We take space for granted. Humans exist within space, objects exist within space, and we even say that memories are stored somewhere inside our heads. Space looks like the backdrop against which everything is placed.

But what if space is not something that exists first, with things then set inside it — what if the form we call space arises afterward, as a byproduct of expressing relationships efficiently?

An LLM does not store text as it is; it turns words and sentences into vectors of hundreds to thousands of dimensions. Words with similar meanings are placed close together, different ones far apart, and pairs like king and queen, or Seoul and Busan, are bound by a consistent distance and direction. From the single objective of predicting the probability distribution of the next word, a space that organizes language by meaning emerges inside the model.

If you have N objects and want to store every relationship among them one by one, the amount of relational data scales with N². But if you give each object a single set of coordinates and express the relationships as distances, the amount to store shrinks to something that scales with N. That is because once you have coordinates, distance, direction, clustering, and neighborhood relations all follow at once from geometry. This is the essence of what space does. Space is not a backdrop but a compression device that folds countless relationships into compactly represented positions.

This compression is not free. Mathematically, folding relationships into coordinates is efficient only when those relationships were generated by a low-dimensional geometric structure in the first place. If the relationships are random, no amount of coordinate assignment reduces anything. So the very fact that embeddings work is itself a discovery — it means that language, as data, really does harbor a low-dimensional regularity. In machine learning this property is called the manifold hypothesis: the observation that data which looks complex actually lies on a surface of far lower dimension.

A similar form appears in the brain. Cognitive neuroscience shows that even objects with no physical location — such as social relationships or abstract concepts — are handled as spatial structures in the hippocampus and entorhinal cortex. The cells that fired while finding a physical path behave similarly when handling the distances between concepts. This spatial machinery seems to have arisen first for navigation — foraging for food and returning to the nest — and was later co-opted (exaptation) for handling abstract relationships. A spatial tool in the brain that evolved to interpret space was extended to relationships that fit the shape of that tool.

The phrase “fit the shape of that tool” reveals both the usefulness and the limits of space at once. Distance is symmetric, it obeys the triangle inequality (when you draw a triangle within that space, no side is longer than the sum of the other two), and it is continuous. Relationships with these properties belong naturally in space, but relationships that fall outside them do not fit space well. A representative example is the tree structure. Placing a tree into flat Euclidean space distorts the distances, which is why trees are now often placed into hyperbolic space, whose curvature is negative. The fact that another geometry succeeds where Euclidean space fails shows that space is not universal but a tool chosen to match the nature of the relationship. Likewise, asymmetric relationships that do not hold when reversed — such as unrequited love — or logical implication cannot be expressed by a symmetric distance. So we can say that space appears only for the class of relationships that possess distance, neighborhood, and continuity.

Despite these limits, when viewed from the standpoint of space, three different fields overlap into a single heuristic. Artificial intelligence compresses meaning into space, the brain organizes concepts into space, and some modern physics holds that spacetime itself emerges from a more fundamental information structure. All of them start from the shared mode of representation we call space. We cannot regard these three concepts as entirely the same thing, yet there is a strange sense that some principle runs through the middle of them.

Looking into it, I found that this idea was not entirely new in philosophy. Leibniz saw space not as a container holding things but as the relations among things themselves, and Kant called space not a property of the world but a form of intuition through which we receive the world. Even so, it feels deeply interesting that a common character of space shows up across various fields outside of philosophy.

We are animals that evolved within space, and our mathematics and intuition are spatial down to their roots. The observation that space seems to constitute the world may be a property of the world, or it may be a property of us, the observers. Do we live in a world that is well suited to being represented as space? Or is it simply that we cannot interpret the world in any other way? I find myself wondering.