Search for: All records

General mathematical reasoning is computationally undecidable, but humans routinely solve new problems. Moreover, discoveries developed over centuries are taught to subsequent generations quickly. What structure enables this, and how might that inform automated mathematical reasoning? We posit that central to both puzzles is the structure of procedural abstractions underlying mathematics. We explore this idea in a case study on 5 sections of beginning algebra on the Khan Academy platform. To define a computational foundation, we introduce Peano, a theorem-proving environment where the set of valid actions at any point is finite. We use Peano to formalize introductory algebra problems and axioms, obtaining well-defined search problems. We observe existing reinforcement learning methods for symbolic reasoning to be insufficient to solve harder problems. Adding the ability to induce reusable abstractions (""tactics"") from its own solutions allows an agent to make steady progress, solving all problems. Furthermore, these abstractions induce an order to the problems, seen at random during training. The recovered order has significant agreement with the expert-designed Khan Academy curriculum, and second-generation agents trained on the recovered curriculum learn significantly faster. These results illustrate the synergistic role of abstractions and curricula in the cultural transmission of mathematics. 

Euclidean geometry is among the earliest forms of mathematical thinking. While the geometric primitives underlying its constructions, such as perfect lines and circles, do not often occur in the natural world, humans rarely struggle to perceive and reason with them. Will computer vision models trained on natural images show the same sensitivity to Euclidean geometry? Here we explore these questions by studying few-shot generalization in the universe of Euclidean geometry constructions. We introduce Geoclidean, a domain-specific language for Euclidean geometry, and use it to generate two datasets of geometric concept learning tasks for benchmarking generalization judgements of humans and machines. We find that humans are indeed sensitive to Euclidean geometry and generalize strongly from a few visual examples of a geometric concept. In contrast, low-level and high-level visual features from standard computer vision models pretrained on natural images do not support correct generalization. Thus Geoclidean represents a novel few-shot generalization benchmark for geometric concept learning, where the performance of humans and of AI models diverge. The Geoclidean framework and dataset are publicly available for download.