Objects Do Not Determine Structure: A Symmetry Rigidity Theorem for Mathematical Structuralism

Mathematical structuralism holds that mathematical objects are determined by their structural roles — the pattern of relations they bear to other objects — rather than by intrinsic properties (Benacerraf, 1965; Shapiro, 1997). We prove the converse side of this thesis in the categorical setting: bare objects do not canonically determine structure. Our central result is the Symmetry Rigidity Theorem. Formalizing "bare objects" as the image of the object functor Ob: Cat_sm → Set and "canonical reconstruction" as a functorial section of Ob, we show that any such section must produce categories on which the full symmetric group Sym(X) acts as automorphisms. Since Sym(X) acts transitively on ordered pairs of distinct elements, every functorial section yields structurally uniform categories — all non-diagonal hom-sets are isomorphic, all diagonal hom-sets are isomorphic, and composition is permutation-invariant. No functorial section can produce the non-uniform hom-sets that characterize virtually all algebraically or topologically interesting categories. We prove the two adjoint reconstructions (the discrete and codiscrete categories) are extremal — initial and terminal among categories with a fixed object set — and both are structurally degenerate. The theorem sharpens the identity problem for ante rem structuralism (Keränen, 2001): canonical reconstruction produces maximally non-rigid structures, so the identity problem is a necessary consequence of canonicality, not an accident of specific mathematical examples. A geometric interpretation via the nerve construction identifies the non-recoverable content: for connected groupoids, it is precisely the holonomy of the categorical connection (π₁ ≅ Aut).