Hello again, everyone! Feeling good for Halloween? Well, I posted this one on Patreon yesterday, and it's scary because it has MATH IN IT! OOOOOOOO!

It's not really that bad, but if you're the odd weirdo or two that likes that kind of punishment, consider giving me a visit over at my Patreon. For a cup of coffee a month, YOU TOO can help me write a kick ass book to SAVE HUMANITY! Okay, maybe not. BUT CHECK IT OUT HERE!

Anyway, with the Introduction and shameless shilling out of the way, let's take a look at what we got for today.

Analyzing Systems! And its LIMITS!

This is a little foray into a cybernetic-adjacent idea that I have been working on for some time now. I think it is interesting and will play a big role how I view systems related to the cybernetic community, as well something to establish fundamental limitations on how we APPROACH the understanding of systems.

So, a few things we need to lay out before we get to the REALLY big things. First, cybernetics (if you don’t already know) deals with the regulation of systems and their methods of communication. This is the earliest form of cybernetics and consists of how systems behave and is also known as FIRST ORDER CYBERNETICS!

Second order cybernetics deals with how the OBSERVER of systems well.. OBSERVE IT. It also has to do with how an observer of a system can actually regulate the system in question. Observers of a system can be anyone from say, an authoritarian leader to the actual constituents of the system itself in a direct democratic fashion. The observer and their tools play a crucial role in this flavor of cybernetic thought.

So, let’s say we have a system, as illustrated with some potato draw.io visual aids:

Ignore the clipped parts on the left. Shit, I pointed it out.

I’m depicting a system here as a Set S, with the members of S = {m_1, m_2, … m_18}. Each “m_x” is an element or “member” of set S, and each member has a particular number of possible states, which for example, could be another set we could call P = {0,1}. In this situation, binary logic is being used, but it doesn’t necessarily matter. You could have a thousand different states in set P. It also bears mentioning in this system we constructed, that the members of S are the “atoms” of this system (Think the Greek word ἄτομον or atomon, which means ‘indivisible’). There is nothing larger. There is nothing smaller.

Each of these states can exist in every element of this system we have put together. Additionally, in this example, we can determine the possible number of states for the whole system! This would be called the variety of system S.

This example is pretty straightforward if you have the right tools. This requires a TEENSY bit of math (I know, SPOOOKY!), but really, it’s just exponents. You can skip it if your eyes glaze over, as long as you understand what it means when I say, “ALL possible states”.

The total number of states (variety) of the system of this construction is 2^18 = 262,144. WHEW! That’s a big number. So, we define variety as:

V_S = |S|^n                       (NOTE: Those bars arond S denotes cardinality, or the number of members in S)

Where V_S is the Variety of System S, with p number of members m:

S = {m_1, m_2, … m_p}.       (ANOTHER NOTE: That little line denotes "subscript", which I apparently can't do on this platform. IGOR HELP!)

With n number of states:

P = {p_1, p_2, … p_n}.

OKAY, WHY THE HELL IS THIS IMPORTANT?

These numbers get really big, REALLY fast. The more members and numbers of states you add, the larger the numbers get. Now, this is really a simple model, but if we were interested in say, modeling the interactions between each member of the system or something else depending on what we need, we can see that the variety of the system can become absolutely stupid.

Imagine if we did a “mesh” configuration of connections between each member of the example system above, and add in an additional number of states each connection can have between each member, you can see things becoming more and more complex. Wanna see what that looks like? Well, using draw.io is a bitch trying to do this, so I pulled this from the internet: A mesh configuration between 16 nodes:

Credit goes to x-engineer.com And holy hell that’s a lot of connections.

Now, not every system is going to be like this, there are different types of ways that members of a system can be connected (or disconnected). This model shows you the maximum variety, given members, connections and possible states.

This tells us two VERY IMPORTANT THINGS about systems:

1.) The variety of a system is directly related to the complexity of a system. In fact, there is some contention about variety as complexity, but I think it is clear what I mean in this context. Hopefully.

2.) The more variety a system has, the more difficult it becomes to analyze. When I say analyze, I mean to understand the workings of the system, numerically, qualitatively, etc.

Analyzing systems down to the “nitty gritty” is difficult. When you have lots of variables to account for, you end up having to use computers to analyze how different systems operate and behave and so on. If the granularly analyzed system gets too complex, analyzing it can practically become an intractable problem. For example, simulating a brain down to the atom would be a gargantuan task. It would take more processing power than we are capable of adequately providing.

BUT… IF we could perform that analysis, we could know pretty much anything we want about the system. Our simulations would be absolutely faithful.

Of course, we are in the real world, and analyzing everything down to a frazzle like that is simply not possible. So, we have to compromise. How do we do that?

LEVEL OF ANALYSIS

Assume for a moment I asked you to care for a fish tank in some way. If I asked you to simply clean the outer glass with Windex every day, you wouldn’t need to know the temperature or pH of the water (assuming all other functions are indeed adequately taken care of, and this is your only duty). Say I needed to add in a new fish to the tank, suddenly, those two numbers become important. Simply assessing the glass for streaks or dirt wouldn’t be sufficient to make sure you could get those fish in the tank safely.

What I mean here is the objective of an observer with respect to a complex system is to adequately consider the variables necessary to make the system reliably work towards that objective with minimal probability of that knowledge being inadequate. In other words, the harder the job, the more you need to know.

We see throughout the different sciences that as our investigations go deeper, we find that we must account for and control for more variables in a system. Our Resolution of the system must be considered.

In classical notions of cybernetics, the idea is to consider the inner workings of a system as a black box. Input goes in, output comes out, and we use feedback (error correcting negative feedback) to change the subsequent outputs.

We don’t need to know exactly what’s going on in the box if we can have a controlling influence “steer” the output to something more desirable. Hence the name “Cybernetics” or "kybernētēs”, or  κυβερνήτης in Greek, which means “pilot” or steersman. That being said, systems are complex, and perception of the black box may not be sufficient, depending on the task at hand.

Sometimes we can make our black box bigger. Sometimes we need it to be smaller. In the latter example, we need to know about the possible “sub workings” or subsets of our system. This helps us understand it more clearly but shrinks our black box to something of higher resolution.

In short, we are making the system S black box into a bunch of tinier little black boxes.

Our Members are "Subdivided" Into Tiny little "Sub-Systems" With Inputs and Outputs. Sexy.

So, you can see that we are not dealing with only one input and output, as we would if we treated the entire system as a black box, but with multiple inputs and outputs. Because of this, our model of the system becomes better, while our analysis of it become more difficult because of the number of inputs and outputs we need to consider.

This is actually an important aspect of systems and is directly related to what’s called the good regulator theorem. The Good Regulator Theorem (in so many words) states that proper regulation of a system requires our model of the system to “faithfully reflect” its inner workings. The better our model, the better we can account for things and the better we can tune our negative feedback to approach stability. This makes sense, right? Because a common saying in systems thought (and statistics) is that all models are wrong, but some are useful.

PUTTING IT TOGETHER

So, we have covered these three prongs of system analysis – and by association, regulation:

1. Level of Analysis, or LA. Depending on our resolution of our observation of a system, our observation of the variety of that system can be limited, or complete. This pertains to our models in practice because there is almost always a difference between the apparent variety of a system and the actual variety.

2. “Analyzability”, or A. As our level of analysis gets deeper into a system, our ability to practicably analyze it becomes increasingly difficult.

3. Model Mapping Efficacy, or MME. Our understanding of a system (i.e. the models we make of it) become more efficient in their predictive and regulatory power as our level of analysis deepens.

In short:

As A goes up, LA and MME goes down

As A goes down, LA and MME goes up.

Or, in a picture:

Very Potato Quality. You're Welcome.

WHAT DOES ALL THIS MEAN MAN?

Well, it means that for any type of cybernetic analysis we do, there is going to be some limits. A steersman doesn’t need to know the exact position of every water molecule in a bay when they point their boat towards the lighthouse. Depending on the purpose of the analysis we do, such details may become imperative.

Additionally, it suggests an empirical limit in fields such as Artificial Intelligence, where there is a demarcation between the rigorous and formal understanding of an AI system we make, and the empirical results of a constructed AI system. This will be important not only in the way we look at cybernetic models, but also how we look at the system domains of economics, governance and society within the cybernetic community.

It is a tool that we can use to optimize say, computational resource allocation, as well as giving us a starting point on how to analyze a system. For example, if the purpose of your analysis is rather complex, then our apparent variety must meet the variety implicit in the application of the purpose in question.

If that seems a little prosaic to you, let me also tell you this: You could in fact use this to understand HOW look at your investments in cryptocurrency. For example, consider the cryptocurrency market as a system. If you wanted to say, build a program that modelled the market so that you could say, buy low and sell high for profit on a particular altcoin or something, you have to consider exactly what you can model with respect to how much processing power it would take to analyze it. In short, you could create a cybernetic system that MAKES YOU MONEY! That's a spicy meatball.

AND if you heed that advice and make a mint, check me out over on PATREON. Because Shillers Gotta Shill.

Keep your eye on the markets!