Dimensionality — More Axes Than You're Watching
Systems have more degrees of freedom than any single view captures. Most mistakes come from forgetting which dimensions you projected away.
Every map of the globe distorts something. Mercator keeps angles but inflates areas — Greenland looks the size of Africa. Equal-area projections fix that but warp shapes. There's no cheat code: you can't flatten a sphere without losing information. The only question is which information you lose.
This is dimensionality. Systems have more degrees of freedom than any single view captures. A neural network's weights live in millions of dimensions; you watch a loss curve in one. An economy has millions of interacting agents; you read GDP. A patient has a whole physiological state; you see a symptom checklist. The view is always a projection, and the projection always discards something.
That's fine — you can't operate in the full space. But most mistakes come from forgetting that you projected at all.
00. — This is what makes interpretability hard. A model's behavior is determined by its full weight configuration — millions of dimensions. Any explanation ("it's using this feature") is a projection onto human-readable dimensions. Two models can look identical under one projection and behave completely differently under another. The circuit description is a projection, and the discarded dimensions are doing work.Failure modes
Mistaking the map for the territory
The classic version: treating your low-dimensional summary as if it captured everything that matters. Mercator isn't wrong — it's a valid projection. But planning a polar expedition using Mercator distances will get you killed. The map became the territory in your reasoning, and the discarded dimension (area distortion) was load-bearing.
This shows up any time a metric becomes a target. A benchmark score is a projection of a model's capabilities onto one number. Optimizing the benchmark is optimizing the projection, not the thing it was projecting. The capabilities that didn't contribute to the score can degrade freely. 00. — Goodhart's law — "when a measure becomes a target, it ceases to be a good measure" — is a dimensionality statement. The measure is a projection; targeting it decouples the projection from the structure it was summarizing.
Collapsing the possibility space
A subtler version: taking a structure with many possible states and reporting a single one. A decision tree has a topology — some paths converge, some diverge. Reporting the single path you took ("the answer is yellow") discards the topology. It might be that every path leads to the same answer, or it might be that the answer flips depending on a single node. The projection from "all possible traversals" to "the traversal we did" destroyed the useful information.
This is the difference between handing someone a map and reporting a route. The route is one path through a space. The map is the space. One is a projection; the other preserves dimensionality.
00. — Whether something looks sudden or gradual depends on which dimensions you're watching. Water "suddenly" boils — if you're watching temperature and state. Watch the distribution of molecular kinetic energies and the transition is smooth. Emergence looks dramatic because we're watching a projection, not the full state.