Take a look at some object near you and ask yourself the question: “What do I know about this object, and how do I know that?” At first we probably all tend to say something about physics, like something about colour and light, something about mass and gravity and so on. If we then think a little deeper, it becomes a typical question from philosophy, specifically epistemology. Therefore it might come as a surprise that it is actually mathematics that teaches us a lot about this question.
The further you go in mathematics, the more abstract and general it gets. One of the more abstract and general branches is one called category theory: a theory that studies mathematical structures called categories (you wouldn’t have guessed it, I know). These categories consist of two parts: mathematical objects and morphisms that relate two objects to each other. It probably won’t surprise you that it’s not an easy topic and that you can't simply explain it without first bombarding others with abstract definitions. However, what I want to talk about is a lemma from category theory named after the Japanese mathematician and computer scientist Nobuo Yoneda, and fortunately, the general idea this lemma tells us isn't that complicated.
As mentioned, a category has two parts: the objects and the morphisms. What the Yoneda lemma essentially comes down to is that we can completely understand an object by looking at all the morphisms that relate it to something. In other words, you can completely understand an object by how it is related to every other object. Or, if two objects have exactly the same relationships to all other objects, they are essentially the same object.
We can get an intuitive idea of what the Yoneda lemma entails through a completely non-mathematical example. Imagine, for example, you're a particle physicist working at a particle accelerator, and you have some particle you don't know anything about. What you can do is launch all sorts of other known particles at it and see what happens. Once you've done this with all the other particles at all sorts of different energies, you'll know everything there is to know about your mysterious particle. The particle is completely understandable by its relations to other particles.
So, the Yoneda lemma tells us objects are their relationships. If it were up to me, I would say mathematics here gives us something very close to a theory of identity.
Yoneda first came up with the idea central to his lemma somewhere in the 20th century, but already in 1814 Michael Faraday, one of the greatest physicists of all time, took a very similar view but on physical matter.
Say we want to distinguish a particle of matter we call a, from the system of powers or forces in and around it we call m. Faraday then wrote in an article:
“To my mind … the a or nucleus vanishes, and the substance consists of the powers or m; and indeed what notion can we form of the nucleus independent of its powers? All our perception and knowledge of the atom, and even our fancy, is limited to ideas of its powers: what thought remains on which to hang the imagination of an a independent of the acknowledged forces? Now the powers we know and recognize in every phenomenon of the creation, the abstract matter in none; why then assume the existence of that of which we are ignorant, which we can not conceive, and for which there is no philosophical necessity?”
In other words, the only thing we know about a particle is how it interacts through its forces. The link with the Yoneda lemma is obvious: the object - in this case physical matter - is entirely knowable by the relations/interactions with all other things.
But now, through mathematics and physics, we end up at a hardcore metaphysical point. Because what the Yoneda lemma together with Faraday’s vision tell us is that there is no ground to assume there exists some impenetrable piece of matter in which the forces reside. There is no essence to matter, but there are only the interactions it has with all other things. (Interactions which are described in physics).
In a way, everything is one whole, in that every distinction is made by us. There is no intrinsic distinction between a chair and a table, only the one we impose on it. And we make these distinctions for the greater part intuitively and solely for practicality.
Like when discussing the clothes we wear, for instance, we distinguish shoes from pants, simply because in this context this is more useful. But if we switch the context to a chemistry lecture, we distinguish the molecules that make them up because that way material qualities are explained the best. A few steps smaller we end up distinguishing elementary particles.
At this point, the distinction between shoes and pants has vanished: shoes are particles arranged in a way we call a shoe, and pants in a way we call pants. And these particles are known entirely by their relations. In each situation we distinguish things in a way that is most practical, and there is no deeper essence in that. An idea like that of a Ding an sich (thing in itself), as Kant and Schopenhauer believed in, is unfounded, because if I take away all the interactions some object has, nothing remains according to the Yoneda lemma.
Now let’s return to the question we started with: “What do I know about this object, and how do I know that?”
I have a pencil here. I know its colour through the light that interacts with it and reflects into my eyes. I know its weight by how it interacts on a scale. I know its dimensions by how it relates to a ruler, and I could go on this way. If I zoom in on the pencil, I will see the different microscopical structures it has. And if I go smaller and smaller, I find only the elementary particles, which are nothing more than some radial force fields. Do this for any other object and you end up at the same.
Everything I know about this pencil is all through interactions with it. And if we all let the pencil interact with everything else, then that is the pencil, and there is nothing more to know about the pencil. And if we break the pencil down to smaller parts, there is no ground to speak about a pencil anymore, but we have individual particles that interact in their own way. That are in turn made up of elementary particles which are merely some forces, but nothing in essence.
Ultimately, we see that, metaphysically, it is complete nonsense to speak of this pencil.