文 | 数智奔流，作者 | 田渝，编辑 | 沈校
10月26日，OpenAI又公布了数学、计算机科学和物理学领域的三项新发现。
这次，OpenAI没有把它们包装成尘埃落定的科学成果，公司将相关手稿称为有待进一步理解的原始研究材料，邀请外部专家组织研究团队，核查、解释这些结果。
与此同时，人类数学协会（AHM）最新公布的名单显示，协会已经有全球1008位成员加入，其中654人加入“无AI研究小组”，他们承诺不发表由AI生成的论文、不与前沿实验室合作、不在研究中使用AI。
如果AI能在短时间内生成数百份手稿，数学家却要花几个月甚至几年去理解和验证，那么，研究速度变快意味着什么？答案越来越容易得到，人类对知识的理解也变深了吗？
菲尔兹奖得主陶哲轩，正是提出这一问题的数学家之一。在10月初OpenAI一次性发布722份AI生成数学手稿时，陶哲轩转载了AHM的客座声明，号召全球数学界集体行动，呼吁数学家停止与OpenAI合作。
在《新科学家》最新一期对他的专访中，他承认AI在数学解题方面取得的惊人进步。他担忧的，是数学研究中某些更重要的东西，在追逐速度和成果的过程中被忽略。
AI目前已经能解决人类长时间苦苦追寻的难题，我们是不是也正在失去从解决问题过程中学习、创造和理解的机会？
问：数学和AI会如何相遇？我们该如何看待数学蕴含的力量与希望？
陶哲轩：对于数学来说，这是一个剧烈动荡的时代。数学有许多不同的侧面，AI正在极大强化其中的一些侧面，却几乎没有触及其他侧面。
数学有很大一部分是解答问题，这一面恰好非常适合AI。可是，数学也关乎理解。我们不想只让黑箱给出答案，自己却不知道答案从何而来。
数学教会我们认识世界：某些事物彼此关联，某些事物彼此相似。数学是把其他所有科学连接起来的粘合剂，这一方面还没有受到AI的影响。
问：OpenAI宣称解决了纳维—斯托克斯千禧年难题的一个版本。当这样一项重大成果由AI生成，而不是由数学家多年研究得出时，这份证明究竟缺少了些什么？
陶哲轩：纳维—斯托克斯问题是：方程能否预测奇点的出现，或者说，能否形成奇点，也就是所谓的“爆破”？你能不能从某种非常平滑的水的初始状态出发，让水不断运动，并在某个时刻使速度或压力变成无穷大？长期以来，人们一直认为这种情况不会发生。
大约十年前，学界共识开始转变。我们发现一些类似方程，那些方程确实会形成奇点。也许可以大量使用机器学习和AI，寻找那些会发展成奇点的解的良好候选。我们一度已经非常接近解决这个问题。
一些数学家只差一点就能构造出奇点，大概就在OpenAI宣布结果的前一天。实际情况似乎是，OpenAI团队启动了一项紧急攻关计划。他们启用了自己最大的模型，投入价值数百万美元的算力，只为抢在前面完成。我从未见过如此残酷的竞争。
这项成果留下一份长达100页的AI生成证明，令人失望的是，我们几乎没有从中获得数学洞见。一位同事借助AI钻研数日也只能解释一半，后续研究方向更不明确。一些研究者还担心成果被AI抢先，原本活跃的领域陷入混乱。
那份证明最终会被理解，但这次的做法很不理想，这是一种自动化掠夺式开采。
问：在思考数学时，有哪些容易落入的陷阱？
陶哲轩：我认为世界有三个层次：现实本身、观察到的数据、以及用来解释数据的模型。模型再精确也是对现实的近似。我们要知道模型能解释什么，还必须清楚它何时失效。
2008年金融危机的例子非常典型，许多金融模型低估了不同地区房贷违约之间的关联，风险爆发时模型假设不再成立，原本看似已经对冲的风险集中暴露，导致金融工具大规模失效。因此，理解模型的边界，与理解模型本身同样重要。
问：数学为什么需要暂时脱离现实？
陶哲轩：数学像解题的游乐场，我们经常使用“玩”的语言，我把数学看作成年人的游戏，一种确实有用的游戏。我们从“玩”一个问题中学到的东西，往往又能帮助我们解决更实际、更贴近现实生活的问题。有时候，解决一个真实问题的最好办法，恰恰是先解决一个非常不现实的问题，它会给你一个基准。
问：对于刚开始学数学的学生，你会如何向他们证明，枯燥的数学仍然值得学习？
陶哲轩：数学教育要经历三个阶段。第一个是“前严谨阶段”，对应小学。在这个阶段，你只是学习数和图形等概念，很多孩子喜欢这个阶段，因为它与现实世界相连，而且不用精确得出答案。随后是“严谨阶段”。在这个阶段，要学习各种古怪的公式，必须把东西算对。最后是“后严谨阶段”。一旦你已经具备正确进行计算的严谨训练，就可以重新回归更偏直觉的思考方式。
到最后阶段，数学才真正变得有趣。很遗憾，人们几乎要到研究生阶段才开始看到这一点。至少如今网上有很多机会，你可以找到由喜欢解各种谜题的人组成的社群。人们在用一种非常有趣、容易亲近的方式接触数学。要做到这一点，往往需要某种非线性、非传统的教育环境。
问：现实世界如何影响数学家的思考方式？
陶哲轩：的确，我们思考抽象的数学对象时，往往要找到现实中的对应物，而我们对那个对应物已经有所经验。有时，人们也会借助游戏。你可以把某些类型的数学问题看成游戏。
通常情况下，在平面上沿一个方向一直走下去，并不会从另一侧绕回来。但有些数学空间是弯曲的，一些游戏也一样。就像《吃豆人》：你从一边走出去，就会从另一边回来。
游戏可以教会你各种奇特的几何。现实世界中有很多经验，都能为我们思考抽象数学对象提供新的方式。
问：AI本身能否成为一个数学问题？
陶哲轩：当然，AI的理论还非常不完整。以语言模型为例，它们通过预测下一个词就能生成连贯的文本，甚至解决数学问题。我们还没有真正理解模型究竟如何从海量数据中学到这些规律。它们有时表现出惊人的能力，有时却会犯下难以预料的错误。
所以，我们需要发展新的数学理论，解释AI的能力与局限。随着AI越来越重要，研究它本身也将成为数学家的重要工作。
问：有哪些数学工具能推动这个过程？
陶哲轩：这涉及几个不同的分支，最优化理论帮助改进模型训练，拓扑数据分析可以研究数据的结构，不确定性量化能评估模型预测的可靠程度。此外，信息论和计算复杂性理论也能帮助理解模型需要多少参数、多少数据和多少计算资源。
现在我们主要依赖经验法则，比如香农的信息论为通信建立了成熟的数学基础，让我们能够计算信息容量、分析传输极限。相比之下，对于智能或数据复杂度却没有相应概念，不知道AI的极限在哪。
问：AI的风险在哪？
陶哲轩：AI非常擅长解数学谜题，而事实证明，在网络安全领域，攻击者试图侵入一个不安全的系统，其实与解数学谜题非常相似。系统有一定的规则，那些擅长数学的AI，也擅长侵入系统。现在人们给AI布置任务时，往往会说：“不惜一切手段解决这个问题。”我认为，AI开始变得危险。
Lean可以用来检验证明，但它不是一门完美的语言。人们已经在Lean本身发现一些可被利用的漏洞。AI开始不再老老实实地解题，而是通过寻找Lean中的漏洞来作弊，构造一份假证明，再让它蒙混过关。如果我们没有防护措施与监督，只是交给它们任务，那迟早会出问题。
问：我们掌握的数学，足以建立良好的防护栏吗？
陶哲轩：不够。AI技术发展的太快了。单就数学而言，利用现有的工具已经能够做出很多优秀的数学研究。从我们的角度出发，实在没有必要进一步加速；再考虑到所有安全风险，这样做不值得。
现在的科技巨头陷入一种囚徒困境：“我们必须加速，否则竞争对手就会抢在前面。”这是一种极其有害的互动。没有人愿意走得这么快，但每个人都觉得自己别无选择。
问：哪些领域的研究人员可以帮助数学家解决这个问题？
陶哲轩：这会是一门高度跨学科的科学。首先，你需要从数据中总结经验，所以需要数据科学家。还有神经科学家、哲学家、教育家等等。如今，数学将触及一切。尤其是在AI时代，我们必须与所有人交流。
问：除了AI，还有什么数学问题是你希望能在未来取得进展的？
陶哲轩：我个人最喜欢的问题之一是孪生素数猜想：是否存在无穷多对相差恰好为2的素数？我喜欢它，不只是因为问题本身，还因为解决它可能需要一种理解素数的新方法。我不想要一份长达1000页，却不能给我带来任何洞见的证明。
问：素数问题吸引数学家数千年，为什么这类抽象问题能在漫长岁月里让这么多人深深着迷？
陶哲轩：有些问题很容易理解，却很难解决。1900年，希尔伯特提出了著名的23个数学问题，其中包括黎曼猜想。他相信人类能够不断推进数学知识，并以一句话表达这种信念：“Wir müssen wissen, wir werden wissen.”——“我们必须知道，我们必将知道。”
AI 正在解决越来越多的问题，也终将遇到现有方法无法跨越的障碍，而新的数学前沿可能就在那里。
这个世界上还有更紧迫的事情，是我们可能来不及做、又必须去做的。不过，数学嘛，我们完全没有把它做完的危险。
陶哲轩的这句话听起来充满乐观，但AI正在改变数学的游戏规则，人类数学家的角色，也需要重新定义。
10月22日，陶哲轩在《Nature》发表文章《数学能否在AI时代生存？》。文章里阐述了他对Math 2.0的构想。
他认为，传统数学思路难以维系，数学家需要更整体的视角评价对学科的贡献。
与此同时，AI有望催生数学研究新范式，比如让它生成分析一个包含数百万个难度各异的数学问题的数据集，从中挖掘意想不到的规律，利用规律提出有价值的新问题。
但陶哲轩尤其强调，数学不能完全依赖大型AI企业，数学共同体要建立自己的开放模型和研究工具，让研究者能够参与训练、检查和改进模型，并保留对AI研究过程的监督和干预权。
这不是一种空想，早在9月份他就在博客中倡导推动SAIR开放数学模型计划，试图让数学家掌握自己的AI研究基础设施。
另一条技术路线也在快速推进：10月26日，AI数学研究公司Axiom Math公布重大研究成果，AI系统AxiomProver在Lean 4中完成了有限单群分类定理（CFSG）的形式化证明，这是数学史上规模最大的证明之一。
简单地说，这就像让AI接手一套由上百位专家、数百本说明书共同编写的超大型工程手册。它要读懂所有内容，还要把每一个步骤都转换成计算机能检查的指令，核对前后逻辑是否成立。
令人意外的是，团队发现了34处原有文献中需要处理的显著差异。其中有一处涉及不平衡群定理的关键论证，确实需要新的数学推导。
这项工作的意义早就超出了证明一道数学题，这一次，AI开始尝试把分散的知识连接，检查它们之间的逻辑关系，甚至发现人类研究中长期遗留的问题。
SAIR与Axiom，两条路径指向一个答案：AI，逐渐成为数学研究基础设施的一部分。
那么，人类数学家又该把精力放在哪里？
多伦多大学数学系助理教授丹尼尔·利特提出另一种可能：AI生产成果和人类理解可以同时增长，过去数学界常把创造数学成果和培养高质量数学家两个目标混在一起，AI在改变前者的实现方式，不一定意味着后者失去意义。教育与其要学生反复完成AI擅长的计算，不如重视口头答辩、学术讨论和独立解决新问题的能力。
对于下一代数学家，他们面对的是一张不断扩展、随时可以检验和探索的知识地图，数学的下一段历史正在展开。
原视频链接（内容有删减）：https://www.youtube.com/watch?v=EYrlyxuyFbs
本文由 AI宝库自动抓取并翻译，版权归原作者所有；如需查看原文请点下方链接。
♪ We've got a lot of ideas ♪
On 26 October, Open AI published three new discoveries in the fields of mathematics, computer science and physics.
On this occasion, Open AI did not package them into dusted scientific results, and the company referred to the manuscripts as original research materials to be further understood and invited external experts to organize research teams to verify and interpret the results.
At the same time, the newly published list of the Association of Human Mathematics (AHM) shows that it has 1008 members worldwide, of whom 654 are members of the “No-AI Study Group”, who undertake not to publish AI-generated papers, not to cooperate with forward laboratories and not to use AI in research.
If AI produces hundreds of manuscripts in a short period of time, and mathematicians take months or even years to understand and validate, what does a faster pace of research mean? Answers are becoming more and more accessible, and does human understanding of knowledge become deeper?
The Philtz Prize winner, Tao Chul-chul, was one of the mathematicians who raised the issue. At the beginning of October, when Open AI had one-time publish722 AI-generated math manuscripts, Tao Chul-chul had reproduced AHM ' s guest statement, calling for collective action by the global mathematics community and for mathematicians to stop working with Open AI.
In his latest interview with the New Scientist, he acknowledged the remarkable progress made by AI in the mathematical solution. He was worried about something more important in mathematical research, which was ignored in the pursuit of speed and results.
Has AI been able to resolve the long-sought-after dilemma of humanity, and are we also losing the opportunity to learn, create and understand from the solution?
Q: How will math and AI meet? How do we look at the power and hope of mathematics?
Tao Cheol-chul: It's a turbulent time for math. Mathematics has many different sides, and AI is greatly strengthening some of them, but barely touching others.
A lot of math answers questions, and that's exactly what's right for AI. But math is about understanding. We don't want the black box to give answers, but we don't know where they come from.
Math teaches us the world: things are connected, things are similar. Mathematics is the adhesive that connects all other science, and this aspect has not been affected by AI.
Q: Open AI claims to have solved a version of the Navi-Stokes millennium dilemma. When such a significant result is generated by AI rather than by years of research by mathematicians, what is missing from this evidence?
The question is: Can the equation predict the emergence of a singularity, or can it create a singularity, the so-called “explosion”? Can you start from the initial state of some very smooth water, keep it moving and at some point turn speed or pressure into infinite? It has long been thought that this will not happen.
About 10 years ago, the academic consensus began to change. We found some similar equations, and those equations do create a singularity. Perhaps a lot of machine learning and AI can be used to find good candidates for solutions that can develop into wonders. We were once very close to solving this problem.
Some mathematicians were just a little bit close to building a singularity, about the day before Open AI announced its results. The fact seems to be that the Open AI team initiated an emergency clearance plan. They used their largest model and invested millions of dollars in calculations, just to get ahead of them. I've never seen such a cruel competition.
This result leaves behind a 100-page AI generation proof that, disappointingly, we have little mathematical insight from it. One colleague had been able to explain for only half of the days with AI, and the direction of the follow-up study was even less clear. Some researchers are also concerned that the results are ahead of AI and that areas that were already active are in disarray.
The certificate will eventually be understood, but this is not an ideal exercise, an automated predatory exploitation.
Q: What are the easy traps when thinking about mathematics?
Tao Cheol-chul: I think there are three levels of the world: reality itself, observed data and models used to interpret data. Modelling is also a approximation of reality. We need to know what the model explains, and we need to know when it will fail.
The example of the 2008 financial crisis is very typical, with many financial models underestimating the linkages between mortgage defaults in different regions, with a model assumption no longer valid at the time of the risk outbreak and a risk concentration exposure that appeared to have hedged, leading to a large-scale failure of financial instruments. Understanding the boundaries of the model is therefore as important as understanding the model itself.
Q: Why does math need to be out of reality?
Tao Cheol-chul: Mathematical-like playgrounds, where we often use “playing”, I see math as an adult game, a really useful game. What we learn from “playing” a problem can often help us to solve problems that are more realistic and closer to reality. Sometimes the best way to resolve a real problem is to first address a very unrealistic problem, which will give you a benchmark.
Q: How will you prove to students who have just started studying mathematics that boring mathematics is still worth learning?
Tao Chul-chul: Math education goes through three stages. The first is the “pre-cautionary phase”, which corresponds to primary schools. At this stage, you're just learning the concepts of numbers and graphics, and many kids like it because it's connected to the real world and doesn't have to be precise. This was followed by a “severe phase”. At this stage, to learn all the weird formulas, things have to be right. Finally, there is the “post-cautionary phase”. Once you are properly trained to calculate, you can return to a more intuitive way of thinking.
At the final stage, math becomes really interesting. Unfortunately, it is almost at the post-graduate level that people begin to see this. At least there are many opportunities on the Internet today to find communities of people who like to solve puzzles. People are getting in touch with mathematics in a very interesting, accessible way. This often requires a non-linear, non-traditional educational environment.
Q: How does the real world affect the way mathematicians think?
TIJ: Indeed, when we think about abstract mathematical objects, we often have to find the counterpart in reality, and we have some experience with that counterpart. Sometimes people use games. You can see certain kinds of mathematical problems as games.
As a rule, the plane continues in one direction and does not return from the other. But some mathematical space is curved, and so are some games. Like Bean Eater: You walk out of one side, you come back from the other.
The game teaches you all sorts of strange geometry. There is a lot of experience in the real world that can provide us with new ways to think of abstract mathematical objects.
Q: Can AI itself be a mathematical question?
Tao Cheol-ho: Of course, AI's theory is very incomplete. In the case of language models, they generate consistent text by predicting the next word, and even solve mathematical problems. We still don't really understand how models learn from big data. They sometimes display amazing capabilities and sometimes make unexpected mistakes.
So we need to develop new mathematical theories to explain AI's capabilities and limitations. As AI becomes more important, research itself will become an important job for mathematicians.
Q: What are the mathematical tools that drive this process?
Tao Cheol-chul: This involves several different branches, optimizes the theory to help improve model training, expands data analysis to look at the structure of data, and uncertainty quantization assesses the reliability of model predictions. In addition, the theory of informatics and computational complexity can help to understand how much parameters, data and computing resources the model requires.
Now we rely primarily on empirical laws, such as Shannon's informatics, to create a mature mathematical basis for communication, allowing us to calculate the amount of information, analyse the transmission limits. In contrast, there is no equivalent concept of intelligence or data complexity and it is not known where the limits of AI are.
Q: What are the risks for AI?
Tao Cheol-chul: AI is very good at solving mathematical puzzles, and in the area of cybersecurity, it turns out that the attackers are trying to hack into an insecure system, which is very similar to the one that solves mathematical puzzles. There's some rules in the system, and those AIs that are good at math, they're good at hacking into the system. Now, when people give AI the task, they often say, “Do whatever it takes to solve this problem”. I think AI is getting dangerous.
Lean can be used to test the proof, but it is not a perfect language. People have found some exploitable loopholes in Lean itself. AI began not to solve the problem in an honest manner, but to cheat by looking for a loophole in Lean, to construct a false certificate and then to confuse it with cleared. If we do not have protection and oversight, we simply entrust them with the task, and sooner or later something will happen.
Q: Do we have enough mathematics to build a good fence?
Dao Cheol-chul: Not enough. AI's technological development is too fast. Mathematically alone, many good mathematical studies can already be done using existing tools. From our point of view, there is no real need for further acceleration; it is not worth it, given all security risks.
Now the technology giant is in a prisoner's predicament: “We have to speed up, or the competition will be ahead.” This is an extremely harmful interaction. No one is willing to walk so fast, but everyone feels they have no choice.
Q: What areas can researchers help mathematicians to solve this problem?
Tao Cheol-ho: This is a highly interdisciplinary science. First, you need to learn from data, so you need data scientists. There are neurologists, philosophers, educators, etc. Now, math will touch everything. Especially in the AI era, we must communicate with everyone.
Q: What are the mathematical questions besides AI that you want to make in the future?
Tao Cheol-chul: One of my personal favorite questions is the twin numbers guess: Is there an infinite difference of two? I like it, not only because of the problem itself, but also because it may require a new method of understanding the prime. I don't want a 1,000-page proof, but I can't get any insight.
Q: Why do these abstract questions, which attract mathematicians for thousands of years, become so fascinating to so many people over a long period of time?
Tao Cheol-chul: Some problems are easy to understand but difficult to solve. In 1900, Gilbert raised the famous 23 mathematical questions, including Lehman's guess. He believed that humans could continue to advance the knowledge of mathematics and express it in one sentence: “Wir müssen wissen, wir wwerden wissen.” — “We must know, we must know.”
AI is addressing an increasing number of problems and will eventually encounter obstacles beyond the reach of existing methods, where the new mathematical frontier may be.
There are more urgent things in this world that we may not have time to do, and we must do. But, math, we're in no danger of doing it.
Tool Cheol-chul sounds optimistic, but AI is changing the rules of mathematics and the role of human mathematicians needs to be redefined.
On 22 October, Tao Chul-hyun published in Nature, " Can mathematics survive in the AI era? " The article sets out his vision of Math 2.0.
In his view, traditional mathematical thinking was difficult to maintain and mathematicians needed a more holistic perspective to evaluate their contribution to the discipline.
At the same time, AI is expected to generate new paradigms for mathematical research, such as enabling it to generate a data set that analyses millions of difficult mathematical issues, from which unexpected patterns are extracted, and to use them to raise valuable new questions.
But Tao Chul-hyun emphasizes in particular that mathematics cannot be entirely dependent on large AI firms, and that the Mathematics Community needs to develop its own open models and research tools to enable researchers to participate in training, inspection and improvement of models, and to retain oversight and intervention in the AI research process.
It's not an empty mind, and as early as September he advocated in his blog the promotion of the SAIR Open Mathematical Model project, trying to get mathematicians to take control of their AI research infrastructure.
Another technical route is moving fast: on October 26, Axiom Math, AI Mathematics, published major research results, and AxiomProver completed the formalization of the limited fusion taxonomy (CFSG) in Lean 4, one of the largest in mathematical history.
To put it simply, it is like having AI take over a super-large engineering manual prepared by hundreds of experts and hundreds of this note. It has to read everything and translate every step into a computer-readable command to check whether the logic is valid or not.
Surprisingly, the team found significant differences to be addressed in 34 original literature. There is a key argument for an unbalanced group theorem that does require new mathematical reasoning.
This work has long gone beyond proof of a mathematical question, and this time AI began to try to connect dispersed knowledge, examine the logical relationship between them and even discover the long-term legacy of human research.
SAIR and Axiom, two paths point to one answer: AI, becoming part of the mathematical research infrastructure.
So, where do human mathematicians put their energy?
Daniel Litt, Assistant Professor of Mathematics, University of Toronto, suggested another possibility: AI production and human understanding could grow at the same time, and in the past the mathematics community used to mix the goals of creating mathematical results and developing high-quality mathematicians. Together, AI is changing the way the former is achieved, which does not necessarily mean that the latter is meaningless. Education does not require students to repeat AI ' s best calculations, but rather to focus on oral pleadings, academic discussions and the ability to solve new problems independently.
For the next generation of mathematicians, they are confronted with an ever-expanding, ready to be tested and explored knowledge map, and the next history of mathematics is under way.
Original video link (reduced): https://www.youtube.com/watch?v=EYrlyxuyFbs
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