Peano: learning formal mathematical reasoning

Abstract

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 five 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. This article is part of a discussion meeting issue ‘Cognitive artificial intelligence’.

Evaluating language models for mathematics through interactions

“…We do not explore plug-ins ( 53 ) in this paper, nor alternative hybrid neuro-symbolic approaches (e.g., refs. 54 , 55 , 56 , 57 , 58 , 59 ), which may prove a useful salve for some of these failure mode.…”

Introduction to ‘Cognitive artificial intelligence’

“…Goodman & Poesia [ 5 ] continue the theme of how machines can learn to engage in rich, structured representation and reasoning, now focusing on mathematics, in their paper ‘Peano: learning formal mathematical reasoning'. They note that while mathematics is created slowly, involving a huge collective intellectual effort over many centuries, it can relatively rapidly be taught afresh to each generation of students, who can learn to apply it successfully also from a very limited set of training examples.…”