We use models to help us understand the world around us. Models come in many forms: architectural models, mechanical drawings, free body diagrams, software object models, maps, and even art, are some of the types of models that we use to represent the real world. We use models to represent many different types of systems, such as electrical, mechanical, environmental, cultural, and economic systems.

Real-world systems are complex. They have many interacting variables, feedback mechanisms, and linear and nonlinear behaviours. Real-world systems often have ill-defined boundaries, with external factors acting on the system that may not be well understood. In order to develop models that can help us to understand certain aspects of real-world systems, we necessarily simplify our representations of the real world in order to isolate the aspects that we’re attempting to understand. Because of these simplifications, models have limitations.

Essentially, all models are wrong, but some are useful.
George Box

Consider the following model, which is a free-body diagram that we can use to help us to understand the motion of a mechanical system:

By Hill.rick.c (Own work) [CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0)], via Wikimedia Commons
A simple free body diagram.

To understand the motion of this system, we can develop a system of equations (a mathematical description of the model) derived from Newton’s laws of motion. This model should describe, fairly accurately, how the system will behave at scales that humans can readily observe. However, we know that Newton’s laws of motion have limited applicability outside of the scales and speeds that humans can readily observe, so this model has a limited scope of usefulness.

Similarly, other types of models have limited scopes of usefulness. The London tube map, first designed by Harry Beck, is a well-known example of effective map design:

The London tube map.

The London tube map, though a classic example of schematic map design, has a limited scope of usefulness. It’s only useful for navigating the London Underground system in the time period for which it’s published. It’s not useful for, for example, navigating London’s surface streets, and it may have limited usefulness 20 years from now.

Prehistoric humans created art that may also have been used as maps:

Prehistoric Rock Paintings at Manda Guéli Cave, Ennedi Mountains, Chad

Paintings like these may have served as maps, as models instructing the viewer where game animals or predators might have been found. These paintings are also art, models depicting how the artists’ people interacted with their environment. Again, these models have limitations. The climate in Chad is today much different than when this painting was created, and most of the animals depicted here can no longer be found here. As a map, this model has limited usefulness because it doesn’t include a scale; it’s not easy to determine from this painting how far hunters might have to travel to find food.

Salvador Dali’s The Disintegration of the Persistence of Memory is supposed to contain representations of concepts in quantum mechanics:

A model of quantum mechanics?

As with most of Dali’s famous works, it is a very abstract model of its subject matter and, as such, has limited usefulness in describing and understanding quantum mechanics.

In economics, the formula for Gross Domestic Product (GDP) is:

GDP = C  + I  + G + (Ex – Im)

This formula is a model that describes a country’s GDP in terms of consumption (C), investment (I), government spending (G), exports (Ex), and imports (Im). While it can describe the size of a country’s GDP, this model does not describe the sectors of the country’s economy and how they interact, nor does it describe how changes in one component over time affect the other components over time. Other models must be used to provide a more complete picture of a country’s economy.

All models have limitations. Whatever type of model we’re dealing with, including and maybe especially maps, we need to be aware of the scope of usefulness and limitations of the model that we’re using.