Tag: General Systems Theory

Distinguishing between sets and systems through mathematics, economics, and chemistry

A set with objects

So far, we’ve presented the (3) systems’ axioms and the notions of system’s behavior and system’s boundary. We have also explored these ideas via different examples. And we’ve touched on the idea of a set. However, we now want to differentiate between what a set is and what a system is. Once we show the difference between the two, then we will be able to demonstrate the difference between a subset and a subsystem. And most importantly, we will be able to better observe, analyze, and make sense of different kinds of systems, albeit economic systems, political systems, or political systems.

So how can we differentiate between a set and a system? First, we can address this question by referencing back to the (3) systems’ axioms:

  1. A system consists of a set of elements.
  2. Elements in a system interact.
  3. A system has a function, or purpose.
Set A is a subset of set B

The difference between a set and a system is that a set satisfies the first axiom; whereas, a system satisfies all three axioms. More specifically, a set B is a collection of well-defined objects (we will use Naive Set Theory for now), for instance B = {2,4,6,8,10}. Further more, the elements in this set interact with each other. For example, the element ‘2’ interacts with element ‘8,’ or element ‘4’ interacts with element ’10,’ or some combination of possible interaction. And finally, there is a function that is produced, or purpose, via the interactions.

As we can see, a set satisfies the first axiom; whereas, a system satisfies all three axioms. Now we have the tools to delve into the subsets and subsystems. We will see that subsets satisfy the first axiom while subsystems satisfy all three axioms.

As stated before, a set B is a collection of well-defined objects, for instance B = {2,4,6,8,10}. However, a subset of B can be partitioned and observed. For instance, a subset A is a subset of set B if all of the elements in the set A are contained in the set B. That is, A = {2,4,6} so since all of the elements in the set A are contained in  the set B, the set A = {2,4,6} is a subset of set B = {2,4,6,8,10}.

Thus, this subset or any combination of subsets with any of the five elements – 2,4,6,8,10 – satisfies the first system’s axiom.

To illustrate the second axiom with respect to a subsystem, we want to show that if elements interact in a subsystem, then they interact in a parent system. There are a few ways we can do this. For this article, we can do this by observing the interactions in set A = {2,4,6}. Thus if ‘2’ interacts with ‘4’ and ‘6,’ and ‘4’ interacts with ‘6’ in set A, then these elements also interact in set B because set A = {2, 4, 6} is a subset of set B = {2, 4, 6, 8, 10} because set A is contained in set B.

The final step is to show that a subsystem has a function, or purpose. It could be the case that a subsystem has the same function as its parent system, or it could be the case that it has a function different from its parent system. But either way, it ought to have a function no matter if it is the same or different from its parent system. So how can this be illustrated?

Example 1

As Donella Meadows conveyed in her book Thinking in Systems: A Primer identifying the function of a system can sometimes be difficult. Indeed, there are instances where the function or a system is fairly obvious.

One way this can be done is by mapping the elements in set B to the elements in set A. In other words, the elements in set B will go to the elements in set A.

The sketch in Example 1 illustrates this point. For instance, 1 goes to 3, and 2 also goes to 3; 4 goes to 7; and 5 goes to 8.

And so something is imputed through 1, 2, 4, and 5, and something is outputted through 3, 7, and 8. This means the elements in set B = {1, 2, 4, 5} would be the inputs of the system and the elements in set A = {3, 7, 8} would be the outputs.

To illustrate this point further, one could view a system that includes labor and wages as the elements. That is, a person exchanges their labor, hours worked, for a wage. If, for example, the wage was set at $30 per hour, then a person would obviously make more for every hour worked as Graph 1 shows.

That is, if 5 hours are imputed into the system, then $150 will be outputted from the system; if 6 hours are imputed into the system, then $180 will be outputted from the system; and if 7 hours are imputed into the system, then $210 will be outputted from the system. And of course this game could be played over and over again. Thus, as the number of hours imputed into the system increases, the number of dollars outputted from the system increases.

Example 1

Another demonstration of a function can be illustrated through an interaction between an oxygen molecule, O2, and two hydrogen molecules, 2H2. If a gaseous oxygen molecule interacts with two gaseous hydrogen molecules at a high temperature, these molecules are known as the reactants in chemistry, then two gaseous H2O molecules, known as the products in chemistry, will be produced. In other words, if one gaseous oxygen molecule and a two gaseous hydrogen molecules are imputed into a system, then the system will output two gaseous H2O molecules as Example 2 demonstrates.

Example 2

These systems’ functions and purposes are obviously not what we often think of as a function or purpose of a system. They are in one instance somewhat familiar and in another instance esoteric.

In this article, we have used mathematics along with a couple of examples from economics and chemistry to distinguish the difference between a set and a system. Moving forward, we will be able to continue building off of these axioms, notions, and examples as we begin to apply these ideas to more familiar systems such as economic systems, political systems, and social systems.

Let us now, as we have done before, attempt to disprove our notions and work in the tradition of natural philosophy until the next blog.


Matt Johnson is a blogger/writer for The Systems Scientist and the Urban Dynamics blog. He has also contributed to the Iowa State Daily and Our Black News.

Matt has a Bachelor of Science in Systems Science, with focuses in applied mathematics and economic systems, from Iowa State University. He is also a professional member of the Society of Industrial and Applied Mathematics and the International Society for the Systems Sciences and a scholarly member of Omicron Delta Epsilon, which is an International Honors Society for Economics. 

You can connect with him directly in the comments section, and follow him on Facebook

You can also follow The Systems Scientist on Twitter or Facebook.


Photo Credit: Pixabay






Copyright ©2017 – The Systems Scientist

A quick view of an economic system

By Matt Johnson

In this short blog, I will illustrate one way an urban dynamicist, i.e., systems scientist, looks at an economic system and its data.

Diagram 1

Diagram 1 is hierarchical, derives from the U.S. Census Bureau, and represents a few of the many levels of an economic system. Moreover, each level of the economic system in Diagram 1 is further a sub-system, or sub-economy, of the general United States economy.

This means that a zip code, for example, can be examined as an economic system, and then it can be compared and contrasted with a city’s economic system. And this examination will illustrate similarities and differences between a sub-system, a zip code, and a general system, a city, for instance.

Thus, an urban dynamicist can partition out each level of the economic system and analyze each level as a distinct entity, although one system is still a sub-system of the one superior to it in the hierarchy. Within each level, differences, relationships, perspectives, dynamics, and models can be examined through data.

As stated before, each level of the system can be analyzed against the other levels of the system through data, because data provides a picture at each level of the system. For example, the State can be illustrated and compared to the Division, Zip Code, or Census Tract via crime densities, demographic comparisons and migration patterns, and economic variables such as median household incomes, unemployment rates, the labor force and labor participation rates.

Here is the stochastic (probabilistic) behavior of the labor force in Minneapolis over the past 10 years as seen here in Graph 1.

Graph 1

And here is the stochastic (probabilistic) behavior of the Minnesota labor force over the past 10 years as illustrated in Graph 2.

Future articles will delve deeper into the specifics of the behavior and dynamics of these two systems and their respective data sets. For now, the main point is that data can provide a picture of the economic systems at their respective levels of the system.

One last thought, Diagram 1 does not illustrate the interactions or dynamics that take place within each level of the system by itself, nor does it account for a lot of things. This is why the data is needed. So assumptions and conclusions should be limited.

As this focus on data continues, I will be utilizing the hierarchical model and other systems models to help illustrate and explain how economic systems can be better understood. In addition, I will be using systems theory along with applied mathematics to explore the complexity of systems. But I will also be working diligently and meticulously to convey this information to you the best I can.

As I get better at explaining this stuff to you, I hope your knowledge of systems, mathematics, and economics increases as well.


Matt Johnson is a writer for the Urban Dynamics blog; and is a mathematical scientist. He has also contributed to the Iowa State Daily and Our Black News.

You can connect with him directly in the comments section, and follow him on Facebook

Photo credit: Pixabay






Copyright ©2017 – The Systems Scientist



A Starting Point for Systems Language

By Matt Johnson

One thing I haven’t done very well on this blog is talk about systems and the language of systems. This is important because it provides a common language for those who come from different backgrounds and political affiliations. It is no secret that there is a political divide in this country and it has existed for sometime now.

Liberals view the world one way and conservatives view the world another way. This in turn influences policy decisions and applications. But systems are neither liberal or conservative. Systems do not care if you like Star Wars or if you like Star Trek (you’re writer likes both). Systems do not care if you like chocolate ice cream or vanilla ice cream. The point is that systems have no persona and no agenda. Systems are self-differentiating, complex entities. And very few people understand systems in general.

Why is a common language important? It’s important because it helps to clarify what is meant by “system,” “structure,” and “environment.” We often hear terms like “systemic oppression,” “systemic racism,” and “the patriarchy,” which is another way of saying, “It’s the system.” These terms, when used and maybe for the best of intentions, really just muddy the waters. There is no corollary or definition behind them. They can mean anything. They are ambiguous.

But it makes sense that the current systems language in everyday use is ambiguous. This is because everyday citizens have been forced to adapt and improve in their language. They see something and it’s complex. How does one explain something that seems to possess an absurd amount of variables (or things going on)? So people have to improvise. They have to make sense of it somehow.

Another reason why systems language is ambiguous is because the scientific version of it just doesn’t exist, at least not at the level of physics, chemistry, or biology. Those sciences have been around for so long that the lexicon of those sciences has had time to migrate out to the populace. For example, people use infinity, quantum, calculus, derivatives, atoms, electrons, planets, comets, black holes, gravity, psychoanalysis, and the list goes on and on. These things make at least some sense because there have been scientific practitioners to help aid in the dissemination and understanding of such scientific language, for instance, Carl Sagan and Neil deGrasse Tyson. But systems science has not had this relationship with the general public, nor has it had its Michio Kaku or Bill Nye.

The science of systems is young. It’s origins can be traced back to Ludwig Von Bertalanffy and his book General Systems Theory (see the Mark Twain page for the link) in the middle of the 20th century. Compared to physics and mathematics, this is an extremely young science. But that’s one of the reasons why this blog exists.

It’s here to provide corollary (a proposition established from truth) and definition to the conversation of systems science. It’s here to explain what a system is and what it does. It’s here to explain the difference between what a system is, what a structure is, and what an environment is. And it’s here to explain why the vast majority of systems discussed on Urban Dynamics are probabilistic systems and not causal. It’s here to provide access to an otherwise esoteric field of science.

I will do my best to make sense of these ideas I have shared with you. But it will take time for these ideas to make sense. and I won’t promise that it will happen over night. Habits will need to be broken and some knowledge will need to be accumulated on the part of the reader. But in the end, it will payoff because I will have provided you with a new way to look at the world and a language to describe it.