Which shadow? PCA as a projection

In Book VII of Plato’s Republic, prisoners see shadows on a wall and take them for the objects themselves. Give them a movable wall: which viewing direction reveals the most variation?

PCA chooses an orthogonal projection that keeps as much squared variation as possible. That is the notebook’s own mathematical adaptation — Plato’s fire casts perspective shadows, and his allegory concerns knowledge, not an algorithm.

The analogy ends before PCA becomes a claim about truth. Whether the shadow keeps what a task needs is a separate question, and you answer it by measuring the task.