Toby Ord's interpolation-extrapolation-hyperpolation trichotomy: why AI fails off the hyperplane
tobyordoxford · x · 2026-10-08
In a discussion with mathematician Timothy Gowers, philosopher Toby Ord proposes a trichotomy for describing generalization in AI: interpolation (what lies between known examples, i.e. the convex hull), extrapolation (what lies beyond them, the affine hull), and hyperpolation (what lies in directions that cannot be defined by existing examples, i.e. off the hyperplane). He argues current AI systems are poor at the third category — reasoning along directions not spanned by training examples — a limitation humans also share. The framework offers a geometric lens for thinking about where large models' generalization breaks down.
Related event: Philosopher Toby Ord Proposes Three-Way Framework for AI Generalization(2 posts)→
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