本期Babbage节目探讨了人工智能给数学领域带来的深刻变革。大型语言模型（LLMs）曾以不擅长算术而臭名昭著，但如今已迅速进步，能够解决复杂的数学问题，这改变了数学家们的看法。 前数学家、《经济学人》记者Anjani Trevedi指出，传统的研究数学旨在通过严谨、优美的证明来证实基本真理，这与学校里教授的死记硬背式算术截然不同。几个世纪以来，数学家们一直依赖逻辑演绎，并常在前人工作的基础上进行构建，由此形成的知识体系虽然坚实，但也极为分散，难以驾驭。长期以来，计算机一直用于大规模计算和检查复杂配置，但现代人工智能代表了一个全新的前沿领域。 最初，鉴于人工智能在基础数学方面表现不佳，数学家们对其能力持怀疑态度。然而，在过去一年中，情况发生了巨大变化，LLMs解决了许多长期存在的难题，如部分埃尔德什问题，现在正着手处理更复杂的证明。菲尔兹奖得主、数学家陶哲轩将人工智能描述为一名“非常聪明的本科生”，它与人类不同，能够不倦地将数百种方法应用于数千个问题。虽然人工智能通常解决的是“唾手可得的成果”——即那些相对容易但数量众多且未曾受到人类关注的问题——但它能够将数学中不同子领域的方法结合起来，这是人类专家鲜少能做到的壮举，这为发现新的联系和见解提供了巨大潜力。 数学领域的一个主要瓶颈是需要对证明进行严格验证。人工智能模型虽然强大，但仍会犯错，陶哲轩估计其错误率高达98%。这使得建立一个可靠的系统来确保信任变得至关重要。正在出现的解决方案是“自动形式化”，这是一个将自然语言数学证明自动转换为计算机代码的过程，这些代码可以进行形式化验证以确保其正确性。DARPA的Patrick Shafto将当前庞大的数学知识体系描述为由论文和思想组成的“一团糟”。自动形式化旨在创建一个统一、一致且可由机器验证的知识库，这将加速进展、减少重复工作，并使高等数学更容易获取。 在此领域的一个里程碑式成就，是玛丽娜·维亚佐夫斯卡（Marina Vyazovska）因解决八维球体堆积问题而获得菲尔兹奖的证明被形式化。Math Inc. 公司开发的名为Gauss的人工智能系统，成功地将她耗费数年时间才完成的证明转换成可由计算机检查的代码。维亚佐夫斯卡指出，虽然人工智能的输出可能会冗长（例如，明确证明“2+2=4”），但这一过程提供了对所用复杂方法更深入、更形式化的理解。 人工智能在数学领域的愿景包括由Harmonic等公司创建“数学超级智能”，或像GitHub那样的平台（如谷歌的Defined），研究人员可以在这些平台上发布、下载并即时验证证明。数学家们将人工智能视为一个强大的工具，可以提高他们的生产力，探索更多想法，并连接不同数学领域之间的点，最终拓宽他们的智力视野。 然而，挑战依然存在。人工智能模型可能会“幻觉”或过于轻易地接受人类的提示，这可能会损害原创性。此外，人工智能系统摄取并随后形式化或在此基础上构建的人类原创工作的归属和署名权问题也至关重要。维亚佐夫斯卡与Math Inc.的合作经历凸显了，随着人工智能越来越多地介入数学家的工作，制定明确的伦理准则的必要性。 最终，共识认为，人工智能的作用将是处理数学证明和形式化中的“繁重工作”和技术细节，从而创建一个高度组织化且可验证的知识库。反过来，这将使人类数学家能够专注于数学中真正的创造性、直觉性和概念性方面，推动探究的边界，并可能在其他科学领域带来突破。尽管人工智能最终可能会自主生成新颖的证明，但其效用以及由谁来“消费”和解释这些证明，仍然是一个社会挑战。

This episode of Babbage explores the profound transformation underway in the world of mathematics due to artificial intelligence. Once notoriously bad at arithmetic, large language models (LLMs) have rapidly advanced to solve complex mathematical problems, changing mathematicians' perspectives. Anjani Trevedi, a former mathematician and correspondent for The Economist, highlights that traditional research mathematics focuses on proving fundamental truths through rigorous, aesthetically pleasing proofs, unlike the rote arithmetic taught in school. For centuries, mathematicians have relied on logical deduction, often building upon prior work, creating a body of knowledge that, while robust, is also widely dispersed and can be difficult to navigate. Computers have long been used for large-scale calculations and checking complex configurations, but modern AI represents a new frontier. Initially, mathematicians were skeptical of AI's capabilities given its struggle with basic math. However, the landscape shifted dramatically in the past year, with LLMs solving long-standing puzzles like some of the Erdős problems and now tackling more complex proofs. Terence Tao, a Fields Medal-winning mathematician, describes AI as a "really bright undergraduate" that, unlike humans, can tirelessly apply hundreds of methods to thousands of problems. While AI often solves the "low-hanging fruit" – problems that are relatively easy but numerous and haven't received human attention – its ability to combine methods from disparate sub-fields of mathematics, a feat rarely achieved by human specialists, offers significant potential for discovering new connections and insights. A major bottleneck in mathematics is the need for rigorous verification of proofs. AI models, though powerful, still make mistakes, with Tao estimating an error rate of 98%. This necessitates a robust system to ensure trust. The emerging solution is "auto-formalization," a process that automatically translates natural language mathematical proofs into computer code that can be formally verified for correctness. Patrick Shafto of DARPA describes the current body of mathematical knowledge as a "hot mess" of papers and ideas. Auto-formalization aims to create a unified, consistent, and machine-verifiable repository, which would accelerate progress, reduce redundant efforts, and make advanced mathematics more accessible. A landmark achievement in this space is the formalization of Marina Vyazovska's Fields Medal-winning proof for the 8-dimensional sphere packing problem. An AI system called Gauss by Math Inc. successfully converted her proof, which took her years to develop, into computer-checkable code. Vyazovska notes that while the AI's output can be verbose (e.g., explicitly proving "2+2=4"), this process provides a deeper, formalized understanding of the complex methods used. The vision for AI in mathematics includes the creation of "mathematical superintelligence" by companies like Harmonic, or GitHub-style platforms (like Google's Defined) where researchers can publish, download, and instantly verify proofs. Mathematicians view AI as a powerful tool to enhance their productivity, explore more ideas, and connect dots across different mathematical domains, ultimately broadening their intellectual aperture. However, challenges remain. AI models can "hallucinate" or agree too readily with human prompts, potentially undermining originality. There are also critical issues concerning credit and attribution for human-generated work that AI systems ingest and then formalize or build upon. Vyazovska's experience with Math Inc. highlights the need for clear ethical guidelines as AI increasingly engages with mathematicians' work. Ultimately, the consensus suggests that AI's role will be to handle the "grunt work" and mechanics of mathematical proof and formalization, creating a highly organized and verifiable knowledge base. This, in turn, could free human mathematicians to focus on the truly creative, intuitive, and conceptual aspects of mathematics, pushing the boundaries of inquiry and potentially leading to breakthroughs in other scientific fields. While AI may eventually generate novel proofs autonomously, the question of their utility and who will "consume" and interpret them remains a societal challenge.