Welcome to the next stage of our journey into cybernetics. In the previous module, we navigated the landscape of Shannon's information theory and built a crucial bridge to cybernetics through the work of W. Ross Ashby. We concluded that while Shannon's theory provides the physics of communication channels, Ashby's work generalizes these ideas to address the fundamental cybernetic problem of regulation and control.
To truly grasp Ashby's profound contributions, we must start with his foundational concept, a simple yet powerful tool for measuring complexity. This lesson is dedicated to that single idea. Our goal is to define variety in Ashby's sense as the number of distinguishable states of a system. This concept will become the bedrock for everything that follows, from designing control systems to managing complex organizations.
1. The Architect of Variety: W. Ross Ashby
Before we dive into the concept itself, it's worth taking a moment to introduce its architect. William Ross Ashby (1903-1972) was a British psychiatrist and a pioneering figure in cybernetics, second only to Norbert Wiener in his influence. His background in studying the complexities of the human brain gave him a unique perspective. He sought universal principles that could explain how any system—be it biological, mechanical, or social—maintains stability in a changing environment. His 1956 book, An Introduction to Cybernetics, remains a seminal text in the field.
Ashby realized that to talk rigorously about control and regulation, one first needs a way to quantify the complexity of the situation that needs to be controlled. This measure he called "variety."
2. Defining Variety
In everyday language, "variety" implies a range of different things. Ashby's definition is more precise but carries the same intuitive meaning. It is a measure of the number of possibilities, the complexity, or the uncertainty associated with a system.
Let's watch a short video clip that gives a very clear and simple working definition using a familiar example.
Management Cybernetics: The Law of Requisite Variety
This clip from a lecture on management cybernetics provides a concise and practical introduction to the concept of variety.
Please watch the segment from the number of different states. The speaker uses a traffic light to illustrate how to count a system's variety.
As the video explains, variety is simply the number of different states a system can be in.
- A coin has a variety of 2 (heads, tails).
- A standard six-sided die has a variety of 6.
- A traffic light, as discussed, has a variety of 3 (red, yellow, green) or 4 if you include the "off" state.
For a more formal and detailed definition, let's turn to a key resource.
Variety (cybernetics) - Wikipedia
This Wikipedia article provides a concise summary of Ashby's concept. We will focus on the initial overview which contains the formal definition.
Please read the first two sections of the article. Start with the introduction and continue through the Overview section, ending just before the example calculation for a sequence, Ashby's definition. As you read, focus on two key points: The formal definition of variety. The two different ways variety can be measured.
As the reading confirms, Ashby defined variety in two related ways:
- As a count: The total number of distinct, distinguishable states. This is the most straightforward interpretation.
- As a logarithm: The logarithm to the base 2 of the number of states. The result is measured in bits.
This dual definition is deliberate and powerful. Measuring variety in bits directly connects Ashby's cybernetic concept to Shannon's information theory, which we studied in the last module.
If a system has equally likely states, its variety is bits. You may recognize this formula—it's identical to the formula for Shannon's entropy () for a source with equally probable symbols.
- A light switch with 2 states (On, Off) has a variety of bit.
- A system with 4 states {a, b, c, d} has a variety of bits.
This means that for Ashby, variety is information. A system's variety, measured in bits, tells you the amount of uncertainty you have about its state before you observe it. It is the number of binary questions (yes/no) you would need to ask, on average, to determine the system's exact state. For someone with your background in computer science, this should be a very familiar way of thinking about states and information content.
3. "Distinguishable by Whom?": The Role of the Observer
Ashby's definition contains a subtle but critical word: "distinguishable." This raises an immediate philosophical question: distinguishable by whom?
The variety of a system is not an absolute, intrinsic property. It is relative to an observer and their capacity to tell one state from another.
Consider a dimmer switch for a lamp.
- To a person who only cares if the light is on or off, the switch has a variety of 2.
- To a lighting designer trying to create a specific ambiance, the switch might have a variety of 100 distinguishable levels of brightness.
- To an electrical engineer measuring the resistance, the switch might have a nearly infinite variety.
The variety depends on the frame of reference and the discriminating power of the observer.
Let's read a short passage that explores this crucial idea.
Variety (cybernetics) - Wikipedia
This section of the Wikipedia article explicitly addresses the observer-dependent nature of variety.
Please read the paragraph that begins "Since the number" if the variety is to be well defined. Then, continue reading the remainder of that section, which explains Gordon Pask's contribution to this idea, ending at the end of the section.
This idea that the observer is part of the system description is a profound and recurring theme in cybernetics. It moves us away from a purely objective view of the world and toward one where knowledge and complexity are constructed through interaction. This is a concept that we will explore in much greater depth when we discuss second-order cybernetics later in the course, but its seeds are planted here in Ashby's foundational definition.
By defining variety, Ashby gave cybernetics a mathematical tool to quantify complexity, a tool that is inextricably linked to both information theory and the role of the observer. As he himself argued, this concept is "inseparable from that of 'information'".
Conclusion
In this lesson, we have focused on a single, fundamental building block of Ashby's cybernetics. We have moved from the general concept of communication to the specific, measurable concept of variety.
Here are the key takeaways:
- Variety, in the cybernetic sense coined by W. Ross Ashby, is the number of distinguishable states a system can possess. It is a measure of its complexity.
- Variety can be expressed either as a simple count of states or, more formally, as the logarithm base 2 of that number, measured in bits.
- This logarithmic measure directly connects variety to Shannon's entropy, framing variety as a measure of uncertainty or information content.
- Crucially, variety is not an absolute property of a system; it is relative to an observer and their ability to discriminate between states.
You now have a precise definition of variety, the essential prerequisite for understanding Ashby's most famous contribution. In our next lesson, we will use this definition to unlock the Law of Requisite Variety, a principle that dictates the fundamental requirements for any successful regulator, from a simple thermostat to the management of a national economy.