Large Systems Theory Needs Something Other than a Set

Large system theory studies how things make up things, and how things are made up of things. It’s a theory of data properties. Traditional mathematical structures fail to represent the plurality of perspectives we use to describe systems. This article introduces the pluristate set.

Set Theory is the study of collections of things. More properly, set theory is the set of rules mathematics has adopted for organizing and analyzing collections of things. Systems theory is another, less precise way of looking at collections of things. It’s defined as an interdisciplinary approach to studying collections of things that work together or are in some way dependent upon each other.

Large Systems Theory is the study of transitional states within systems (and this is not category theory, which attempts to generalize mathematics – a very different set of ideas and much more abstract than large systems). At least, that is how I define it. There is no formal (academically accepted) definition for large systems theory. It’s more-or-less my thing. Large Systems Theory is the foundation of my work in studying and analyzing systems built on the Internet.

The basic tenet of large systems theory is simple: any system can be defined and measured, whereas a large system cannot be well defined or precisely measured. You can kind of explain what a large system is and give an approximate measurement.

The Paradox of Large Systems

The Paradox of Large Systems is one of the most interesting aspects to this field, in my opinion. The paradox is simple: Every large system can be redefined as a system. I like to use the universe as an example because most people can relate to that. The universe is so large we have no way of precisely measuring it. We simply lack the technology and an understanding of the laws of science (or nature or whatever you want to call them) to be able to accurately measure the universe. And so that means the universe is a large system.

But by its very definition the universe is a single thing. From that perspective, the universe is a system because we can measure it according to the scale of known universes. There is only 1 that we know of, but even if we can confirm there are other universes we’ll only be able to do that by differentiating this universe from those universes. So we may never be able to measure the large system of all universes but we’ll still know that we exist in this universe. Hence, the universe is merely a system.

The Paradox of Large Systems is a universal law of large systems theory. Every system that we can define can be redefined as a large system. The perspective that tells you a thing is a system is the external perspective. You are an outside observer. The perspective that tells you a thing is a large system is the internal perspective. You are an inside observer.

The external perspective can see the boundaries of the system.

The internal perspective cannot see the boundaries of the system.

How To Measure a System

If you cannot see the boundaries of the system then you cannot measure it. A large system has scope (the properties of its components or constituent parts) and boundaries (the limits of what can be included in the system).

Chewing gum is not part of any combustion engine. Your car and your rich uncle’s private airplane do not run on chewing gum. Hence, chewing gum is not part of the systems that comprise their engines.

If you can measure a system then you can count its component parts. If you can measure a system then you can differentiate between what kinds of component parts it contains and what are not components.

You know an orange doesn’t grow on an apple tree, so oranges are not part of the apple tree system.

Why You Cannot Measure a Large System

If everything can be looked at from the inside (as a large system) and the outside (as a system), then it follows that we must be able to see some boundaries of the system from the external perspective.

So why can’t we see those boundaries from the internal perspective?

The answer is quite simple – and complicated. The boundary of a system is a single thing, or part of the atomic nature of the thing. It’s part of what makes the system a system.

The outside observer sees the full system. The external perspective may be able to look inside the system, identify all its constituent parts, classify them, and even count them.

The inside observer simply doesn’t know how big the universe is. You can never be sure from the inside perspective if there is something else hiding out just over there. You can never be sure if everything included in the system is made of all the things you believe they should be made of. There is no way an internal perspective can confirm the nature of every component of the large system.

It’s that uncertainty that makes the large system unmeasurable. You can switch between perspectives an infinite number of times but you can never reconcile them to each other.

Systems Exist In a Plurality of States

The Duality Principle in mathematics says that you can use a true statement from one perspective to define a true statement in another perspective. So in a Cartesian plane you use two points to define a line and you use two lines to define a point.

The line and point are not the same thing, so what we’re seeing with large systems is not a duality. Rather, we’re looking at a plurality of states. There are, in fact, at least three states for every system (so far as we can determine).

There is the infinite state, defined by the unmeasurable internal perspective.

There is the finite state, define by the measurable external perspective.

And then there is the atomic state or component state, where the (large) system is merely a component in another, larger system.

This third state leads us to superstructures or data superstructures. I sometimes use a handful of sand to explain superstructures. The sand can be measured as a single thing (a handful) and as a collection of things (grains of sand). But there is a difference between a handful of sand (it’s loose and fluidic) and a grain of sand (it’s rigid and cohesive). Ignore the fact there may be different types of sand. For the purpose of the theory there is only one kind of sand.

*=> A superstructure alpha-prime is composed of a pluraity of the structures of alpha, where an alpha structure is composed of a plurality of alpha components such that the properties of the alpha structure are distinct from but a product of the interactions of the properties of the alpha components, and the properties of the alpha-prime superstructure are a product of the interactions of the properties of the structures of alpha.

What makes sand sand is its collective state, the “handful”. If you simply isolate a single grain of sand it’s not really sand. It’s just a hard little grain of something that – if you bring a lot of them together – comprises sand.

You Cannot Define a System by a Set

In mathematics a set is any collection of objects together with precise criteria for determining whether or not a given object is in the collection.

You cannot use that definition to describe a universe because we don’t know exactly what is in the universe. And yet you can use that definition to describe the list of universes we know about. So we may not be able to precisely define a universe but we can precisely define the set of universes we know about.

If a system is both measurable and unmeasurable at the same time then it follows you need a mathematical concept that describes both of these states. Better yet, you need a mathematical concept that describes the three known states of systems.

Call It A Pluristate Set

Frankly, I think this is a terrible name but all the good names are taken. You could call it a pluriset but that’s been trademarked. You could call it a quantum set as a play on the Heisenberg Uncertainty Principle but that’s already used for something else and I don’t want to define systems in terms of quantum-anything.

Perhaps a case can be made for calling it a Heisenberg Set but the ambiguity of a (large) system doesn’t depend on the Uncertainty Principle (which states that you cannot know the location and momentum of a particle at the same time). Although you could redefine a system as a series of contiguous states moving through a framework.

That’s kind of what Einstein did with his theory of relativity. But the problem with applying that to Large Systems Theory is that you’re really only moving one perspective through the framework.

We need something that allows us to grapple with all three perspectives (and any others waiting to be discovered) at the same time.

So, for lack of a better name, I’ll use Pluristate Set for now.

The Definition of a Pluristate Set

A pluristate set defines a system’s properties from one of three perspectives:

  1. The atomic perspective in which the system’s properties derive from its singular, coherent state
  2. The collective perspective in which the system’s properties derive from the interactions of its atomic components
  3. The superstructure perspective in which the system’s properties cannot be predicted from either of the atomic or collective perspective

Large systems theory predicts the existence of superstructures but doesn’t provide any mechanism for predicting the properties of superstructures.

We don’t have to know that there are other universes in order to predict a superstructure that consists of multiple universes. We’ve got several theories that propose such a superstructure could exist. But those theories cannot begin to explain what the properties of the proposed superuniverse structures may be.

We can’t fully define what other universes would look like in terms of their physics at any level. Hence, we don’t even know what the atomic properties of a universe would be, given that it’s in a superstructure consisting of two or more universes.

And that means you cannot predict anything about the states beyond the superstructure perspective. You must first be able to measure that superstructure before you can begin to understand how it interacts with similar superstructures at an atomic level.


Inadequate as the idea of a pluristate set may be, it’s the only way to think about systems in terms of their multiple concurrent perspectives. And it’s easier to say “pluristate” than it is to say “pluriperspective”.

A pluristate set is unique enough that it can define a specific kind of thing. We can count pluristate sets on our fingers if we wish.

It’s not a set in a traditional mathematical sense. You might expect to define and perform operations on the members of a traditional set, like a group of numbers. But if anyone defines operations for a pluristate set they won’t look anything like the additive, subtractive, multiplicative, or divisive properties of traditional number sets.

You don’t combine states of a system, or divide them by each other. The functions of a pluristate set will be transformations and reverse transformations. There may be transitional functions that work for one kind of system but which don’t work for other kinds of systems.


People have been discussing this theory on Facebook (and perhaps elsewhere). Several attempts have been made to identify the theory with something in recent literature. That’s perfectly reasonable but so far everything I’ve seen suggested has been a mismatch.

For example, category theory attempts to redefine all of mathematics. The way I would describe category theory to someone is to say it is a philosophical hammer that attempts to beat everything into the shape of an algebra.

An algebra is a set combined with a group of rules for operations to be conducted on the set. In computer science we tend to call these things classes and/or objects (depending on what generation you’re from and what language you first learn to program in). I’m sure there are other names for these things now.

Someone else brought up branch algebras, which theory Laurent Bartholdi introduced in 2005. Bartholdi says branch algebras are “infinite dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves.”

*=> Large systems are not algebras. There is no morphism or isomorphism (fancy words for functions or operations associated with a specific set). There is literally no collection of functions that you would use to describe a large system (not in the classic mathematical sense because we don’t have a branch of mathematics – that I’m aware of – that describes transitions of states of data). Large Systems Theory describes the relationships (and/or) transitions between states of complexity of things. In other words, Large Systems Theory is a theory of data properties.

If it could be explained with something as simple as a set or an algebra then there would be no theory of large systems, but rather a set of a special type or an algebra of a special type.

To think of this as an algebra is equivalent to thinking of the elemental composition of a star as an algebra. It makes no sense. The inventory in a shoe store is not an algebra but it is a system (and from an internal perspective it’s a large system).

In Large Systems Theory everything is data. The universe is data. The grains of sand on a beach are data. The molecules that make up the atmosphere of a gaseous planet are data. All physical things that interact with each other in a systemic way are data – and these data have properties.

The properties of any system of data change at the level where the system becomes an atomic thing (a single item of data) in a larger system (a data superstructure). Every data superstructure is atomic data for some higher order data superstructure. And conversely every data superstructure is composed of lower order atomic data that are themselves data superstructures.

There is no way to reduce the progression to a smallest order of atomicity. Nor is there any way to expand the progression to a final order of superstructure.

Which led someone to suggest infinities. This theory is distinct from the theory of infinities to the extent that I understand the theory of infinities, because infinities are numbers. A number is not a system and therefore a number cannot be a large system.

I realize the people who have already commented on this article may never see the addendum, but if you’re here because you’ve seen some of those discussions, maybe you’ll want to mention this addendum to the people who would be most interested in it.

Related Articles about Large Systems Theory

How to Almost Measure a Large System on this blog

Why It Is So Hard to Describe Large Systems Using Math 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