Core Viewpoint - The article discusses the limitations of large language models (LLMs) in mathematical reasoning, particularly in proving inequalities, and introduces a new framework called IneqMath to evaluate their reasoning capabilities [1][4][28]. Group 1: Challenges in Mathematical Reasoning - Current LLMs often provide seemingly correct answers but lack rigorous reasoning processes, raising questions about their true understanding of logical proofs [1][18]. - Formal systems like Lean and Coq can verify proofs but are complex and not easily scalable for intricate problems [1][4]. Group 2: IneqMath Framework - Researchers from Stanford, Berkeley, and MIT propose breaking down inequality proofs into two informal tasks: Bound Estimation and Relation Prediction, creating a bridge between natural language and formal logic [4][8]. - The IneqMath dataset consists of 1,252 training problems with detailed solutions and 200 test problems annotated by International Mathematical Olympiad gold medalists [8]. Group 3: Evaluation of Reasoning - An AI mathematical judging system was developed to assess the logical soundness of each reasoning step, achieving a high F1 score of 0.93, indicating strong agreement with human evaluations [15][17]. - The judging system includes various evaluators to check for logical gaps, numerical approximations, and computation accuracy [16]. Group 4: Model Performance Insights - Despite high answer accuracy, many models fail to provide logically sound reasoning, with Grok 3 mini showing only 6% of answers having a rigorous process [18][20]. - Larger models do not necessarily improve reasoning rigor, and simply increasing the number of tokens does not lead to significant enhancements in logical clarity [20][23]. Group 5: Effective Strategies for Improvement - Two effective methods identified are self-critique, which improves accuracy by about 5%, and theorem hints, which can enhance accuracy by up to 10% for complex problems [25]. - These findings suggest that improving reasoning in models requires more than just computational power; it involves teaching models to self-reflect and utilize tools effectively [25][28].

白交 一水 发自 凹非寺 量子位 | 公众号 QbitAI 快来围观，陶哲轩当视频博主了。 第一个产出就很炸裂： 人类需要写满一页纸的证明，结果借助AI 33分钟就搞定了？！ 整个过程看起来一气呵成，还是全程 "盲证" 不用过脑子那种。 对于这一操作，网友们惊呆：这具有足够的历史意义。 在没有明显引导、宣传之下，他的订阅数一天时间已经有900+，观看数超两千，目前仍然在高速增长中。 大家赶在爆火之前留言： 今天我们相聚在这里，就是为了见证伟大数学频道的诞生。 具体来看看是如何做到？ 33分钟盲证定理 陶哲轩这次选取了泛代数中的一个命题，即 证明Magma方程E1689蕴含E2 。 方程具体是什么不重要，我们只需要了解，即使是方程理论项目的合作者Bruno Le Floch，也足足人工花了一页纸才完成证明。 而用上AI后，整个证明过程仅用时 33分钟 ： 具体而言，陶哲轩尝试完全基于Bruno Le Floch的草稿，逐行进行形式化。 他将草稿拆分为微小逻辑单元，交由 GitHub Copilot生成代码骨架，再以Lean的canonical策略匹配填补细节 ，过程中也涉及部分手动补 全。 最终，整个形式化证明 ...

Group 1 - The book "The Story of Proof: From Pythagorean Theorem to Modern Mathematics" discusses the evolution of mathematical proofs and their significance across various branches of mathematics, featuring notable mathematicians like Euclid and Gödel [2] - "Extraordinary Numbers: The Truth of the Universe in 9 Magical Numbers" explores how specific numbers play crucial roles in understanding concepts like black holes and quantum mechanics, as presented by physicist Antonio Padilla [3][4] - "DK Timeline of the History of Science" presents a chronological overview of scientific advancements over 3 million years, highlighting 1,400 significant moments in science [6] Group 2 - "The Primacy of Doubt" by Tim Palmer examines the inherent uncertainties in various fields, including quantum physics and climate change, emphasizing the importance of a skeptical mindset in decision-making [8] - "Earthrise: Humanity's First Complete View of Earth" reflects on the new perspectives gained by humanity after seeing Earth from space, emphasizing the interconnectedness of life on Earth [11] - "The Astronomer's Chair: Science, Design, and Visual Culture in the 19th Century" analyzes the significance of specially designed chairs for astronomers, linking them to broader themes of scientific practice and cultural identity [13] Group 3 - "Reading Tang Poetry from a Physical Perspective" interprets classic Tang poetry through the lens of physics, illustrating the relationship between physical principles and poetic expression [15] - "Asteroid Hunter: The Complete Record of the OSIRIS-REx Mission" details the mission to collect samples from the asteroid Bennu, which holds clues to the origins of life and potential threats to Earth [17] - "A Natural History of Human Civilization" by Mark Bertness argues for an integrated view of human history and natural history, tracing the evolution of civilization alongside ecological changes [19] Group 4 - "The World Mineral Atlas" serves as a comprehensive guide to over 500 minerals, detailing their properties and origins, supported by high-quality images [23] - "Birds: A Report on the Dramatic Lives of Avian Actors" categorizes 59 bird species with humorous descriptions, providing insights into their behaviors and characteristics [25][26] - "Animal Architecture" explores the relationship between human architecture and animal building practices, suggesting a collaborative approach to design that considers ecological impacts [28] Group 5 - "Snow Leopard Family: The Zhuoma Dynasty" documents the life of a wild snow leopard family over five years, showcasing their social behaviors and survival strategies [30] - "The Story of Leaves" examines the science and history of leaves, discussing their growth, diversity, and ecological significance [32] - "How to Read a Tree: Exploring the Life Secrets of Trees" reveals the hidden messages in trees, offering insights into their growth and environmental indicators [34] Group 6 - "Going into Nature: The Emotional Connection Between Humans and Nature" discusses the deep ties between human well-being and the natural world, supported by scientific research [36][37] - "Rebuilding the World: Adventures of Engineers" narrates the stories behind significant engineering feats, highlighting the challenges and innovations in the field [40] - "A Brief History of the Mind" explores the evolution of consciousness and intelligence, addressing philosophical questions about human existence and the impact of artificial intelligence [42]