Isomorphism is a concept derived from mathematics, particularly from abstract algebra and graph theory. The word “isomorphism” comes from the Greek “isos,” which means equal, and “morphe,” which means form or shape. The term applies when two complex structures can be mapped onto each other in such a way that each part of one structure corresponds to a part in the other, where corresponding means that the two parts play a similar role in each. Isomorphism, at its core, is about structural similarity. Even though the concept originates in math, it can be applied in a variety of fields. For example, consider two musical instruments, like a piano and a guitar. If you play the same scale or chord on both instruments, even though you use different techniques, the notes and musical structure stay the same. This makes the scales or chords isomorphic across the two instruments, providing the same musical patterns in different forms. Isomorphism is a powerful concept that bridges the gap between different domains. Understanding how different systems can be fundamentally the same in structure, even if it doesn’t seem so from their external shape, can significantly expand our knowledge about them.
This structural similarity is particularly significant when studying complex systems, which are composed of interconnected parts or smaller systems that exhibit one or more properties not obvious from the properties of each individual element. These properties, often referred to as emergent behaviors, arise from the interactions and relationships between the components of the system. For instance, traffic congestion is an emergent behavior that arises from the individual decisions of drivers. The pattern of traffic jams is not evident when looking at a single car but emerges when considering the interactions of many cars on the road. Several distinctive features characterize complex systems: emergence, nonlinearity, embedded feedback loops, adaptability, capacity to learn, self-organization, and network structure. Studying them often involves interdisciplinary approaches, combining insights and methods from different fields and searching for isomorphisms between them. The goal is to understand how the interactions among components lead to the emergent properties and behaviors of the system as a whole and how it is structurally organized. If we have already developed an understanding of a system and it has some structural similarities to a new one, the overlapping principles can give us valuable insights into the latter.
The perception of an isomorphism between two known structures gives a significant advance in knowledge. Douglas Hofstadter argues that such perception is what creates meaning in the minds of people. It is not easy to notice such correspondence at first glance, as often the external shape of different systems might not seem to have any commonalities. However, when it occurs, some clarity arises in relation to each system, helping to further decipher the parts of it as well as clearly map the overall structure. Relating parts of systems to each other and dividing them into known categories is called interpretation, and it has both advantages and disadvantages. Mainly because it often results in being highly subjective, interpretation can vary significantly depending on the observer's perspective and knowledge base. While isomorphism can provide an objective basis for comparison by focusing on structural similarities, the subjective nature of interpretation can still influence how these relationships are perceived and applied. Thus, the challenge lies in balancing the objective identification of isomorphic structures with the subjective aspects of interpretation to achieve meaningful and accurate categorizations that help build a better understanding and do not contribute further to biases and misconceptions.
The human organism is one of the most complex systems in the known universe, and it has a unique quality to study itself. The concept of isomorphism can be a useful tool for understanding the complexity of our behavior and movement. It first leaked out of the area of math into the field of Gestalt therapy, where the term isomorphism is used to refer to the structural similarity between perceptual experiences and neural processes. The idea is that the patterns of activity in the brain mirror the patterns of perception. Using this concept helped us comprehend how we perceive objects as whole forms rather than just a collection of parts. It was a serious leap in understanding how the brain organizes sensory information. This led to insights into various perceptual phenomena, such as figure-ground organization, where the brain distinguishes objects (figures) from their backgrounds (grounds) based on their structural relationships. It also contributed to the understanding of cognitive processes such as problem-solving, where the reorganization of perceptual fields can lead to sudden insights (often referred to as the "Aha! moment").
The concept of isomorphism influenced later psychological theories and research, including cognitive psychology and neuroscience. The idea that mental representations are structurally similar to external stimuli laid the groundwork for mapping neural networks and understanding how the brain processes information. Development and application of this principle have boosted cognitive psychology and created such gems as Jean Piaget’s notion of Schema, Lev Vygotsky’s concept of the Zone of Proximal Development, Howard Gardner’s idea of multiple intelligences, and other outstanding works that helped humanity to understand how finding isomorphic relationships can significantly improve the learning process.
In relation to human movement in general and motor learning in particular, there exists a term called “transcription”. It is applied to situations when the system breaks down the existing pattern and builds up a new one from what it already knows. For example, when learning to play tennis after mastering badminton, a player might use the existing knowledge of racket handling and specific footwork, adapting them to the new sport. On a more general level, we can say that almost all our movement patterns are constructed from standing, which will be an initial point for an adult to transcribe new movements from. It is called “original movement form” and is a meeting point that, if addressed, can unlock the problem in other areas where movement efficiency struggles. Transcription and isomorphism are closely related through the idea of structural correspondence and the representation of complex systems. When learning a new skill, if I can recognize what constitutes it, I can search in my existing “library” for an isomorphic structure and use it as a reference point to start constructing a new entity, so I do not have to start from scratch every time.
Transcription is an unconscious process that is accompanied by stages of recognition of patterns on different levels of complexity corresponding to a given movement. This is the foundation for human motor learning: it is based on the associative quality of cognition and is encoded in the system. It is both a positive and negative trait because when we have already built a pattern, it might result in not being optimal for a new situation but will still interfere with the learning of a novel skill. Making the unconscious processes more available to our conscious awareness can be a determining factor in improving how fast we learn new things. This is why creating a common terminology and processing cognitively every new movement situation can help pull information out of this “library” and make the progression quicker. On the other hand, it can also lead to “paralysis by analysis”, so building the learning process must rely on a deep understanding of this paradox. There is no doubt that it is helpful to involve theoretical scaffolding but when we talk about movement it has to be mainly built on actual physical practice that gives references that are embodied, and are not purely cognitive.
Using an interdisciplinary approach gives us a huge advantage in learning and developing a new understanding of complex systems. It is not only applicable horizontally within a field, like different physical disciplines to understand human movement, for example, but also multidimensionally, pulling and mapping areas that seemingly have nothing to do with each other. If I want to study something as complex as a human, I better develop a grasp of a wide collection of subjects, be it math, economics, physiology, psychology, philosophy, etc. The wider my knowledge base is, the more isomorphisms I can perceive, and thus I possess more comprehensive maps of all the complex systems within which I have to operate. Seeing myself as a part of a bigger whole and operating from a place of understanding of the relationships I form inside different systems is a better way to navigate life and make decisions based on the reality of things rather than my subjective perceptions of them. It allows seeing the world as a bunch of connections and opportunities that cannot ever be fully mapped but at least give the endless possibility to be trying.