Homomorphism

2 min read

Homomorphism

A homomorphism is a many-one mapping, not an isomorphism, arranged so that “certain operational characteristics concerning the relationship of elements are preserved” (DC 1966: 108). “A scientific model is a homomorphism on to which two different situations are mapped, and which actually defines the extent to which they are structurally identical” (DC 1966: 113).

What Beer says

A model is the “mental representation of the world that is not direct perception” (DC 1966: 100); “explicitly made available for experimentation” (DC 1966: 101). Mapping: each element has an “image” (DC 1966: 107). Isomorphism: one-one correspondence that also “preserves operational characteristics” (DC 1966: 108). First comparison is analogy, “always open to disputation” (DC 1966: 112); both models then mapped homomorphically — “completely free of the bedevilling disputation that attends analogy” (DC 1966: 113). Dropped elements cost utility, not validity. The manager “can at best settle for a homomorph” (DC 1966: 124). Precursor of the Viable System Model: “a homomorphic mapping of the brain which is by definition a simplification” (DC 1966: 136).

Where it sits in the model

Why brain–firm is not analogy: the VSM maps structure (Brain of the Firm; Part II — The Viable System Model). Recognizing a system is forming a conceptual model (Observer and recognition; DC 1966: 246).

Textual notes

Jacket vs body vs method (Pure VSM — Appendix): jacket “should not be thought of as ‘analogy’” (BOTF 1981, jacket); body “Models are more than analogies” (BOTF 1981: 75); method: scientific model as homomorphism (DC 1966: 113). MS restates for a general readership (MS 1967: 63–69).

Punch and Judy

Punch and Judy (Mathis and Heath, v0.5.0 draft) turns on the construction-secured validity this page describes: it reads Beer as holding that structural identity is secured by construction so that a model losing utility “cannot lose in validity,” and argues that dropping the prediction test he also attached leaves the claim exposed to nothing (Chs 20–21). Chapter 1 charges that the anatomical shorthand taught has drifted from the homomorphic mapping Beer actually wrote — a charge the manuscript states it cannot yet prove — while recording that Beer took the mapping to be a simplification whose failure would cost usefulness, not validity. Chapter 21 adds that a community may hold the representational reading, but not take it when teaching and selling and the performative reading when asked for evidence.

Open points

DC 1966: 136 e-book reads “simplificationand”; print character unchecked. Draft not canon (Pure VSM — Overview).

Part V — Method and epistemology · Decision and Control · Management Science (1967) · Stafford Beer · Punch and Judy — Overview