Eugene Nivorozhkin

Triadic epistemologies—frameworks structured around Subject, Object, and Observer— face a structural difficulty: the regress of observation generates indefinite recursion without a principled stopping point. We argue that this difficulty has a precise mathematical expression. Formalising the triadic structure as the attractor of a three-contraction iterated function system (IFS) yields a Sierpi´nski triangle with Hausdorff dimension dH = ln 3/ ln 2 ≈ 1.585—fractional, and therefore lacking a canonical measure compatible with the ambient space of epistemic possibility. Adding Context as a fourth epistemic role, formalised as a fourth IFS contraction, produces a Sierpi´nski tetrahedron with Hausdorff dimension dH = ln 4/ ln 2 = 2 exactly. This phase transition from fractional to integer dimension is the central result. At integer Hausdorff dimension the knowledge surface acquires a well-defined measure compatible with standard surface measure on 2-manifolds, making Shannon entropy over epistemic states well-defined in the standard sense. Context is therefore not an optional enrichment of the triadic structure but a mathematical necessity: the condition under which the epistemic surface becomes measurable, and under which claims about the quantity of knowledge and uncertainty become formally stateable.

Triadic epistemologies—frameworks structured around Subject, Object, and Observer—face a structural difficulty: the regress of observation generates indefinite recursion without a principled stopping point. We argue that this difficulty has a precise mathematical expression. Formalising the triadic structure as the attractor of a three-contraction iterated function system (IFS) yields a Sierpi´nski triangle with Hausdorff dimension dH = ln 3/ ln 2 ≈ 1.585—fractional, and therefore lacking a canonical measure compatible with the ambient space of epistemic possibility. Adding Context as a fourth epistemic role, formalised as a fourth IFS contraction, produces a Sierpinski tetrahedron with Hausdorff dimension dH = ln 4/ ln 2 = 2 exactly. This phase transition from fractional to integer dimension is the central result. At integer Hausdorff dimension the knowledge surface acquires a well-defined measure compatible with standard surface measure on 2-manifolds, making Shannon entropy over epistemic states well-defined in the standard sense. Context is therefore not an optional enrichment of the triadic structure but a mathematical necessity: the condition under which the epistemic surface becomes measurable, and under which claims about the quantity of knowledge and uncertainty become formally stateable.