Chapter 28: Abstract Objects and Structural Realism

Mathematical entities are positions in structures, not independently existing substances in a separate Platonic realm. Mathematical structuralism—the thesis that mathematical objects have no intrinsic nature beyond their structural relations—is the natural ontology of Category τ. The Platonism-nominalism debate presupposes substance ontology; τ removes the presupposition, and the debate dissolves. The question “Is mathematics discovered or invented?” dissolves in parallel: when objects are structural positions, investigation is traversal of the address space, and the discovery-invention distinction collapses. Abstract objects are not mysterious extras added to a physical ontology; they are structural positions accessed through the diagrammatic register ℝeg_D.