In chapter 7 of Decision and Control (1966) Beer reports an electroencephalogram of an industrial company, worked out by hand. The experiment only makes sense against the theory of the model that he builds in chapter 6 (“About Models”) and chapter 7 (“Models in Aspic”). Decision and Control is Beer’s account of what operational research (OR) is and why it works, and much of this long book deals with techniques and cases. The theory of the model is the strand his later work rests on. When Beer insists in Brain of the Firm that the comparison between the firm and the nervous system is not an analogy (see chapter 4.1), he is relying on the argument reconstructed here.
24.1 Everyone has models
Chapter 6 opens with a quotation from the Persian physician Rhazes, and Beer paraphrases his point: “before we do something, we contemplate doing it” (DC 1966: 100). He starts, then, from an ordinary faculty of the human mind, the ability to anticipate, and leaves science for later. Rational conduct therefore depends “not only upon knowing what is really happening and being able to interpret it, but on having present in our minds a representation of what is going to happen next” (DC 1966: 100). This representation, Beer adds, “is not an account of what is the case, but a continuous prognosis of what is about to be reported to us as being the case. It is a prognosis continuously corrected by feedback” (ibid.). From this he takes his definition: “Let us call this mental representation of the world that is not direct perception of the world a model of the world” (DC 1966: 100). He then separates the model from two neighboring concepts. It is not a hypothesis, because “the model does not postulate the causal mechanisms that underlie events”; nor is it a theory, because “it has no explanatory content” (DC 1966: 101). What remains is this: “A model is simply a reflection of whatever is the case, which is explicitly made available for experimentation” (DC 1966: 101). The possibility of experiment separates a model from perception. A healthy person, Beer writes, distinguishes between his sensations of reality, “with which he must not tinker”, and the “‘spare copy’” of the perceptual pattern which he takes off for experiment (ibid.). Science continues the same activity. “Now science is a carefully thought out, professional, version of this rigorously rational behaviour” (DC 1966: 101). The scientist differs from the ordinary citizen only in being “professional about its use. He is self-conscious about the model he uses” (ibid.). Beer is open that this is his own position: “this account of scientific methodology is my own, and must not be taken to be an agreed account of the matter” (DC 1966: 102). The same passage takes a position on mathematics. In Beer’s usage “a mathematical model is an algebraic statement of the representation already discussed” (DC 1966: 102). He rejects the narrow sense of “mathematical model” as “simply an equation purporting to relate one set of variables […] with another set”, calling it “the very much debased sense in which the term ‘mathematical model’ seems often to be used” (DC 1966: 102). An equation linking a few variables is, on this view, not yet a model. A model represents the situation as a whole, and mathematics is one language in which that representation can be written down. Beer then asks how the manager and the scientist handle their models. The manager compares his present model with remembered models of similar situations; this is “a way of saying formally that ‘he brings his experience to bear’” (DC 1966: 102). The OR scientist cannot do this, because “he has by definition no historical models of the situation” (DC 1966: 103). He brings models from his own science instead. A thermodynamicist brings a model of systems characterized by changes in entropy. A neurophysiologist brings a model of the nervous system, and “there are obvious resemblances between the controls used in an enterprise and those used in the human body. For example: both are hierarchical, both are redundant and both incorporate sub-systems of greater and lesser autonomy” (DC 1966: 104–105). No such transfer is ever wholly useless: “In the last resort, any physical system resembles any other to the extent that each endures” (DC 1966: 105). Here Beer finds his formal reason for the interdisciplinary OR team. Each scientist sheds light from a different direction, and Beer compares the whole process “by a simile, with the process of triangulation in a survey of terrain” (DC 1966: 105).
24.2 Mapping, isomorphism, homomorphism
The neurophysiologist’s resemblances expose a weakness, and Beer names it himself. As described so far, the method “seems to depend on the examination of similes, which is well known to be dangerous and highly suspect” (DC 1966: 105). To say that a company’s control system is “like” the central nervous system is, “At the worst, […] a glib literary metaphor” (ibid.). If nothing better can be done, he concedes, “the developing methodology is futile” (DC 1966: 106). The remedy is a vocabulary taken from three branches of mathematics: “group theory, set theory and algebraic topology” (DC 1966: 106). Beer builds it up from the idea of a set, “no more than a properly specified collection of items, called elements of the set” (DC 1966: 106), and from partitions of sets, which he illustrates with the organization chart (DC 1966: 106–107). To make “correspondence” between models precise, he then introduces three terms (DC 1966: 107–108):
- Mapping: a correspondence in which each element of one set has an “image” in another (DC 1966: 107).
- Isomorphism: a mapping that “not only involves a one-one correspondence of elements, but which also preserves operational characteristics”; “isomorph means ‘having the same form’” (DC 1966: 108).
- Homomorphism: a many-one mapping that is not isomorphic, but “the transformation can be arranged so that certain operational characteristics concerning the relationship of elements are preserved. When this is done, the result is a homomorphism” (DC 1966: 108). Beer’s examples show what is at stake. Two engines from the same assembly line map on to each other element for element and function for function: “if the first engine works, so will the second” (DC 1966: 108). That is isomorphism. For homomorphism he uses arithmetic to the base five. The infinite set of natural numbers is mapped many-one on to the five digits 0–4, and the test is whether addition still gives consistent results after the transformation: adding 11 and 19 and then transforming the result must give the same answer as transforming 11 and 19 first and then adding them under the new rule (DC 1966: 109–110). Information is lost, since many numbers share one image, but a structure (here, addition) is preserved. Everything that follows depends on this combination of a mapping that loses detail with a structure that is kept.
24.3 The scientific model
Manager and scientist each have a conceptual model of their situation, and with the new terms Beer can say what OR does with the two. Compared directly, they yield at first an analogy, whose validity “must always be open to disputation” (DC 1966: 112). Beer distinguishes three levels at which such a comparison can be made. The first is metaphor, which “is a poetic device, and its validity is aesthetic; it can offer science no more help than its verbal facility” (DC 1966: 111–112). The second is analogy, whose validity is logical, but “the validity of a particular analogy must always be open to disputation, and its relevance in a particular instance cannot be formally demonstrated in a conclusive way” (DC 1966: 112). The third is identity; but between two things that are not in fact one and the same, identity has “a mystical validity” (ibid.). In practice, Beer notes, the scientist mixes all three, and the process is “self-defeating” (ibid.). In more modern terms, he is “attempting to establish a mapping between the two conceptual models, under some transformation which he would like to be isomorphic” (ibid.). Beer’s way out has two stages. The conceptual models are “literary things”, and the scientist first translates each of them into “the languages of mathematics, statistics and logic” (DC 1966: 112). The translation has a price: “some of the conceptual richness must inevitably be lost” (ibid.). The transformations from contents of mind to formal propositions are many-one, and Beer glosses this: “This is what a loss of richness means”. If the result is to represent its source it “should be homomorphic, too” (DC 1966: 113). The scientist therefore maps both conceptual models homomorphically into a formal language, and the two resulting models “may well be isomorphic with each other” (DC 1966: 113). “So he has achieved an identity between the managerial and the scientific situations which is shorn of mysticism and of poetry, and which is completely free of the bedevilling disputation that attends analogy” (ibid.). Then comes the definition: “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 is left out can damage usefulness but never validity: “If the transformation has ignored as irrelevant elements which are in fact relevant, then the model will lose in utility, but it cannot lose in validity” (ibid.). Utility is tested by prediction (ibid.). If the predictive value is low, “we had better start again”; if it is high, the model can be used “without further worry as to the appropriateness or otherwise of the analogy drawn at the conceptual level” (DC 1966: 113). The definition has a critical edge. University teaching in OR, Beer observes, concentrates on mathematical techniques and treats modeling as the direct algebraic formulation of relationships believed to exist in the managerial situation. The scientific half of the procedure is ignored and the mappings are not studied. “Such activity should be referred to as applied mathematics, and not as operational research” (DC 1966: 114). He had made the same objection to the “debased” sense of “mathematical model” (section 24.1). Beer’s later insistence that the VSM is not an analogy (chapter 4.1) rests on this procedure. The brain–firm comparison is meant as a homomorphic mapping of structure. MS restates the argument for a general readership (MS 1967: 63–69; see chapter 27.3). The claim needs to be read precisely. Beer does not say that the firm is a brain. He says that there is a formal structure onto which both can be mapped, and that the structural identity extends exactly as far as that shared model. Anything outside it is simply not claimed.
24.4 The neurophysiological model of a company (1960s)
The electroencephalogram belongs to the case studies of DC’s chapter 7, where the precursor of the VSM already appears. The company concerned was in unusual difficulty: production needed long, planned runs, while demand “proved impossible to forecast”; raw materials were scarce and labor relations tense (DC 1966: 134–135). Beer calls the approach chosen ecological: “The company is an organism which must adapt to its environment” (DC 1966: 135). Among possible organisms, only those with highly developed nervous systems seemed able to cope with such an environment, and “If the entire management process of the company under discussion is conceived of as a brain-like activity, the possibility of a successful outcome is assured” (DC 1966: 135). For an industrial company, “The conceptual model of this situation is inevitably taken from the science of neurophysiology” (DC 1966: 136). Beer anticipates objections from brain scientists and does not wave them away. Those who took part, “and certainly myself”, were “neither underrating the task, nor laying an overweening claim to a knowledge they do not possess” (DC 1966: 136). The method protects the attempt: “The object is to construct a homomorphic mapping of the brain which is by definition a simplification”, and, “by definition again, a simplification of which the limitations are understood” (ibid.). The danger lies elsewhere: “The really pejorative term here would be over-simplification; but that stage is not reached so long as a certain structure is preserved in the homomorph, and so long as we know exactly what that structure is” (ibid.). The formal language was set theory, and the submodels were “homomorphic models, worked out in set theory, of the sensory cortex, the motor cortex, the thalamic level and the reticular formation” (DC 1966: 137). Beer gives no mathematical detail, since “the mathematics is extremely involved” (ibid.). He does describe the experiment, which he calls “surely the first attempt to record the brain rhythms of an industrial company” (ibid.). An electroencephalogram is a many-one transformation of neural activity, and “Yet what is inscribed is a homomorphic record” (ibid.). The pens preserve structure, such as the difference between sleep and waking, while discarding everything about individual neurons, just as base-five arithmetic keeps addition and drops the numbers. “Successful management decisions were taken using this model of the company’s ‘brain’” (DC 1966: 138). Accounting offers the comparison. A financial expert also reads rhythms, from balance sheets and profit-and-loss accounts, but these are not generated by truly homomorphic mappings: “the attempt to map every feature of the business on to a unidimensional monetary scale does not succeed, and this is why wrong decisions are sometimes taken” (DC 1966: 139). The brain model, by contrast, has at least the right order of complexity (ibid.). Beer is careful about the status of what was done. The “electroencephalogram” of ten areas of company activity was “all worked out by hand”; “In short, it was an experiment” (ibid.). MS gives a non-technical version of the related reliability argument for redundant decision structures (MS 1967: 99–100; see chapter 27.4).
24.5 Models, laws and conventions
In chapter 7 (“Models in Aspic”) Beer carries the argument over to natural laws. Language is a mapping of the world, but “in practice, the mapping will be homomorphic”, able to preserve some structure “but committed to losing some information”; our account of nature is therefore “‘true’, but defective” (DC 1966: 121). From this he draws a strong conclusion: “a law of nature is derived from what is no more than a model of such general utility that it is never even recognized as such” (DC 1966: 121). Conversely, “What we usually call a natural law is a statement about a particular model of the universe; what we here call a model of a problem situation entails its own localized natural laws” (ibid.). OR, on this view, does nothing peculiar. Its inductions are no harder to justify than those of any other science; “they simply make a more agile use of models than is customarily necessary in science” (DC 1966: 122). What counts as a valid model rests on “social conventions” (DC 1966: 122). Models can therefore harden and block discovery. “Progress in science might well be defined as the overthrow of a model, and its appurtenances, that has exhausted its usefulness” (ibid.). “The great scientist is one who sees the need and the moment to destroy a model”, and Beer adds: “Perhaps the same is true of the businessman and his business” (ibid.). Science, however, has protections that management lacks. It is founded on observation and experiment, it intends to expose a reality beyond its own desires, and it knows that it deals in models (DC 1966: 122–123). Management has none of these: “Management is not founded on observation and experiment, but on a drive towards a set of outcomes” (DC 1966: 123), and it “is not normally aware of the conventional nature of its intellectual processes and control procedures” (ibid.). The enterprise is therefore “accustomed to confuse […] its models of reality with reality itself” (DC 1966: 123). Yet it cannot do without models: “management operates through a model of the business, a model in which the organizational structure, the structure of costs and prices, the structure of labour relationships, the structure of production itself, are all homomorphic mappings of the real thing” (ibid.). No manager can handle a complete isomorph of his enterprise “unless he is the only employee”; he “can at best settle for a homomorph” (DC 1966: 124). Delegation is itself a mapping: “To delegate is to embark on a series of one-many transformations” (ibid.). The chapter closes on what the method is for. The systems of management “are not susceptible to total dissection, total analysis, total measurement” (DC 1966: 125), but their mechanisms can still be found. “The process of homomorphic modelling is at the least a heuristic method for inferring the existence and structure of systems of which the complexity defeats isomorphic modelling” (DC 1966: 125). Beer’s first example is historical: Henry Cavendish’s wooden model of the torpedo fish, published in 1776, of which Beer says “it might be fair to call what he did operational research” (DC 1966: 125–126). The company’s electroencephalogram stands in the same line, a homomorphic record of a system too complex to be mapped whole. Three results carry forward. A model is judged by utility and prediction. A homomorphism is a deliberate loss of information that keeps a specified structure. And what counts as a system, and which model is valid, involves the observer and the conventions he works within. Chapter 25 takes up the third result in DC’s own treatment of systems and variety.
24.6 The formal languages
Chapter 8 of DC (“The Formal Languages”) turns from the model to the languages in which it is made rigorous: mathematics, statistics and logic. Beer does not teach the techniques; his object is “to explain the meaning of these techniques” (DC 1966: 142). His first point ranks them below the model. A formal language “is a deductive machine: it investigates itself, rather than the sensual world”, and its manipulations must be “properly embedded in its surrounding model” (DC 1966: 143). “Hence models take pride of place over the formal languages used to make them rigorous. For a good homomorphic model that is less than rigorously expressed may be very useful; but a rigorous demonstration that derives from a poor model, improperly mapped from the problem situation, is highly dangerous and may be irrelevant” (ibid.). Mathematics earns its place because a pattern of decision can recur even when no single decision does. Equations can “adequately (by which is meant nothing more nor less than homomorphically) represent a wide variety of managerial situations” (DC 1966: 144). Beer’s example is linear programming for a production shop (DC 1966: 144–149), and he then turns the example against the technique. Its conceptual model is thin: “In our nomenclature, the mapping from real life is not strongly homomorphic” (DC 1966: 150). In one case the computed optimal mix of products could not be sold in exactly those proportions, and nobody had examined the stockholding that would have uncoupled production from the market (DC 1966: 150–151). Beer’s verdict: “it was defective operational research. And the trouble lay in the failure to formulate a good model” (DC 1966: 151). For a retail distribution network, “The model is now all-important; the technique of linear programming is worthless until a homomorphic mapping of this retailing world is found” (DC 1966: 152). Logic gets the same treatment. Its qualitative rigour is “methodologically prior” to the treatment of quantity (DC 1966: 161), and Beer asks for “a trained logician in the interdisciplinary team” (DC 1966: 162). His two cases, a military security problem and the flow of ingots in an alloy steelworks, each begin with an orthodox model, a map and a flow chart. The map is a good homomorph of the terrain, “the model is not homomorphic with the operational facts” (DC 1966: 162), and the same holds for the layout of the steelworks (DC 1966: 166). Enriched by an elastic geometry and by a model from acoustics, the two models produced new concepts that could be measured, “thresholds of coverage and reverberating ingots” (DC 1966: 171). Beer then reveals that “the logical structures of the military and civil arguments are identical with each other” (ibid.). On my reading, this is the procedure of section 24.3 shown on a small scale: two unlike situations mapped onto one formal structure.
24.7 Models in the control of operations
Chapter 13 of DC (“Controlling Operations”) applies the theory of models to control. It opens with a critique of orthodox practice. Whoever would control a high-variety situation must first reduce its variety by making a model of it (DC 1966: 305). In orthodox practice the model is made by subdivision and classification, and writing it down is “a variety-killing exercise. It has to be” (DC 1966: 306). It can take years to build and is out of date when finished, its categories never fill with data, and forecasts drawn from it are amplified without new information, which amplifies the noise as well (DC 1966: 306–309). The arrangements “are typically so simple that they cannot go right” (DC 1966: 311). To the cybernetician, “the shop floor is a control system generating variety for the purpose of controlling the planning office, and not vice versa” (DC 1966: 310). The cybernetic alternative also reduces variety, but “in a way which will allow at least some of the lost variety to be regenerated later on” (DC 1966: 313). Its first model has two parts. The structural model, M1a, records the relationships of the situation without numbers, in mathematics, statistics and logic; where even this is too large, “the ordinary operational research technique of homomorphic transformation must be used to keep it down” (DC 1966: 315). Beer’s image is an account of every note on a piano replaced “by an account of the relationships obtaining for a single octave” (DC 1966: 316). The parametric model, M1b, records performance optima, not averages, because averaging can “kill variety so finally that it can never be regenerated” (DC 1966: 317). Applied through each other like the co-ordinates of a map (see chapter 25.5), the two parts generate predictions (DC 1966: 318–319). Between prediction and event Beer places a black box that records their ratio. The ratios fall into statistical populations, and these supply a classification of events that “is quite unlike any conventional breakdown of the total system which people happen to think of” (DC 1966: 324). The ratio is not treated as an error. It is “an attempt to complete the incomplete language of the analytical model through the real world”, which is “essentially a process of completion from without” (DC 1966: 323; on the principle see chapter 25.2). One conclusion concerns the models behind the specialist functions. Production control and cost control “are isomorphic systems” (DC 1966: 341), and “the notion that every specialist control function should be an isomorphic mapping of every other, and a homomorphic mapping of a basic control theory which expresses no particular specialist function but is modelled on the manager’s own role, is seen as a valid and economical concept” (DC 1966: 341–342).
24.8 Models of the enterprise
Chapters 14 and 15 carry the argument from operations to the enterprise as a whole. Chapter 14 (“Self-Organizing Systems”) holds, for a reason Beer calls “perhaps idiosyncratic”, that “organization is an attribute of the observer of a system rather than of the system itself” (DC 1966: 350), and that for a system that already exists “The answer is not to design the control, but to constrain the system” (DC 1966: 354). In chapter 15 (“Controlling Enterprises”) the choice of model becomes the issue. The laissez-faire and the mandatory approaches to control rest on partial models: “the doctrinaire laissez-faire man is backing a model of the ecological system, and the doctrinaire controller is adopting a model from the neurological system” (DC 1966: 375). Beer recalls the triangulation of section 24.1 and concludes: “If we are to adopt a useful model of an organic system, we must use a model taken from an organic system. Only a complex creature, such as a man himself, provides an adequate model, for he has problems of both internal and external adjustment to contend with” (ibid.). Asked within that model, the question of centralization versus decentralization shows itself to be absurd (DC 1966: 375–376). Partial models also explain why political opponents do not meet: “the conceptual models we adopt are far more potent than the propositional assumptions we put forward in the course of debate. The latter can be challenged and perhaps falsified; the former structure the universe for the speaker and make it impossible for him to know what the opposition is saying” (DC 1966: 377). The model Beer then builds (DC 1966: 383–397) is a set of interlocking homeostats. Information about the enterprise enters the control system by a transduction that “is an isomorphic mapping” and then passes through black boxes that reduce its variety many-one (DC 1966: 384–385). At the centre, “H2 is a homomorphic, homeostatic model of the moment-to-moment state of affairs” (DC 1966: 393), the “‘still centre’” of the control system (ibid.). H2 and H4 corresponded in the original brain model to the sensory and the motor cortex (DC 1966: 394). H5 amplifies variety again into day-to-day plans, H7 balances the internal and external situations for years ahead and largely determines policy, and an algedonic loop, H9, lets policy and the world’s reaction to it bypass the routine machinery (DC 1966: 394–396). Beer says the account is “based on the set-theoretic model of some parts of the brain referred to as far back as Chapter 7” (DC 1966: 391), the model of section 24.4, and he marks its status: “This is the only case, of the very large number of studies and models and approaches put forward in this book, in which what is no more and no less than a theory has been advanced” (DC 1966: 397). On my reading, several elements of the later VSM are present here in 1966 form: a homomorphic model of the present at the centre, a long-range model that produces policy, and an algedonic bypass. DC does not use the later terms, and the correspondence is my interpretation.
References
- DC — Beer, Stafford (1966). Decision and Control: The Meaning of Operational Research and Management Cybernetics. Chichester: John Wiley & Sons. Cited from the Stafford Beer Classic Library reprint, 1994.
- MS — Beer, Stafford (1967). Management Science: The Business Use of Operations Research. Garden City, NY: Doubleday, in association with Aldus Books, London.