The Great Mental Models Volume 3: Systems and Mathematics

by Rhiannon Beaubien

A constitution is one means of making that happen. It can be thought of as a high-level algorithm to limit the power and define the responsibilities of those charged with governance.

Part of the definition of an algorithm is that it uses the same input every time and produces the same output every time.

Where algorithms can become really interesting is when seemingly innocuous, standard inputs create entirely brand-new outputs. Algorithms seem to be a natural consequence of repetitive actions. For most humans, doing the same thing in the same way over and over gets boring. We thus wonder if there is a way to codify those repetitive actions to streamline the process. A lot of modern math seems to be a result of the codification of the processes used to manipulate numbers.

Algorithms are developed to get a certain output. As we’ve discussed, you start with inputs, you follow a process, and you end up with expected outputs. However, sometimes it’s not obvious which inputs will result in the desired outputs. So one way to use this model is to help you determine and refine what kind of inputs to feed into it in the first place. You can consider it “algorithmic thinking.” You may not have the luxury of a completely closed system where you can implement complete end-to-end automation, but the lens of algorithms can show you how to organize your system to leave as little to chance as possible.

At their core, algorithms are a clear set of rules that provide instruction on what to do. We can also conceptualize them as if–then processes that are useful because they can help us ignore variables that don’t matter and focus on requirements. Algorithms as a model suggest a way of thinking that explores what processes can be put in place to get us the results we want.

Often, scholars distinguish between complex systems—systems in which the entities follow fixed rules—and complex adaptive systems—systems in which the entities adapt. If the entities adapt, then the system has a greater capacity to respond to changes in the environment. » Scott E. Page1

Complex adaptive systems have properties that are greater than the sum of their parts. You cannot understand them from studying their individual components, which may be simple but which interact in unpredictable, nonlinear ways. A few, often basic rules enable the parts to self-organize without centralized control.

A system's ability to change in response to its environment and in pursuit of a goal makes it adaptive.

Complex adaptive systems have “memories”—they are impacted by what has happened to them before.

Traffic changes its behavior based on information from its environment. Focusing on one car won’t teach you about the entire system because what matters is the interactions between them.

Within complex adaptive systems, components are all interdependent. They can directly or indirectly influence the behavior of the entire system. If one car breaks down on a main street, it can have knock-on effects for the traffic in the rest of the city. Interactions between parts amplify the impact of tiny changes.

In a complex adaptive system, we can never do just one thing.4 Any time we intervene, unintended consequences are almost inevitable. Often when we try to improve a complex adaptive system, we end up making things worse because we overestimate our degree of control.

To work with complex adaptive systems, we cannot expect them to be governed by predictable rules. Nor can we expect to understand the macro by examining the micro. To handle complex adaptive systems, we need ...

We can still learn from complex systems; we just need to be humble and use the scientific method. We must not mistake correlation for causation, and we should always be open to learning more about the system and accepting that it will change. What we learned yesterday may guide us, but it can change tomorrow. We shouldn’t give up just because a system is complex.

From the outside, complex adaptive systems can look chaotic, but they tend to work best when slightly disorganized, as this allows for mutations and experimentation. In the long run, deviations tend to cancel out into more coherent patterns of functioning.

No gluing together of partial studies of a complex nonlinear system can give a good idea of the behavior of...

A system becomes “critical” when it is on the verge of changing from one state to another. The final unit of input before the change has a disproportionate impact. It is the proverbial straw to break the camel’s back. Before a critical mass is reached, the camel can support the amount of weight it’s required to carry. Then the weight passes a threshold where any additional amount is disastrous, and the final straw tips the camel into another state. Once a system passes a certain threshold and enters a critical state, it only takes a tiny nudge to change it.

In social systems, critical mass tends to mean when enough people have adopted something, such as a belief or product, that its growth can sustain itself.

The amount of energy required for a system to achieve critical mass is variable. Different systems have different properties and thus require varying amounts of inputs to tip from one state to another.

Using critical mass as a model helps us understand the effort required to achieve sustained change. Systems have certain inflection points where they change from one state to another. It doesn’t help us to focus solely on the tipping point and ignore the work required to bring a system there. Because the inputs that tip a system into a new state tend not to have a linear effect—the final unit of input that leads to the change in state has an outsized marginal impact—we are disproportionately impressed by them.

We are also disproportionately affected by them. But the straw only breaks the camel’s back when there is already a lot of weight on it. Putting one piece of straw on a camel isn’t always going to have the same effect.

The critical mass lens also helps us identify the parts of a system we can ta...

In social systems, for example, we don’t need to spend equal effort changing everyone’s mind. We can instead focus our efforts on changing the minds of opini...

The Overton Window refers to the range of ideas considered acceptable for politicians to propose as policy. Ideas outside of that, no matter how good, cannot gain widespread support. They’re too extreme for the current climate and are best avoided lest they harm one’s chances of re-election.

Over time, the Overton Window shifts. Some politicians may advance far-out ideas in a deliberate attempt to move the Window further from the norm and make more moderate ideas more palatable.

Politicians must prioritize the Overton Window over their personal beliefs.2 It’s important to recognize that the Overton Window isn’t universal. The conservative political positions of one country may be considered liberal in another.

We like to tell stories about the tipping points. We look at the landmark cases or individual actions that sparked a cascade of change in the past and wonder how we can re-create them to push our current system into a new state. Using the mental model of critical mass, however, reminds us that it is equally important to pay attention to the effort involved in the buildup.

We can learn from the mental model of critical mass that changing a system doesn’t require changing everything about it. Changing a small percentage of its parts can shift the whole thing into a new state. Getting people to alter their beliefs doesn’t mean convincing everyone; once you pass a threshold, the change perpetuates itself.

Sometimes, people change their minds a lot in a short time. Although it can seem as though this shift occurs overnight, what really happens is that things change slowly until a critical mass of people hold a viewpoint. Interestingly, a majority is not required for things to tip and result in almost everyone changing their minds.

Targeting opinion leaders can accelerate getting to the tipping point.

Unprecedented legal decisions are often the most visible sign of significant social change. They can appear sudden and dramatic, but they are usually the result of a slow build of changing opinions that becomes impossible to ignore at a certain point.

While we could look at the case as a turning point, it was really the result of changing opinions building up until a critical mass was reached.

Cities are complex systems where planners have often misidentified the elements required to create enough density to produce self-sustaining interactions.14 In cities, it’s not the amount of infrastructure that produces interactions, it’s how that infrastructure is laid out. A certain number of interactions are required for a city to function well and adapt to meet the needs of those living in it. What makes a city safe, interesting, prosperous, and creative isn’t the buildings or streets. It’s how the infrastructure fosters interactions and relationships between people.

She argued that when we isolate parts of cities, we miss the many interconnected functions that they perform. For example, “A city sidewalk by itself is nothing. It is an abstraction. It means something only in conjunction with the buildings and other uses that border it, or border other sidewalks very near it.”15

The system formed by a sidewalk and its users is what makes an area both safe and interesting. When the area around a sidewalk is subject to active mixed uses—homes, cafés, shops, and so on—there are always eyes upon it and people passing through. These people do not need to know each other or even talk to each other. It is enough that they see each other, are aware they are watching and being watched, and observe each other’s behavior.

It is the interplay of people that ensures a sidewalk is safe and places limits on antisocial behavior. People moderate how they act knowing someone is or might be watching. Any antisocial behavior that does break out is likely to be swiftly halted by the interventions of bystanders. An organic, unorganized system of control enforced by social norms is more immediate and effective than the use of police, although the threat of them being called plays a role.

we need to consider a sidewalk as needing a minimum number of interactions in order to function as part of a city system instead of just a piece of concrete.

You feel safer on a sidewalk that is part of a whole system of self-sustaining interactions.18 The more people that are using a sidewalk, at different times and for varied purposes, the better it functions as a safe space.

This is true outside of cities, but other factors are likely to be relevant for safety in towns or rural areas. It is this “intricacy of sidewalk use, bringing with it a constant succession of eyes” that also makes an area lively and interesting, and therefore a desirable destination.

Activity attracts more activity. Many people using an area brings economic benefits, which further attracts more businesses, especially more unusual and speciali...

Visual activity is appealing to the eye. We like to watch things happening and other people going about their days, so crowds attract more crowds. People who watch attract more people who do by making an area safer, and people who do attract more people who watch by making an area interesting.20...

In Happy City, Charles Montgomery says, “The richest social environments are those in which we feel free to edge closer together or move apart as we wish. They scale not abruptly but gradually, from private realm to semi-private, to public; from boardroom to living room to porch to neighborhood to city.”

One instance of an environment that allows individuals to regulate their level of interaction but also accommodates a wide variety of users is a plaza or square. Present at varying scales as a standard feature of the city, squares are mixed-use areas in which one could, among other things, meet a friend or date, enjoy a coffee at a sidewalk café, watch a street performer, exercise a dog or child, attend a protest, or strike up a conversation with a stranger. The surrounding bustle allows for both safety and interest.

You feel safe meeting a date or talking to a stranger because there are enough people around to take note if something goes wrong. But you also feel comfortable having a private conversation with a friend or writing in a notebook because no one is likely to pay much attention to you in particular with so much going on.

Street culture is not specific to certain cultures; it is about having the right kind of spaces for it. These spaces recognize that a minimum of interactions must be maintained in order for them to function in a self-sustained way. The actual architecture should be invisible because the focus is on the people and on bringing out their best qualities.

If you focus on the infrastructure first and just build a list of requirements—like houses, stores, and streets—you increase the chance that your city won’t function well. Rather, the infrastructure needs to be designed to facilitate a critical mass of interactions—something that planned cities often miss. Planned cities may try to design out those interactions because designers see them as a waste of time, or they may even want to discourage any type of organized action.

Planned cities often segregate different functions, like workplaces and homes, ignoring the benefits of mixed-use areas, which are the standard in natural cities. Visually this looks ordered and pleasing, but it doesn’t promote the interactions cities require. It’s useful for people to be able to access resources close to where they live. It cuts down on commuting and increases time people can spend on relationships.

The mental model of critical mass gives us insight into the amount of material needed for a system to change states. We can categorize material as interactions, connections, or efforts. When enough material builds up, systems reach their tipping point. When we keep going, we get sustainable change. Using critical mass as a lens on situations where you want different outcomes helps you identify both the design elements you need to change and the work you need to put in.

When we look at systems on the macro scale, they sometimes exhibit capabilities that aren’t present on the micro scale. This is known as emergence: when systems as a whole function in ways we can’t predict by looking at their parts. As Aristotle put it thousands of years ago, “The whole is something over and above its parts, and not just the sum of them all.”1 The mental model of emergence reminds us that new capabilities are often produced from seemingly innocuous elements.

You look at where you’re going and where you are and it never makes much sense, but then you look back at where you’ve been and a pattern seems to emerge. And if you project forward from that pattern, then sometimes you can come up with something. » Robert M. Pirsig2