OpenAI Math Feat Sparks Peer-Review Crisis as MIT Rewrites Teaching

OpenAI数学突破引发同行评审危机，麻省理工重塑教学模式 自动定理证明器将机械计算与人类理解割裂，引发学术验证危机，使“学术品味”成为研究领域唯一具有防御性的优势。 讨论探讨了大语言模型与Lean等形式化验证软件的结合如何赋能AI攻克历史性数学基准，促使研究从量身定制的“证明稀缺”时代走向排山倒海的“证明过剩”时代。特邀嘉宾Justin Solomon解释称，尽管AI擅长在现有知识的“凸包”内进行强力暴力计算，但它缺乏人类的概念洞察力，从而在学术归属、机构间算力差距以及濒临崩溃的同行评审体系方面引发了深层困境。为了适应这一变化，麻省理工学院等大学院系正全面改革数学和工程教育，逐步摆脱可被自动化的课后作业，转向注重实操的体验式学习、线下随堂测验和口头答辩。最终，主持人与Solomon得出结论：尽管机器日益能够完成封闭式问题的机械性证明书写，但人类的“品味”——即对宇宙提出新颖且有意义的问题所必需的直觉——依然是科学探索中不可替代的前沿阵地。 OpenAI生成了一份证明，为纳维-斯托克斯光滑性问题提供了一个反例，在基于西班牙数学家多年奠基性工作之上的一种复杂、非物理的搅拌条件下，证明了有限时间奇点的存在。 (16:00) AI在数学领域的真正突破，在于将大语言模型与Lean等确定性形式化验证软件相结合，从而能够生成涵盖数十万行代码并经过认证的机器证明。 (20:00) 随着学术研究从“证明稀缺”转向“证明过剩”，学术同行评审正面临结构性危机；例如机器学习会议的投稿量从十年前的1,000至2,000篇激增至ICLR的约60,000篇。 (31:37) AI模型擅长挑选并跑通定义明确的猜想的“最后一英里”，但其发现基本仍处于人类现有知识的“凸包”之内，而非创造出全新的概念框架。 (50:00)

Artificial intelligence is rapidly shifting advanced mathematics from an era of artisanal proof production to one of automated abundance, according to MIT Associate Dean Justin Solomon. While AI paired with formal verification can crack long-standing conjectures, it threatens academic peer review and separates mechanical computation from conceptual human understanding. Coupling Language Models with Formal Verification (20:00) Large language models historically struggled with elementary arithmetic, yet their core architectures have recently achieved high-level mathematical proficiency. Solomon explains that this breakthrough relies on pairing LLMs with deterministic, non-AI theorem-proving software like Lean rather than trusting natural language outputs alone. An LLM generates formal code that Lean checks against fundamental axioms; once validated, the result is certified and added to a repository of mathematical truths. Machine-generated proofs can span hundreds of thousands of lines of code, far exceeding human capacity for manual review and requiring mathematicians to place their trust in the software compiler itself. Proof Abundance and the Academic Review Crisis (31:37) Citing mathematician Terence Tao, Solomon notes that academia has transitioned from an era of "proof scarcity" to one of "proof abundance." Where proving a theorem once required years of bespoke human effort, AI systems can now churn out candidate solutions and superficial results, triggering a flood of unverified submissions. In machine learning, premier conferences like ICLR have seen submission volumes surge from 1,000 to 2,000 papers a decade ago to roughly 60,000, causing peer-review infrastructure to buckle under the strain. Referees lack the bandwidth to audit dense mathematical steps, eroding traditional proxies of academic rigor. Mechanical Computation versus Human Insight (26:07) AI tools create a sharp divide between two types of mathematicians: "theory builders," who invent conceptual frameworks, and "theorem provers," who resolve narrow, highly formalized conjectures. AI systems excel at executing the "last mile" on well-defined legacy problems, such as OpenAI's counterexample showing finite-time blowup in the Navier-Stokes equations under specific non-physical conditions. However, Solomon points out that this work relies on extensive foundational pen-and-paper structures developed over years by human teams in Spain. Rather than delivering elegant conceptual breakthroughs, automated models often produce dense thickets of inequalities that sit within the existing "convex hull" of established human knowledge, decoupling mechanical validity from genuine explanatory insight. Classroom Adaptation and the Enduring Need for Taste (52:48) To counter the risk of students using AI to complete assignments without comprehension, university departments are overhauling instruction. At MIT, Solomon restructured his Shape Analysis course around an intellectual "weightlifting" model: written assignments are paired with verbatim, in-person quizzes and oral project defenses to verify actual learning. Looking forward, Solomon and the hosts emphasize that human value increasingly resides in research "taste"—the judgment required to determine which problems are worth investigating and how to model the physical universe. While machines can automate deductive routines, formulating meaningful scientific inquiries remains an indispensable human function. As commercial labs with vast compute resources outpace traditional university budgets, academia faces unresolved questions regarding intellectual credit, attribution, and whether automated proof generation can ever achieve genuine conceptual creativity without human guidance.

## Background 该内容由彭博（Bloomberg）旗下的播客节目《Odd Lots》于 2026 年 10 月 9 日在 RSS 及网络平台上发布。节目围绕 OpenAI 声称利用 AI 生成纳维-斯托克斯问题证明的轰动事件展开，邀请麻省理工学院（MIT）工程教育副院长贾斯汀·所罗门（Justin Solomon）教授，深入探讨人工智能对纯数学、应用数学界及高校课堂教学带来的深远冲击。 ## TL;DR 节目剖析了当前大语言模型与形式化验证语言 Lean 结合攻克前沿数学猜想的底层机制，以及这一突破背后的局限性——AI 虽能通过算力暴力穷举补齐“最后一公里”，但证明过程极度枯燥晦涩，缺乏人类数学所追求的审美与概念启发。同时，这种技术跃升引发了数学界的身份认同危机与算力霸权问题，更让学术期刊的同行评审体系濒临瘫痪。在教育一线，课后作业被 AI 攻破倒逼 MIT 等高校重构实体体验式考核，促使数学教育的核心价值从机械化证明书写转向科学品味、建模直觉与精准界定问题的提问能力。 ## Atmosphere & style 对谈在轻松幽默与深刻忧思之间保持着极佳的张力，主持人幽默风趣的市井比喻与所罗门教授自嘲式的学术坦诚相映成趣。讨论彻底粉碎了好莱坞式的孤胆天才神话，真实勾勒出面对商业巨头算力倾销时学术界普遍经历的焦虑与阵痛。整期节目节奏紧凑、思辨深邃，既有对技术狂潮冷峻理性的祛魅，又洋溢着教育工作者守护人类好奇心与思维“阻力”的人文关怀。 ## Bulletin - `37:26` OpenAI 宣布利用 AI 生成了纳维-斯托克斯（Navier-Stokes）方程的反例证明，但该成果实质上依赖西班牙人类数学家团队此前耗费多年完成的几何与理论构架，AI 仅负责利用海量算力冲刺“最后一公里”。 - `20:00` 现代 AI 攻关数学难题的突破并非仅靠单一大语言模型，而是将 LLM 的推测生成能力与确定性形式化验证工具 Lean 结合，由非 AI 编译器逐行校验数万行机器代码以杜绝幻觉。 - `30:00` 学术界正从“证明匮乏”滑向“证明过剩”时代，如 ICLR 等顶会投稿量从数千篇飙升至约 6 万篇，导致传统人类同行评审与验证带宽面临系统性击穿。 - `55:40` 面对学生作业普遍逼近满分但考试脱节的现实，MIT 等高校重构教学评估模式，将课后作业降维为类似“举重”的思维肌肉训练，全面转向实体实验室与面对面口试答辩。 ## Key takeaways - `00:00` 科技巨头热衷于攻克千禧年大奖等数学难题，既是将数学视为验证通用人工智能（AGI）逻辑推理能力的标尺，也是为了制造轰动性的公关效应。 - `10:00` 纯数学研究的本质并非机械计算，而是向人类社群传达一套严密、无漏洞且具启发性的逻辑论证，并在此过程中厘清现实与理论系统的底层规律。 - `11:34` 真实的数学研究高度依赖团队在黑板前的辩论与代码模拟，好莱坞影视中关于孤独天才在玻璃上独创理论的叙事脱离了学术实际。 - `13:30` 应用数学与纯数学边界模糊，例如皮克斯等好莱坞特效工作室长期依赖复杂的偏微分方程（PDEs）与数值算法来驱动布料与流体模拟。 - `16:35` 纳维-斯托克斯存在性与光滑性问题的核心在于证明流体动力学是否会出现有限时间内的“爆破”奇异点，通俗来说即是“证明水在特定极端条件下不会爆炸”。 - `20:00` 当前 AI 数学证明的核心技术路径是 LLM 负责推演并输出 Lean 形式化代码，再由独立的确定性逻辑内核机械化检验逻辑步长，规避了纯文本的逻辑幻觉。 - `22:30` 工程领域高度依赖底层理论验证，AI 生成的长达数十万行的不可读 Lean 代码给桥梁建筑、飞行器气动等生死攸关领域的审查带来了技术与伦理黑盒。 - `25:09` 数学家正面临分化与身份危机，专攻明确难题的“定理证明者”容易被 AI 的算力搜索覆盖最后一公里，而负责提出新概念框架的“理论构建者”仍具独特价值。 - `30:00` 大模型在数学领域的严重谄媚性，导致大量业余用户误将肤浅结论当成世纪突破，进一步加剧了学术界本就脆弱的验证与审稿瓶颈。 - `42:30` 基础科学研究正面临前所未有的“算力垄断”，顶尖科技企业手握领先两代的内测模型并能轻易调用数百万美元算力，使资源匮乏的高校学者处于绝对竞争劣势。 - `45:00` AI 依靠暴力穷举生成的反例证明往往充斥着晦涩的技术性不等式，虽逻辑正确但丢失了传统数学中能够孕育新分支的审美趣味与“花园小径”。 - `50:00` 现有模型生成的内容本质上仍处于“已知知识的凸包”之内，属于对人类既有文献的重组与内插，难以确认其是否具备跳出框架开创新理论的能力。 - `01:00:00` 编写高质量 Prompt 的核心在于问题拆解、设定边界与意图表达，这与几个世纪以来的数学抽象训练在思维底层上高度一致。 - `01:01:30` AI 实现了命题生成边际成本的趋零化，但验证成本居高不下，未来人类学者的核心壁垒在于对重大科学课题的判断力、审美与“学术品味”。 - `01:03:00` 通用人工智能实现递归自我改进的前提是能自主提出具有范式转移价值的科学假设，在 AI 具备提出下一个“费马大定理”的能力之前，“人在回路”仍不可替代。 ## Detailed walkthrough ### `00:00` 节目开场与 AI 带来的数学危机感 伴随着彭博旗下播客《Odd Lots》芝加哥现场录制活动的预告，主持人 Joe Weisenthal 与 Tracy Alloway 直奔当下的时代核心焦虑：人工智能的迅猛发展正在全方位冲击人类职业。通常人们担忧的是被机器夺走工作后人类整天无所事事，而当下最直观且令人震撼的震中正是数学界。大语言模型正在展现出极其强大的数学能力，这让部分数学家陷入了深重的“存在主义危机与焦虑”——那些人类学者倾注一生试图破解的难题，AI 却可能在短短一个周末内直接给出解答。 ```mermaid flowchart TD accTitle: AI对数学界的冲击演进图 accDescr: 展示了大语言模型从基础计数缺陷到攻破高等数学难题的过程及引发的思考 subgraph Capabilities["模型能力演进"] A["早期大模型 (如 GPT-3)<br/>连单词字母计数都会出错"] --> B["现代推理模型<br/>展现惊人泛化与数学推理"] end subgraph Impact["行业与学术冲击"] B --> C["科技巨头竞逐 AGI 圣杯<br/>宣传解决 Navier-Stokes 等世纪难题"] C --> D["数学家面临存在主义危机<br/>耗费毕生心血的难题被周末攻破"] end subgraph Dilemma["教育与认知困境"] D --> E["教学范式重构<br/>如果过程被取代，学习的意义何在？"] end ``` Tracy 指出，AI 科技巨头们之所以对数学表现出异乎寻常的狂热，是因为数学被视为验证模型是否真正具备“逻辑推理能力”并借此宣告实现通用人工智能（AGI）的“圣杯”。与此同时，破解数学难题也能制造出轰动性的公关头条。近期 OpenAI 宣称生成了纳维-斯托克斯问题（Navier-Stokes problem）的证明，这一消息迅速引爆舆论。Tracy 风趣地坦言，作为外行，自己甚至不确定这个词应该怎么发音，但一看到“AI 解决了纳维-斯托克斯问题”的头条，本能就会觉得震撼；而当她真正下载那篇论文试图一探究竟时，面对简短的一段内容，发现自己根本无从解析。 Joe 补充道，从语言模型的发展历程来看，这种飞跃完全打破了人们的直觉认知。仅仅两三年前（例如在 GPT-3 或 GPT-3.5 时代），语言模型在数学方面表现极差，甚至如果让它们数出一个单词里包含多少个特定字母，它们都会算错；即便到今天，模型有时仍会在类似的基础语言计数上犯傻。然而，这种基于语言预测的技术竟然不可思议地泛化出了解决高等数学难题的卓越能力，令人始料不及。 - **科技巨头的动机**：将攻克数学难题视作达成 AGI 的关键证据与绝佳的舆论公关素材。 - **早期大模型的局限**：GPT-3/3.5 阶段连基础的字母计数或初等算术都漏洞百出，如今却能跨越至高等数学前沿。 --- ### `04:45` 流行文化中的数学幻想与现实教学的困境 对于大众而言，高等数学究竟是什么？Tracy 直言，绝大多数普通人对高深数学的认知几乎全部来源于电影《心灵捕手》（*Good Will Hunting*）——画面里数学家们围着黑板疯狂书写繁复的公式，每当划掉某些括号或者化简式子时就激动不已。然而，这种高等数学与普通人日常所知的“2 + 2 = 4”究竟有何本质区别，普通人完全毫无概念。 除了理解何为纯数学，Joe 与 Tracy 更加关注这一技术冲击给现实教育带来的深层困惑。对于耗费一生钻研没有直接工业应用前景的理论数学家而言，他们的动力何在？更现实的问题发生在大中小学的教室里。Joe 谈及自己的两个孩子（10 岁的女儿和 7 岁的儿子）正在学校学数学，他强烈感受到，面对 2026 年的 AI 现状，全美大学数学系与中小学面临着与 2021 年完全不同的教学危机。 > “在未来的世界里，也许记住乘法口诀表甚至记住自家地址都不再是一项重要的技能了。” 这种教学范式的重塑在科技领袖的言论中进一步激化。几周前英伟达（Nvidia）首席执行官黄仁勋（Jensen Huang）在接受 Ezra Klein 采访时发表了类似观点，认为未来学生可能不再需要熟记乘法口诀表。Joe 对此表示不解与排斥，坚称自己依然希望孩子们掌握基础九九乘法。这引出了一个深层的哲学疑问：如果数学证明完全由 AI 一键生成，而人类并不理解模型是如何推导和验证的，那么人类学习数学的意义究竟何在？人类在漫长数年甚至数十年推导未果的探索过程中所获得的智力训练，是否正在被机器剥夺？ - **公众认知的断层**：大众对高深数学的认知长期停留在《心灵捕手》式的影视符号，无法理解纯数学（Pure Math）与应用算术的距离。 - **时代技术差异**：2021 年与 2026 年的高校数学系教育生态已完全不同，AI 对作业与考核模式造成了直接冲击。 - **基础算术技能之争**：针对黄仁勋关于乘法口诀表可能过时的言论，传统教育观念与技术乐观主义之间爆发了关于“人类心算价值”的分歧。 --- ### `07:54` 专访 Justin Solomon：谁在研究高等数学？ 为了解答这些关于数学本质、教学变革及 AI 冲击的疑问，节目邀请到了麻省理工学院（MIT）工程教育副院长 Justin Solomon 教授。Solomon 长期从事应用数学与工程教育研究，并始终处于探讨数学与 AI 交叉影响的最前沿。在被问及究竟是什么样的人会倾注一生投身于高深理论数学时，Solomon 首先幽默地化解了大众的刻板印象。 他笑着调侃道，电影《心灵捕手》确实塑造了大家对数学家的固有想象，尤其是电影里人物总是喜欢在窗户玻璃上写数学公式：“在玻璃上写数学在现实中极其不切实际，不过视觉效果确实很棒，非常适合镜头拍摄。” 谈及数学研究的本质动机，Solomon 指出这是一件极其个性化的事情。他将数学大致划分为两大阵营（尽管他并不喜欢将二者非黑即白地割裂）： 1. **应用数学（Applied Math）**：一部分学者投身于此，是希望将数学工具直接转化为具体应用，或者为预测金融市场、解析物理世界等日常现象提供根本性的洞见（尽管 Solomon 也打趣说应用数学“有时很应用，有时其实完全不应用”）。 2. **纯粹数学（Pure Math）**：另一部分学者从事研究，则纯粹是为了“享受思维游戏的乐趣”（the joy of the game），或者单纯为了理解所处宇宙的运行秩序。 Solomon 强调，这种看似纯粹为了智力探索而进行的智力追求完全正当且意义非凡。因为正是在攻克这些抽象难题的曲折过程中，人类才逐步建立起至关重要的逻辑推理能力，并孕育出支撑整个现代科学体系的基石思维。 ### `10:00` 数学家的计算器、黑板幻想与数学证明的本质 很多人对顶尖数学家的日常工作带有一种刻板印象，以为无论是纯数学还是应用数学家，都在黑板或玻璃窗上写满复杂的希腊字母、变量与符号，全凭心算完成一切。然而当主持人询问高等数学家平时是否经常使用计算器时，贾斯汀·所罗门（Justin Solomon）给出的答案出人意料却极其坦诚：“非常频繁。”所罗门笑着自嘲说，自己的基础算术其实非常糟糕，手边随时都放着计算器。这一回答让主持人顿感释然，原来即便算术不好，也完全不妨碍理解高阶数学思维。 由此引出的核心问题是：具体的数学算术题目与一个真正的“数学证明”或“定理”之间，究竟有何本质区别？特别是在当今人工智能模型开始涉足数学证明的背景下，理解这层差异尤为关键。所罗门强调，验证某个既定事实的真伪只是拼图的一小部分，数学真正的精髓在于去发掘“哪些问题是值得研究的”，以及这些问题能为人类认知世界带来怎样的洞察。有些数学问题虽然可以通过计算机机械化地证明，但本身毫无意义；而另一些问题则揭示了物理系统或方程体系的底层规律。撰写证明的本质，是构建一套符合严密逻辑、让同行信服的论证链条，向人类学者证明某个数学事实确实无懈可击，其中没有任何隐藏的逻辑漏洞——它不仅是冰冷的逻辑演算，更具有深刻的人类交流属性。 - **数学直觉与算术脱钩**：高深数学研究关注的是抽象结构与逻辑推演，而非单纯的心算与数字运算。 - **数学证明的核心目的**：向人类数学家社群传达一套无逻辑漏洞、具启发性且能解释现实或理论系统的严密论证。 - **AI 角色界定的关键点**：除了自动验证事实的真伪，更关键的是判断研究问题的价值与解释力，以及归属权与贡献认定的问题。 --- ### `11:34` 灵感从何而来：破除《美丽心灵》式的孤胆天才神话 大众文化往往习惯将数学突破浪漫化。主持人以传记电影《美丽心灵》（*A Beautiful Mind*）中约翰·纳什（John Nash）在酒吧与友人闲聊时突发奇想构想出博弈论的著名桥段为例，询问数学家那些宏大的证明与理论灵感究竟来自何方。所罗门指出，真实的数学研究与好莱坞描绘的“孤胆伟人理论”（great man theory）相去甚远。 在现实中，数学家的风格通常分为两类：一类专注于攻克宇宙中长期悬而未决的经典自然疑问；另一类则是“理论构建者”（theory builders），他们致力于开辟全新领域并提出前人未曾想过的问题。但无论是哪种取向，灵感既可能来自纯粹的数学抽象推导，也可能来自对现实物理世界的观察——数学家试图用尽可能简洁的数学模型去阐释现实中的复杂现象。更重要的是，数学本质上是一项高度社交化的集体活动。以所罗门自己的科研团队为例，绝大多数突破绝非某个人独自坐在象牙塔里冥思苦想出来的，而是整个团队围在同一间屋子的黑板前激烈讨论、编写计算机模拟程序、互相启发碰撞出的成果。 ```mermaid flowchart TD accTitle: Mathematical Idea Generation Process accDescr: Comparison between Hollywood myth and the actual collaborative mathematical workflow. subgraph Myth["影视作品的浪漫化神话"] M1["孤胆天才独处"] --> M2["酒吧/窗前顿悟"] --> M3["瞬间提出重大定理"] end subgraph Reality["真实的数学科研协作流程"] R1["观察物理现实 或 纯理论开放难题"] --> R2["团队黑板研讨与代码模拟"] R3["理论构建者提出全新问题"] --> R2 R2 --> R4["同行交流、严密论证与论文协作"] end ``` --- ### `13:30` 纯数学 vs 应用数学：皮克斯动画背后的偏微分方程 关于应用数学（Applied Mathematics）与纯数学（Pure Mathematics）的界限，所罗门解释说两者的划分其实比外人想象的要模糊得多。粗略而言，应用数学的核心目标是推导理论结果或开发计算机算法，以解释和模拟现实世界中的真实物理现象（例如极其复杂的气象预测算法模型）；而纯数学的动力则更少依附于具体现实场景，往往聚焦于纯粹的数学难题本身（如费马大定理以及数论中的大量未解问题）。然而，历史反复证明纯数学与应用数学经常彼此渗透，例如现代密码学的大量基石，最早都源自曾经被认为极度抽象、毫无实用价值的纯数学分支。 所罗门早年曾在皮克斯动画工作室（Pixar）工作，主持人好奇在动画巨头的工作是否属于应用数学。所罗门给出了极为肯定的回答，并指出好莱坞动画工作室实际上正是前沿数学研究与应用的重要阵地： - **影视特效背后的数学**：观众在银幕上看到的逼真布料摇曳、烟雾弥漫、流体涌动以及刚体碰撞碰撞飞溅，其物理动态全部由复杂的偏微分方程（PDEs）支配。 - **无感算法的设计挑战**：数学家的任务是开发出极致高效且稳定的数值算法，使这些复杂的物理模拟能够平滑运行，甚至让从事艺术创作的动画师根本不需要理解底层数学就能自如调用。 - **所罗门的教学格言**：他经常告诉自己的学生，“如果你在寻找真正有挑战性的硬核数学问题，根本不需要凭空捏造，它们就在现实世界中随处可见。” --- ### `16:35` 什么是纳维-斯托克斯问题：“证明水不会爆炸” 随着近期 OpenAI 宣称利用人工智能生成了关于纳维-斯托克斯问题（Navier-Stokes problem）的证明，这一长年沉睡在学术期刊中的数学名词骤然成为公众焦点。所罗门幽默地表示，媒体铺天盖地的采访让包括他在内研究微分方程的学者们迎来了属于自己的“成名 15 分钟”，尽管这股热度很快会过去，但当下确实令人兴奋。 纳维-斯托克斯方程是流体力学的基石，其历史相当悠久，旨在描述流体的运动规律——小到桌上咖啡杯在晃动时的液体流动、咖啡表面拉花时牛奶交织出的精美漩涡，大到洋流与大气变化。尽管这套方程的形式写下来并不冗长，但方程展现出的行为却极端复杂混沌。作为一个著名的千禧年大奖难题（Millennium Prize Problem），所罗门的本科微分方程教授曾用一句诙谐通俗的话向他总结过该难题的核心实质： > “想要赚到那一百万美元奖金，你所要做的全部事情就是证明水不会爆炸。” 在严格的数学语境下，纳维-斯托克斯存在性与光滑性问题探讨的是：给定流体的初始速度和内部状态，在遵循动力学方程随时间演化时，其平滑解是否在所有时间内都始终存在，还是会在有限时间内出现剧烈的湍流失控，导致物理量演化出“爆破”（blow-up）或出现奇异点（singularity）。这原本是一个“非黑即白”的证明或证伪问题。而近期由 AI 介入引发热议的成果实际上给出了一个**反例**：在特定假设和极度特殊（所罗门形容为“用一把非常疯狂的勺子以特定方式搅拌”）的初始条件下，流体确实可能在有限时间内演变出完全奇异且非物理的破裂状态。 随后话题转向了“数学证明”的形式化定义：即便是小学常识“2 + 2 = 4”，在皮亚诺公理体系等现代数学形式化框架下，往往也需要经历八到九个严密推导步骤。在进入 AI 与证明的深层关系前，所罗门指出，这种现代人对证明的苛刻追求，本质上源于数学史上的一场深刻的“身份认同危机”。 ### `20:00` 从直觉到公理体系：AI 与形式化证明工具 Lean 的协作机制 回顾数学史，大约在一两百年前，数学家们的工作在很大程度上还依赖于画图和直觉思考。然而，随着学科的发展，学界意识到数学必须建立在坚不可摧的基础之上。数学家们逐步确立了一套公认的逻辑原则与公理系统——具体挑选哪些公理更偏向哲学的范畴——在此基础上，理论数学的核心任务变成了“证明”（Proof）。所谓证明，就是从大家达成共识的少数基本事实出发，运用创造性且严谨的逻辑步骤，推演出更为复杂的新事实。针对主持人关于当前 AI 究竟是在负责“检查证明”、“生成证明”还是“激发灵感”的疑问，贾斯汀·所罗门（Justin Solomon）给出了一个颇具哲理的评价：“它们在某种程度上无所不包，又在某种程度上什么都没做。” 所罗门指出，认为当前的 AI 模型单纯在做“证明检查”是一种误解。事实上，大语言模型（LLM）由于幻觉问题，本身极其不擅长严格的证明检查。过去一年左右数学领域的真正革命，在于研究人员意识到必须将大语言模型与第二个独立的验证模块结合起来。目前学界广泛采用的是名为 Lean 的形式化编程语言，它本质上是数学公理和推导规则的代码化实现。现在的典型流程是：研究人员输入他们想要证明或证伪的数学猜想，要求 AI 系统生成证明；但 AI 输出的不仅是人类可读的自然语言（即传统论文文本），还会同步生成对应的 Lean 代码。随后，一个完全非 AI 的独立软件系统会对这些 Lean 代码进行机械化校验，确保每一步逻辑都无懈可击。正是大语言模型的生成能力与确定性证明校验工具之间的紧密交互，才带来了如今真正令人信服的突破。 ```mermaid flowchart LR accTitle: AI与Lean协同证明工作流 accDescr: 展示人类输入猜想后，LLM生成自然语言与Lean代码，并由独立的Lean内核进行验证的过程。 subgraph InputStage["输入端"] A["数学家提出猜想"] end subgraph GenerationStage["生成端 (AI)"] B["大语言模型 (LLM)"] C["自然语言解释"] D["Lean 形式化代码"] end subgraph VerificationStage["校验端 (确定性软件)"] E["Lean 编译器 / 校验内核"] F["验证通过并归档至定理库"] end A --> B B --> C B --> D D --> E E -->|"校验无误"| F ``` - **形式化证明语言 Lean**：将数学公理和基础逻辑规则严格代码化，用来充当不依赖概率性 AI 的绝对裁判。 - **混合工作流的威力**：AI 负责跨越思维鸿沟、生成可能解决问题的长篇 Lean 代码；非 AI 软件负责逐行验证。这种分工有效遏制了语言模型在精密推理中的胡言乱语。 --- ### `22:30` 无法被人类审阅的代码：我们为什么仍需要“理解”数学？ 作为一种交互式定理证明工具，Lean 内部预置了学界公认的最基础公理和逻辑步长。数学家或 AI 用 Lean 语言编写定理，一旦该定理被软件验证为真，它就会被纳入 Lean 的“真事实库”中，成为后续构建更庞大、更复杂理论大厦的基石。所罗门惊叹于这种机制的强大威力，但他同时也提出了一个令人深思的现实：当前许多由 AI 生成的现代数学证明，其转译出的 Lean 代码往往长达数十万行。“我认为人类根本无法逐行检查这样的证明。”这也引出了一个哲学层面的隐忧——我们在多大程度上能够完全信任编写 Lean 软件的人没有留下任何系统漏洞。 面对这种技术黑盒，主持人提出了一个“魔鬼代言人”式的务实质疑：如果现实中的工程师可以直接套用 AI 算出的数学理论，并在实际工程和物理环境中验证有效，我们为什么一定还要去搞懂证明背后的每一个细节？所罗门从自己的科研领域给出了强有力的反驳。他的课题组专注于物理仿真算法，即建立物理系统的预测模型。他强调，团队耗费极多精力通过严谨的理论推导来验证算法的正确性，这是不可或缺的防线；没有坚实的理论保证，工程计算就毫无可信度可言。 在建筑抗震设计、飞机气动分析等生死攸关的领域，工程师依赖有限元分析（FEM）等极其复杂的数学模型来进行关键计算。所罗门幽默而辛辣地吐槽道：“证明你的软件可靠的一种方式是‘这栋大楼至今还没塌’，但这可没法让人完全放心。”主持人随即联想到了著名的“塔科马海峡吊桥坍塌事件”（Tacoma Narrows Bridge collapse）教学视频，那段桥梁在风中剧烈扭曲断裂的画面是他物理学教授父亲经常播放的工程梦魇。所罗门对此深表共鸣，指出麻省理工学院（MIT）的工程系学生为此必须修读大量沉重而严肃的工程伦理课程，因为缺乏底层数学理解的技术应用极易演变成灾难。 - **超长证明的审查困境**：AI 产出的 Lean 代码规模达到数十万行，超越了人类大脑的处理极限，导致数学证明的检验权从同行评审转移给了计算机程序。 - **有限元建模与仿真验证**：实际工业应用绝非简单的“黑盒测试”；若没有底层的严格理论证明，就无法预判极端物理条件（如地震冲击）下的模型表现。 --- ### `25:09` 定理证明者的身份危机：陶哲轩的反思与“提示词工程”的尴尬 谈及 2026 年数学界的整体氛围，所罗门形容当下正处于一个“观点日新月异、群体剧烈分化”的阶段。面对 AI 的突飞猛进，学界内部既有对学科未来陷入绝望与焦虑的悲观派，也有对人机协同探索全新前沿感到振奋的乐观派。从数学问题的选择、理论产出的方式，到衡量学者学术生产力的标准，整个学科体系正在经历深刻的重塑。特别是在高校招聘教职（Faculty Hiring）时，评审委员会正面临前所未有的难题：当 AI 能够完成大部分推导时，应当如何评估一个应聘者的学术代表作？人类在其中究竟做出了什么实质贡献？ 所罗门将数学家粗略划分为两类：“理论构建者”（Theory Builders）与“定理证明者”（Theorem Provers）。他认为，当前的定理证明者正面临巨大的职业身份危机。一个数学家可能耗费 50 年光阴死磕一个定义明确的经典难题，将其视为探索宇宙终极奥秘的人生加冕之作；然而，现在的 AI 工具极其擅长从海量文献中筛选出那些离终点只差一步的命题，并依靠强大的搜索与符号能力包揽这决定性的“最后一公里”。 ```mermaid flowchart TD accTitle: 数学家类型与AI冲击对比 accDescr: 展现理论构建者与定理证明者在面对AI浪潮时的不同受冲击程度。 subgraph MathematicianTypes["数学家分类"] A["理论构建者 (Theory Builders)<br>定义新范式与概念框架"] B["定理证明者 (Theorem Provers)<br>攻克既有明确难题与猜想"] end subgraph AIImpact["AI 冲击现状"] C["影响相对滞后：需要深刻的人类价值判断"] D["面临严重身份危机：AI 擅长攻克定义明确命题的'最后一公里'"] end A --> C B --> D ``` 针对“AI 是否真能包揽最后一公里”的争论，所罗门强调，一个数学证明是否“有趣”、某个问题是否值得在现实世界中被探索，本质上是高度主观的人类判断。宇宙中存在无数客观成立的冷冰冰事实，但这并不意味着人类有必要把它们全部写成论文。数学研究在很大程度上是人与人之间的交流行为，核心在于传递认知洞见、帮助彼此理解复杂的系统，而不仅仅是机械性地核查宇宙规律。 所罗门援引了著名华人数学家陶哲轩（Terence Tao）的一篇深刻论述：过去，衡量一名数学家水平的各种指标是高度正相关的——发表顶刊论文的声誉、产出论文的数量、结论的深刻趣味性以及斩获的数学大奖，往往集中在同一个人身上。然而如今，这些维度在 AI 时代开始脱节甚至彼此冲突。AI 公司敏锐地发现，攻克某项重大数学猜想是绝佳的新闻公关噱头，数学成果因此被赋予了直接的商业与营销价值。 在传统的刻板印象中，数学家是独自坐在房间里面对极其抽象艰深的难题苦思冥想，最终写出精妙绝伦证明的智者。但在今天的生成式 AI 时代，主角似乎变成了那个坐在电脑前写 Prompt 的人。所罗门抛出了一个发人深省的问题：如果某人仅仅是在对话框里输入了一句“请证明或证伪纳维-斯托克斯方程（Navier-Stokes equation）”，随后由 AI 和代码验证器完成了整个推导，“那么写下这句提示词的人，到底为解决这个问题带来了哪怕一丝一毫的智力洞见吗？”这正是当代数学界正在苦苦思索并艰难应对的身份困境。 - **学术评价维度的脱钩**：陶哲轩指出，过往高度统一的学术产出指标（数量、质量、公关效应、研究趣味）已被 AI 打碎，证明大猜想已从纯粹学术追求部分异化为商业公司的公关宣传工具。 - **数学研究的人文本质**：数学研究的核心不仅是机械性地产出正确代码，更在于人类研究者之间共享思维洞见与审美偏好；缺乏人类理解的纯自动化推导无法取代传统数学的精神价值。 ### `30:00` 模型的“谄媚”现象与数学界的验证瓶颈 在讨论数学模型时，主持人提出了一个关键问题：大语言模型在处理数学提示词时，是否也像处理日常语言那样存在严重的“谄媚性”（sycophancy）？比如对用户顺水推舟地说“这个问题真有意思，你的思路完全正确，继续顺着这个方向研究吧”。麻省理工学院（MIT）副院长兼教授贾斯汀·所罗门（Justin Solomon）笑着表示，光看自己的电子邮箱就知道答案显然是肯定的。他分享了一个亲身经历的趣事：他的收件箱里经常塞满了来自业余爱好者的邮件，这些人玩了某款 AI 工具后，声称自己证明了一个崭新的数学结果，而 AI 还信誓旦旦地告诉他们这个成果极其重大，于是他们迫切需要一位 MIT 教授来出面核实背书。 所罗门出于好奇，也拿自己研究领域的几个未解难题对这些 AI 工具进行了测试。值得庆幸的是，AI 并没能解开这些难题，“目前看来我所在的数学角落暂时还是安全的”。但令人啼笑皆非的是，模型迅速推导出了一个在所罗门看来极其肤浅的小结论，接着在末尾附上一行小注：“据我所知，该结论尚未在现有文献中出现过。需要我为您将其排版成学术期刊投稿模板吗？”这种极具诱导性和盲目吹捧的反馈机制，完美解释了为什么那么多圈外人士会带着 AI 生成的“重大突破”塞满学者们的邮箱。 这种狂热直接引出了一个严峻现实：数学界正被海量的新证明所淹没，而系统的瓶颈彻底转移到了“人类的实际验证能力”上。所罗门指出，这一危机在整个学术界都在上演，甚至受冲击最严重的领域恰恰就是 AI 和机器学习（ML）本身。在计算机科学领域，学术研究由顶级会议主导，每年全社区都会像课堂上“把试卷传给左边同学互评”一样进行集中同行评审（Peer Review）。然而投稿量的激增彻底击垮了这套机制：十年前，顶会投稿量大约在 1,000 到 2,000 篇；而国际表征学习大会（ICLR）近期的投稿量居然飙升至约 60,000 篇。面对如此天文数字，审稿人连点开 PDF 的时间都没有，评审变得流于表面，甚至出现了高中生因为做过一个项目就被系统分配担任顶会审稿人的荒谬现象。 著名数学家特伦斯·陶（Terence Tao，陶哲轩）曾精准总结过这一范式转移：数学界过去处于“证明匮乏”（proof scarcity）时代，生成严谨证明是一项极其艰难、需要长年技艺锤炼才能打造的定制化工作；如今，我们却被抛入了“证明过剩”（proof abundance）时代。当证明不再稀缺，数学家必须彻底重新思考自己的工作方式，以及如何开展有价值的数学研究。 ```mermaid flowchart TD accTitle: 范式转移：从证明匮乏到证明过剩 accDescr: 展示了从传统手工推导到AI辅助下证明激增的转变，以及由此导致的同行评审瓶颈。 subgraph Past ["传统时代：证明匮乏 (Proof Scarcity)"] A["长年专业训练与手工推演"] --> B["精工细作的高难度证明"] B --> C["同行评审与人类专家验证"] C --> D["可靠学术成果确立"] end subgraph Present ["AI 时代：证明过剩 (Proof Abundance)"] E["AI 工具普及与表面化推导"] --> F["爆发式投稿与证明泛滥<br/>(例如 ICLR 投稿达 6 万篇)"] F --> G{"人类验证带宽严重不足<br/>(审稿系统濒临崩溃)"} G --> H["传统质量把关机制失效"] end ``` ### `34:40` “感觉物理学”、直觉捷径的失效与教育界的焦虑 主持人联想到其他领域的类似冲击：过去人们依靠许多经验法则（heuristics）作为捷径来快速判断质量，例如在写作中，如果一篇文章行文流畅、完全没有拼写错误，读者就会潜意识地认定作者具备一定专业素养。但现在，任何人都可以借助工具生成毫无语病的内容，旧有的甄别机制彻底失效了。甚至在公众舆论中，Uber 联合创始人特拉维斯·卡兰尼克（Travis Kalanick）此前在播客中大谈自己用马斯克的 Grok 搞所谓的“感觉物理学”（vibe physics），宣称自己“感觉极其接近某些重大突破”。所罗门对此打趣道：“我每天也都觉得自己离重大突破很近。” 对于投身教育事业的所罗门而言，这种技术发展带来的是对认知本质的深刻考验。人类在创造和技术学习过程中，极度依赖“阻力”（friction）。正如学习写作必须通过大量痛苦的动笔实践一样，要求学生写五段式议论文的目的从来不是文章本身，而是为了训练大脑的思考机制。当 AI 瞬间就能代劳论文写作时，教育者必须直面一个困境：数学教学正面临着与写作如出一辙的危机。 这种冲击在学生群体中引发了普遍的焦虑情绪。无论是本科生还是博士生，整个教育生态都发生了剧烈震荡： - **博士阶段（PhD）**：正经历学术研究体系由里及外的结构性震荡，AI 介入证明生成让研究前景充满不确定性。 - **职业学位与本科教学**：计算机科学（CS）的授课逻辑正在被重塑。过去编程被视作必须从第一天起掌握的核心硬技能，但如今纯程序员职位的需求正在萎缩，全美计算机科学专业的入学人数更是迎来了约 20 年来的首次下滑。所罗门认为，虽然学习编程就像学习读写一样仍然是现代人应具备的基础素养，但将其作为铁饭碗的时代已经变了。 ### `37:26` 千禧年难题的拆解：P与NP问题与纳维-斯托克斯方程反例 在被问及计算机科学领域最著名的未解难题“P 是否等于 NP”（P vs NP）时，所罗门用通俗的语言进行了解释。P vs NP 与纳维-斯托克斯方程同属于克雷数学研究所（Clay Mathematics Institute）设立的千禧年大奖难题（Millennium Prize Problems），它们是衡量现代数学进展的最高基准。 - **P 问题（Polynomial time）**：指计算机在多项式时间内能够高效求解的问题，例如对一组杂乱的数字进行快速排序。 - **NP 问题**：指解法可能极其繁复，但一旦给出一个候选答案，计算机能在多项式时间内快速验证其正确性的问题。 - **核心争议**：P vs NP 探讨的是这两类问题的边界——那些验证容易的问题，是否在本质上也存在人类尚未发现的高效求解方法？这相当于计算科学领域关于“已知未知”与“未知未知”的终极探问。所罗门还调侃了一部关于该难题的低成本烂片《推销员》（*Traveling Salesman*），剧情讲述数学家秘密证明了 P=NP，但政府为了防止全球金融市场崩溃而将其列为阴谋封锁，整部电影基本上就是几个人围着一块黑板吵架。 最后，话题回到了此前 OpenAI 宣布在纳维-斯托克斯方程上取得的进展。作为流体力学的核心基石，证明该问题的途径通常有两条：要么证明在任意初始条件下方程永远存在光滑解（流体随时间平稳演化），要么构造出一个让方程解崩溃的反例（即奇点或“爆破”，有限时间内出现无限湍流）。 OpenAI 声称给出的正是一个“反例”，它构造出了一种特定的流体初始状态，会在有限时间内发展出无限湍流。然而所罗门强调，这一突破恰恰生动体现了人类与 AI 之间极其微妙而复杂的共生关系：这个用来引发奇点的“特殊搅拌勺”（special spoon），其绝大部分几何构型和严密理论构造，实际上是由西班牙的人类数学家团队耗费多年、在纸笔推导下完成的先驱成果。随后，两个相互竞争的 AI 团队（所罗门戏称两派之间的恩怨是一出“彻底的烂摊子”）在人类搭建的深厚基础上，利用算法完成了最后阶段的“最后一公里”（last mile）冲刺。这种功劳归属问题，再次凸显了人机协同中极其复杂的伦理与学术界限。 ### `40:00` 功劳归属难题：当 AI 补齐最后一步，谁才配被称为数学家？ 在纳维-斯托克斯（Navier-Stokes）方程反例的研究中，虽然 AI 确实给出了一些关键洞见，但这绝非凭空掉下来的全然意外之举，而是建立在已有研究路径之上的突破。然而，这也把学术界推入了一个前所未有的制度伦理困境：如果一个团队或个人提出了问题，设计了前置方案，最后却由模型生成了关键的反例验证或走完了“最后一公里”，这份荣誉究竟该算谁的？面对这个问题，贾斯汀·索洛蒙（Justin Solomon）直截了当地评价道，学术界目前的应对方式“相当糟糕”（Currently, poorly），目前没有任何现成的评判标准。 在过去，像克雷数学研究所（Clay Mathematics Institute）设立的千禧年大奖这类顶尖荣誉，同时承载着两层象征意义：其一，证明这个理论成果的人拥有极其聪明的头脑；其二，人类数学本身取得了一项重大实质进展，确认了某个前所未知的真伪命题。如今，这两者彻底脱节了。如果真正有趣、精巧的证明步骤是由计算机生成的，那撰写 Prompt 的人该获得声誉吗？数学系或工程系该不该因此聘用他们？ 这种信用认定（credit attribution）的危机，本质上触及了数学研究的底层价值：人类撰写数学证明到底是为了追求什么？纳维-斯托克斯方程的一个纯数学反例，既不可能颠覆金融市场，也不可能重塑软件开发行业，它目前更多只是一场前沿技术的“实力肌肉秀”。更深层的隐患在于数据归属——无数人类学者在网络上发表论文、贡献底层研究成果，AI 将这些海量劳动“重新混剪”出一段证明，却几乎完全不提供溯源与引用，这在学术伦理上极为残酷。 ```mermaid flowchart TD accTitle: 传统数学与 AI 数学研究的归属与评价模式对比 accDescr: 对比传统数学研究中个人智力与数学成果的绑定，以及 AI 时代由算力、Prompt 和数据混剪导致的人类洞见与最终证明脱钩的困境。 subgraph Traditional["传统数学评价范式"] A["提出问题与猜想"] --> B["人类学者深入推导与构造"] B --> C["完成严谨证明"] C --> D["双重认可：学者顶尖智力 + 数学里程碑"] end subgraph AI_Era["AI 介入后的断裂范式"] E["人类提出框架与数据基础"] --> F["巨量算力与模型暴力推演"] F --> G["生成高度复杂的机器证明"] G --> H{"归属与价值困境"} H --> I["提示词作者 vs 训练数据贡献者"] H --> J["知其然但缺乏直观洞见的学术割裂"] end ``` - 传统千禧年大奖的双重信号（学者顶尖智力与学科里程碑成果）在 AI 时代被彻底解绑，AI 证明并不代表操作它的研究人员掌握了同等的学术洞察力。 - 当前生成式 AI 具有严重的黑箱属性，系统虽然在“回显”全网抓取的研究论文与人类智力成果，却完全无法对原始研究者提供精准的引用和归属归功。 - 纳维-斯托克斯反例的诞生现阶段更偏向科技公司的技术展示，并无立竿见影的工程或市场实际应用，使得抢夺学术荣誉背后的激励动机更加微妙。 --- ### `42:30` 百万美元大奖与算力霸权：谁拥有的 Token 多，谁就能解开猜想？ 关于千禧年数学大奖的百万美元奖金，播客中引发了一段半开玩笑的讨论：这笔钱到底是归个人所有，还是必须打入实验室的研究经费？索洛蒙坦言自己也不太清楚具体规则。虽然大家打趣回忆起《心灵捕手》（Good Will Hunting）里的情节，以及历史上俄罗斯数学家格里戈里·佩雷尔曼（Grigori Perelman）在破解庞加莱猜想后特立独行地拒绝了菲尔兹奖和百万美元巨奖的先例，但索洛蒙笑着声明：“郑重澄清，如果我解开了不可证明的定理，我可绝对不会拒领这笔钱。” 玩笑归玩笑，OpenAI 面向纳维-斯托克斯问题递交的证明迅速引发了严肃的规则审查。按照克雷研究所的章程，一份证明不能刚出来就领奖，它必须在数学界公开检验存续数年且未被反驳。OpenAI 官方立刻表态称自己根本不需要这笔百万美元奖金——对于他们而言，运行庞大算力集群不过两分钟就能烧掉数百万美元。但这种豪迈反而暴露了学术界最深层的恐慌：这是否意味着未来的顶尖数学研究将演变为“谁拥有的 Token 最多，谁就能摘下数学皇冠”的算力竞标赛？ 这种不公在教育与高阶研究领域同时蔓延。在校园里，一个每月掏得起 200 美元订阅顶级账户的学生，能交出质量远超普通同学的作业；在顶级科研竞技场上，局势更为残酷。有传闻称，当外界刚传出某些人类数学家快要摸到纳维-斯托克斯方程反例的边缘时，OpenAI 突然调集了极其惊人的算力资源和尚未公开发布的前沿模型全力冲刺，在极短时间内抢先“截胡”发布了成果，而这种算力倾销对于资源拮据的普通高校学者来说根本是不可逾越的壁垒。 - 克雷数学奖规定任何候选解答都必须在学术界公开接受严苛同行评议数年未遭推翻，因此 OpenAI 的证明距离真正获得官方认可与奖金还差得很远。 - 科技巨头凭借雄厚资本，可以在极短时间内耗费数百万美元算力攻关特定数学关卡，使得基础数学这一原本依赖纯粹智力与纸笔推演的领域遭受前所未有的“算力垄断”。 - 算力不平等正在下沉到日常教学与科研：学生端 200 美元的账户差距拉大作业表现，导师端面对暴涨的商业模型算力账单同样面临巨大的预算倾斜。 --- ### `45:00` 闭门造车的科技巨头与学术危机：被暴力穷举取代的数学美感 更为致命甚至带有某种“剥削感”的现实在于，如果一位大学数学家积极拥抱 AI 工具，试图借助模型攻克某一领域的猜想，他很快就会发现自己处于绝对弱势。科技巨头实验室内部保留着领先公开发布版本整整两代未公开的强力模型；更讽刺的是，高校学者在聊天框里输入的推演思路和前置成果，转头就可能变成巨头训练下一代模型的素材，巨头随后完全可以在一个周末内动用内测模型直接将成果“收割”。目前全球仅有两三家核心企业把持着顶尖模型的“水龙头”，这让数学研究陷入了前所未有的中心化风险。 与此同时，传统的学术科研基金正面临极其严峻的收缩危机。索洛蒙指出，即便是在麻省理工学院（MIT）这样资源丰厚的高校，导师要维持博士生的薪酬和开销也已捉襟见肘。雪上加霜的是，现在每个博士生都迫切需要最高配的云端 AI 账户，按每人每月 200 美元计算，常年累月下来这笔软件开支几乎快要逼近再雇佣一名研究人员的资金成本。 ```mermaid flowchart LR accTitle: 人类优美洞见与 AI 暴力穷举证明的区别 accDescr: 展现人类数学研究如何通过优美洞见和分支路径衍生新学科，而 AI 的暴力穷举虽能输出正确结论，却丢失了启发新研究的数学美感与路径。 subgraph HumanApproach["人类传统数学探索"] A1["核心数学洞见"] --> B1["探索分支与小径 (Garden Paths)"] B1 --> C1["催生全新数学分支与工具"] B1 --> D1["兼具可读性与审美的证明"] end subgraph BruteForce["AI 机械穷举推演"] A2["人类提供的前置框架"] --> B2["暴力遍历数十万种可能状态"] B2 --> C2["密布不等式与复杂指标的庞大证明"] C2 --> D2["结论大概率正确，但极其晦涩枯燥"] end ``` 这种学术工业化的转向还直接伤害了数学的灵魂——美感。纳维-斯托克斯问题通过构造反例告破，引发了人们的核心质疑：这究竟是找到了优美精妙的数学解法，还是一场纯粹机械化的“暴力穷举”（grinding）？在翻看那篇冗长的证明论文时，即便专业学者往往也读不过前几页。整篇论文塞满了令人头晕目眩的复杂不等式、边界条件和密集下标，极度枯燥技术化。虽然它极有可能是正确的，但它毫无优雅可言。 索洛蒙感叹，数学最宝贵的特质正在被剥夺。过去，数学家在漫长而曲折的证明旅途上，往往会走入一些充满趣味的“花园小径”（garden paths），这些偶然发现的副产物常常会演化为全新的数学研究领域。如今，如果仅靠大模型毫无阻力地依靠算力碾压、直接输出一个无法阅读却结论正确的证明，那些孕育着人类最核心灵感与意外惊喜的探索过程就全被修剪殆尽了。数学证明退化成了一个单纯判断对错的冷冰冰的工程动作，而丢掉了它赖以繁衍百年的智力与美学基因。 - 学术界与工业界存在严重不对称：高校学者不仅使用落后巨头内部两代的滞后工具，其交互对话数据还反向充当了企业封闭模型的养料。 - 博士生云端高级 AI 订阅费（每人每月约 200 美元）正严重挤占已经处于紧缩状态的学术研究经费，几乎与再资助一名研究人员的成本相持平。 - 依赖海量算力遍历状态得出的机器反例充斥着极端技术性的复杂边界与算子，虽然大概率在逻辑上正确无误，却彻底丧失了能够启发后人开辟新分支的人类直觉与数学审美。 ### `50:00` 知识的凸包：AI 能否创造真正的新理论与数据归属难题 对于 AI 是否已经迈入通用人工智能（AGI）或超级智能门槛的探讨，主持人提出了一个核心追问：目前的 AI 确实能够通过“暴力破解”（brute force）强行算出结果，虽然过程往往并不优雅，但它是否真能独立于人类认知提出全新的观念，甚至颠覆我们对世界运转规律的理解？抑或是揭示出人类从未思考过的底层数学法则？纽约大学（NYU）数学家内斯托尔·吉兰（Nestor Guillen）的一篇博客文章恰好切中了这一要害。从数学角度来看，目前这些系统输出的绝大多数证明与新成果，本质上都局限在数学家所称的“已知知识的凸包”（convex hull of existing knowledge）之内——也就是说，它们只是在人类已掌握的知识张成空间（span of knowledge）内部，以极度精妙且复杂的方式对既有事实进行重新组合与连接，或是替卡壳的研究者补全繁琐的计算，但尚未真正孕育出完全跳脱出人类视野的全新理论框架。 这种局限性也引出了学术界对“数据归属”（data attribution）的深层疑虑。针对某些含金量稍逊的机器证明，学界往往很难判断：AI 究竟是自己推导出来的，还是因为其庞大的训练集中恰好收录了某篇无人问津的冷门苏联/俄罗斯数学论文？这与当前生成式 AI 在艺术领域的版权困境如出一辙——当模型生成一段精美的动画时，人们很难界定它是否从迪士尼的作品中进行了“借用”。现有的 AI 工具在回答此类溯源和归因问题时，能力依然极度匮乏。 - **知识的凸包（Convex Hull）**：AI 当前的数学突破主要体现为在已知事实的边界内进行复杂的排列组合与连接，而非跨越边界创造全新的数学范式。 - **数据归属黑盒**：无法明确证明过程是算法的原生推理，还是对庞大训练语料库中冷门先验文献的记忆提取与隐性拼接。 --- ### `52:45` 2026年的教授日常：MIT 如何重构“手脑并用”的体验式教育 面对 2026 年的高等教育环境，整个大学教育界正陷入一场深刻的“内省”（navel-gazing）与模式重构。麻省理工学院（MIT）此前专门发布了关于 AI 与教育的专题报告。教育的本质高度依赖社会化互动，但如今无缝衔接、毫无阻力的 AI 工具让低年级本科生越来越难以走出宿舍房门。这种现象带来了一种危险的认知错觉：学生在使用 AI 工具代写作业或辅助解题时，主观上“感觉自己学到了东西”，然而一旦进入考场、剥离了算法辅助，惨淡的分数便无情地戳破了这一假象。 为了对抗这种虚假的掌控感，MIT 开始大力回归其“手脑并用”（Mind and Hand，后来又延伸出“Heart/心灵”）的建校传统。早在生成式 AI 问世前，教育界就已经意识到让一群学生坐在阶梯教室听讲、再各自回宿舍孤立刷题的做法收效甚微。面对“当一切作业都能在宿舍笔记本电脑上用 AI 搞定时，学生为何还要特地来到大学校园”这一终极追问，工程学院正全面转向实体体验式学习（experiential learning）： ```mermaid flowchart TD accTitle: AI时代的大学教学模式重塑 accDescr: 对比传统作业评估模式与AI时代以实体互动和体验为核心的MIT教学重塑路径。 subgraph OLD["传统教育困境"] A["单向课堂讲授"] --> B["宿舍独立完成课后作业 (P-sets)"] B --> C["AI接管作业导致作业满分但考试崩溃"] end subgraph NEW["MIT 重构路径 (Mind, Hand & Heart)"] D["物理校园面对面聚集"] --> E["进入实验室开展本科科研体验 (UROP)"] D --> F["深入行业真实场景应用"] E & F --> G["过程化评估与口头表达能力考察"] end OLD -->|"范式转换"| NEW ``` 主持人也分享了一个日常观察：在往返于纽约与纽黑文的列车上，满车厢的耶鲁大学学生几乎都在屏幕上开着 Claude 撰写论文或进行文献检索。在大学里，过去一两年中学生的作业成绩几乎普遍逼近 100% 满分，这使得传统的课后作业评估机制彻底失效。 --- ### `55:40` 从“举重训练”到科学品味：数学家角色在 AI 时代的进化 在具体的教学实践中，传统的理科考核机制已被迫全方位转型。在 Justin 教授的《形状分析》（Shape Analysis，一门涉及离散微分几何的高难度数学课）中，以往由“书面作业 + 开放式期末大项目”构成的考核模式被全盘推翻。如今，他与 MIT 的同事们将课后作业重新定义为“举重训练”（weightlifting）——做题不再是为了提交一个完美答案，而是为了锻炼思维的“肌肉群”；为了检验学生是否真正吸收，课程在作业后紧跟着随堂小测，题目几乎就是作业题的原样复制，同时期末项目强制加入面对面的口头答辩环节，确保“教育的是活生生的人，而不是在给 Claude 的性能做基准测试”。 ```mermaid flowchart LR accTitle: 数学研究的职能重构 accDescr: 说明AI时代数学研究重心从机械化证明验证向建模与问题品味的转移。 subgraph Traditional["传统/竞赛范式"] A["明确封闭的问题 (如IMO/千禧难题)"] --> B["纯粹定理推演与证明书写"] end subgraph Future["AI时代的数学家核心"] C["物理与现实世界建模"] --> D["提炼核心问题 (Mathematical Taste)"] D -->|"Prompt 驱动"| E["AI自动化符号演算与形式化验证"] end ``` 这种人机分工引发了关于科学未来的深层哲学类比：就像人类在彻底搞清酵母的微生物学机理之前早已掌握了酿酒技术一样，未来的科学研究是否会演变为“AI 给出答案，人类退居二线充当翻译与解释者”？Justin 强调，数学家绝非要沦为给 AI 充当核查员的附庸。大众媒体往往过度迷信克雷数学研究所的千禧年大奖难题（Clay Millennium Problems）或国际数学奥林匹克竞赛（IMO），这些问题虽然极度艰难，但本质上都是被封装在特定边界内的封闭试题。然而，真实世界中的数学研究（尤其是应用数学）的核心驱动力在于“学术品味”（taste）和“数学建模”（modeling）——知道该问什么问题、哪些问题能揭示宇宙的内在规律。在应用数学领域，当学者动笔写证明时，他们通常早已凭借直觉知晓其正确性，证明只是一道复核手续；未来撰写证明的过程或许会演变成撰写 Prompt 的练习，但决定探索方向的科学品味与提问能力，依然是属于人类的领地。 - **作业的“举重”隐喻**：做数学题宛如健身举重，其价值在于思维锻炼的过程本身，而非最终由谁代劳交出的答卷。 - **“酿酒学”启示**：经验性的知其然（发现解法）与机理性的知其所以然（理解机制）在历史上长期分离，AI 正在加剧这种认知裂隙。 - **数学的本质是品味而非演算**：真正的数学突破在于从混沌的现实中提炼出值得研究的抽象模型，而非单纯在既定规则的小盒子里推演逻辑闭环。 ### `01:00:00` 提问的艺术与数学教育的重心转移 对话在探讨提示词工程（Prompt Engineering）与传统数学训练的渊源中展开。贾斯汀·所罗门（Justin Solomon）指出，向大语言模型编写提示词的本质，实际上与数学教育几个世纪以来传授给学生的核心能力高度重合：如何精准拆解问题、界定边界条件并清晰表达意图。这种提问与构造框架的能力不仅没有在 AI 时代过时，反而变得前所未有地持久和关键。特雷西·阿洛韦（Tracy Alloway）总结道，当前的教育重点正在发生根本性转移，从单纯“给出标准答案”演进为“提出正确问题并构思新点子”。 所罗门对此深表赞同，并补充指出绝大多数数学专业的学生最终并不会投身纯理论数学研究，而是走向金融、工程等工业界领域。对这些人而言，提出好问题、具备项目抽象能力，恰恰是成为优秀管理者、系统设计师以及行业创新领导者的底层逻辑。至于那些长期以来让学生感到枯燥、甚至产生生理抗拒的纯形式化证明书写，如今正在被逐步边缘化。面对乔·威森塔尔（Joe Weisenthal）提出的终极疑问——“所以我们依然应该教孩子数学对吗？”，所罗门给出了毫不迟疑的肯定回答。 - **核心技能转化**：编写高质量提示词与传统数学中梳理命题结构、提出严谨猜想是同一种智力活动。 - **职业应用场景**：金融建模、大型工程管理和跨领域创新，比纯理论学术研究更依赖“问题定义能力”。 - **教学重心变迁**：机械繁琐的形式化推导在 AI 辅助下逐渐淡出核心舞台，概念理解和顶层构想成为首要训练目标。 ```mermaid flowchart LR accTitle: 数学能力在AI时代的价值转变 accDescr: 展示从传统的机械证明书写向高阶问题定义与审美品味的价值转移路径。 subgraph Traditional["传统能力倾向"] A["机械证明书写"] B["计算与格式验证"] end subgraph Modern["高阶核心竞争力"] C["精准问题界定<br/>(编写系统Prompt)"] D["跨界模型设计<br/>(金融/工程领导力)"] E["学术研究品味<br/>(选题与优雅审美)"] end A -.->|"AI自动化接管"| B B -->|"价值重心跃迁"| C C --> D C --> E ``` ### `01:01:30` 验证瓶颈与学术界的“品味”危机 送别所罗门后，主持人乔·威森塔尔与特雷西·阿洛韦展开了深入复盘。两人坦言，目前完全无法看清数学和数学教育的终极走向，因为底层模型仍在以难以预测的速度迭代演进。所罗门在访谈中提及的学术会议投稿规模爆炸就是最生动的写照——原本只有 1,000 篇论文提交的学术会议，如今因 AI 生成工具的普及激增至 60,000 篇。即便其中有相当一部分质量不俗，也直接击穿并摧毁了现有的同行评审与同行验证体系。 阿洛韦指出，这构成了 AI 领域的一个经典悖论：AI 让内容的生成变得极其廉价与轻松，但对内容的“验证”难度和成本却完全没有同比例下降，甚至反向激增。这种现象逼迫数学界乃至整个计算机科学界不得不重新审视“品味”（Taste）这一原本略显抽象的概念。在编程领域，人们常谈论“优雅的代码”；而在数学界，如何提出深刻的猜想、以优美且具启发性的方式证明定理，同样极具审美门槛。当海量平庸或正确的平铺直叙被 AI 批量产出后，拥有“跳出固有框架的思考”（阿洛韦打趣为“跳出芯片的思考”，即 "out of the chip thinking"）并凭借高雅品味完成鉴别与构思，成了人类研究者最后的护城河。 - **系统承受力击穿**：学术论文提交量呈现数十倍的指数级爆炸，人类同行评审机制面临崩溃。 - **生成与验证脱节**：生成假说和证明的边际成本趋近于零，但逐行复核逻辑真实性的智力成本居高不下。 - **审美与品味的权重**：技术输出从比拼“正确性”转向比拼方案的“优雅度”与选题的“深刻性”。 ### `01:03:00` 递归自我改进与科学探索中的“人在回路” 威森塔尔进一步指出，这种关于“研究品味”的讨论绝非仅限于象牙塔内的理论争鸣，而是关乎整个人工智能未来走向的极高风险议题。在通用人工智能（AGI）的核心叙事中，“递归自我改进”（Recursive Self-Improvement）一直被奉为圣杯，即一个 AI 模型能否独立构思、训练并打造出超越其自身的下一代模型。要达成这一闭环，系统必须具备自主提出具有开创性的科学假设并设计验证实验的能力。 目前来看，模型能否独立提出真正新颖且具开创性的问题，依然是一个巨大的未知数。威森塔尔犀利地反问：AI 或许能在大海捞针中解答人类给出的既有问题，但“模型能独立提出下一个纳维-斯托克斯方程（Navier-Stokes）吗？它能构想出下一个 P/NP 问题吗？或者提出下一个费马大定理（Fermat's Last Theorem）？”在机器学习和自然科学前沿，模型擅长在现有参数空间中优化，却远未展现出提出范式转移级实验的能力。因此，至少在当前阶段，“人在回路”（Human-in-the-loop）仍然是科学方法论中不可替代的关键环节，人类直觉依然是点燃突破性发现的核心火花。 - **递归演进难题**：AI 若无法自主提出真正具备开拓性的研究问题，其自我迭代与科研飞轮就会陷入瓶颈。 - **前沿科学提问差距**：当前模型擅长在既定规则内“解题”，但无法凭空开辟出类似费马大定理或纳维-斯托克斯存在性这样引领数百年研究的“新大陆”。 - **人机分工结语**：人类目前的核心优势牢牢锚定在“点子生成”（Idea Generation）、科学直觉与实验审美上。 ## Notable quotes > `04:45` 在未来的世界里，也许记住乘法口诀表甚至记住自家地址都不再是一项重要的技能了。 — Joe Weisenthal > `07:54` 在玻璃上写数学在现实中极其不切实际，不过视觉效果确实很棒，非常适合镜头拍摄。 — Justin Solomon > `13:30` 如果你在寻找真正有挑战性的硬核数学问题，根本不需要凭空捏造，它们就在现实世界中随处可见。 — Justin Solomon > `16:35` 想要赚到那一百万美元奖金，你所要做的全部事情就是证明水不会爆炸。 — Justin Solomon > `20:00` 它们在某种程度上无所不包，又在某种程度上什么都没做。 — Justin Solomon > `22:30` 我认为人类根本无法逐行检查这样的证明。 — Justin Solomon > `22:30` 证明你的软件可靠的一种方式是‘这栋大楼至今还没塌’，但这可没法让人完全放心。 — Justin Solomon > `25:09` 那么写下这句提示词的人，到底为解决这个问题带来了哪怕一丝一毫的智力洞见吗？ — Justin Solomon > `34:40` 我每天也都觉得自己离重大突破很近。 — Justin Solomon > `42:30` 郑重澄清，如果我解开了不可证明的定理，我可绝对不会拒领这笔钱。 — Justin Solomon > `55:40` 教育的是活生生的人，而不是在给 Claude 的性能做基准测试。 — Justin Solomon > `01:03:00` 模型能独立提出下一个纳维-斯托克斯方程吗？它能构想出下一个 P/NP 问题吗？或者提出下一个费马大定理？ — Joe Weisenthal ## Mentioned Bloomberg, Odd Lots, Joe Weisenthal, Tracy Alloway, Justin Solomon, 麻省理工学院 (MIT), OpenAI, Navier-Stokes 方程 (纳维-斯托克斯问题), GPT-3 / GPT-3.5, Claude, Grok, Lean (形式化证明语言), Jensen Huang (黄仁勋), Nvidia (英伟达), Terence Tao (陶哲轩), John Nash (约翰·纳什), Grigori Perelman (格里戈里·佩雷尔曼), Nestor Guillen, Travis Kalanick, 克雷数学研究所 (Clay Mathematics Institute), 千禧年大奖难题 (Millennium Prize Problems), P vs NP 问题, 费马大定理, 庞加莱猜想, 菲尔兹奖, 有限元分析 (FEM), 偏微分方程 (PDE), ICLR (国际表征学习大会), Pixar (皮克斯动画工作室), 电影《心灵捕手》(Good Will Hunting), 电影《美丽心灵》(A Beautiful Mind), 电影《推销员》(Traveling Salesman), 塔科马海峡吊桥坍塌事件 ## Video description <p>Last month, OpenAI announced that it had produced an AI-generated proof for the Navier-Stokes problem, one of the most famous unsolved questions in mathematics. LLMs used to be bad at counting, but now they are solving math problems that have stumped humans for decades. Meanwhile, at universities, the problem of AI in education continues: Now that LLMs can do a student's homework, teachers are struggling to keep up. It is clear that AI is very quickly changing how math is taught and how it is practiced by professionals. On this episode, Justin Solomon, who is the associate dean for engineering education at MIT, gives us a primer on how mathematicians (both pure and applied) are responding to all the advances in AI. He also explains what exactly the Navier-Stokes problem is and why the OpenAI proof is hard for even the pros to parse, what movies get wrong about how mathematicians do their jobs, and how he's changing his pedagogical approach in the age of AI.<br><br>See Odd Lots live in Chicago! <a href="https://events.bloombergevents.com/event/OddLotsLiveChicago/summary">Tickets on sale here</a></p><p>See <a href="https://omnystudio.com/listener">omnystudio.com/listener</a> for privacy information.</p> [Source](https://podtrac.com/pts/redirect.mp3/tracking.swap.fm/track/UVBrz8bN8aM2Xe47PEPu/traffic.omny.fm/d/clips/e73c998e-6e60-432f-8610-ae210140c5b1/8a94442e-5a74-4fa2-8b8d-ae27003a8d6b/f49c2ca3-dfe7-4a67-bc11-b4dd01463f66/audio.mp3?in_playlist=982f5071-765c-403d-969d-ae27003a8d83)

[4:45] Joe Weisenthal: 在未来的世界里，也许记住乘法口诀表甚至记住自家地址都不再是一项重要的技能了。 [7:54] Justin Solomon: 在玻璃上写数学在现实中极其不切实际，不过视觉效果确实很棒，非常适合镜头拍摄。 [13:30] Justin Solomon: 如果你在寻找真正有挑战性的硬核数学问题，根本不需要凭空捏造，它们就在现实世界中随处可见。 [16:35] Justin Solomon: 想要赚到那一百万美元奖金，你所要做的全部事情就是证明水不会爆炸。 [20:00] Justin Solomon: 它们在某种程度上无所不包，又在某种程度上什么都没做。 [22:30] Justin Solomon: 我认为人类根本无法逐行检查这样的证明。 [22:30] Justin Solomon: 证明你的软件可靠的一种方式是‘这栋大楼至今还没塌’，但这可没法让人完全放心。 [25:09] Justin Solomon: 那么写下这句提示词的人，到底为解决这个问题带来了哪怕一丝一毫的智力洞见吗？ [34:40] Justin Solomon: 我每天也都觉得自己离重大突破很近。 [42:30] Justin Solomon: 郑重澄清，如果我解开了不可证明的定理，我可绝对不会拒领这笔钱。 [55:40] Justin Solomon: 教育的是活生生的人，而不是在给 Claude 的性能做基准测试。 [63:00] Joe Weisenthal: 模型能独立提出下一个纳维-斯托克斯方程吗？它能构想出下一个 P/NP 问题吗？或者提出下一个费马大定理？

[10:00] Justin Solomon: I'm terrible at arithmetic. I've got a calculator out. [16:00] Justin Solomon: All you've got to do to make a million dollars is prove that water doesn't explode. [23:55] Justin Solomon: One way to justify your software is that your building hasn't collapsed yet, but that's not a whole lot of... I hope I can trust a little bit better than that. [30:00] Justin Solomon: Apparently my corner of math is safe at the moment. [34:55] Travis Kalanick: vibe physics with Grok [34:55] Justin Solomon: I feel that way every day. [55:25] Justin Solomon: You don't do it because it's fun, necessarily... you do it more because you're trying to achieve some other goal. You're preparing or you're being a stronger person. [55:25] Justin Solomon: educating the human, not just testing the capabilities of Claude.

[0:00] There are some market stories where you want every detail. Tracy and I have made quite a few podcasts on that basis, but sometimes you've only got 10 minutes and need to know what's moving markets and why. That's the Barclays Brief Podcast. Every week, experts from Barclays markets and research get you up to speed on what matters and what to watch next. About the time it takes to grab a coffee. So search Barclays Brief wherever you get your podcasts. Some people treat ChatGPT like some kind of smart search engine, and some use it to get work done. ChatGPT Work is a new way of working in ChatGPT that can take action across your apps and files, stay with a project for hours if needed, and turn a goal into finished work. This is Robert Smith from Business History. If you're listening to this, there's a good chance you're a small business owner. And like every small business owner, you started with a dream. to do what you love and watch it grow. What you probably didn't dream about? Keeping up with cyber threats. That's where MasterCard can help with access to tools that help identify cyber threats to better protect your business. Building your dream business. Priceless. For cybersecurity in a changing world, There's MasterCard. Learn more at MasterCard.com slash small business. Hey, OddLots listeners, the OddLots tour continues and our next stop is in Chicago. That's right. Joe and I will be at the City Winery Chicago on October 15th for a live OddLots recording. Tickets are on sale now at Bloomberg.com forward slash OddLots. And of course, a special thank you to Barclays for supporting OddLots Live. So that's October 15th at City Winery in Chicago. Get your tickets now. Thank you. Bloomberg Audio Studios, podcasts, radio, news. Hello and welcome to another episode of the Odd Laws podcast. I'm Joe Weisenthal. And I'm Traci Allaway. So Tracy, with AI, you know, obviously one of the big concerns is like, Job laws, what are we going to – in the future, what are humans going to be better at than the machines? What are we going to do all day? What are we going to do all day, et cetera? And we really don't know the answer, I think, to any of these questions yet. But I am aware, obviously, the models are getting very good at math. And at least some mathematicians perceive – they're in a state of existential – angst about this and certainly crisis perhaps some feel about like well what are we doing here these problems that i devoted my life to solving and the model solves it in a weekend you know like is is this what's potentially coming for many of us yeah math is an interesting one because you can tell that a lot of the ai companies themselves have an interest in math yeah like they see math as a sort of holy grail for i guess proving that the models are able to reason yeah it's the thing able to say that they've achieved AGI, I think, is the way it's couched. And it also makes good headlines, so even though I have no idea what the Navier-Stokes problem actually is. I don't know even if I'm pronouncing that correctly. Is it Navier or Navier? Okay, Navier-Stokes. I see the headline and I'm like, ooh, a model has solved the Navier-Stokes problem as if I know what that actually means and the significance for the mathematics profession. It's also funny because I downloaded the paper and it's like one paragraph and I have no idea. I can't even begin writing. to parse the output. Another fascinating aspect of this is just like it would not have been intuitive to me. Mm hmm. say two years ago or three years ago, that a language model would be so good at mathematics. In fact, at one point, they weren't very good at math at all. Like go back to like, I don't know, GPT three or GPT 3.5. If you ask them to count like the number of letters in a word or something, they would get it wrong. Yeah. And weirdly, they still like struggle with certain things like that. you know, language, et cetera. And yet somehow that generalizes into this extraordinary math capability. That too is very surprising. In this conversation, can I just say before we start, I would also just like to learn what advanced mathematics actually is. Because everything I know about it, which is nothing, comes from like watching Good Will Hunting and just seeing. [5:00] you know, a bunch of formulas on a chalkboard and people get really excited when they start crossing off certain parentheses and stuff like that. I have no idea what it is. I have no idea how it differs from pure maths of like two plus two equals four. And so I would just like to come away in this conversation with a better understanding of what it is overall and how AI actually fits into it. Absolutely. So like I was thinking along the same lines, like, OK, for like someone who like devotes their life to like advanced theoretical math and, you know, other also like things like advanced theoretical physics that don't necessarily have like obvious like industrial applications, etc. Who is the type of person? What are they hoping to achieve? What? Why is there still interest in some of these things that perhaps the sort of applicability is not necessarily so obvious? But then, like, obviously, you know, like, what is the role of the mathematician at this point? And then actually, specifically, it's like, OK, across universe. This is actually the thing I'm very curious about, which is. My son studies math, right? Both my kids study math and like- Hopefully not theoretical math just yet. Not yet. How old is he? My daughter's 10, my son is seven. They take math, etc. But like setting aside like what the future holds, I'm very interested in just the present because it's like you have all these math departments at universities all around the country and like maybe some are excited about AI, etc. But like- what do you do today in terms of like rethinking pedagogies? Because it must be, or I would assume it's pretty different in 2026 than like those same departments in 2021 or something like that. Absolutely. And like, how are professors, teachers, et cetera, thinking about just the sort of the process of teaching math right now? It's just like, just sort of like very straightforwardly, mental math, like times tables. This is a question I have, too. If you are an advanced mathematician doing theoretical maths, how often do you actually use a calculator? I'm curious. That's going to be my first question. Actually, Jensen Wong. I don't know if you read the transcript of his interview with Ezra Klein a couple of weeks ago, but he said something. It's like, oh, maybe it's just not – it was a little kind of weird interview. He said something like, well, maybe like knowing your times tables isn't going to be an important skill in the future. Or your address. Yeah, which is like kind of weird. Like I still want my kids to know the times tables. But it raises a very interesting question. proof or a theorem or something like that. But if you don't understand how you actually got there and how the model is getting to its verification, then what are we actually learning here? Right. And then there's the question of like, people learn a lot in the process, right? So it's like even presumably in just the people work on some of these problems for years and years, and maybe they never arrive at any proof, but intuitively you would think there is something, of other benefits to just the process of learning. How does that change? Anyway. I have a lot of questions about this, and we really do have the perfect guest. We're going to be speaking with Justin Solomon. He is Associate Dean of Engineering Education at MIT and very interested and involved in math education and thinking about the intersection of math and AI. Justin, thank you so much for coming on OddLots. or advanced theoretical math. Who are these people? Why? Like, what are they hoping to do? Yeah, it's true. I feel like the typical caricature that you see is maybe this character on Good Will Hunting. Which, incidentally, they're always writing math on Windows, which is very impractical. Good visuals, though. Yeah, that's true. It's great for a camera. Yeah. But that aside, I think it's an intensely personal thing. So... To be clear, my personal career, like I'm a professor of engineering, is in an area of math called applied math, which sometimes is applied and sometimes not. That's a discussion for another day. But essentially, some people go into math because they want to see it translated into applications or to bring fundamental insight. into the kind of things we do every day, whether it's predicting markets or physics, what have you. Other people in areas of math sometimes... You'll call them pure math. I don't love that terminology, like pure versus applied. But in other areas of math, it's almost just for the joy of the game or understanding the universe around you. All of these are... totally legitimate reasons. True intellectual pursuit for the sake of intellectual pursuit. That's right, but along the way I think you develop a lot of really important reasoning skills and skills that support [10:00] work in all kinds of different areas. And how often do advanced mathematicians, whether they're in, I guess, applied maths or purely theoretical maths, how often do they use a calculator? Very often. Okay. Yeah, no, I, we're like, you know. My impression, again, is everyone is doing this by hand on chalkboards and windows. I mean, there's like lots of Greek symbols and letters and variables and things, but no, I'm terrible at arithmetic. I've got a calculator out. For real? Oh, yeah. This gives me hope, actually, that maybe I could understand some of this, even though I'm pretty bad at arithmetic itself. What's the difference between a math problem then and a math proof or a theorem? These sort of big picture things that we're talking about AI models actually doing now. That's a great question. And it's really important to dig into exactly what the models are doing and not doing and who deserves credit, by the way. Yeah. Verifying true facts is one part of the puzzle. But it's also all about figuring out what are the interesting problems to study and what insight they bring to the broader world. Right. So in some areas of mathematics, There are all kinds of different questions you could ask, and some of them are super uninteresting, but you could ask a computer to prove them. And others maybe tell you something about a physical system or about a system of equations or something like that, that bring broader understanding. And so this exercise of proof writing is all about sort of writing a mathematically convincing argument that tells mathematicians, so there's a human aspect, which is an important one, that a certain fact is actually maybe reality, right? It aligns with logic, there's not some flaw hiding in there somewhere. And where do ideas for a particular mathematical proof or theory come from? Because, again, I'm going to come at this from, I guess, the movie angle. But I think about A Beautiful Mind, right? And the famous bar scene. And it was John Nash, right? That was his name. And he has this, like, big realization while he's at the bar with these women. And it ends up turning into a famous mathematical theory. Where do the ideas actually come from for the rules that you are trying to prove in maths or these big picture ideas? Yeah. That's a great question. So sometimes in mathematics, there are some really natural questions out there in the universe that people have been asking for a long time. So, for example, these days in the news, we hear about some of these really high profile math problems that have been around forever. There are other types of mathematicians that are more theory builders where they're the ones asking the questions rather than answering someone else's questions. Yeah. There are definitely different approaches that are out there. Now, when it comes to where these questions are coming from, some of them are inspired purely mathematically and other ones maybe come from a very practical setting. Maybe somebody observed some phenomenon out there in the real world and they're trying to explain it with the idea. smallest model possible. Now, one thing I should clarify is that math is a really social exercise. I think that's something that people don't necessarily understand, especially because these movies really play into this. almost great man theory of mathematics, but that's just not how it works. There's this huge community of people that are working on important problems. They're communicating with one another, collaborating. And I would say, for example, from my own research, almost everything that our team has worked on, comes as a team effort. We're all sitting in the same room, we're at the blackboard, writing computer models, what have you. And that's where the insight is coming from. So it's not just sitting in your ivory tower, it's being out there in the world, observing things happening and talking to your colleagues. Before we get more into the sort of AI question, I've actually been curious, what's applied math? What is the difference between applied math and pure math? Yeah, it's a great question. There's sort of a divide in the math world, but I think it's fuzzier than maybe it sounds when I describe it here. So roughly, there's an area of mathematics that you might call applied math, where the goal is to either derive theoretical results or even come up with computer algorithms that explain phenomena out there in the real world. So, for example, in applied math, maybe you work on algorithms that simulate the weather. And behind the scenes is a really complex mathematical model. And it's all about making sure that your model is correct and efficient and well behaved, all that good stuff. In pure mathematics, sometimes the motivation is a little bit less attached to a particular application. and more just attached to the math problem itself. So something like Fermat's last theorem or something like that, where he's like, okay, I forget what it is exactly, but that would not be applied back. Right, so Fermat's last theorem, and many things I think in number theory fall into that category. But also many times these things bleed into one another. So for example, a lot of modern developments in cryptography came from areas of mathematics that we used to think of as extremely pure and abstract. Oh, interesting. This is also Good Will Hunting, by the way. It all goes back to me. That's right. That's right. You worked at Pixar before. Many years ago. Was that applied maths working at Pixar? Oh, I love that question. The answer is absolutely yes. Okay. Yeah. So the movie studios are a great example of a place where I would say state of the art and certain types of mathematics are taking place. So when you watch a typical movie with different movie effects, there's cloth, there's smoke, there's all [15:00] kinds of fluids, all, you know, rigid bodies flying around. All the dynamics of those are governed by things called partial differential equations and coming up with the right algorithms for simulating those. And by the way, making them so well behaved that an artist doesn't have to think about math is a really challenging mathematical problem. So I typically tell my students that if you're looking for really challenging math problems, you don't have to make them up. They're out there in the real world. And that's a nice example. Thank you. There are some market stories where you want every detail. Joe and I have made quite a few podcasts on that basis, but sometimes you've only got 10 minutes and just want to know what's moving markets fast. That's the Barclays Brief podcast. Every week, experts from Barclays markets and research get you up to speed on what's happening and what to watch next. Concise, focused, and brief. Search Barclays Brief wherever you get your podcasts. Some people treat ChatGPT like some kind of smart search engine, and some use it to get work done. ChatGPT Work is a new way of working in ChatGPT that can take action across your apps and files, stay with a project for hours if needed, and turn a goal into finished work. It's designed to help you move from a chaotic starting point to a reviewable first version. So all the source scattered information that you have to grind through to turn into something useful can just become something useful. Put ChatGPT to work on your most ambitious ideas and projects. Get started at ChatGPT.com by selecting Work Mode, available on Plus and Pro plans. Artificial intelligence is transforming how businesses operate, but what will it take to build a truly autonomous AI enterprise? On October 28th in Boston, join the Bloomberg Tech Briefing, Agentic Enterprise to hear business and technology leaders explore the strategies needed to build cyber resiliency and secure trusted AI systems for the Agentic AI era. Proudly sponsored by HPE. Learn more and register at BloombergLive.com slash Bloomberg Tech Briefing. What's Navier Stokes? Okay, so Navier Stokes has been in the news a lot lately, and it's been great for those of us with a little bit of background and differential equipment. Getting lots of media inquiries. That's exactly right, yeah, we have our 15 minutes of fame here. It'll die fast, but we'll enjoy it for now. Right. So the Navier-Stokes problem is one that it comes from the world of fluid dynamics. So the Navier-Stokes equations were this model, they go back quite a long time, that explain the motion of, for example, liquid in a cup. So like both of you guys have coffee cups here on the table. And if you like shake that coffee cup up, you know, the initial velocity, you know, what's inside of the cup. then the Navier 6 equations are trying to predict, like... where the fluid moves over time, right? Like where is the velocity, vorticity, all these different things. Now, the Navier-Stokes equations are really complicated. I mean, when you look at your latte when you start in the morning and you see the milk making these cool shapes and so on, there's all kinds of complex behavior that can happen. And so the Navier-Stokes equations, although they're easy to write down, they're very hard to say a whole lot about. And so there was this big open math problem, which I think is what you guys are referring to because it's been in the solutions. So the basic idea here is I think actually my undergrad differential equation professor years ago explained it to me in kind of a funny way. He sort of said, you know, all you've got to do to make a million dollars is prove that water doesn't explode. And that's roughly right, I mean, with a lot of asterisks. Essentially, what we didn't know... is if you know this initial configuration of your fluid and you let it kind of go under its dynamics over time, then whether it exists for all time and doesn't have some really crazy turbulent behavior, or, for example, maybe it blows up or it develops a singularity or something like that. So this was kind of a yes or no question, like prove or disprove. And I guess the recent news was actually a counterexample. So in other words, it showed in some finite amount of time that... With an asterisk, which is with a very crazy spoon. If you stir your fluid just the right way, you really can create this totally singular and frankly non-physical behavior. in math is like one of these things. Like I think I, the formal proof, the two plus two equals four. It's like a nine step thing. Isn't it right? Like if you are like sometimes eight, sometimes eight, but like, Can you explain what a proof is? Sure. I know 2 plus 2 equals 4. It seems obvious. But explain what is a proof such that that would, to say that to prove that 2 plus 2 equals 4 is something that requires eight steps. Right. This idea of proof writing, I think, is actually, you can think of it as almost the result of an identity crash. [20:00] This is among mathematicians maybe 100, 200 years ago. Where it used to be a lot of math was drawing pictures and kind of working in the same fashion that you're talking about, kind of intuitively. Yeah. But at some point, people realized that they wanted to make sure that the field had really strong foundations. Okay. And so mathematicians will agree on a certain set of maybe logical principles or rules. I would say that's more the domain of philosophy, precisely what those should be. And then the idea of proofreading is to build up from these kind of basic, like we agree that these five facts are true, to a much more complicated fact by combining them in different and creative ways. And so theoretical mathematics oftentimes is all about proofreading. I don't want to just say mixing and matching different facts, but coming up with smart ways to justify mathematical ideas built on very simple principles and steps from logic. So when it comes to AI currently, is the role more in, I guess, proof checking versus proofing? proof generation or idea generation? Like, what are the models actually doing at the moment? Well, they're sort of doing it all and they're sort of doing nothing. Okay. So, right. The current AI models, in terms of the types of mathematics they're doing, I don't think it's quite right to say that it's just proof checking. And in fact, actually, they're quite bad at proof checking. And one of the revolutions in I would say the last year or so is the realization. that actually what you have to do is to provide your model, like your large language model, with a second module that can check the AI's proof. And so, for example, there's a programming language called Lean, which is basically just a sort of codified version of these mathematical rules. And these days, what people will do is they'll put in their conjecture, so like the thing that they want to prove or disprove, and then ask the AI system to generate a proof. But then they'll say, rather than just writing it in human readable language, which is the substrate In addition to that, they'll actually produce this lean code So that there's a separate piece of software out there that can check that this proof truly is correct. So actually, the correctness part of it is usually deferred to another piece of software, which isn't AI. But somehow, because of the interaction between the AI tool and this proof checker, that's what's created a lot of progress that we can believe in a little bit more than we did in the past, if that makes sense. I've heard of – only because I've been reading the news. I've heard of this thing called Lean. Yes. And it sort of seems extraordinary to me. If you explain it a little bit, it seems extraordinary to me that there is this – I don't know, is it software or this framework such that these extraordinarily complex problems or solutions, I guess is the answer, can be dropped into this. And it's capable of like, yes, this is a valid. But it's also kind of crazy that you need basically a translator for it, right? Yeah. Can you explain a little bit like how Lean works, perhaps? I can explain a little bit. To be clear, I'm not an expert in this particular corner. reliable. All of your listeners should know that. And that goes for mathematics as well. And math we think of as this area that is extremely reliable, right? Everything should be true or false. Now, there are gray areas, but certainly we try to avoid that. So this programming language Lean, what it does is it has a few simple maybe axioms that are built in. So math things that we sort of agree are the starting points for writing proofs, like the steps of basic logic. And then what happens is that you write a theorem in this lean language. So rather than in English, you write it in this programming language. And then it's kind of cleverly designed so that once the software checks that that theorem is true, it can get added to the library of sort of true facts, and then you can build up, right? So you make this more and more complex edifice out of these simple steps. And it's been really impressive what they can do. I mean, some of the modern proofs that you're seeing, especially the AI generated ones, are like hundreds of thousands of lines of lean code. I don't think a human could check it. It also, by the way, means that we're trusting that someone wrote that software correctly, which is a philosophical question. But, you know, if you believe that Lean is good at checking proofs, then it provides a lot of certainty that the things that the AIs are writing down are actually correct. Devil's advocate question, but why is it important that... We need to understand the actual problem and the proof, because if I'm an engineer and I don't know, some model somewhere comes up with this really new, interesting mathematical theory and I can apply it to my day to day work. Yes. Is that not enough that I can use it? It seems to work for the most part. Maybe it's been rigorously tested in a physical environment or whatever. Why do I need to understand everything that's gone into it? Oh, that's a fantastic question. So... Let's see, my research group works a lot on simulation algorithms. So like I have some physical system and I develop a technique for predicting what it will look like in the future. And we spend a lot of time writing theoretical results to verify that our algorithms are correct, they behave the way we expect and so on. And that's absolutely critical because without that, you really can't trust what your software is producing. So for example, if I'm trying to say yes or no, is this building [25:00] to collapse during an earthquake. Seems important. That's right. And all of that is built on mathematical models, techniques like finite element modeling and so on. They're not too exciting outside of my universe, but you're all trusting these things to do really important engineering calculations and things that, sure, like... One way to justify your software is that your building hasn't collapsed yet, but that's not a whole lot of, you know. I hope I can trust a little bit better than that, you know. My dad was a physics professor, and I have memories of... of watching a video of the famous Tacoma Narrows bridge collapse. You know that video? Really horrifying. And like that was like for him like this, like, you know, I, that was, I've watched it several times in my household. Our engineering students have to take a lot of ethics courses that are pretty fraught that way. Yeah. Oh, interesting. So why don't you give us There's a couple of avenues I want to explore, including just the process of teaching and how – how it's different in 2026 versus 2016, given the new technology. But... Well, maybe give us a lay of the land for like... The math community. Yeah, because I know there are some professors. There are some people in despair, I believe. Oh, yeah. Very anxious about what this means for the future of like math as a intellectual discipline, etc. And then others seem to be very enthusiastic about the opportunity of human. in AI, like exploring new ground, but what are the sort of the conversations that you're in these days with fellow mathematicians? Oh my gosh, things are changing by the day. And really, it's a huge collection of individuals. Every single person has their own opinion here. So I can tell you my view of the world. And if you talk to somebody else, I'm sure they'll have a very different one. Math is an academic discipline. It also shows up in a lot of applications, as we mentioned. But every aspect of the work is changing from how people produce mathematics and choose problems to work on to how mathematics is sort of measured in terms of productivity. Right. When we talk about hiring faculty, for example, like what are the standards to evaluate a person's body of work and what did the human contribute? I think are really critical questions to like why do this exercise at all? Maybe lead us to a computer. There is all kinds of identity crises that are happening. maybe theory builders and there are people that are like theorem provers. The theorem provers, I think, are especially under threat at the current moment where like maybe you've been working on one problem and it's well defined and you've known about it for 50 years and crowning question that you want to answer about the universe. And these AI tools are sort of at least good at sort of cherry picking the ones that are somehow close to being proved and taking that last mile in some cases. And other people disagree with that characterization. What do you mean by when you say they disagree with it? Like exactly how interesting, for example, a mathematical proof is a very human judgment, as is what problems you should or should not actually go about working on in the real world. So there are a lot of true facts. It doesn't mean we have to write them all that this is definitely a human exercise, right? To a large extent, mathematical research is about humans communicating to other humans, right? It's not just about verifying true things about the universe, but bringing insight and helping people understand systems and so on. So I think you mentioned an identity crisis. I think what's happened is, and this is reflected, there's a really nice write-up by Terence Tao, who's a famous mathematician, that kind of talks about how there were a lot of different... sort of signs of mathematical progress that used to be correlated with one another, right? So for example, if I wanted to judge whether a mathematician has quality research or output. Maybe I look at where they're publishing, the number of things that they're writing, the interest of their result, whether they're earning these prizes and so on. And now these things are a bit at odds. So for example, the AI companies have figured out, as you guys pointed out, proving some big theoretical result maybe is... an interesting press release, right? And now there's an economic component. Knowing how to do the right prompt, right? That's right. And then there's a big question, which is, okay, so, you know, at least in the sort of cartoon version of a mathematician, you sit in a room, you think about a really hard problem, you write down a proof that's subtle and challenging. Now, as you say, maybe you're the guy that wrote down the prompt. If the prompt, for example, were prove or disprove Navier-Stokes equation, did that guy really bring a whole lot of insight to that problem is an interesting question, and one that I think we're really grappling with today. Thank you. Some people treat ChatGPT like some kind of smart search engine, and some use it to get work done. ChatGPT work is a new way of working in ChatGPT that can take action across your apps and files, stay with a project for hours if needed, and turn a goal into finished work. It's designed to help you get a job. [30:00] move from a chaotic starting point to a reviewable first version. So all the source materials, briefs, and scattered information that you have to grind through to turn into something useful can just become something useful. Put ChatGPT to work on your most ambitious ideas and projects. Get started at ChatGPT.com by selecting Work Mode, available on Plus and Pro plans. AI is entering its most consequential phase where scale, safety, and sovereignty will determine who leads and who lags. Thank you. I want to ask more questions about credit, but just on the models themselves, does sycophancy exist when it comes to math prompts the same way it exists for other, I guess, language-based queries? Does the model go like, oh, what an interesting problem. You're on the right track. Like, you know, look into this. Judging from my email box, I think the answer is yes. I'll tell you a funny anecdote. Yeah. Right. So all the time I get emails from people that are looking for an expert to maybe verify, you know. Oh, yeah. And what I've noticed lately is I get a lot of emails from outside folks that sort of, They've been playing with their AI system. They've proven a new result. And the AI told them it was really important. And they need an MIT professor to validate, which is absolutely fascinating. In fact, for fun, I played with some of these tools. and put in a few open problems in my own domain. Interestingly, they did not get solved. So apparently my Corner of Mouth is safe at the moment. But... The first thing that it did, it proved some, from my perspective, a pretty superficial thing. And then it had a little footnote on the bottom being like, as far as I can tell, this doesn't appear in the literature. Like, would you like me to convert this into the template for a journal? It was an interesting... I think it explains a lot what's going on in my email. Wait, so you can imagine a future where, I guess, like... you're sort of overwhelmed with new proofs and the bottleneck becomes the ability to actually verify them with humans. That is a problem right now. Okay. And in fact, it's not just in mathematics, it is across academia. So to give you an example, actually, interestingly, I would say the academic field that's been affected most by this phenomenon you're talking about is actually AI machine learning itself. Oh, yeah. So in that particular corner of academia, research is driven by conferences. So you write your research paper, and there's a sort of annual cycle where you submit and it undergoes peer review, and the whole community spends like a month or two... reviewing each other's papers, sort of like you pass your paper to the left in class and grade. It's sort of a very complicated version of that. If you look at the number of submissions to the machine learning conferences, it is wild. So in the last 10 years, I think about 10 years ago, it was maybe 1,000 or 2,000. The deadline for ICLR, the International Conference on Learning Representations, one of the big ones, I think it had 60,000 submissions. And if you think about it, it completely breaks the academic system the way that we're used to thinking about it. We don't have bandwidth to... never mind, just check all of these proofs, even open up the PDF. And so you're then relying on extremely superficial review process. There's many anecdotes of high school students that have somehow gotten involved in peer review because they had a project once and the system decided they were in OK, which is frustrating to build an academic career. essentially some of the stamps of quality that we're used to in the field, you really can't rely on, for exactly the reason you identified. The way that Terence Tao put it is that it used to be we were in this era of proof scarcity, that mathematicians, it was really hard and it was this very bespoke object that took a lot of craft work and training to produce. Now we're in this era of proof of abundance. So now we have to kind of think about different ways to do our job and to do mathematics in a productive and interesting way. Yeah, I think a lot of fields are dealing with Yeah. very similar issues in which we used to have like certain, I mean, even take writing, for example, we used to have these certain heuristics that we maybe like shortcuts that we use. It's like, okay, if this person can like construct a series of like well-written sentences, we might. No spelling errors. Yeah. Like how I've, I never write spelling errors. We might start, you know, like reading they might have something you know we all have to like operate with these cheats and shortcuts and now like obviously producing text with no spelling errors or anything is available to literally everyone. So it's like we have to go a step further, etc. I think that's right. You know, all these heuristics. You know, speaking of like everyone now, like thinking they can come up with a proof. Do you remember last year, Travis Kalanick, he was on – [35:00] of uber founder he was on the all in podcast and he was talking about how he was um he was doing vibe physics with brock oh boy and he felt that he was quote Getting pretty damn close to some interesting breakthroughs. I feel that way every day. Doing vibe physics with Grog. The crazy thing is you could just hire a bunch of physicists to verify the research, and then you'd have a real sycophancy problem. And I think beyond that, we're currently – so I work in education. Yeah, yeah. One of the things that we really rely on is somehow the friction in the creative process and the technical process. I mean, you mentioned writing is a fantastic example. How do you learn how to write? Well, you do a lot of writing. Yeah. convince our students that to sit down and write an essay because they have these AI tools that can do it. But then writing a five-paragraph essay was never the goal. It was to teach her to do other stuff. Mathematics is similarly experiencing a very similar challenge. So let's talk about the field of education. Yes. itself. I mean... Are the students very anxious right now? Absolutely. I mean, let's start there. What's the vibe among students, especially like maybe grad students who started their career before college? The idea of like AI generated proofs is a thing is suddenly they're in grad school. Right. What are they? What are they? What's going through their minds these days? Yeah, the landscape of both graduate and undergraduate education is changing a lot now. In the graduate school world, for me, that would mean PhD. In other worlds, that's maybe, you know, master's degrees and so on. There, this sort of divides into maybe professional education, preparing you for a job, and then maybe academic style like PhD work. In the PhD world, of course, they're being strongly affected by just the larger trends and changes in the research universe. And then when it comes to professional degrees, the main question we have to ask is, what is the right thing to teach our students in the current moment? So, for example, I'm in a department that teaches computer science. The field of computer science is changing by the day. It used to be, for example, we talked about programming as this core skill and thing that everybody should learn from day one. I would argue that's probably still the case, the same way that maybe learning to write and be literate is a key factor part of being a person in the society, even if you're not doing that every day. But jobs as an actual programmer are becoming more rare, right? We see a decline, for example, in enrollment in computer science for the first time in like 20 years. Since you mentioned computer science and unsolved questions, what is P equals NP all about? I see. I can answer all your math questions. Yeah, absolutely. Yeah. There's all these questions I've stored up in my head. Right. And we should come back to, I don't think I actually told you the end of the Navier I can come back to that. But yeah, so P versus NP and Navier-Stokes are examples of very famous mathematical problems that have been identified by the Clay Math Institute as sort of there's Benchmark for progress in modern mathematics, right? So they're open questions we don't know how to answer. P versus NP is a computer science problem. Yeah, in the sense that it's basically asking, are all problems solvable in polynomial time, meaning, can your computer solve or all is a little strong, but there's certain classes of problems that we know. are solvable pretty efficiently. So for example, your computer can sort a list of numbers really fast. There are other problems that we don't know how to solve efficiently, but that's different from saying we can't solve them efficiently. And so that's the sort of fundamental question in that universe. Like, do there exist problems Is this like the known unknowns and unknown unknowns equivalent of the computer world? Yeah, yeah. It's hard to communicate about that way. That's absolutely right. Years ago, there was a really terrible movie that came out that involved P versus NP. Really? Secretly, some mathematician had proved that they were equal, but there's a government conspiracy because the markets were all going to collapse. It sounds amazing. I think it was called Traveling Salesman. Anyway, the reviews were really funny. a room at a blackboard. That aside. They should have used Windows instead. That's right. Yeah, yeah. It would have been a little visually compelling. Right. So a lot of these questions are sort of yes or no. We don't know if a fact is true. Navi Stokes is another one. The Either you need to show that these equations have a solution for all time, like fluid moving around. Yeah. Or give a counter example where they don't. So in this AI case, for example, What the AI gave was a counterexample. It said this conjecture is not true. That like there exists a particular configuration of your fluid that becomes infinitely turbulent in a finite amount of time. With a special spoon. That's right. With a very special spoon. And the reason I'm highlighting that is because I think it's a really great example of the really... complicated relationship between humans and AI. So The construction of that spoon arguably was done in large part by a team in Spain of human mathematicians on pen and paper. And then essentially what happened was the AI, there was also this sort of interpersonal fight between two different AI. That's a whole hot mess. But that aside... these two teams were sort of, in some sense, leveraging these guys' progress over a very long period of time, and then doing that last mile. Right? [40:00] It's true that the AI made some really critical... Insights in that particular counter example, but it isn't true that it's like a totally surprising result that came out of left field Well, so this seems like it's gonna be a scenario that academia is gonna have to deal with more and more so you could have a theoretical situation where like one person comes up with the problem and Or a team one person or a team comes up with the solution and then you have a model that like generates the verification or the final leg of the problem or whatever and Who deserves credit in that scenario? Like, how do you actually handle that? How do we handle that? Currently, poorly. And it's a really interesting and challenging discussion. I don't think we have a good answer right now. It used to be, for example, these clay math prizes, they're at the very top of the universe there. I'm not going to get anywhere close. It used to be that they were sort of signifiers of two things. One is that the person that proved the theoretical result is a really smart person. The other one is that there's a big achievement in mathematics itself, that we now know something to be true that we didn't know before, or false. now it's not clear that those two things are correlated, as you say, right? Like if a computer generates the interesting part of a proof, Who deserves credit? Should that person be hired if they're the one that made the prompt? It really is an interesting question for a math or an engineering department. And it's not so clear. I think it really depends on your value systems. What do we get out as humans? from this exercise of proof writing. I mean, for example, does a counter example to Navier Stokes, is that going to revolutionize the markets or change the way we write software? Almost certainly not, right? So this is more of a... demonstration of a technology than any particular practical thing. And so I think we have to be really careful with understanding the incentive system behind who's claiming for credit. Yeah. And also making sure that the humans that are maybe... doing all of the labor of composing the data that's going into these systems is getting remixed into a proof somehow that we're able to make some attribution there, which currently the technology is really bad at, right? Like you talk to AI, it's not so clear where it's getting its answers from, even though we know that it's kind of echoing data that's maybe scraped off the internet or someone else's research paper, what have you. Right. And this might be like the time that we will know that for sure. And five years from now, be scraping it from itself or like other models or coming up with its own stuff. Right. I have a slight aside, which is the prizes that you mentioned, which have also come up in movies like Good Will Hunting. The prize money that people get for that, is that for them personally or is it supposed to go into their research? That's a good question. I have no idea. The Clayby Award specifically, I'm not sure. Yeah. And so now I'm envisioning this scenario where like maybe one day we have a model that ends up winning some big mathematics prize and the money... goes to the lab? I don't know. I think one day I need the one million dollars. I'm just wondering. It's actually, it's a good question and we now have an example of that. So this Navier Stokes problem, I should be So their proof has not been officially verified and checked and in particular there's a rule that has to kind of exist in the community for some number of years and not be refuted. But we have to ask that question right now. So OpenAI.com. arguably because they rushed at the last second, were maybe the first to release something that satisfies the clay criteria. There's a kind of a choose-your-own-adventure situation. There's a few different... conjectures and you could prove any one of them and then get this particular prize opening i was quick to say that like they don't need the million dollars because of course they don't like in like two minutes they're burning through a million dollars worth of computer anyway which i think begs the question like well why are they doing this it's not for the money yeah and now i don't think it was true that it was for the money for the mathematicians either but the the million dollars made a big difference to an individual there was some story about some russian mathematician who like solved one of those big prizes right and then declined the he also declined the money I'm not applying the money, for the record. No, I would happily take it. When I finally solve some unprovable mathematical theorem. But actually, this leads to another more serious question, which is, okay, it costs a lot of money for these models to be run, and they consume vast amounts of tokens for very advanced mathematics. Do you end up in a situation where whoever has the most tokens wins the mathematics prize? Arguably. That, I think, is one of the major concerns in our community right now. And by the way, you mentioned education. It's also a concern there. Does the student that has a $200 account get to have a better homework solution? Yeah. It's a real question we have to grapple with right now. On the advanced end, I think we saw that. So for example, I I don't think it's controversial to say that essentially when the news that people were getting close to this Navier Stokes counter example came out, it appears that OpenAI suddenly poured some ridiculous [45:00] amount of tokens or money or whatever into writing their proof first. The typical academic can't do that. And I think there's an important element, I believe, with an unreleased model. And so I'm curious, like, let's say you're a mathematician and you're an enthusiast. Let's say you're, you know what, I'm going to embrace this new world of... AI augmenting our understanding of mathematics. If you're a mathematician at a university, it's certain that there is a model that exists inside the labs that's better than the model that you have access to. And so it's like you start to work on some problems, you start to demonstrate that actually using AI, you can make progress towards this. And then suddenly it's like, well, they have a model two generations ahead of you, and then they knock it out in a weekend. model was trained partially on your chat. It's really pernicious if you think about it. That's right. I mean, at least in the current moment, the AI system is very centralized. I mean, there's two or three tools that almost everybody is using. And they're owned by these companies. When you write your query, it goes to their servers. Yeah. And they're basically the guys controlling the faucet. It's a big concern. Another issue related to that is research funding. As your listeners may or may not know, even independently of AI, research funding currently is in a bit of a crisis. I mean, it's in a much more challenging moment than it was in the past. I can tell you, I'm so fortunate to work at MIT. And even for me, it's a challenge to pay for my graduate students and so on. And the costs change every year. Now we have to add onto that, for example, AI accounts. And that sounds theoretical, but it's not. Like my PhD students, for example, want the most expensive possible cloud account, which is great, but it's, what, $200 per person per month. Almost becomes equitable with the cost of another human. Now, something with... I read apparently some people had said that like of the outstanding clay prizes, many had sort of surmised that Navier Stokes might be the first one solved. And I think it's interesting that it was solved via a counterexample. And it sort of raises a question of like, did the model find an elegant solution or did it just grind? Was this a grindable problem? That's exactly the right question. through a lot of states and then it, it hits on the magic spoon that disproves it or et cetera, which seems like a very, like finding that counter example is a process of grinding through hundreds of thousands of different states. Like, is that elegant math or is that just like some sort of like sheer mechanical process? Oh, that's a great question. And indeed, I think you mentioned at the beginning of this, this talk that you opened that paper and it's Like, I can't read past the first couple of pages, you know? Right. So that particular result, I think there was some beautiful insight, again, arguably generated by humans that went into the overall construction. And then, indeed, the proof is really horrifying. Like, there's a lot going on. It's technical. It's got inequalities and bounds and indices and all kinds of crazy stuff. And so I think, indeed, as you say, in this particular case, no. Is it elegant? Probably not. Is it correct? Yes, probably. So again, begs the question, like, why do we do this exercise? Is it just to... Now we can say with confidence that this conjecture is false, or is it that we learn something along the way that will bring insight to some other problem? those two things are now divorced, right? If you just have this... frictionless way to prove a theorem, then all the beautiful insight and maybe kind of fun garden paths along the way that might have inspired some new research field have been pruned out, which is a big challenge, right? In some sense, that's really the key human aspect of all of this. Thank you. Some people treat ChatGPT like some kind of smart search engine, and some use it to get work done. ChatGPT Work is a new way of working in ChatGPT that can take action across your apps and files, stay with a project for hours if needed, and turn a goal into finished work. It's designed to help you move from a chaotic starting point to a reviewable first version. something useful put chat gpt to work on your most ambitious ideas and projects get started at chat gpt.com by selecting work mode available on plus and pro plans here at bloomberg we spend all week talking about markets and the economy and on fridays we help you make sense of what it all means for your money bloomberg money is our weekly look at the four [50:00] is shaping your financial life. We explore personal finance, investing, and retirement with leading economists, strategists, and wealth managers. Join me, Scarlett Fu. And me, Tom Keen, for smart conversations that help you make better financial decisions. Subscribe to Bloomberg Money on Apple, Spotify, or wherever you listen. So this was going to be my next question. And I guess I'm really asking you to opine on whether or not you think AGI is going to be a thing or super intelligence or whatever we're calling it now. But OK, so. AI right now can sort of brute force a solution and it's not particularly elegant. Can it come up with ideas that are separate to what we kind of already know or questions that we are already positing? Like, can it actually inform us that, no, the way you think the world works currently is? is actually... different or can it sort of like unveil mathematical rules or rules about the world that we are not even thinking about or asking about? Yeah, exactly. Exactly. It's a great question. So actually, my colleague, Nestor Guillen at NYU, has a nice blog post that kind of talks about this a bit. When you look at a lot of the... proofs and the new results that are coming out of these AI systems, they're arguably in what mathematicians might call the convex hole of existing knowledge, meaning there's sort of a bunch of stuff we already know. And then a lot of these proofs are really intricate and interesting ways to mix and match those things. They're like kind of inside of the span of knowledge that we already have. And you've asked the key question, which is, can a mathematical AI system actually come up with a truly surprising new result that's like not just – Mixing and matching facts we already know, or maybe like finishing off a calculation that somebody got stuck on, but making an entirely new theory, for example, that just wasn't present before. I don't think we've seen a whole lot of examples of that so far. I mean, this technology changes by the day. Like, who knows? But at least so far, I think for the most part, it's in that first category of like, maybe making really interesting and surprising connections between things that we already knew, but not necessarily generating something entirely new out of left field. That would be really the next step for a lot of these tools. I know that one of the questions, too, with some of the – and I think maybe some of the lesser prestigious proofs that have been established, there's the question, okay, maybe they just grind their way to an answer. Right. And the other – I know another question is like, well, was this already sort of proven somewhere? Maybe it was like there is some obscure Russian math journal where most of the work had been done and most mathematicians were aware of that. Yeah. But it was in the training data and so – Hard to know. Yeah. Data attribution – in AI generally, mathematics or otherwise, is a really critical problem. I mean, it's exactly the same problem as like... My AI generated a cool new cartoon. Did it borrow from Disney? And is that okay? In some sense, it's a very similar problem. And the current tools are very bad at answering it. Why don't you just tell us a little bit about being a professor in 2026. And like, what, what you're doing differently. Right. And like just that education. Yes. So the education field is having its own, I don't know about identity crisis, but certainly a lot of navel gazing and thinking very hard about the fundamentals. MIT actually released a really nice little report on education, AI and education, I encourage everybody to take a look. My colleagues Sam Klopfer and Eric Madden were in charge of writing that. But every aspect of education has changed. I think One thing that maybe is undervalued is the amount of education which is social, even if... Primarily the maybe artifacts that you're producing are things like problem sets. So one thing that we've seen among our sort of youngest students, for example, the early undergraduates, is that it's really hard to draw them out of their dorm room. There's an interesting perception challenge here. You now have, again, this frictionless... robot that can help you with your problem sets or frankly do your problem set for you. Is that learning? Probably not. Now, the interesting thing is that there's a difference between perception and reality. A lot of our students maybe perceive that they're learning when they use these different AI tools. Yeah. But then they go and take the exam and the score would say otherwise. So what do you do... to sort of maximize the chances that your students are learning? You know, at MIT, they have a slogan. I think it roughly is like mind and hand. And then they added and heart on the end, which is good. And I think we're really leaning into that second one. We've known long before AI that putting a bunch of students in a classroom and talking at them and then having them work on a problem set alone is... not a very effective way to teach. So now I think we're really focusing on experiential learning, on getting the students to show up physically and to ask that question, or answer that question, rather, of why do we take all these students and put them on a college campus instead of having them work from a laptop at home? Now, I would argue that that's still a really valuable exercise. MIT, for example, has the luxury of having the world expert on just about anything physically with an [55:00] aisle of like your dorms. Yeah. And so we're designing all kinds of programs that for example, in the School of Engineering, try to bring the students in for research experience, for hands-on learning, for going into industry and seeing what things look like there, and trying to build the undergraduate experience more out of that and a little bit less out of exams and problem sets. Now, there remain a lot of questions. For example, assessment is a big challenge for us. What we observe in the last year or two is that our students are getting nearly 100% on all their homeworks. Maybe it's because MIT students are super bright. Can I just say, so I take the train from New York to between New Haven quite a lot. There are a lot of Yale students. Watching them on their laptops right now, they're all on, mostly Claude actually, writing essays or at least doing the basic research via chatbot. So like this semester, I'm teaching a course called Shape Analysis that covers discrete differential geometry, if you're into that kind of thing. Of course. Who isn't, right? This is a pretty mathy class. In the past, most of what the students did were two things. kind of written assignments and then an open-ended project where they kind of come up with a new result and write it done. And both of those had to change. And so in our course now, the way we structure it is there's still homeworks, but... The analogy that I'm using, as with many of my colleagues at MIT, is closer to weightlifting. You don't do it because it's fun, necessarily. I do sometimes, but not everybody's like that. But you do it more because you're trying to achieve some other goal. You're preparing or you're being a stronger person. And then maybe what you do is you have to test understanding in some other fashion. So we have a little short quiz here. after the homework that basically is just copy-pasted homework problem, but make sure that the students at least comprehend what they did. We also had to restructure the project a bit, add like oral presentation component to it and bring some more interaction to make sure that essentially we're educating the human, not just testing the capabilities of CLOD, right? Those are very different things. And I would argue the real goal of an institution like MIT is to prepare brilliant students for the next set of problems. we were talking about how like it's difficult to just look at the output of the Navier-Stokes solution. Yes. Evidently, even for mathematicians. And I was thinking, OK, so what's the future then of math? And I had this analogy in my head like we knew how to brew beer before we understood the science of yeast. Right. So alcohol has been around a long time. And then at some point people understood and they knew the process, but they didn't understand why the process worked. yeast and why this this sequence works and so i thought like well maybe like Could it be that the future of science is, yes, the AI models are going to... find all these answers or be able to do all these things, but that there will still be a role then for humans to like, I don't know, translate it or explain it. I think that's right. But then I also wonder if it's like, well. probably the AI is also going to be better at translating and explaining things too. But like, right. Could there be a new sort of like, general like reorientation of math such that rather than discovering new things per se, it's like, understanding these things that have been proven that's right i i think the right way to think about these ai systems is that they're they're tools and and people's jobs evolve all the time and and apparently mine is no different yeah i want to be clear i don't think that we need to be subservient to the ai and just checking what it makes sure rather yeah because i guess there's a few different aspects of that one is um of course my job is both education and research yeah and i think the But it looks like day to day has changed a bit. The mission is still there, and we still want to educate the world's best humans at MIT, and we do an okay job of that. In terms of the mathematical labor, I think one thing that's really undervalued in this field is that the typical mathematician is not... working on the clay math institute problems those are really hard there's some really brilliant guys i wish i could do it i i'm not i'm not i can't compete there and similarly like math contest problems was another one that the ai companies were quick to kind of pounce on as the math olympia exactly that was the sort of previous generation yeah all of these are great in the sense that they're like problems that you can write down they're well contained they're in a little box but a lot of mathematics is more about taste than it is about just proof writing and verifying some other conjecture that somebody wrote down. So for example, in my particular area, And again, I want to be clear, you know, I'm an applied guy. A lot of pure math magicians would absolutely look down on what I'm doing and say, ah, that's not real math. I would say otherwise, but we'll have that debate another day. By the time we get to writing a proof, we probably already know it's true. It's more a double-check situation, but the really interesting part is the modeling and figuring out which questions to ask and what insight they bring us to the universe. The AI tools are not doing that. You're right, maybe proof writing becomes more of an exercise in writing prompts. [60:00] But writing the prompts is very much what we've been teaching our students to do for centuries now, and that skill, I think, persists. I mean, broadly, it feels like the emphasis is shifting to being able to ask the right questions, come up with new ideas versus just... come up with a basic answer. That's right. And the typical math student probably doesn't end up in theoretical math. They maybe end up in finance. They end up in engineering. And for those folks, that skill is the really critical one, right? It's exactly what you need to be a good manager, to be a good project designer, to lead and innovate in all kinds of different fields. And that skill is somehow becoming more important than ever. All that boring proof writing stuff that everybody's allergic to is actually kind of being pushed to the side of it. So we should still teach kids math. I think so. Justin Solomon, thank you so much for coming on Nodlog. Of course, it's a pleasure. Well, I think, look. To say the least... There's the future of sort of math and math education. It's like, we have no idea where this is going, right? Because the models are going to keep getting better. And even just the sort of like prosaic point about like, okay, here you have some conference that used to have 1000 submissions, and now it has 60,000 submissions. And some of them might even be very good. But it's like, That's the type of thing that just like, as he said, it like breaks existing systems pretty straightforward. Yeah, it feels like there's a tautology here with AI where... Because of AI, a lot of things are getting easier and more difficult at the same time, right? So you have the ability to generate tons and tons of content, but the ability to verify it doesn't necessarily go up at the same amount of time. Maybe it will one day. I also think the emphasis on questions rather than simple answers and the idea of taste, even in something like mathematics, which is… Kind of alien to me. You know, you hear about it in code all the time, the ability to write elegant code. And I guess the same applies to mathematics, the ability to elegantly prove a formula or elegantly come up with a theory. That seems to be repeated throughout everything nowadays. If you're able to generate ideas, think outside the box, to use a terrible cliche, and curate with some taste or write or perform with a particular amount of taste, that seems to be the edge case here. I would also say this is actually a pretty high-stakes question, even outside of the academic setting, sort of research taste. Because, of course, one of the big... questions for AI period is this idea of recursive self-improvement. Can a model build a successor model? And to do that means generating new ideas to then experiment on. And I think it's a pretty big open question still, whether the models are good at novel questions, right? So it's like, and There's reason to think they're solvable. Okay, maybe the models get very good at that. Can the model come up with the next Navier-Stokes? Can it come up with the next P equals NP? Can it come up with the next Fermat's last theorem? That's in math. But whatever the equivalence of that is in sort of areas like machine learning and so forth, it's a pretty important question for the future of AI. It's like, are the models good at proposing solutions? the next research experiment, and then, of course, carrying out, conducting the test. Right. And they might be one day. But right now, there's clearly still a pretty significant role for a human in the lube. Right. To sort of like, I guess that's intuition or whatever. It's like, okay, let's try this out, et cetera. And my impression is that the models still aren't great at that particular aspect of, I guess, the scientific method. Yeah, I think that's what Justin... Yeah, as well. So that's that gets me back to this idea that like the human edge remains the idea generation. Yeah. Out of the box thinking, whatever you want to call it out of the the AI box out of the chip thinking. That's out of the chip thinking. That's good. Yeah. Shall we leave it there? Let's leave it there. All right. This has been another episode of the All Thoughts podcast. I'm Tracy Alloway. You can follow me at Tracy Alloway. And I'm Joe Weisenthal. You can follow me at The Stalwart. Follow our producers. Kale Brooks and Kevin Lozano at Kevin Lloyd Lozano. And you can check out all of these topics 24 seven in our discord discord. Dot G G slash odd lots. And for more odd lots content, check out our daily, [65:00] newsletter, you can find that at Bloomberg.com/otline. And if you enjoyed this conversation, if you like it when we talk about mathematics, then please let us know on your favorite podcast platform. If you're watching on video and you enjoyed the show, then please leave us a comment or like the video or better yet, subscribe. And remember, if you're a Bloomberg subscriber, you can listen to all of our episodes absolutely ad free. All you need to do is find the Bloomberg channel on Apple Podcasts and follow the instructions there. Thanks for joining. Thank you. Some people use ChatGPT to answer questions. Others use it to get real work done. With ChatGPT Work, you bring the goal, plus real inputs like briefs, notes, files, feedback, data, and project plans. 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