By definition it’s impossible to measure a large system but scientists and mathematicians have experimented with many ways to come close. Here is another method.
The difference between a large system and a (small) system is that you cannot measure the large system, except by redefining the large system as a single unit or smaller system. If you scoop up a handful of sand you can measure that system as a “single handful of sand” or as the sum of the grains of sand in your hand. But how may molecules does it contain?
When you look at a handful of sand as a collection of molecules it becomes a large system. You know there must be a finite number of molecules in the sand but you have no way of measuring them. And you cannot cheat by assuming you have an electron microscope or some equivalent technology that can “see” the molecules. No matter how far down you go, you’ll eventually reach a point where we lack the technology to measure the handful of sand at some scale – so let’s just use molecules as a metaphor for that too small number.
Another way to illustrate a large system is to think of all the computers that make up the Internet. This number changes constantly because every time you turn off your PC or smartphone it leaves the Internet. The same is true for your refrigerator or television set, if they are IoT devices that connect to the online world.
Although your handful of sand may be a static system the Internet is a dynamic system. It’s constantly changing. You can estimate how many grains of sand or even molecules your handful consists of with some reliiability but you cannot estimate how many computers are connected to the Internet at any given moment.
In either case you can estimate the limits of the sizes of these systems with confidence.
The Limit of a Large System Is Its Boundary
A system may have more than one boundary but every boundary is a limit In 2014 I wrote “Why You Will Never Be Able to ‘See’ A Large System” for the Science 2.0 site. There I said that a system possesses both scope and boundaries: “Scope = identifiable properties that help us to recognize them as components of a system; Boundaries = limits which determine whether things are parts of the system or not.”
If you can calculate the maximum number of components that could possibly fit into a system’s boundaries, you have determined the potential boundaries of the system.
According to Wikipedia, the total mass of the universe is about 1.5×1053 kg. I’ll take their word for it. That estimate represents the potential mass boundary of the universe, meaning the universe cannot have more mass than that (allowing for the possibility that our future knowledge of the universe changes the terms of the estimating formula).
We don’t actually know how much mass is in the universe but we think we have a pretty good idea of how much there is. So we can adjust that potential boundary upward by adding some value (call it E’) that exceeds our estimate. This is a safe zone that doesn’t need to exist.
You Can Always Measure Part of a Large System
If you know a large system exists then you can measure part of it. The measurement verifies its existence.
How many gorillas are alive right now (as you read this)?
No one has the correct answer but we can estimate that it’s probably less than 250,000 living gorillas. That limit assigns an arbitrarily high E’ but that’s okay. If we want to study the system that consists of all living gorillas, we know that we won’t have to work with numbers greater than 250,000. We have a large safe zone.
If you needed to inoculate all the gorillas in the world against some new disease (assuming you have the capability), you know you don’t need more than 250,000 vaccine doses (plus a few extra to allow for accidents and such).
We may not know how many gorillas there are but we’ve counted thousands of gorillas so we know they exist and that therefore there is a system that consists of all living gorillas.
You only need to measure part of a large system to know that it exists. If you can see only 1 star in the universe then you know that a universe containing stars exists.
Large System Localities May Act Like Small Systems
People have argued that if you can measure part of a large system you can estimate how big the system is. While this may be a useful math and statistics methodology it doesn’t work for large systems.
Every system has what I call eddys. An eddy is any part of a system that doesn’t behave like at least one other part of the system. Every large system has at least 2 eddys.
How do we know this?
Because if we could accurately extrapolate what the large system looks like from any one section of it we would be measuring the system – and therefore it wouldn’t have any eddys. Even if you assume that a system contains eddys, if you can extrapolate from what you know of the system what it looks like (with any accuracy) then you know what all the eddys are and therefore have measured the system – so it’s not a large system. It’s just more complex than a simple system that doesn’t have eddys.
A locality is any measurable part of a system. It could be an eddy or just one of a large group of homogenous components. A university data center is both a locality and an eddy of the Internet. If all the computing devices in your home connect to the Internet through the same router then your home is an Internet eddy, and a data center, but it’s quite unlike a university data center.
Every system is the sum of all its localities, but you don’t know how many localities or eddys a large system has.
For the sake of defining a system, think of each eddy is a class of localities, so each eddy is a distinct type of locality (for which there may be many instantiations).
That means you don’t know how many eddys a large system has. But you can count the localities of a large system that you see and you can identify at least 1 of its eddys.
You Can Map a Sub-System of a Large System
A sub-system may consist of multiple localities and multiple eddys. You can’t measure the entire system but you can map what you see, and on the basis of that map you might be able to guess what some of the uncharted eddys look like. You refine your guesses as you learn more about the makeup of the large system.
There are two ways to visualize this process. One way is to see the system the way the rebels in Star Wars tracked the approach of the Death Star when it attacked their base on Yavin IV. The countdown clock gradually colored in a curve of visibility as the Death Star moved around the planet toward the moon.
The other way is to think of the system as a series of concentric spheres. Neither method is perfect and they may not even be equivalent, but there are reasons why you could use both methods to visualize any type of system.
We don’t really know how large the universe is although scientists estimate it could be about 92 billion light-years in diameter. So let’s think of the universe as a massive sphere about 92 billion light-years across. It doesn’t matter if the universe is really shaped like a sphere because we’re only looking at its E’. And it doesn’t matter where the Earth is located inside this massive sphere.
Now, we can actually see part of the universe – a spherical region centered on Earth (or our Solar System) that is about 27.6 billion light-years across. We’re still mapping the observable part of the universe but for all intents and purposes we’ve mapped this huge section of the universe.
It would be an oversimplification to consider everything we can see to be a single locality or eddy, so let’s just call it a sub-system of the large system that is the universe.
We Can Estimate What the Universe May Look Like Today
If you only go by what we see, you have no idea of what the universe looks like today. Everything we see that is 13 billion light-years away has been moving around and evolving for 13 billion years. It doesn’t look that way now and it’s not in the same location.
The task of estimating what all the things we see could possibly look like today is a monumental task – really beyond our computing power – but we can create models that sort of approximate what all that stuff looks like now.
We could do that using models built from concentric layers. In other words, the observable universe is like an onion. We’re at the center of the onion and we have a pretty good idea of what that center looks like.
We can look at the next layer out and estimate – based on what we know about that layer and any similarities between it and our inner core layer – what that layer probably looks like today. In other words, we know sort of where the Andromeda galaxy is in relation to the Milky Way because we know where it was about 2.5 million years ago and what its velocity was relative to the Milky Way.
In other words, the Andromeda galaxy is no longer 2.5 million light-years away – it just looks to us like it’s 2.5 million light-years away because it takes light a long time to travel across the universe.
Now, obviously the farther out you go the more errors your estimates of the universe’s current contemporary state accumulate. But for the sake of building a model we don’t need to worry too much about how accurate it is. The model allows us to make testable predictions and as we learn more about the universe we can revise the model.
In theory we could create a star map that shows the current positions (relative to the Milky Way galaxy) of all the galaxies we’ve been able to observe as they are today. That map would be much larger than 27.6 billion light-years in diameter but it would be smaller than 92 billion light-years in diameter.
And if we compared that star map to the standard astronomical star maps you see at your local planetarium, the estimated universe would look very, very different.
There Is a Function for Estimating What a Large System Looks Like
The function, call it Fn(E’), is a projected sum of Eddys and Localities. Each Eddy is a set of 1 or more localities. So using E for an eddy and L for the known number of localities, the function would look something like this:
Fn(E’) = E1(1..L1) + E2(1..L2) + … En(1..Ln)
This function defines what I call a Pluristate Set. We cannot estimate what the system looks like but we can estimate what its E’ looks like. That’s about as close as you can get to measuring a large system.
Now eventually we find ways to measure large systems, and the E’ function is no longer useful. Each locality is itself a component-like set (in fact, they are the atomic components of the Pluristate Set). An eddy defines the properties of a class of atomic components.
Two eddys may be equivalent except for 1 property (such as location within the system). So you could have a left cog and a right cog and they both look and function like cogs, but they each have distinctive roles or functions within the system.
The E-sets are not mathematical sets, although you could treat them as such. The elements of an E-set are its localities. The localities have the same properties so it follows that any functions you define for a specific E-set work the same for all of its localities.
E’ Is Not a Derivative
I don’t have a better notation to use for this. In calculus a derivative represents the rate of change in a function. In Large Systems Theory an E-function is the sum of all the E-sets of a system, and the E-function has no definite value.
You can crunch an E-function to produce a scalar value if you wish the value is only relative to time Tx in the system’s Naturality Curve. The Naturality Curve is defined by the formula 1 = Ny + Ty + Oy. Basically, the Naturality of a System tends to decline over time, although it may never reach 0.
So there is, in fact, an N’ derivative if you want to compute fn(N) for a system across a timeline T. You could use that to determine how long a system (or sub-system) has until it stops behaving according to the rules of your E-function. At that point you would have to compute a new E-function.
Conclusion
We can only guess at what a large system looks like, and we have no way to confirm how good our guesses are (or to calibrate them) except to tediously map the large system. At some point we’ll map enough of the large system that it is no longer large to us.
Although you cannot predict how a large system will behave, you can make predictions about how its localities behave. These predictions may even be consistent based on the properties defined by the eddys. You just need to correctly document the properties of each eddy.
You may be able to predict the properties of neighboring eddys from known eddys, although that’s not guaranteed. You’ll have to define what “neighboring” means for each system you’re analyzing.
A concentric E-function might work for this kind of estimation, as long as you can live with an unknown rate of change in your margins of error. But given enough thought and practice, someone may come up with a way to calculate the rates of change for margins of error in concentric E-functions.
Related Articles about Large Systems Theory
Why It Is So Hard to Describe Large Systems Using Math on this blog
Large Systems Theory Needs Something Other Than A Set on this blog
Large Systems Theory for Web Marketers and Analysts on SEO Theory
Why Data Superstructures Impede the Analysis of Search Results on SEO Theory
Why You Will Never Be Able to ‘See’ A Large System on Interwebometry at Science 2.0
Can We Prove That A Large System Is Self-organizing? on Interwebometry at Science 2.0
