Where does isomorphism collide?

The process by which organizations operating in the same field come to resemble one another over time, driven less by efficiency than by pressures toward conformity: DiMaggio and Powell's 1983 The Iron Cage Revisited named three mechanisms, coercive isomorphism from legal and political dependence, mimetic isomorphism from copying peers under uncertainty, and normative isomorphism from shared professional training. The referent is a sociological tendency toward homogeneity across a population of organizations, seen as convergence in structure and practice, and it is a borrowed metaphor of resemblance rather than the exact structure-preserving map of the mathematicians, the shared crystal form of the chemists, or the mind-to-brain correspondence of the Gestalt theorists.[1]

Unit

A field's drift toward organizational sameness, by degree of convergence

Recovery means

Not applicable

Measured by

Assessed by tracking convergence in organizational structures, policies, and practices across a field over time

A structure-preserving bijection between two mathematical structures whose inverse also preserves structure, so that the two are identical in every property expressible in their shared language and differ only in the names of their elements: for groups it is a bijective homomorphism, and for graphs a vertex bijection that preserves adjacency. Dummit and Foote's Abstract Algebra defines isomorphic structures as the same up to relabeling. The referent is an exact correspondence in a formal system, either present or absent with nothing in between, and it carries none of the loose resemblance of the sociologists, the physical form-sharing of the crystallographers, or the mind-body matching of the perceptual theorists.[2]

Unit

An exact structure-preserving bijection between two structures

Recovery means

Not applicable

Measured by

Established by exhibiting a bijection that preserves the relevant operations or relations, or ruled out by an invariant

The property, named in Mitscherlich's law of isomorphism, whereby two chemically distinct substances crystallize in the same form and can substitute for one another in a shared crystal lattice to form mixed crystals: the classic cases were the inorganic compounds that Mitscherlich found to share a crystal shape despite their differing composition. Salvia's 2013 Emil Wohlwill's Entdeckung des Isomorphismus traces the discovery and its place in nineteenth-century crystallography. The referent is a relationship between real substances that share an outward crystalline form, verified by measuring crystal angles and forming mixed crystals, and it is unrelated to the exact map of algebra, the organizational resemblance of sociology, or the perceptual correspondence of Gestalt theory.[3]

Unit

Two compositions sharing one crystal form, able to form mixed crystals

Recovery means

Not applicable

Measured by

Confirmed by measuring interfacial crystal angles and by growing mixed crystals across the two compositions

The proposed structural correspondence between the organization of conscious experience and the organization of its underlying brain process: Kohler's psychophysical isomorphism held that a percept and its neural substrate share the same functional structure, so that the form of experience mirrors the form of the physiological field. Wagemans and colleagues' 2012 A Century of Gestalt Psychology in Visual Perception: II reviews the doctrine and its fate. The referent is a hypothesized topological, not literal, matching between mind and brain, argued within perceptual theory, and it shares nothing with the algebraist's exact bijection, the chemist's shared crystal form, or the sociologist's converging organizations beyond the single word.[4]

Unit

A structural match between a percept and its neural substrate

Recovery means

Not applicable

Measured by

Argued from the correspondence between the structure of experience and hypothesized brain fields, rather than directly measured