I’m sure everyone has heard of Plato’s Theory of Forms at least in some capacity. The Theory of Forms states that beyond the physical realm exists an eternal realm filled with unchanging “Forms” which are the true essence of all things. For example, the form of Beauty is perfect beauty, the form of a Rock is the ideal “rock-ness” that all rocks in the world participate in, and the form of a circle is a perfect, abstract circle that is the ideal in which all real circles attempt to be. This notion is seemingly simple to grasp at first glance, but, like many theories in philosophy, it unfolds into something quite profound. Plato was using it to help explain why some things feel more “real” than others, or how we can describe something as “table” or “rock” without needing to give detailed explanations; in these cases, the object is reflecting a single abstract Form that allows for an understanding of said object. He also uses Forms to explain why we have certain knowledge in an ever-changing world, such as that found in mathematics. The Forms function as an anchor for the human condition. They allow us some level of footing in a world that is endlessly expanding and changing. This creates issues when thinking about the world under the lens of a physicalist because it implies there is some other realm of existence beyond the physical reality we have all grown accustomed to. It implies that Reality is not what we see, but what we can conceptualize philosophically.
As I thought more about this theory, I realized there was a straight line to the field of topology and how this conceptual universe of shapes functions. Topology offers us a way to understand the world through abstraction and continuity instead of materials. What if the field of topology isn’t just a branch of mathematics, but is an echo of Plato’s Forms?
First, let’s explain some ways in which the field of topology overlaps with the perspective found in Plato’s Forms.
1. Topology famously disregards material properties.
Topology, though labeled as “rubber sheet geometry,” doesn’t care at all what material an object is made of. Topology deals with the potential of an object, not its physical substance. To a topologist, a circle made of wood is the same as one made of Silly Putty. Plato tackles this in his Allegory of the Cave, in which Truth isn’t found in the immediate sensory experience, but only after abandoning that modality in favor of rational thought.
2. Topology is about relationships, not appearances.
Further, to a topologist, a circle made of wood is the same as a triangle made of Silly Putty. According to topology, these two objects are both just 1-dimensional loops. Either can be smoothed into the other without any tearing or cutting. Topology is concerned with homeomorphism, or the structural relationship that shows how one shape can be morphed into another, instead of a shape’s immediate appearance. Similar to how the Form of Beauty remains across all beautiful things, a topological property remains across all representations of a certain space.
3. Topology doesn’t measure.
Topology is concerned with connections, not angles, lengths, or distances. The topologist endlessly studies a space’s potential, not its measurements. To a topologist, the measurement of a space can change under various transformations and therefore doesn’t help explain the essential structure of a space. Topology looks past the surface-level measurements to consider what persists through change. Likewise, Plato wasn’t concerned with the illusion of permanence, but rather the eternal Truth.
4. Topology applies to imagined spaces.
Topologists frequently study spaces that don’t exist in our 3D space, such as Klein bottles, 4D manifolds, and infinite-dimensional function spaces. Topology isn’t accessed through a sensory experience, but rather, is conceptualized. It is concerned with the eternal structure that exists beyond the appearance of a structure in 3D space. Like Plato’s Forms, we don’t see these structures in 3D space, but we can conceptualize them and prove their existence through mathematical theories. Plato also saw mathematics as a stepping stone between the physical world and the world of Forms. In The Republic, he says, “The branch of study which deals with geometry and the related disciplines… draws the soul towards truth and forms a stepping-stone to the higher realm… They compel the soul to turn around and look away from the world of becoming toward the world of being” (Plato, Republic, 533d).
This leads to an exciting question about the ontological nature of topological spaces. Are they real, or do they exist only as a cognitive tool? Without getting into a conversation about the nature of reality just yet, I think it’s safe to say that the study of topology reveals something real that exists beyond sensory experience. In a way, the field of topology is the language of Forms, but made precise through mathematics. It is the search for the metaphysical in the rational, an exercise in finding truth through philosophical reasoning, a path Plato envisioned. Both Plato’s Forms and the field of topology see Truth through studying structure; they dive into the metaphysical ocean of theory and discover the eternal in its deepest recesses.
Citations:
Plato. The Republic. Translated by G.M.A. Grube, revised by C.D.C. Reeve, Hackett Publishing Company, 1992.