In the study of topology, mathematics goes beyond the reaches of measurable lines or angles found in Euclidean geometry and moves into a realm of abstraction that focuses on potential. As I began my journey into this world, I realized there were so many parallels to my philosophy background, particularly when focusing on notions of identity, change, and form, and the field of Metaphysics.
The donut and the coffee mug
Topology is known for its insights about the coffee mug and the donut. Topologists see these two objects as the same because they have the same number of holes. The handle of the mug and the hole in the center of a donut allow the two shapes to be individually stretched to become the other. See, to topologists, the form of an object is more defined by its qualitative aspects than its quantitative ones; the presence of a hole is a defining structural feature, not some measurement. This may seem intuitively false given that what appears to be a “counting” of holes occurs, but it doesn’t function in the same way as it does when we measure the angles of a triangle in Euclidean geometry. In this way, the hole is a defining quality of the shape, making it homologically equivalent to every other shape possessing the same number of holes, even if they differ in other ways. Instead of “counting” the holes, homology detects and classifies the holes using algebraic tools. As we understand how counting isn’t the same as classifying, we can see how homology is something more qualitatively interesting. Another aspect of topology is the notion of homotopy, or the study of how paths, functions, or spaces can be continuously transformed into each other. As long as you aren’t tearing a shape, you can stretch it endlessly into other shapes. This is what gives topology the name “rubber sheet geometry.” In topology, a shape can be manipulated without cutting or fusing to become another shape. It suggests that spaces can be classified not only by their features but also by their ability to be transformed. Homotopy, in this sense, focuses on the paths to becoming, and this is where the investigation really begins.
The Ship of Theseus
I am immediately reminded of the Ship of Theseus paradox. In this thought experiment, a ship is modified over time due to its wood rotting, and eventually, it is replaced with new materials. We are then asked, “Is this the same ship as before its materials were replaced?” This brings about interesting questions about identity over time and how much of the “sameness” needs to be physically maintained for any object (or person) to remain the “same.” Like our coffee mug and donut example, is there something that would make the ship qualitatively the same, insofar as it can have all of its wood replaced and still be considered the same ship?
To begin this analysis, I think about the function of the ship. This ship, created by some person at some time, serves some purpose. The marking of a specific time in which the ship was created helps identify the ship as one particular ship. But then the question can be asked, “Well, does that mean that every ship made at the same time is the same as every other ship?” Yes and no. Yes, in the sense that all ships built in the same way are also built at the same time and share that sameness. However, no, in the sense that the ship’s identity at the time it was built is unique to it. This is the Continued Identity Theory, or the idea that the identity of the ship remains even if parts are replaced over time, because it is established in its moment of making as a specific ship.
Another angle to answering this paradox is Four-Dimensionalism, or the idea that objects extend themselves outward in time and space, creating a fourth-dimensional object that would be maintained no matter if the pieces of the object were changed temporally. The changing of the various pieces of the ship would be a temporal part of this overarching four-dimensional object. While this may seem to be more intellectualized than the intuitive Continued Identity Theory, it still holds merit as a strong argument. Still, others say that an object is identified by its relationship to the past, or explain linguistically that the ship is determined by its definition. Then, of course, we have those who argue that it’s subjective whether the ship is the same or not, and we are wasting time even having the argument.
The Self
I think about these things in the context of the human condition. We often hear that “change is the only constant” and that we as human beings are changing constantly. On a cellular level, our skin undergoes regeneration over weeks, our red blood cells over months, and even bone cells over years. Yet our “sameness” still hasn’t changed. In topological terms, we are homologically and homotopically constant, our essential structure persists, and our transformations over time preserve some sense of Self. No matter how often pieces of us are replaced or how frequently we undergo new senses of identity, a qualitative aspect of the Self remains constant. If we draw a parallel, the hole in the center of a donut is the Self, the qualitatively defining factor that maintains our identity, makes us who we are, and allows us to take various forms without losing our defining factor.
I think the field of topology is fascinating and shines a light on how the world of mathematics, which is often seen as dull or dry, has a vibrance all its own. Philosophically, I find myself only at the tip of the iceberg and eager to pursue this field more deeply. The metaphysical hum generated by the field of topology is something we all should pay more attention to as we exist in an increasingly quantified world. Topology orients us towards a world that isn’t defined by its measurable traits, but towards one that is concerned with continuity amongst distortion and identity amongst the eternal. Topology allows us to understand ourselves as dynamic, fluid identities that exist within the realm of possibility; it reminds us that we are poetry in motion.