人类认知的数学及其应用：问题求解瓶颈的资源理性搜索
Mathematics for and by human cognition: A resource-rational search for bottlenecks in problem-solving
- Salk Institute of Biological Studies（索尔克生物研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出数学抽象是资源理性搜索问题求解瓶颈的过程，认知约束驱动知识重组与新形式体系产生，并探讨其对机器数学发现的启示。
AI中文摘要：
人类认知约束通常被视为问题求解中的限制因素。我们认为，这些约束反而可以在推动数学及其他领域的发展中发挥关键作用。我们提出了一种数学抽象理论，将其视为对问题求解中瓶颈的资源理性搜索。由认知约束产生的瓶颈会促使人们重组现有知识，从而可能催生新的形式体系，其应用范围可能超越最初激发这些体系的问题。通过借鉴数学史上的案例，我们阐述了此类瓶颈如何推动新抽象的发展，并考察了认知约束和情感反应如何塑造这一过程。最后，我们讨论了这一理论对机器数学发现的启示，并认为融入类似人类的约束可能有助于发现有用的数学抽象。
英文摘要：
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.
