The statistical nature of geometric reasoning
Geometric reasoning has an inherent dissonance: its abstract axioms and propositions refer to infinitesimal points and infinite straight lines while our perception of the physical world deals with fuzzy dots and curved stripes. How we use these disparate mechanisms to make geometric judgments remains unresolved. Here, we deploy a classically used cognitive geometric task - planar triangle completion - to study the statistics of errors in the location of the missing vertex. Our results show that the mean location has an error proportional to the side of the triangle, the standard deviation is sub-linearly dependent on the side length, and has a negative skewness. These scale-dependent responses directly contradict the conclusions of recent cognitive studies that innate Euclidean rules drive our geometric judgments. To explain our observations, we turn to a perceptual basis for geometric reasoning that balances the competing effects of local smoothness and global orientation of extrapolated trajectories. The resulting mathematical framework captures our observations and further predicts the statistics of the missing angle in a second triangle completion task. To go beyond purely perceptual geometric tasks, we carry out a categorical version of triangle completion that asks about the change in the missing angle after a change in triangle shape. The observed responses show a systematic scale-dependent discrepancy at odds with rule-based Euclidean reasoning, but one that is completely consistent with our framework. All together, our findings point to the use of statistical dynamic models of the noisy perceived physical world, rather than on the abstract rules of Euclid in determining how we reason geometrically.