Eugene Nivorozhkin

Ontic structural realism faces a familiar problem: if relations are fundamental, what are the relata, and how are they individuated? A common formulation of the problem assumes that a determinate relatum must possess an intrinsic identity independent of the relations in which it stands. This paper examines that assumption from a non-eliminativist structural-realist perspective. It develops the Sierpiński gasket as a formal model of recursive relational individuation. The model contains determinate elements and a recursively specified structure, while its incidence profiles provide a model-relative criterion for distinguishing elements without assigning them intrinsic philosophical identities independent of the construction. The gasket is also the final coalgebra of a gluing functor on tripointed metric spaces, a known result which fixes its points and topology by its gluing structure without coordinates in an ambient plane. The mathematical results establish the coherence of this architecture; they do not show that physical reality has fractal geometry or that ontic structural realism is true. Rayleigh–Bénard convection is used as a physical illustration of the distinction: convective cells are dynamically real and measurable, while their exact boundaries depend on the fields and scale of analysis used to describe them. The paper argues that model-relative individuation need not be arbitrary or observer-created. Its narrower conclusion is that the relations-require-relata objection does not by itself establish that non-eliminativist structural realism is incoherent. Relata may remain real while their relevant individuation conditions are structurally dependent.

Recursive models of knowledge that include the observer are often triadic, built on Subject, Object and Observer. Formalised as an iterated function system on a simplex, a triadic model yields the Sierpinski gasket, of fractional Hausdorff dimension ln 3/ ln 2; a fourth role yields the Sierpinski tetrahedron, of dimension 2. This invites an objection: any power of two gives an integer, so nothing in the arithmetic singles out four roles or the value 2. The paper addresses the objection under explicit modelling conditions. If each role’s perspective is a homothetic copy of the whole shrunk towards that role, and every pair of perspectives meets in exactly one point, the contraction ratio is forced to one half and a fourth role in the triad’s own plane is excluded. Under these conditions the attractor of r roles has Hausdorff dimension at least the dimension of the space spanned by r − 1 of its roles only for r ≤ 4, and equal to it only for r = 4. Four is therefore a threshold: the last number of roles at which the structure keeps pace with the configuration it extends, and the only one at which it matches it exactly, with finite positive measure in that dimension. The paper also counts the independent loops of contact carried by the construction’s voids, shows that boundary elaboration cannot do the work of a fourth role, shows that the contact condition decides whether Context extends the triad’s space or saturates it, and states the limits of the result.