trajectory of states is modeled as a continuous mapping on a topological state space (a potentially infinite-dimensional Banach space) and is further conceptualized intuitionistically as a choice sequence not fixed in advance. We establish fundamental properties of these state flows, including existence and uniqueness of evolutions under certain continuity conditions (via a Banach fixed-point argument), non-measurability results (demonstrating the impossibility of assigning a classical measure to all subsets of the continuum of states), and logical semantic frameworks (defining a Tarskian truth definition for propositions about states). Throughout, we draw on ideas of Brouwer, Banach, Tarski, Poincaré, Hadamard, and others--blending intuitionistic perspective with classical analysis--to rigorously capture a continuous, ever-evolving process of an abstract cognitive state without resorting to category-theoretic notions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02613 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear Continuum of States and Intuitionistic Flows in a Cognitive Space Alpay, Faruk Alpay, Taylan Logic Dynamical Systems 37Cxx, 03F55, 54C35, 28A05, 28A05, 28D05, 03B48, 53B12 F.4.1; G.3 We construct a rigorous mathematical framework for an abstract continuous evolution of an internal state, inspired by the intuitive notion of a flowing thought sequence. Using tools from topology, functional analysis, measure theory, and logic, we formalize an indefinitely proceeding sequence of states as a non-linear continuum with rich structure. In our development, a trajectory of states is modeled as a continuous mapping on a topological state space (a potentially infinite-dimensional Banach space) and is further conceptualized intuitionistically as a choice sequence not fixed in advance. We establish fundamental properties of these state flows, including existence and uniqueness of evolutions under certain continuity conditions (via a Banach fixed-point argument), non-measurability results (demonstrating the impossibility of assigning a classical measure to all subsets of the continuum of states), and logical semantic frameworks (defining a Tarskian truth definition for propositions about states). Throughout, we draw on ideas of Brouwer, Banach, Tarski, Poincaré, Hadamard, and others--blending intuitionistic perspective with classical analysis--to rigorously capture a continuous, ever-evolving process of an abstract cognitive state without resorting to category-theoretic notions. |
| title | Nonlinear Continuum of States and Intuitionistic Flows in a Cognitive Space |
| topic | Logic Dynamical Systems 37Cxx, 03F55, 54C35, 28A05, 28A05, 28D05, 03B48, 53B12 F.4.1; G.3 |
| url | https://arxiv.org/abs/2509.02613 |