--- title: "Ch 24 — Operational research, models and homomorphism (Decision and Control)" description: "The methodological chapter: what a model is, why a model of a firm can only be a many-one mapping, and how homomorphism lets two situations be compared…" url: "https://wiki.cybernetic-futures.org/pure-vsm/part-05-method-and-epistemology/ch24-operational-research-models-and-homomorphism" modified: "2026-11-12T15:26:26.113Z" language: "en" site: "Cybernetic Futures" --- # Ch 24 — Operational research, models and homomorphism (Decision and Control) [Chapter content](https://wiki.cybernetic-futures.org/pure-vsm/part-05-method-and-epistemology/ch24-operational-research-models-and-homomorphism/ch24-operational-research-models-and-homomorphism-content) · Overview diagram # Ch 24 — Operational research, models and homomorphism (Decision and Control) **Part V — Method and epistemology.** The methodological ground of everything before it: what a model is, what kind of correspondence between two situations is defensible, and why the brain–firm comparison is not an analogy. ## The question What is a model — and what licenses one situation to stand for another? ## The reading Beer starts from ordinary thinking: the mental representation of the world that is not direct perception of it is a model of the world (DC 1966: 100), and a model is a reflection of what is the case made explicitly available for experimentation (101). The scientist differs from the citizen not in having models but in being professional about their use and self-conscious about the one in hand (101). The narrow sense in which the term mathematical model is often used — an equation relating one set of variables to another — is rejected as debased (102). To make correspondence precise he takes three terms from mathematics. A mapping is a correspondence in which each element of one set has an image in another; an isomorphism is a one-one mapping that also preserves operational characteristics, being literally the same form; a homomorphism is a many-one mapping in which the transformation is arranged so that certain operational characteristics survive (DC 1966: 107–08). The scientific procedure follows: manager and scientist each hold a conceptual model, their comparison begins as an analogy whose validity is always open to disputation, and the scientist maps both models homomorphically into a formal language, where the two results may be isomorphic (112–13). What that achieves is an identity between the managerial and the scientific situation shorn of mysticism and of poetry, and free of the disputation that attends analogy (113). Hence his definition: a scientific model is a homomorphism onto which two different situations are mapped, and which defines the extent to which they are structurally identical (113). What the transformation ignores cannot damage validity, only usefulness — utility being tested by prediction (113). That is the basis for the later claim that the VSM is not an analogy (chapter 4): the brain–firm relation is meant as a homomorphic mapping of structure, not a literary comparison. *Decision and Control* already reports the precursor of the model, for an industrial company whose conceptual model is inevitably taken from neurophysiology — a homomorphic mapping of the brain, by definition a simplification whose limitations are understood — built in set theory from submodels of the sensory cortex, the motor cortex, the thalamic level and the reticular formation, and used to take successful management decisions (136–38). The chapter's last movement extends this to laws and to self-knowledge. A law of nature is derived from what is no more than a model of such general utility that it is never recognized as such (DC 1966: 121); what counts as a valid model rests on social conventions, and progress in science might well be defined as the overthrow of a model that has exhausted its usefulness (122). Enterprises, Beer adds, are accustomed to confuse their models of reality with reality itself (123). No manager can handle a complete isomorph of his enterprise unless he is its only employee; he can at best settle for a homomorph (124) — and homomorphic modelling is at least a heuristic method for inferring the structure of systems whose complexity defeats isomorphic modelling (125). ## Beer locators DC 1966: 100–08, 112–13, 121–25, 136–38, 142–71, 299–398; MS 1967: 63–69, 99–100. ## Status Draft, *Pure VSM* v0.8 (2026-09-23). Beer's texts govern where the draft and they differ. Checked: the *Decision and Control* text layer drops em-dashes in places, so the quotations were cut to avoid depending on a missing character; the one place where the e-book itself reads "simplificationand" (136) is now quoted as two fragments rather than one sentence. ## Open points - What character the print has at DC 1966: 136 remains unchecked; a printed copy is needed. - §§24.6–24.8 (formal languages; orthodox versus cybernetic models in the control of operations; partial and organic models of the enterprise) were added as items for review — about 1,350 words, because the item spans four chapters of *Decision and Control*. Whether the handover paragraph ("Three results carry forward …") belongs at the end of §24.5 or after §24.8 is a structural question for Krishan. ## Related [Part V — Method and epistemology](https://wiki.cybernetic-futures.org/pure-vsm/part-05-method-and-epistemology), [Ch 23 — Syntegrity and the VSM](https://wiki.cybernetic-futures.org/pure-vsm/part-04-team-syntegrity/ch23-syntegrity-and-the-vsm), [Ch 25 — Systems as an act of recognition](https://wiki.cybernetic-futures.org/pure-vsm/part-05-method-and-epistemology/ch25-systems-as-an-act-of-recognition), [Homomorphism](https://wiki.cybernetic-futures.org/pure-vsm/concepts/homomorphism), [Viable System Model](https://wiki.cybernetic-futures.org/pure-vsm/entities/viable-system-model), [Decision and Control](https://wiki.cybernetic-futures.org/pure-vsm/entities/decision-and-control), [Management Science](https://wiki.cybernetic-futures.org/pure-vsm/entities/management-science-1967), [Brain of the Firm](https://wiki.cybernetic-futures.org/pure-vsm/entities/brain-of-the-firm).