Here's something you've probably noticed but never formalized: you can't really know what anything is - you can only know how it behaves in different contexts.
This insight has a name in mathematics: the Yoneda lemma. It states that objects are completely determined by their relationships to everything else. No hidden essence, no inner nature - just patterns of connection.
You see this everywhere once you notice it.
Here's the mind-bending insight: every object is completely determined by the morphisms (arrows) pointing into it from all other objects.
Think about your job. What makes your position unique? It's not some abstract "essence of being a software engineer" or "inner nature of being a manager." It's the specific ways other things in your company connect to your role: what tasks flow to you, what resources you can access, what decisions require your input.
Change those incoming connections—different responsibilities flowing to you, different access permissions, different decision-making authority—and the job becomes something entirely different, even if the title stays the same.
The Yoneda lemma formalizes this insight mathematically. It states that if you know all the morphisms from every object in a category to some target object A, then you've completely characterized A. More precisely, it establishes a natural correspondence between the object A and the "hom-functor" that maps each object X to the set of morphisms from X to A.
This isn't just saying these morphisms are important—it's proving they're sufficient to distinguish any object from any other.
For centuries, philosophers have been stuck in a false choice about how we learn things.
Option 1: Induction - Start with specific examples and try to find the general rule. You see white swans, more white swans, and conclude all swans are white. But how can you be sure? You've only seen a tiny sample.
Option 2: Deduction - Start with general rules and apply them to specific cases. All humans are mortal, Socrates is human, therefore Socrates is mortal. This feels certain, but you're just unpacking what was already assumed in the first place.
Both approaches assume there's a gap between your mind and the world—that knowledge means building accurate mental representations of external reality. But what if that's the wrong way to think about it entirely?
The Yoneda perspective offers a completely different approach. Instead of asking "What is this object?" you ask "How does this object relate to everything else?"
Think about understanding any person—say, a new coworker. You might think they have some fixed essence—"Sarah is an introverted engineer who's good with data." But the Yoneda insight suggests Sarah is the network of connections: who seeks her advice, what projects flow to her, which teams she collaborates with, how she relates to different departments, what expertise others draw from her.
When Sarah starts leading cross-functional meetings, you're not discovering something hidden about her "true nature." You're seeing new connections activate—and those connections are changing who Sarah actually is in real time.
This eliminates the gap between subjective knowledge and objective reality that's troubled philosophers for centuries. There's no "view from nowhere"—every perspective is a position within the relational network. But this doesn't make everything relative. Objectivity emerges from the structure of relationships themselves.
Here's how this reframes the traditional problems:
Induction works differently when you stop trying to bridge finite experiences to infinite truths. There are no universal laws floating outside the network of relationships. What we call "laws of nature" are stable patterns in how things relate. You're not making unjustified leaps—you're mapping relational structure.
Deduction becomes relational too. "Socrates is mortal" doesn't follow from abstract logical forms. It follows from how the concepts relate within our network of understanding. The "necessity" comes from preserving the relational structure.
In this view, knowledge becomes like a function that preserves relationships. When you understand something, you're mapping the patterns in your experience onto the patterns in the world.
Think about why math feels so certain. It's not because numbers exist in some Platonic realm. It's because math is pure relational structure. When you prove 2 + 2 = 4, you're exploring how objects relate in the category of arithmetic. The certainty comes from the constraints of how these relationships compose.
Science works the same way. A good theory doesn't just mirror reality—it maps the relational structure of experiments onto theoretical categories. When a theory gets falsified, it means the mapping failed to preserve important relationships.
This way of thinking transforms how you approach fundamental questions:
Who are you? Not some unchanging essence, but the consistency of relational patterns over time. You're the stable connections between your past and future selves, your relationships to others, your position in social and physical networks.
How should you act? Ethics becomes about harmonizing relational structures rather than following universal rules. Moral action preserves and enhances the patterns of relationship that let communities flourish.
What is consciousness? Not a mysterious inner theater, but your capacity to map relational structures—to recognize patterns of connection and respond appropriately. Awareness is your brain doing category theory.
Here's the most radical implication: the traditional split between subjective experience and objective reality dissolves completely. You are an object in the category of social relationships. Your coffee cup is a subject in the category of physical interactions. What we call "subjective experience" is just one type of relational pattern within the larger structure of reality.
This doesn't eliminate consciousness, meaning, or intentionality. It reveals them as emergent properties of complex relational networks. Consciousness happens when the categorical structure becomes reflexive—when the mapping turns back on itself.
Going back to Sarah: we'd be wrong to think we could understand her as an isolated object with fixed properties. But we'd also be wrong to think that makes knowledge impossible. The Yoneda perspective reveals a third option—understanding Sarah through the ever-changing web of connections that constitute who she actually is.
We live within the categories we study. And that turns out to be the source of both knowledge and being.