A Mathematical Definition of System
What exactly does it mean for something to be a system?
I gave a basic definition in my post on systemness.
“A system can be defined as a set of things that are connected in such a way that they produce a pattern of unique behavior over time.”
This informal definition is great for doing intuitive systems thinking. However, a rigorous science of systems will require something more formal.
George Mobus provides a mathematical definition meant to serve as a starting point for a formal definition of system in chapter 4 of Systems Science: Theory, Analysis, Modeling and Design.
I will only cover the definition at a very high level in this post in order to provide a feel for its general structure and purpose. Future posts will cover these concepts in greater detail.
A Mathematical Definition of System
Mobus’ definition is inspired by the work of two pioneers in systems engineering and systems science, Wayne Wymore and George Klir. Wymore and Klir took a purely mathematical approach to defining “system” and used math to explore the implications of their definitions. This approach was grounded in a radically constructivist philosophy which holds that systems don’t exist in the real world independent of the human mind.1
In contrast, this approach starts with a definition based on principles of systems science. It is grounded in a realist perspective which contends that systems do have an independent existence outside of the human mind. Mathematics are applied as a way to provide a structure for “holding” the details of a system description. We are using math to organize information about a system that we’ve captured through real-world observation. We are not depending on mathematical abstractions to reason about a theoretical system’s behavior.
With this in mind, let’s take a look at the definition.
System of Interest
A system, S is defined as a tuple (an ordered list) with seven elements.
S is our system of interest. The index i indicates which subsystem we are dealing with. The index l tells us which level we are operating at within the hierarchy of systems and subsystems .
Components
C represents the set of the system’s components.
e indicates the component’s type and m represents the component’s membership function (in the event the set is fuzzy.)
Component
Each individual component c in the set of components C can be treated as either its own system S or an atomic component c that can’t, or doesn’t need to be further decomposed.
Internal interactions
N is a graph (from graph theory) that defines the interactions between all of the components in C.
N has a set of vertices made of system components, C:
N has a set of directed edges between the components, L:
External interactions
G is a graph that defines interactions between the components of the system and the environment.
Interactions with the environment happen via external sources, Src, and sinks, Snk.
F is a set of directed flow edges between components.
Boundary
B is a boundary consisting of a set of properties, P, and a set of interfaces, I.
One example of a property would be “porosity,” with 0 meaning the boundary is completely non-porous.
Interfaces are a special type of system that facilitate flows across boundaries.
Transformations
T represents a set of transformation rules for each of the components in S.
Each component, c, has a formula, equation, or algorithm that describes how that component transforms inputs into outputs.
Memory
H is an object that records the history of the system, or its record of state transitions as it develops or evolves.
Time
Finally, “Delta_t” is a time interval relevant to the level of the system of interest.
In plain english, we could read this whole equation as:
“A system, which may a subsystem of a larger system, consists of a set of components, internal interactions, external interactions with the environment, and boundaries.
The system follows a set of transformation rules or functions that describe how its subsystems process inputs into outputs. The system also keeps a record of its state transitions. All these aspects of the system can evolve over certain time intervals.”
Remember that this is meant to serve as a starting point for a formal definition of system. It is far from a finished product and will require intensive research to reach its potential.
Applying the Framework
The beauty of this definition is that it is:
Flexible enough to describe any type of system — biological, social, or technological.
Comprehensive enough to formally capture many of the important aspects of systemness.
Accessible to people, like myself, who haven’t had to do any serious math in over a decade (and may have a slight math phobia).
This mathematical framework is meant to serve as the foundation that will allow us to build a systems knowledgebase that can support understanding, design, policy generation, and decision making among researchers for any system.
Imagine being able to efficiently organize all existing knowledge about a cell, a robot, a corporation, or the complete wiring of a fruit fly’s brain. Then, with the click of a button being able to generate a wide variety of models that empower stakeholders from different backgrounds to analyze and reason about the system from multiple perspectives.
A robust mathematical framework is an important first step towards making this vision real.
When thinking about this work, I like to keep in mind the saying that “all models are wrong, some are useful.”
The goal isn’t to build a perfect representation of a system that will grant us absolute control and predictive powers. It is to produce models that are as realistic and useful as possible. To create tools that help us reason about and effectively deal with complex systems that aren’t well suited to being deeply understood with the traditional scientific method.
Read George Klir’s Facets of Systems Science to see how he approached defining system
This is ambitious, but I do believe that every system primarily exists in the mind's eye of an OBSERVER...complex systems especially can't be truly understood without reference to the cognition and perspectives and worldviews of the people who are participating in and trying to understand them...no idea how to begin turning that into maths though!