Cognitive limits help drive advances in mathematical problem solving
Mathematics for and by human cognition: A resource-rational search for bottlenecks in problem-solving
Artificial Intelligence
Summary
Problem-solving can be hard because our brains have limits, but this paper shows that these limits can actually push people to find better ways to think about math. The authors explain how struggling with tricky problems can lead to new ideas and ways of organizing knowledge that help solve not only the original problem but others too. They look at moments in math history to show how these mental bottlenecks led to important breakthroughs. They also suggest that building AI with similar human-like limits might help it find new useful math concepts.
What this means in practice
- •For machine learning engineers: Develop AI systems that mimic human cognitive limits to discover new mathematical abstractions more effectively.
- •For software architects: Design software tools that support identifying and restructuring bottlenecks in complex problem-solving workflows.
A position paper. It proposes an approach and reports no results.
Authors
Sneha Aenugu
Abstract
Human cognitive constraints are generally viewed as limiting factors in problem-solving. We argue that these constraints can instead play a critical role in driving advances in mathematics and beyond. We propose a theory of mathematical abstraction as a resource-rational search for bottlenecks in problem-solving. Bottlenecks arising from cognitive constraints create pressure to restructure existing knowledge, potentially giving rise to novel formalisms with applications beyond the problems that originally motivated them. Drawing on episodes from the history of mathematics, we illustrate how such bottlenecks can drive the development of novel abstractions and examine how cognitive constraints and affective responses shape this process. Finally, we discuss the implications of this account for machine mathematical discovery and argue that incorporating human-like constraints may facilitate the discovery of useful mathematical abstractions.