Part 1 — Toolkit for reasoning in the language of complex structure
People are often parts of complex structures, form them, and live in them. In a series of articles about thinking in language of structures, I will share experiences from practicing qualitative mathematics. The aim is to encourage understanding complex reality in terms of the emergent structures and relations between them.
The research on topics related to complex phenomena inspired me to apply methodology as a meta-language in cases which might require from practitioners to clarify, define, and sort out the complexity. Through imaginary case studies, I will sketch the versatility of possible applications that can assist in transforming the perception of complex situations through the development of multidimensional worldviews, to communicate complex concepts and efficiently solve problems.
Standpoint that everything is interconnected in a complex system, with distinct parts of the system affecting the behavior of each other, can be approached from different methodological frameworks.
‘Thinking comes before language!’ says Ulf Grenander in A Calculus of Ideas, and the language which I chose to present is the one of structure, called the Q-analysis, developed by Ron Atkin with a purpose to describe and explain things through understanding the structural complexity. As a result, the Q-analysis emerged as an algebraic language for the discussion, definition, and description of structure in a holistic view of part-whole relationships.
Q-analysis is based on two fundamental concepts: a set of objects in relations (of any nature and origin) and mapping it into a suitable mathematical framework. The former reconstructs the shape of objects via relations between them, whereas the latter underlies observation of connectivity patterns, multidimensional classification, and treatment of the structure of objects.
In the background of the method is the following paradigm:
- Identification of objects (expressed in words, drawings, or whatever)
- Their integration into blends of emergent entities through relation which binds them into a whole
- Identify and/or build new patterns in structure
- Run stories through scenarios, and then, if necessary, iterate the process.
Interconnected events in a complex system impose a task to have versatile viewpoints.
Big sporting events may have a positive impact on the financial system of the city, and a negative effect on the urban transportation system by causing congestion on the traffic flow over the streets surrounding the place of the event. Also, these two changes are happening at different time scales and different system-size scales.
I will introduce some basic concepts of Q-analysis which don’t require formal mathematical knowledge, are rather intuitive, and are easy to comprehend.
RELATION
The other way to say that things in a complex system are connected is to say that there is a relationship between them. Actually, we could say that the relation is a necessary condition of emergent property. Understanding the relation between things brings us closer to reconstruction and interpretation of complex system’s behavior. But what the ‘relation’ is in the first place? Well, it is a rule by which one element interacts with another and which binds elements into a whole.
WHOLE
Once an object has been assembled, it is usually treated as a single object or entity with a meaning. This new object may possess emergent properties not possessed by its components. It is a kind of ‘package of relations between elements’. A way how we choose to aggregate elements (or terms or descriptors) into a whole is neither right nor wrong, but merely more useful than others, and depends on our interest in a complex system in the first place.
Say you like sweetened cappuccino. An experience of drinking a sweet warm cappuccino can’t be guessed knowing the taste and textures of all ingredients separately. Combining hot water, ground coffee, milk, and granulated sugar, into a cup produces a relation between ingredients. Accordingly, it seems to be consistent with the “Gestalt” experience of drinking sweet cappuccino. In complexity science, this is called the emergent property, though the point is the same.
In the Q-analysis (and, of course, in the algebraic topology) objects or elements of a set are called vertices, whereas their assemble via relation into an entity is called simplex, which is a multidimensional object living in multidimensional space, and their collection forms the so-called simplicial complex. These are just technical terms for presenting the notions of Gestalt whole and connections between different wholes. Geometrically simplex is represented as a polyhedron, vertex as a point, and relation as a line.
Sweetened cappuccino lives in the complex system of hot beverages, and it is placed in 3-dimensional space.
STRUCTURE
The notion of structure is so prevailing in everyday speech that we usually don’t notice its importance, and that it is not easy to define structure. We talk about the structure of a complex system, a building, a novel, a sonnet, a financial system, a company, religious beliefs, a political system, a country, an urban area, human relationships, a play, a galaxy, a symphony, … and as can be presumed the list is endless.
For example, Jeff Johnson postulates the Fundamental Property of Systems: ‘The behavior of systems depends on how their elements are connected.’, i.e. its structure.
So, the simplest and fairly general property of structure is that it stores information about the way how things are connected together. And further, in order to describe and explain things in our everyday life we have to relate something to something else, and build the structure.
We can agree that the cappuccino is one ‘Gestalt’ experience, yet hot coffee with sugar and cinnamon is completely different, and not to mention hot milky tea with honey. These three hot beverages are connected through shared ingredients and hence build the structure, or at least part of the structure, of the complex system of hot beverages.
Remove an element from a whole, and the whole ceases to exist.
Remove a whole from the structure, and the structure changes its shape.
When considering changes in complex socio-technological systems sometimes we are in a position to identify missing parts and if necessary, can design them. It will alter the structure of the system toward the desired future.
TRAFFIC AND TREATMENT
Formation of the structure of wholes doesn’t mean that it exists just for itself, but rather has some purpose, or function. So, the structure sets a stage for various manifestations or occurrences emerging from it. In other words, the structure forms a kind of map, and like in the case of any map we can consider flow over the parts of it or the ways these parts are treated.
For example, it can be the support for the flow of traffic (of anything) through some of its parts, which now depends on its complex structure. In the case of the system of hot beverages, traffic can be the flow of a number of persons who drink different beverages, or traffic can be changes in prices. On the other hand, making a cappuccino as a hot beverage is the outcome of treatment of the relation between ingredients. But, treating the same ingredients in a different way by applying another preparation method will give us different hot beverages as an outcome.
MULTIDIMENSIONAL FORCES AND EVENTS
Terms ‘multidimensional force’ or ‘multidimensional event’ are expressions that describe different kinds and types of causes that govern changes over the structure, or changes of the structure itself. In other words, the action of complex structural forces on the system are responsible for changes of flows over and are strongly affected by the structure. On the other hand, a change of structure of a complex system is a result of an event when a new whole or new entity is formed due to assembling elements.
When a complex system of stores in shopping malls is overwhelmed with the flow of customers on Black Friday various attracting forces act. Not to mention a sudden increase in cash flow on that occasion which is affecting the complex financial system.
Getting an idea is pretty much an event of assemblage of previously unrelated (or partially related) chunks of knowledge or experience. The design process is accompanied by the occurrence of events of assembling either in the ideation stage where mental structures are formed or in the prototyping stage where tangible models are built.
Briefly, considerations about complex systems start with identification of their elements, relations between them and the emergent wholes that they form. It is followed by analyzing the structure and various traffics over it, alongside with the forces that govern the traffic and events that change system’s structure. Well, the trick is to implement this methodology in a way to make it as much as reliable map of a complex situation whose changes can help make decisions for desired system functioning.
Whenever we have a complex system, well, of nearly anything, we can apply the methodical approach derived from the above concepts to the design of changes in systems. Interestingly, it doesn’t contradict the other approaches, such as systems thinking, or design thinking, but being rather a complementary approach to understanding systems.
DISCLAIMER: The examples and results presented here are not scientifically rigorous, neither linguistically nor semantically, in the sense of interlacing the generality of terms, or mathematically, in the sense of strict formalism application. It represents just an informal attempt of reconstruction of conceptual maps and getting an insight into the structural relationships between them. The way of reasoning is inspired by outcomes my own works (published with collaborators), which is on the other hand notably influenced by works of Ronald Atkin, Jeffrey Johnson and other Q-analysists.
