---
title: "content"
description: "On page 228 of Brain of the Firm, after the model of the firm (Figure 27) has been reproduced inside its own units, Beer writes: “Hence: Recursive System…"
url: "https://wiki.cybernetic-futures.org/pure-vsm/part-02-the-viable-system-model/ch06-recursion/ch06-recursion-content"
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site: "Cybernetic Futures"
---

# content

## 6.1 The theorem
On page 228 of *Brain of the Firm*, after the model of the firm (Figure 27) has been reproduced inside its own units, Beer writes: "Hence: Recursive System Theorem. If a viable system contains a viable system, then the organizational structure must be recursive" (BOTF 1981: 228). Recursion is what allows one model with five systems to describe organizations of any size. Beer states it as a theorem, and he states it in two versions.
The paragraph that leads up to the theorem explains why Beer thinks it is needed. As I read the passage, the problem is this: a metasystem regulates a system, but then something must regulate the metasystem, and so on without end. Beer's answer is that nature has already solved it: "there is an ecological answer, given in nature, to the abstract logical problem presented by an infinite regress of metasystems. It is the identification of the viable system, its determination to be itself and to survive" (ibid.). On this reading, the regress stops at the viable system because a viable system regulates itself. If viable systems are then organized into a larger viable whole, "we must expect that an organization of viable systems into some large viable whole will be organizationally recursive" (ibid.). The link to chapter 5 is direct. The regress stops at the same identity and intention to survive by which Beer defines viability.
The theorem "finally validates the five-tier hierarchic model we have been using all along. If the viable firm is organized like this, so is its major viable unit" (ibid.). Beer then follows the chain downward: "If the unit is organized like this, so is its viable sub-unit — the individual factory, for example. If the factory is organized like this, so is the individual viable shop of which the basic unit is the section of which the basic unit is the man" (ibid.). The chain ends where the model began: "The shop, the section and the man then are each organized like this, and we know that the man is so organized, since that is where the model came from in the first place" (ibid.). Here the neurophysiological derivation of chapter 4 meets the recursive structure of the firm. The human being is both the source of the model and its lowest level of recursion in the firm.
This chain, from firm through factory, shop and section to the man, serves as the reference case in the sections below.
In *The Heart of Enterprise* the theorem reads: "In a recursive organizational structure, any viable system contains, and is contained in, a viable system" (HOE 1979: 118). Beer calls this "an alternative version of the Theorem as stated in *Brain of the Firm*, which expressed the same point from the opposite angle" (ibid.). The BOTF version goes from containment to recursion: if viable systems contain viable systems, the structure must be recursive. The HOE version starts from recursion and states what it implies: every viable system is both container and contained. In HOE Beer also announces how the theorem is to be established: "Theorems, no doubt, ought to be proved. The proof offered later will be a topological proof" (ibid.). Behind the proof lies the idea that all viable systems "when drawn out as systems diagrams \[...\] will all look alike (they display the topological invariance discussed earlier)" (ibid.). Beer does not later label any passage of HOE as that proof. The nearest candidate is Chapter Twelve, where he states the task: to make the model 'stick', "it is necessary to show that each level of recursion exhibits the viable system as an *isomorphic* mapping of that same viable system at the levels of recursion above and below it" (HOE 1979: 307). That this chapter is the proof announced on p. 118 is my reading, not Beer's statement. Chapter 4.3 already set the two versions side by side.
## 6.2 Recursion is a mathematical, not a metaphorical, term
In *Diagnosing the System for Organizations* Beer explains why he uses the term: the embedded systems "are themselves viable systems. They are RECURSIONS OF THE VIABLE SYSTEM. We shall make use of this mathematical term because, while its meaning in context is evident, it reminds us that we are not talking loosely" (DTS 1985: 2). The sentence ends: we are not talking loosely "about **any** kind of system contained inside another — but about an absolutely precise definition of viability" (ibid.). Recursion in Beer's sense therefore means more than "being part of something larger". A unit that could not maintain a separate existence of its own may well be contained in a firm without being a recursion of the viable system.
In HOE Beer stresses that the levels of recursion "answer to mathematical criteria" and that a precise definition of the viable system provides "a vast attenuator of the variety" of organizational description (HOE 1979: 308). In the same passage he warns against the loose use: "If the term *recursion* is allowed to become a meaningless appendage to the term *organizational level*, then we do not have a valid model of a viable system" (ibid.).
The same page says why recursion attenuates variety. Beer contrasts it with the conventional organization chart, with its "arbitrary scattering of managerial boxes". The alternative "has been to treat a viable system as consisting of two levels alone: the level of the operational elements of the system, and the level of metasystem that organizes those elements" (HOE 1979: 308). Every viable system, at every level, is described with the same two-level scheme. In place of one enormous chart the observer needs one model, applied repeatedly: firm, factory, shop and section each get the same two levels. This is what Beer means by an attenuator of variety in organizational description (on variety see chapter 2).
DTS adds that the recursions can be followed very far in both directions: they "can be extended upward to the terrestrial globe (within the Universe) and downward to the cell" (DTS 1985: 4). DTS also connects recursion with self-reference. Viable systems "are **self-referential**", and in this characteristic lies the explanation, among other things, of "recursivity itself" (ibid.; see chapter 5.2).
## 6.3 Example: Chile
The longest example of recursion in Beer's books is the Chilean state, in the chapters of BOTF on his work in Chile. There Beer applies recursion to the state: "the government is embedded in the nation \[...\]. The government should be concieved \[sic\] as a viable system (System Five being the President of the Republic) in which System One consists of the Headquarters of each major function — health, education, finance, industry". Industry in turn is a viable system (System Five: the Minister of Economics), whose System One consists of the industrial sectors (BOTF 1981: 249).
Beer continues the chain further down. Each sector (with "the Undersecretary for Economics with his appropriate committee" as System Five) contains, as System One, a set of enterprises. Then: "Embedded in the enterprise is the plant; within that the department; within that the social unit of a working group; and within that the individual worker — viable systems all" (BOTF 1981: 249). Above the government, too, the chain continues: "Recursively speaking, the Chilean nation is embedded in the world of nations" (ibid.).
The Chilean chain has the same shape as the one on page 228, from firm to man, on a larger scale. At each level a System Five is named, and the System One of that level consists of the viable systems of the next level down. It ends, like the first chain, with the individual: there the man, here the individual worker.
## 6.4 System-in-focus
Every analysis chooses a level of recursion. DTS calls the system chosen the "System-in-focus"; its embedded units together "will be known as SYSTEM ONE of the System-in-focus" (DTS 1985: 19). Beer's rule of thumb concerns the number of these units: you "might have up to seven or eight: not many more, I hope, or else you may be missing a whole level of recursion" (ibid.). A long list of operational units is thus a diagnostic sign. It suggests that an intermediate level of viable systems has been overlooked.
DTS also advises how many levels to consider at once: "In practice, the best plan is to consider a trio of viable systems at any one time: the organization we wish to study, that within which it is contained, and the set of organizations contained by it — **one level of recursion down**" (DTS 1985: 4). The recursions further down are not ignored. They are treated as viable systems in their own right, but not analysed in detail at the same time.
The choice of the System-in-focus is the practical side of the theorem. Because every viable system contains and is contained in a viable system, the same organization can be System One at one level and the System-in-focus at another. The subsidiary firm of chapter 5.3 is an instance: a System One of its parent corporation and at the same time a viable system with its own Systems One to Five (BOTF 1981: 275). Chapter 7 describes System One of the System-in-focus in detail.

## References

- **HOE** — Beer, Stafford (1979). *The Heart of Enterprise*. Chichester: John Wiley & Sons.
- **BOTF** — Beer, Stafford (1981). *Brain of the Firm*. 2nd ed. Chichester: John Wiley & Sons. (1st ed. London: Allen Lane, 1972.)
- **DTS** — Beer, Stafford (1985). *Diagnosing the System for Organizations*. Chichester: John Wiley & Sons.