From Reentry to Subject: A Topological Approach to the Hard Problem of Consciousness
Yury Berdinsky
Study at a glance
AI-extracted from the abstract| Characteristics | Theoretical or philosophical paper |
|---|---|
| Topics | Philosophy of mind |
| Key points | Argues that the gap between objective description and subjective experience is a quotient construction: when an evaluative loop is active, the quotient space is non-Hausdorff and contains asymmetric doubletons, which mathematically encode directedness and ineffability. Proposes that local sections of a sheaf over this space quantize experience into symbols and actions, while no global section separates the experiencer from the experienced. |
Abstract
Why are physical processes in the brain accompanied by subjective experience at all-by the redness of red, the taste of coffee or the feeling of pain? This puzzle, known as the "hard problem of consciousness", has for centuries been left to metaphysics, because from the outside we see only a mechanism: ionic currents, neuronal discharges and the cogwheels of the connectome. The present work proposes an entirely new, topological approach to its solution, proving that the "interior" of a subject is not a mystical addition to physics but a rigorous consequence of the geometry of the causal cascade itself. We show that when information flows close into a genuine evaluative loop, the physical state space inevitably undergoes factorization. The gap that thereby arises between the objective description and subjective experience becomes not a mystery but an exact mathematical fact. This approach also resolves automatically the classical paradox of Leibniz's mill: within this model, many fundamental contradictions of the philosophy of mind become strict consequences of long-established mathematical theorems. A physical substrate-a brain, a silicon chip, a network of people-is described by ordinary mathematics: its state space is Hausdorff, which means that any two distinct states can be surrounded by disjoint neighbourhoods and therefore told apart by measurement. The interior of a subject is not like that: an experienced content and the act of experiencing it cannot be pulled apart, and no verbal report ever fully separates them. This paper shows that the gap between the two descriptions is not a mystery but a quotient construction. We start from the objective reentry loop of the subject-centred model of K. V. Titov-the closed causal cycle between the intending subsystem D and the intentional subsystem I-and define an equivalence relation on the physical state space: two states are equivalent when the loop itself, with its own internal observables, cannot distinguish them. We prove a Central Factorization Theorem: whenever the loop is alive in the precise sense S > 0 (spectral radius ρ(R) > 1 and cycle complexity C > 0), the quotient space Y = X/ ∼ is not Hausdorff and contains, for every D-I pair, a connected Alexandroff doubleton {d, i} with topology {∅, {i}, {d, i}}. The doubleton is asymmetric: {i} is open, {d} is not, and consequently every continuous map from it into a T 1 space is constant-the mathematical form of directedness (D shapes I, not conversely) and of the ineffability of the interior. We then prove that Y is nevertheless locally Hausdorff away from the glued locus, so that it carries ordinary charts; these charts are identified with the three spaces of Titov's model-subjective SuS, virtual ViS, objective ObS-via the spectral bands of R. Finally we build the reverse bridge: a sheaf F over Y whose sections are discrete external objects (words, motor commands, coordinates). Local sections exist and are exactly the quantization of experience into symbols and actions; a global section separating d from i does not exist. The paper is written as an interdisciplinary text: every notion-topology, open set, quotient, sheaf, spectral radius-is introduced from scratch, with examples, so that a mathematician, a neurophysiologist, a BCI engineer, an AGI developer and a philosopher of mind can all read it. Empirical correlates (τ m ≈ 10-20 ms, Ivanitsky's information synthesis at 100-150 ms, thalamocortical loops), falsifiable predictions, and a machine-verified Lean 4 formalization of the four central topological statements is included.
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