Maths enters its AI era
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本期Babbage节目探讨了人工智能给数学领域带来的深刻变革。大型语言模型（LLMs）曾以不擅长算术而臭名昭著，但如今已迅速进步，能够解决复杂的数学问题，这改变了数学家们的看法。
前数学家、《经济学人》记者Anjani Trevedi指出，传统的研究数学旨在通过严谨、优美的证明来证实基本真理，这与学校里教授的死记硬背式算术截然不同。几个世纪以来，数学家们一直依赖逻辑演绎，并常在前人工作的基础上进行构建，由此形成的知识体系虽然坚实，但也极为分散，难以驾驭。长期以来，计算机一直用于大规模计算和检查复杂配置，但现代人工智能代表了一个全新的前沿领域。
最初，鉴于人工智能在基础数学方面表现不佳，数学家们对其能力持怀疑态度。然而，在过去一年中，情况发生了巨大变化，LLMs解决了许多长期存在的难题，如部分埃尔德什问题，现在正着手处理更复杂的证明。菲尔兹奖得主、数学家陶哲轩将人工智能描述为一名“非常聪明的本科生”，它与人类不同，能够不倦地将数百种方法应用于数千个问题。虽然人工智能通常解决的是“唾手可得的成果”——即那些相对容易但数量众多且未曾受到人类关注的问题——但它能够将数学中不同子领域的方法结合起来，这是人类专家鲜少能做到的壮举，这为发现新的联系和见解提供了巨大潜力。
数学领域的一个主要瓶颈是需要对证明进行严格验证。人工智能模型虽然强大，但仍会犯错，陶哲轩估计其错误率高达98%。这使得建立一个可靠的系统来确保信任变得至关重要。正在出现的解决方案是“自动形式化”，这是一个将自然语言数学证明自动转换为计算机代码的过程，这些代码可以进行形式化验证以确保其正确性。DARPA的Patrick Shafto将当前庞大的数学知识体系描述为由论文和思想组成的“一团糟”。自动形式化旨在创建一个统一、一致且可由机器验证的知识库，这将加速进展、减少重复工作，并使高等数学更容易获取。
在此领域的一个里程碑式成就，是玛丽娜·维亚佐夫斯卡（Marina Vyazovska）因解决八维球体堆积问题而获得菲尔兹奖的证明被形式化。Math Inc. 公司开发的名为Gauss的人工智能系统，成功地将她耗费数年时间才完成的证明转换成可由计算机检查的代码。维亚佐夫斯卡指出，虽然人工智能的输出可能会冗长（例如，明确证明“2+2=4”），但这一过程提供了对所用复杂方法更深入、更形式化的理解。
人工智能在数学领域的愿景包括由Harmonic等公司创建“数学超级智能”，或像GitHub那样的平台（如谷歌的Defined），研究人员可以在这些平台上发布、下载并即时验证证明。数学家们将人工智能视为一个强大的工具，可以提高他们的生产力，探索更多想法，并连接不同数学领域之间的点，最终拓宽他们的智力视野。
然而，挑战依然存在。人工智能模型可能会“幻觉”或过于轻易地接受人类的提示，这可能会损害原创性。此外，人工智能系统摄取并随后形式化或在此基础上构建的人类原创工作的归属和署名权问题也至关重要。维亚佐夫斯卡与Math Inc.的合作经历凸显了，随着人工智能越来越多地介入数学家的工作，制定明确的伦理准则的必要性。
最终，共识认为，人工智能的作用将是处理数学证明和形式化中的“繁重工作”和技术细节，从而创建一个高度组织化且可验证的知识库。反过来，这将使人类数学家能够专注于数学中真正的创造性、直觉性和概念性方面，推动探究的边界，并可能在其他科学领域带来突破。尽管人工智能最终可能会自主生成新颖的证明，但其效用以及由谁来“消费”和解释这些证明，仍然是一个社会挑战。
This episode of Babbage explores the profound transformation underway in the world of mathematics due to artificial intelligence. Once notoriously bad at arithmetic, large language models (LLMs) have rapidly advanced to solve complex mathematical problems, changing mathematicians' perspectives.
Anjani Trevedi, a former mathematician and correspondent for The Economist, highlights that traditional research mathematics focuses on proving fundamental truths through rigorous, aesthetically pleasing proofs, unlike the rote arithmetic taught in school. For centuries, mathematicians have relied on logical deduction, often building upon prior work, creating a body of knowledge that, while robust, is also widely dispersed and can be difficult to navigate. Computers have long been used for large-scale calculations and checking complex configurations, but modern AI represents a new frontier.
Initially, mathematicians were skeptical of AI's capabilities given its struggle with basic math. However, the landscape shifted dramatically in the past year, with LLMs solving long-standing puzzles like some of the Erdős problems and now tackling more complex proofs. Terence Tao, a Fields Medal-winning mathematician, describes AI as a "really bright undergraduate" that, unlike humans, can tirelessly apply hundreds of methods to thousands of problems. While AI often solves the "low-hanging fruit" – problems that are relatively easy but numerous and haven't received human attention – its ability to combine methods from disparate sub-fields of mathematics, a feat rarely achieved by human specialists, offers significant potential for discovering new connections and insights.
A major bottleneck in mathematics is the need for rigorous verification of proofs. AI models, though powerful, still make mistakes, with Tao estimating an error rate of 98%. This necessitates a robust system to ensure trust. The emerging solution is "auto-formalization," a process that automatically translates natural language mathematical proofs into computer code that can be formally verified for correctness. Patrick Shafto of DARPA describes the current body of mathematical knowledge as a "hot mess" of papers and ideas. Auto-formalization aims to create a unified, consistent, and machine-verifiable repository, which would accelerate progress, reduce redundant efforts, and make advanced mathematics more accessible.
A landmark achievement in this space is the formalization of Marina Vyazovska's Fields Medal-winning proof for the 8-dimensional sphere packing problem. An AI system called Gauss by Math Inc. successfully converted her proof, which took her years to develop, into computer-checkable code. Vyazovska notes that while the AI's output can be verbose (e.g., explicitly proving "2+2=4"), this process provides a deeper, formalized understanding of the complex methods used.
The vision for AI in mathematics includes the creation of "mathematical superintelligence" by companies like Harmonic, or GitHub-style platforms (like Google's Defined) where researchers can publish, download, and instantly verify proofs. Mathematicians view AI as a powerful tool to enhance their productivity, explore more ideas, and connect dots across different mathematical domains, ultimately broadening their intellectual aperture.
However, challenges remain. AI models can "hallucinate" or agree too readily with human prompts, potentially undermining originality. There are also critical issues concerning credit and attribution for human-generated work that AI systems ingest and then formalize or build upon. Vyazovska's experience with Math Inc. highlights the need for clear ethical guidelines as AI increasingly engages with mathematicians' work.
Ultimately, the consensus suggests that AI's role will be to handle the "grunt work" and mechanics of mathematical proof and formalization, creating a highly organized and verifiable knowledge base. This, in turn, could free human mathematicians to focus on the truly creative, intuitive, and conceptual aspects of mathematics, pushing the boundaries of inquiry and potentially leading to breakthroughs in other scientific fields. While AI may eventually generate novel proofs autonomously, the question of their utility and who will "consume" and interpret them remains a societal challenge.
中英文字稿 
本期Babbage节目由IDA爱尔兰支持。IDA爱尔兰在欧盟中拥有最高的人均STEM（科学、技术、工程和数学）毕业生比例，能够帮助您找到国际化和发展的所需技能。访问idaireland.com了解更多信息。现在可以大规模地进行数学运算，这是个改变游戏规则的因素。我们逐渐接近一个转折点，事物将会发生重大变化。这至少和互联网的影响不相上下。在数学领域中，一场深刻的变革正在进行。过去我们依靠人工计算，后来有了计算器和电脑，因此能够进行大规模计算。而现在，你有能力去推理非常复杂的事情。在未来一到两年内，人工智能将成为数学家的工具，如果不使用它，实际上就是在无缘无故地让自己处于劣势。
▶ 英文原文 ⏱
This episode of Babbage is supported by IDA Ireland. With the highest share of STEM graduates per capita in the EU, IDA Ireland can help source the skills you need to internationalize and thrive. Visit idaireland.com to learn more. The fact that you can now do math at scale is just a, that's the game changer. We're getting close to this place where like things are really going to shift. It's at least as impactful as the interaction of the internet. There's a profound transformation underway in the world of mathematics. We used to have human computers before and then calculators came and computers came and we were able to do these large calculations. Now you have this ability to reason about very complex things. Within one or two years AI will just be a tool that if you don't use it as a mathematician you're putting yourself at a disadvantage for no reason at all really.
在这个时候，你可能觉得AI改变某个领域的性质已经不再听起来那么革命性了。不过，请记住，就在几年前，大型语言模型在算术方面表现得非常糟糕。然而，随着AI模型的进步，它们的解决问题、推理和数学能力也在提高。我真的希望如果解决千禧年大奖问题需要我们花费一百年，也许当AI和人类结合在一起时，我们可以缩短这个时间。我不知道，可能是五年或者十年。最近，很难不注意到关于前沿AI模型解决越来越复杂的数学问题的新闻。例如，OpenAI的GPT-5就证明了一项数论中的猜想，这个猜想已经困扰了学界数十年。这个问题涉及被称为原始集合的整数群。
▶ 英文原文 ⏱
At this point you're thinking AI changing the nature of everything in a given field it doesn't sound revolutionary anymore. But just remember that only a few years ago large language models were famously awful at arithmetic. As AI models have got better though so too have their problem solving, reasoning and mathematical skills. I do really hope that if it might take us a hundred years to solve millennium price problems, maybe when AI and humans combine together we can shorten the times. I don't know, perhaps five years or ten years. Recently it's been hard to miss the headlines about frontier AI models solving increasingly complicated problems in maths. GPT-5 from OpenAI for example proved a conjecture in number theory that had lain unsolved for decades. The problem concerns groups of whole numbers known as primitive sets.
以下是这段文字的中文翻译，力求表达清晰易读：
这些数字集合中没有任何数字可以被另一个数字整除。AI在一些人类提示的帮助下，采用了一种前所未有的解决方法。如果AI继续沿着这条道路发展，并且对数学家变得更加有用，能够揭示隐藏的结构并加速人类最基础的研究领域的进步，那将会带来怎样的结果呢？我是阿洛克·贾，这是来自《经济学人》的《巴别塔》。今天的话题是数学家为什么突然对AI产生了兴趣。今天与我一起讨论的是阿贾尼·特雷韦迪，她是《经济学人》的一位记者，也是前数学家。阿贾尼，谢谢你的参与。谢谢邀请我。
▶ 英文原文 ⏱
These are where no numbers in the set can be divided evenly by any other. The AI, with some human prompting, used a method to solve it that no human had ever thought to deploy before. If AI continues down this path and becomes even more useful for mathematicians, if it can uncover hidden structures structures and accelerate the progress in the most fundamental of branches of human inquiry, where could that lead? I'm Alok Jha and this is Babbage from The Economist. Today, how mathematicians suddenly became interested in AI. With me on today's show is Anjani Trevedi, a correspondent here at The Economist and a former mathematician herself. Anjani, thanks for joining me. Thank you for having me.
安贾尼，就告诉我你为什么想关注数学和人工智能领域。是什么让这个故事对你而言值得报道？这是个有趣的问题。我在大学时学习过数学，现在无论我们在报纸上如何报道或写作，人工智能就在四处可见，你会想尝试，但又不确定如何让孩子接触这一领域。随后，我开始注意到很多关于人工智能解决问题和数学奥林匹克问题的消息，这让我好奇。这是一个关于我曾经努力学习了四年的领域的存在性问题。如果人工智能能解决这些问题，那我之前所做的一切又有什么意义？于是，我开始深入了解人工智能如何影响数学，数学家对此如何看待，以及人工智能与数学之间到底发生了什么。
▶ 英文原文 ⏱
Anjani, just tell me why you wanted to look at the field of mathematics and AI. What made this story important for you to report on? It's an interesting question. I had studied math in university and so as everywhere we look, as we're reporting and writing for the paper, you hear about AI, you know, you're compelled to try it out, you're not sure what to do with your kids in AI. And then I started seeing a lot of noise about how AI was solving problems and math Olympiad problems and it just made me wonder. It was an existential question over a field that I had toiled over for four years. If AI can solve it, then what was the point of all of that? And then I started getting deeper into how AI was affecting math and how mathematicians were thinking about it and getting deeper into what was really happening with artificial intelligence and mathematics.
所以你是一名经过培训的数学家。我想我们需要在一开始就明确一点，Anjani。在这个播客节目中，我们说的是“maths”而不是“math”，对吧？看看你能否在余下的节目中坚持用“maths”。好的，你是一名数学家，你研究的是数学，这也是我们将在节目中讨论的内容。所以这不同于人们在学校熟悉的算术。我觉得我们有必要为那些认为数学太难因此没继续学习的人，稍微谈谈数学的意义。能否给我们讲一讲研究数学与我们在学校学的数学有什么不同呢？
▶ 英文原文 ⏱
So you're a mathematician by training and I think we should get something very clear at the beginning, Anjani. It's maths rather than math here on this podcast, right? So let's just see if you can keep that going for the rest of it. Maths with an S, right? Okay. So you're a mathematician. You studied research mathematics, which is what we're going to be talking about on this show. So rather than the arithmetic, I suppose people are familiar with at school. And I think it's worth us just talking a bit about the point of mathematics for those who may not have continued with it beyond school, thinking it's too hard. Just lay out for us what is different from the research mathematics we're about to talk about versus the maths we did at school.
我们在学校学习的数学是几百年来数学研究成果的产物。我们在学校学习的那些定理和公理，其实经过了多年的证明和确立，才被认可为真理。这就是数学研究的过程。数学家们就是这样，与他们的同行讨论、花费大量时间研究来确立这些真理。是的，我想我们理所当然地认为一加一等于二，但实际上，数学家花了很长时间来定义加法的意义，并证明一加一等于二。更不用说，一开始"等于"这个概念本身又是什么意思呢？
▶ 英文原文 ⏱
The maths we did at school is a product of all the research mathematics that has happened for centuries. And these kind of theorems and axioms that we, when we are in school and looking at them, have taken years to prove and establish as a truth. And that's the process of what research mathematics is about. And that's what mathematicians, you know, stand around and discuss with their fellow mathematicians and spend hours researching to kind of establish these truths. Yeah. I mean, I suppose we take it for granted, of course, that one plus one equals two, but actually mathematicians have spent a long time defining what addition means and proving that one plus one equals two. What does equals even mean in the first place?
在某种程度上，试图定义这些术语和公理几乎具有哲学意味。正如你所说，我们假设为真的东西必须被证明。那对数学家来说，什么才算是证明呢？这其中有两个方面。第一，它确立了一种真理，并且没有其他更好的方式可以达到数学家所发现的定理或最终结果。因此，这一部分实际上是找到答案。第二部分则是美感的元素。我在与数学家交流时发现这很有趣，他们指出证明不仅仅是把一堆信息写在纸上，而是要以一种方式引导读者或其他数学家，通过证明突出以前未曾形成的想法和联系。
▶ 英文原文 ⏱
It's almost philosophical at some point in terms of trying to define these terms and axioms, as you say, these things that we assume to be true have to be proven. But what does it mean for a mathematician to have a proof? There's two parts of it. The first is that it establishes a truth and that there is no other way that a theorem or a final result that a mathematician has found can be done in a better way. So that's one part of it is actually getting to the answer. The second part of it is there's an element of aesthetic. And I found it really interesting while talking to mathematicians who made the point that it wasn't just any proof, you know, it wasn't just putting a bunch of information on paper. It was really leading a reader or another mathematician through the proof in a way that highlighted ideas and connections that hadn't been made before.
这就是为什么数学家如此重视证明的原因。从根本上来说，证明既是一种艺术，也是一门科学，它取决于数学家如何看待这个证明以及如何推进这个证明。将证明写在纸上并没有一个固定的公式。所以在某种意义上来说，这就是将一个抽象的想法展现出来，证明这是解决某个特定问题的唯一方法，没有更好的方法。正因为如此，数学家可以利用这个证明在将来建立更复杂的思想和公理。
▶ 英文原文 ⏱
And that's kind of why mathematicians care so much about proofs. Ultimately, it's a bit of art and it's a bit of science and it comes down to how the mathematician sees the proof and where they lead that proof to. There's no formulaic process of putting a proof on paper. So in a sense, it's about taking an abstract idea and showing that that's the only way to solve that particular problem. That there is no better way. Exactly. And then you can use that proof to then build further, more complicated ideas and axioms in the future.
好的，数学对很多人来说可能感到非常复杂和抽象。因此，我觉得当我们讨论一个问题的时候，脑海中最好有一个具体问题的例子来帮助理解。我知道你一直在写关于一个叫做“球体堆叠问题”的东西。能给我们解释一下这是什么吗？
想象一下你走进当地的超市，看到一箱箱的橙子。有人已经思考过如何把最多数量的橙子摆放进去。而专业的数学家多年来一直在研究，怎样才能以最有效的方式把球形物体，比如橙子、棒球或其他东西，放进箱子里，并在多维空间中进行排列。
▶ 英文原文 ⏱
Exactly. So, um, maths can feel very complicated and abstract to people. So I think probably it's useful for people to have in the back of their minds, an example of a problem that we will try and solve as we sort of talk through it. I know you've been writing about something called the sphere packing problem. Just explain what that is for us. Well, imagine you walk into the local supermarket and there are crates of oranges. Somebody has thought about how to put the most number of oranges in there. And professional mathematicians have spent years trying to figure out what is the most efficient way to pack spherical objects, whether it's oranges or baseballs or anything else into a crate and in multiple dimensions.
所以，如果你考虑一维的球形填充问题，那实际上就是一条直线，并且在直线上排列圆。你最多能放多少个圆来覆盖整条直线呢？在这种情况下，它的效率是百分之百的，因为你可以通过圆心或直径画一条直线，就完成了一维的球形填充。接下来，我们进入三维空间，这就像是一箱橙子的问题，考虑如何排列橙子以使它们之间的空隙最小。然后就到了四维空间，虽然我们作为人类很难完全理解它。
▶ 英文原文 ⏱
So if you think about the sphere packing problem in one dimension, that's effectively a straight line and organized circles on the line. And what's the maximum number of circles you could put in to cover the entire line? In that case, it's a hundred percent efficient because you run a straight line through the center of the circles or the diameters and you've got your sphere packing in one dimension. And from there, we go to three dimensions, which is the oranges in a crate, example of how can you organize them so there's the least amount of space between the oranges. And so they have gone up the dimension and then four dimensions, which we obviously can't quite grasp as human beings.
他们已经研究到了八维和二十四维。那么，为什么数学家关注八维空间中的球体堆积问题是有用的呢？它有什么实际价值呢？其实，这些看似深奥的维度在数学中有广泛的应用。比如说，当你通过嘈杂的信道发送数据时，无论是Wi-Fi、手机还是一般的电信服务，数据都有可能被背景噪声干扰。为了应对这一问题，工程师会使用纠错码来修复这些错误。而球体堆积问题正是帮助设计这些纠错码的精准数学模型。
▶ 英文原文 ⏱
And they've gone up to eight dimensions and 24 dimensions. So why is it useful for mathematicians to be looking at something like sphere packing in eight dimensions? What practical value does it have? So there's actually loads of uses for these seemingly esoteric dimensions in mathematics. For example, if you were to send data over a noisy channel, whether it's Wi-Fi or a cell phone or just telecoms in general, there's a chance that the data gets corrupted by background noise. And so engineers will use error correcting codes to fix this. And the sphere packing problem is the exact mathematical model that has helped design these.
我们将讨论人工智能如何改变数学。在深入探讨这个话题之前，数学家们以前使用计算机的频率高吗？当然。过去的一些重大数学证明之所以使用计算机，主要是因为需要进行大量的计算工作。我们通常想到计算器，但这些工作量远超一般计算器的能力，几乎是无限次方的计算。一个长达数千页的证明就是如此，因此数学家使用计算机来完成这些繁重的计算任务。
▶ 英文原文 ⏱
So we're going to be talking about how AI is changing mathematics. But before we go into all of that, though, have computers been used a lot by mathematicians until now? Absolutely. I mean, some of the major proofs in the past used computers purely because just the sheer volume and amount of calculations that needed to be done. You know, we think of calculators, but it was just calculators raised to the nth power because it was that much more computation. And a proof that was thousands of pages long required that. And mathematicians have used computers for that.
他们最近也开始使用计算机来检查复杂的配置，因为很多数学内容已经被从自然语言转换成计算机代码，然后计算机可以验证或检查这些内容。因此，这方面已经有一些工作进行了。现有的编程语言，比如在大学里我们用的MATLAB，还有专门用于证明的Lean编程。这些语言存在已久，他们还创建了一个名为mathlib的庞大库，将许多数学证明数字化整理，以便计算机检查。因此，我认为计算机在计算和处理大量数据方面发挥了重要作用。
▶ 英文原文 ⏱
And they've also more recently started using them to check complex configurations, because a lot of math is, you know, they've converted what's written in natural language into computer code, and then the computer can verify it or check it. And so there has been some work done around that. There's programming languages that exist, you know, in university we used MATLAB, there's lean programming for proofs. And so these languages have been around forever. And they've created mathlib, which is a vast library of proofs that have been digitized and put in a way that can be checked by computers. So computers have played a big role, I would say, as it comes to computation and just the volumes we're talking about.
我觉得，这在科学领域似乎是个相对常见的故事。我在大学时学过物理，我们用了很多计算机代码来尝试解决各种情境下的数学方程。这些并不是严格意义上的数学证明，而是对特定情境的答案。我们用这种方式使用计算机代码，也用超级计算机来尝试“蛮力”解决问题，当用纯数学无法轻松得到答案时。这一切都很常见。当然，这些都不属于现代意义上的人工智能。虽然早期可能有过一些使用机器学习的尝试，但传统数学领域并没有使用人工智能。
▶ 英文原文 ⏱
I mean, this seems like a familiar story when it comes to sciences in general. I mean, I studied physics at university and we used a lot of computer code to try and solve mathematical equations in various contexts. And they were not proofs as such, but sort of answers to a specific situation. And it feels like computer code has been used in that way. And also supercomputers have been used to sort of brute force, try and solve equations as well, when easy answers aren't available in a pure mathematical sense. So that's very familiar. None of these, of course, are artificial intelligence in the modern sense. I mean, there might have been some old machine learning going on for a long time ago, but none of the sort of traditional mathematical enterprises have used artificial intelligence.
在现代生成式人工智能时代，数学家与人工智能的关系是什么呢？这很有趣。最初，因为CHAP-JPT在处理基本算术时遇到困难，很多人对其持怀疑态度。但一开始，许多数学家对人工智能持开放态度，因为他们对计算机也保持开放。人们使用人工智能帮助进行文献检索，就像一个出色的研究助理一样。因此，在某种程度上，使用大型语言模型和生成式人工智能来完成我们可能通过复杂的网络搜索所做的事情，比如说，面对大量文献，我们不可能全部读完。这时，人工智能就能派上用场。
▶ 英文原文 ⏱
So in the sort of modern generative era of artificial intelligence, what's the relationship with AI been from mathematicians? So it's interesting. Initially, there was a lot of skepticism because CHAP-JPT had struggled to do basic arithmetic. But early on, a lot of mathematicians, you know, they've kind of been open, I'd say, to AI, because they've been open to computers. And people were using AI to help with literature search. And it was like a good research assistant. Yeah. So in a way, using large language models and generative AI to do the kinds of things that we might have been doing if trying to do a fancy web search or something, which is to say, look, you know, there's loads of literature out there. I can't possibly read it all.
那么，你能为我总结一下，在许多地方的摘要是什么吗，找到那些可能需要很长时间才能找到的联系，但他们并不是为了实际研究或数学而这样做的？因为我们知道，CHAP-JPT 和 Claude 的原始版本甚至无法进行简单的数字加法或表示数字。确实如此。不过，现在似乎情况发生了变化。前沿数学家们对于大型语言模型的改进以及它们现在如何帮助这些数学家的工作感到一种兴奋。请介绍一下这个变化。我是说，似乎有什么东西发生了变化，对吧？是的，我认为在过去的一年中，这些大型语言模型的能力确实有了质的飞跃。
▶ 英文原文 ⏱
So can you summarize for me what the abstracts are in lots of places, find connections that might otherwise take me a long time to do myself, but they're not doing it for actual research or actual mathematics? Because as we knew, the original versions of CHAP-JPT and Claude couldn't even add simple numbers together or represent numbers. Precisely. And it feels like that's changed, though. There is a sort of excitement now amongst sort of frontier mathematicians about the improvements in large language models and what they're now helping those mathematicians to do. Just introduce us to that. I mean, something seems to have changed, right? Yes. I think over the course of the last year, there's been a step change in what these LLMs can actually do.
这件事的起点是，这些大型语言模型（LLMs）解决了一个艾尔德什（Erdős）问题。艾尔德什问题是一组长期以来未被解决的数学难题，虽然它们听起来很简单，却让数学家们苦思冥想多年。尽管人类可能花更多的时间也许能解决它，但LLM成功解决了这个问题。这带来了巨大的突破。从那以后，进展迅速：LLM从解决难题开始，逐渐开始处理数学证明。这也是为什么数学家现在更加认真地研究如何利用这些进展。考虑到几年前这些模型甚至无法做简单的算术运算，这种进步速度令人惊讶。
▶ 英文原文 ⏱
It started with these LLMs solving one of the Erdős problems, which is a group of longstanding and very simple sounding puzzles for mathematicians to ponder over, but it had not been solved. And while a human being could have spent a little more time and potentially solved it, the LLM was able to solve it. So that led to a step change. And from there, there was rapid progression in what was happening. It went from puzzles and then LLMs started taking on proofs. And that's why mathematicians are now taking it a little more seriously to really understand how they can leverage what's happening. That's an astonishing rate of progress, given that, you know, just a few years ago, they couldn't do simple arithmetic.
他们现在在做什么，你是说研究生还是博士水平的工作？这可能有些夸张，但他们确实在做一些相当于优秀本科生的工作，甚至可能在朝着研究生的水平努力。不过，这都是在数学专家的指导下进行的。这是我们必须要理解的一点：他们在各种引导和推动下，仍然能够做出回应。当你提到大型语言模型时，你是指那些我们都能使用的，比如GPT-5和Claude这样的模型吗，还是一些特殊的模型？
▶ 英文原文 ⏱
And now they're doing, what are you saying, postgraduate or PhD level work? Well, that might be a stretch, but they are doing, you know, the work of a talented undergraduate student and perhaps getting further towards maybe a graduate student. But that's also under the guidance of expert mathematicians. And that's one thing we really have to appreciate here is they are being prompted and pushed in various directions, and they're still able to respond to that. When you say large language models, are you talking about the ones available to all of us, like GPT-5 and Claude and those ones, or are they something special?
不，这些正是被数学家和一些开发自身模型的创业公司使用的模型。为了理解为什么数学家对此越来越感兴趣，我采访了特伦斯·陶。他是加州大学洛杉矶分校的数学教授，并于2006年获得了常被称为数学界诺贝尔奖的菲尔兹奖。特伦斯告诉我，他和他的同事们如何在工作中使用大型语言模型。它正在成为一个出色的合作伙伴，它虽不是人类，但具备与人不同的能力。
▶ 英文原文 ⏱
No, it is exactly these models that are being used by mathematicians and by startups that are developing their own models. So to get a sense of why mathematicians are getting more and more excited about it, I spoke to Terence Tao. He's a professor of mathematics at the University of California at Los Angeles, and he won the Fields Medal in 2006, which is often described as the Nobel Prize in math. Terence told me how he and his colleagues have been using large language models in their work. It is becoming a great collaborator, not exactly human, it has different capabilities than humans.
大致来说，你可以把人工智能数学家想象成一个非常聪明的大学生，已经学习了所有的教科书，掌握了每一个人类数学家用过的技巧。当它被要求解决一个问题时，它会不知疲倦地尝试各种方法。而人类数学家会选择一两个自己最感兴趣的问题，花费数周或数月的时间集中精力攻克这些问题，尝试非常困难的方法。相对而言，人工智能可以同时处理上千个问题，并对每个问题尝试比较常规的方法。
▶ 英文原文 ⏱
So roughly speaking, you can think of an AI mathematician as like a really bright undergraduate who has studied every single textbook and like knows every trick that any human mathematician has ever done. And then when asked to work on a problem, it will just try everything tirelessly. So a human mathematician will pick one or two problems that he or she is most interested in, and will spend weeks or months focusing on that problem and trying to extremely difficult approaches. But an AI, you can set it on a thousand problems and it will try fairly standard methods for each one.
机器可以为每个问题尝试上百种方法，而人类可能只会尝试一两种。因此，它在解决注意力瓶颈类型问题上表现得非常出色。有少数我们非常关心的重要问题，也有很多是某个地方有人提出的问题，如果有人能关注这些问题，那将是很好的，但我们没有足够的人类数学家来研究每一个问题。其中一些问题实际上相对简单，只是没有人对它们进行足够的研究。而现有能力已经提高到了一定程度，现在在某些情况下，机器可以自主解决这些相对最简单的问题。
▶ 英文原文 ⏱
But it can try a hundred methods per problem, whereas a human might only try one or two. And so it's proving to be really good at the attention bottleneck type of problem. There's a small number of high profile problems that we really care about. And then there's a lot of problems that somebody posed somewhere, and it would be great if someone looked at them, but we don't have enough human mathematicians to look at every single problem. And some of those problems are actually relatively easy. It's just that no one's actually studied them enough. And the capabilities have improved to a point where they can now, in some cases, autonomously solve the easiest of these problems.
现在，这可能让人产生一种错觉，以为它们能够和人类数学家平分秋色地竞争，但实际上它们是在广度上竞争，而非深度上。AI 的做法基本上是将每一个问题都输入 AI。AI 有机会解决的是所有问题中最简单的部分，即“触手可及”的 “果实”。对，触手可及的果实，而不是最美味的果实。从数量上讲，现在的成果非常令人印象深刻，但我认为很快这些“低垂果实”都会被摘完，这其实是件好事。这样人们就能知道应该把注意力放在哪里。
▶ 英文原文 ⏱
Now, this can give an illusion that they are competing on par with human mathematicians, but they're competing by a scale rather than by depth. So the way the AIs do it is that they basically feed every single problem into the AI. And the easiest of all the problems, the AI has a chance to solve. The low-hanging fruit. The low-hanging fruit, exactly. As opposed to kind of the tastiest fruit. Yeah. So numerically, it's very impressive right now, but I think soon all the low-hanging fruit will be gone, which is great actually. And it will tell humans where to direct their attention.
这基本上是在处理基于大量现有文献的未解决问题。然后你会进入这样一个领域，那里会出现更大的一些问题。这有点像大家常说的数学研究的精髓——你在思考下一个问题和你想要挑战的下一个计算问题。因为你之前提到过，它已经自主解决了一些可能相对简单的问题，但你认为它有能力突破一些界限吗？
▶ 英文原文 ⏱
So this is kind of dealing with unsolved problems based on a lot of existing literature, essentially. And then you kind of move to this area where there's the bigger problems that emerge. And, you know, it's kind of what everyone says, you know, the spirit of mathematical research, where you're thinking about the next problem and the next kind of computational challenge that you want to take on. Because you mentioned earlier that it has autonomously solved some problems, maybe relatively easy, but do you think that that has the ability to push some boundaries?
当然，重新组合现有的方法已经可以带领我们走得非常远。我是说，在现代数学中，几乎每篇论文都是基于过去的文献。没有借助前人的积累，你无法解决任何高级问题。所以潜力很大。有些子领域已经发展出一套工具和想法，而另一些子领域有另一套工具和想法。但是，有时没有人类数学家能够同时精通这两个领域，以使用这两套工具。不过，人工智能可以做到这一点。
▶ 英文原文 ⏱
Certainly recombining existing methods can already take us very, very far. I mean, in modern mathematics, almost every paper is based on past literature. You can't solve any advanced problem without standing on the shoulders of giants. And so there's a lot of potential. There are some sub-fields which have developed some set of tools and ideas, and another sub-field that's sort of a separate set of tools and ideas. But sometimes there's no human mathematician that is expert enough in both of those areas to use both. But AIs could do that.
这已经在推动边界，只是将现有的想法融合在一起。虽然这可能看起来不像是最有创意的活动，但它非常有成效。在你提到的这些方法中，在某种假设的情况下，问题被设置，然后AI给出10种解决方案，这些方案是否涵盖了数学的各个子领域？比如说是否有来自组合数学的一些小众领域的东西，也有来自纯数学其他子领域的一些东西。它将所有这些整合在一起，因为在它的“大脑”中，或者说在它的计算机思维中，它假设所有这些实际上都是一个庞大的文献库。
▶ 英文原文 ⏱
And that already pushes the boundary, just come merging together existing ideas. It may not feel like the most creative activity, but it is very productive. So in these approaches that you mentioned in the kind of hypothetical case in which a problem is set and then the AI, you know, spits out, these are the 10 approaches, are they across like the sub-fields of mathematics? So there's something from like some niche area of combinatorics and there's something from some other sub-area of like pure math. And it kind of brings it all together because in its head, or you know, in its computer mind, it assumes that it's all one big stock of literature effectively.
我们不能确定原因。之所以有效，可能是因为在人类身上存在着一套不同的认知偏差。当一个人研究一个问题时，如果他们在某个领域是专家，他们自然会使用自己熟悉的工具，而不考虑其他人使用的工具。我猜他们可能是在假装自己是专家。所以，如果问题是关于组合数学的，他们会说：“哦，我应该使用组合数学的工具去解决这个问题。”这正是一个组合数学专家会做的事情。不过，有时候他们并不会这样做，因为他们会犯错，但这些错误有时可能会带来意想不到的收获。
▶ 英文原文 ⏱
We don't know for sure. The reason why it's effective maybe is because it has a different set of cognitive biases in humans. So when a human studies a problem, you know, if they're expert in one area, they will just naturally reach for the tools that they're familiar with and naturally not think about the tools that other people think about. I guess they're trying to pretend to be experts. So if a problem is in, say, combinatorics, they will say, "Oh, I should use a tool for combinatorics," which is what a combinatorialist would do. But every so often they don't because sometimes they make mistakes, but it could be that they actually make fruitful mistakes.
你知道吗，他们没有意识到这是一个组合数学的问题。他们认为这是一个几何问题，于是便使用几何学中的方法，而人类通常不会这样做，但有时候这样反而解决了问题。所以，他们的策略偏好与人类不同。由于人类的精力有限，所以在进行非常复杂的计算前，会花更多时间预先规划，仔细考虑计划。然而，一个可以在同一问题上尝试上千次的人工智能，可能会选择尝试全部100种方法，而不是花五分钟去挑选出哪种方法更好。
▶ 英文原文 ⏱
You know, they don't realize that this is a problem in combinatorics. They think it's a problem in geometry, and they reach for two in geometry, which a human wouldn't do, but occasionally that solves the problem. So they have a different mix of strategy preferences than humans. Humans have limited bandwidth, so they will spend more time in the pre-planning process, you know, before doing a very nasty computation, they will think harder about getting a plan. But an AI which can run a thousand different attempts at the same problem, they may just try all 100 methods rather than spend five minutes figuring out which method is actually the best.
当我们观察人工智能在数学领域的发展轨迹，结合已有的经验，你觉得可能会实现哪些突破？我可以想象，例如，有两个不同的数学领域，譬如几何学和数论。也许有一天，人工智能会发现这两个领域中某些结果之间有相似性，它们可能会有某种联系。于是，人工智能可能会注意到这一点，并建议有人去研究这两组结果，看看它们之间是否存在关系。然后，也许某位人类专家会深入研究，并指出：“哦，是的，这两者可能有关联，因为在这两个领域中都有出现某个共同的量。”
▶ 英文原文 ⏱
When you look at the trajectory of AI in mathematics, from what we've seen already, what do you think is possible? I can imagine, for example, that there are two different areas of mathematics, let's say geometry and number theory. And maybe an AI notices that there are some similarities between some set of results in one field, and there's some parallel set of results in another field, and there could be some connection. And so an AI might notice, hey, someone should look at these two families of results, and maybe there's some relationship between them. And then maybe some human expert looks at them and says, oh yeah, maybe the connection is because there is some common quantity which shows up in both.
然后，他们可能会提出一个猜想，接着也许一个人工智能会说，我刚刚对照了前100个简单案例检查了你的猜想，结果是错的。我说，好吧，没问题。于是人类说，那如果我调整一下这个猜想，试试这样呢？然后可能人工智能会说，哦，这样更接近正确了。现在我测试的100个例子中，有98个是对的，但还有两个不对。啊，好，谢谢你指出来。我只需要再做一个最后的修正。然后，啊，好的，人工智能说，好，现在就我所能判断的，我无法反驳你的猜想。这很有可能是真的。那么我们就发表一篇论文，接着其他人会继续深化研究。
▶ 英文原文 ⏱
And then they might make a conjecture, and then maybe an AI says, I've just checked your conjecture against the first 100 easy cases, it's wrong. I said, all right, okay. So then the human says, what if I tweak the conjecture and I do this? And then maybe the AI says, oh, okay, that's closer to correct. Now, 98 of the 100 examples I tried work, and then there's two that don't. Ah, okay, thanks for pointing that out. I just need to make this one final correction. And then, ah, okay, the AI says, okay, now, as far as I can tell, I can't disprove your conjecture. This is a good chance of being true. So now we publish a paper, and then somebody else takes it forward.
因此，我可以想象这样的对话场景：人工智能可能会识别线索或驳回错误的提议，但你需要检查大量的例子才能意识到那些是错误的。你认为哪个领域可以被推进并将数学推向新的领域，从而在物理学、密码学或其他领域引发重大突破，进而迭代产生新的发展和研究？我想这个过程会让我们感到惊讶。传统上，数学家通常专注于一两个领域。
▶ 英文原文 ⏱
And so I can imagine kind of conversations like this, where, you know, AIs might sort of pick up clues or shoot down proposals that are wrong, but you would have to check lots and lots of cases before you realize they're wrong. What field do you think could be advanced and kind of push math into some of these areas where they kind of lead to other big breakthroughs in physics or in cryptography or in other areas that would then iterate a cycle of new developments and research? I think we'll be surprised. So, traditionally, mathematicians specialize in one or two fields.
即使是与其他领域的数学家交流，有时也会存在某种语言障碍。你需要花很多时间向对方解释你所在学科的基本概念。而如果你想与科学家交流，比如地质学家或生态生物学家等，他们可能希望数学家帮助解决一个问题，但却无法用准确的语言表达他们的需求。我认为，人工智能可以在跨学科合作中发挥促进作用。这方面有许多潜力尚未被充分利用。
▶ 英文原文 ⏱
And even talking to other mathematicians in another field, sometimes there's a kind of language barrier. You have to spend a lot of time explaining the fundamentals of your subject to the other person. And then if you want to talk to a scientist, and you know, maybe there's some geologist or ecobiologist or whatever who wants a mathematician help them with a problem, but they don't have the language to enunciate exactly what they want. I could see AI facilitating interdisciplinary collaborations. There's so much untapped potential.
让我感到鼓舞的是，在早期问题中，我看到许多AI进步实际上是由业余爱好者推动的。有很多高中生或计算机科学家只是利用AI工具进行尝试，解决那些专业数学家不曾涉足的问题。他们在尝试我们未曾想到的方法，以创新的方式与AI合作。我希望在未来能够看到更多公众科学和公众数学的蓬勃发展。
▶ 英文原文 ⏱
And one encouraging thing I'm seeing actually from what the early problem stuff is that, actually, a lot of the AI progress is driven by amateurs, that there are high school students or computer scientists just playing around with AI tools, just trying things. I mean, they're working on problems that the professional mathematicians weren't touching. But, you know, they're trying things that we wouldn't have thought of and working with AI in innovative ways. I'm hoping to see sort of a flourishing of citizen science and citizen mathematics in the future.
安简，听到世界上最杰出的数学家之一泰伦斯·陶的讲话真是太有趣了。泰伦斯解释说，人工智能已经在解决许多问题，并且在某种程度上扩大了人类几个世纪以来从事的数学问题解决事业。但我想，如果更多的问题被解决，是不是意味着需要更多的人去核查这些问题呢？我的意思是，你不能只是信任计算机来处理这些事情，对吗？这是数学领域最大的瓶颈之一。
▶ 英文原文 ⏱
Anjan, it was really interesting to hear Terence Tao there, one of the world's most eminent mathematicians. So Terence explained that AI is solving a lot of problems already and, you know, kind of scaling up the mathematical problem solving enterprise that humans have been engaged in for centuries. But I guess if more problems are being solved, does that mean that more humans need to check those problems? I mean, you can't just trust a computer to do this stuff, right? This is one of the biggest bottlenecks for maths.
证明的目的在于确保答案无懈可击，并且这是证明某个特定数学真理的最佳方式。因此，如果数学家每次想使用某个证明时都要回过头来复查，这将会非常耗费精力，等于是在重复这些工作。同样地，如果要检查大型语言模型（LLMs）所做的工作，那也十分繁琐。Terence 告诉我，LLMs 在数学上出错的概率高达98%。
▶ 英文原文 ⏱
And the whole point of a proof is to guarantee that answers are airtight and that this was the best possible way to establish this particular mathematical truth. And so it's a lot of work if mathematicians have to go back and check each proof every single time that they want to use it. And you're essentially repeating that work. So if you can go back and check through what the LLMs have done, that is also very cumbersome. So Terence told me that LLMs make mistakes in math 98% of the times.
但这2%确实是非常了不起的成就。你知道，另一位数学家告诉我，大型语言模型（LLM）重新确认了一些已有的真理，而这些是重要的，但它们还发现了一些拼写错误和其他错误，这也很有帮助。不过，这些大海捞针的任务总要有办法解决。否则，数学家们将面临大量松散无序的数学问题。因此，当他们在处理这些问题时，需要一个框架来指导他们。
▶ 英文原文 ⏱
But the 2% are really great achievements. You know, another mathematician told me that the LLM re-established some existing truths and that those were important, but that they also found a couple typos and errors. And that was helpful. But those needles and haystacks need to be found somehow. And otherwise mathematicians will be inundated with mathematical sloth. And so as they are filtering through a lot of these problems, they need a framework.
所以，如果人类无法亲自验证这些由大语言模型生成的数百万个新证明，那么我们该如何应对呢？研究人员正在努力的解决方案是自动化人类的直觉推理和手动验证过程，这被称为自动形式化（auto-formalization）。形式化验证意味着存在一种证明其正确性的计算机可检查证明。他们的方法是将之前用自然语言编写的证明转化为计算机代码，然后由计算机进行检测和验证。
▶ 英文原文 ⏱
So if humans can't be going around verifying these, you know, millions of new proofs that large language models can now do, what is the sort of framework? What is the way to sort of wade through these haystacks? So the solution that researchers are working through is essentially to, you know, automate the process of humans using their intuition and manually verifying proofs. And so it's called auto-formalization. Now, formal verification means that there is a proof that is computer checkable for its correctness. And the way they've done that is by, as we said earlier, converting proofs that are written in natural language into computer code, and then those are checked against it.
因此, 最终的目标是让人工智能能够逻辑地完成这些步骤并验证证明，而不是依赖引导来完成步骤，或者只是跟随其他数学家处理问题的方式。但在实现这一目标之前，我们面临一个巨大的挑战，即让数学家使用相同的语言交流，这也是我们之前提到的问题。这就是Lean的作用所在。Lean是一种计算机代码，它基本上有助于让所有人达成一致。
▶ 英文原文 ⏱
So the eventual goal would be that AI can logically work through these steps and verify the proofs, rather than being led through the steps, or follow the way of any other mathematician would follow those steps. But before they can do that, there's a big challenge to get mathematicians speaking the same language, which is what we were speaking about earlier. And that's where Lean comes in. And Lean is computer code, and it essentially helps put everyone on the same page.
可以让我打断一下吗？我一直以为数学家们都说同一种语言。我的基本理解是，他们不是使用相同的符号、方程、证明方法和技巧吗？你说在这个领域中存在理解困难这让我感到惊讶。这个问题提得好。数学证明通常被认为是用自然语言表达的，但这其实是数学家们彼此之间的交流方式。
▶ 英文原文 ⏱
Can I just pause you there? And I thought that my mathematicians did speak the same language. I mean, this is my basic understanding. I mean, don't they use the same symbols and equations and proofs, the methods and things? I mean, it surprises me that you're saying that there is this sort of difficulty in understanding across the field. That's a good point. So mathematical proofs, you know, we say that they've been stated in natural language, but that's mathematicians talking to each other.
你知道，在我们想到使用演绎推理来陈述各种引理的最古老数学文本时，它们会用各种符号来表达整数和实数的想法。这是一个非常以规则为基础的文本，而随着逻辑学等领域的研究，这种特性变得更为明显。讽刺的是，虽然我们认为数学非常程式化，但实际上，在证明的世界中，并不是这样的。
▶ 英文原文 ⏱
You know, they're written, like when we think about the oldest mathematical texts that use deductive reasoning to state various kind of lemmas, you know, they'll lay out their ideas with various symbols and for integers and real numbers. And it's a very rules-based text. And then it's been made more so with the study of logic and so on. The irony is that while we think about math as very formulaic, actually, in the world of proofs, it isn't.
一位非常清楚地向我解释了所有这些的人是帕特里克·沙福，他是美国国防高级研究计划局（DARPA）的一位数学家。DARPA正在资助一些项目，旨在利用人工智能加速纯数学的进展率，其中一个方法是简化形式化过程。因此，研究数学，至少在纯数学方面，通常是围绕证明命题而进行的。所以，与其寻找一组方程的解，你需要做的是证明关于某一类事物的一些普遍命题。
▶ 英文原文 ⏱
Someone who explained all of this to me very clearly is Patrick Shaftoe. He's a mathematician at America's Defense Advanced Research Projects Agency, or DARPA, which is funding projects to use artificial intelligence to accelerate the rate of progress in pure math by, among other things, streamlining the formalization process. So research mathematics tends to focus around, at least on the pure math side, proving statements. And so rather than finding a solution to a set of equations, what you want to do is prove some general statement about a class of things.
这意味着你可以坐下来阅读资料，或者尝试画图，或者和朋友站在黑板前讨论这个问题。你也可以阅读以前的工作，试图了解类似的问题是如何被解决的，诸如此类的方法。对吧。当你有一个流程来验证并说，这个证明在我们已知的规则和已有的文献中是有效的或有道理的，那么这究竟能带给我们什么呢？
▶ 英文原文 ⏱
And so what that means is like, you may sit down and read, you may try to draw pictures, you may stand at the chalkboard with a friend and talk about the problem. You may read previous work to try to get a sense of how similar problems have been tackled, these kinds of things. Right. And so when you then have a process to verify that and to say, okay, this proof works or makes sense within the rules that we know of and within the literature that we have, where does that get us then?
这让我们能够更牢固、更快速地构建系统，对吧？比如说，如果我们知道加密是可以被证明正确的，那就太好了。而且，你知道，这是一项正在进行的努力。它包括证明用于加密的数学是正确的，还包括证明实现这些数学的代码是正确的。在这里，你可以看到形式化验证的一个真正优势，就是它能够把这两个问题更加紧密地结合在一起。
▶ 英文原文 ⏱
So that gets us to the point where we can build more robustly and faster, right? So for example, it would be lovely if we knew encryption was provably correct. And, you know, this is an active endeavor, right? Like it involves, you know, proving the math correct, which underlies things like encryption. It also involves proving things about code correct that implement that math. And there you can see one of the real wins of formal verification is you're bringing those two problems closer together.
这就是一个例子。当然，还有很多很多例子，数学在我们日常生活中习以为常的事物中扮演着关键但常常隐藏的角色。上次我们谈话时，我非常喜欢你表达它的方式——你说，数学知识的体系现在就像是一团乱麻，存在于论文、想法、人们的思维中，等等。
▶ 英文原文 ⏱
And so that's one example. Of course, there's lots and lots of examples where math plays a critical, although often quite hidden role in the things that we just take for granted every day. Last time we were speaking, I actually really liked the way you put it, which is that the body of mathematical knowledge is, I think you said it right now exists as a hot mess of like papers and ideas and thoughts in people's heads and so on and so forth.
所以，如果你能稍微展开一点这个话题就好了。数学存在于发表在期刊上的论文中，也可能存在于网上的预印本和教科书中，除此之外，它还存在于人们的头脑中。这意味着，如果你想研究一个特定的数学问题，你可能需要查阅许多不同的来源来找到相关的信息。
▶ 英文原文 ⏱
So if you can kind of expand a little bit on that. Yeah. Math exists in papers that get sent to journals and get published, potentially on the archive on the internet and in textbooks, and otherwise it lives in people's heads. And so what that means is that if you want to study a particular mathematical problem, you might need to look across many, many sources for relevant information.
正因为如此，数学家们常常证明一些之前就已知的结果，只不过这些结果在你所提的问题背景下并不被大家所熟知。而这正是当前的现状。这就需要对文献有深入的专业知识，但即便是博学的数学家，通常也无法知道任何特定问题的所有相关文献。
▶ 英文原文 ⏱
And for this reason, it's not uncommon that mathematicians prove results that were somehow known before, but they just weren't known in the context of the question that you were asking, right? And that's sort of the state of affairs as it exists today. And it requires deep expertise to know a literature, but even the most learned mathematicians don't know all of the relevant literature for any particular question, typically.
在最近几个月里，我们常常听说人工智能已经进入了数学领域，并解决了一些长久以来的埃尔德什问题以及其他小谜题。同时，它也开始解决一些人们要么没有尝试解决，要么尚未完成的数学问题。这让我好奇，人工智能是如何介入并解决这些挑战的呢？它在这些方面能发挥什么有意义的作用？
数学中的核心挑战实际上在于信任。这是因为如果你想证明某件事情的正确性，并且需要建立在他人研究结果的基础上，你就需要确保这些结果也是正确的。在过去的一年中，我们在数学领域看到了一些令人印象深刻的人工智能进展。然而，这些情况都没有建立在信任的基础上。所以虽然我们能够进行一些新的、更有趣的数学研究，但从根本上来说，我们受到人类和信任的限制。
▶ 英文原文 ⏱
And so I guess in all of this, you know, we've been hearing, especially in recent months, that AI has now come in and it is solving a long kind of, you know, the Erdős problems and other kind of little puzzles and so on. But it's also starting to do other types of mathematical problems, if you will, that people have not either attempted to solve them or they just haven't completed them. And so I wondered, how does AI step in and solve these challenges? And where does it have a role to play in a meaningful way? : I mean, the central challenge in math is really focused on trust. And that's because if you want to prove something correct and you need to build on other people's results, you need to know those are correct also. Over the past even just year, there have been some really impressive advances in AI for math. All of those kinds of situations are really things that don't build on the trust. So while we might be able to do some new and more interesting math, we're fundamentally bottlenecked by humans and by trust.
想象一下，对吧？如果你无法信任以前的工作成果，那会意味着什么呢？那就意味着你可能要花费数小时，甚至数天的时间去重做工作。这样是不对的。对。 我认为这里有另一个关键点，这可能会真正引发数学领域的变革。那就是通过与所谓的形式验证编程语言的连接。而这里的潜在创新在于，假如我们可以自动将自然语言的数学内容翻译成可以被验证的形式语言。然后，我们可以利用现存的大量数学文献，这些文献目前都处于一种混乱状态，比如论文和教科书等等。我们可以把所有这些内容编码进计算机语言并加以验证。这样会产生一系列后续的影响。首先是确认内容的正确性，其次是更清楚地展示论文之间的联系，第三是降低进入数学领域的门槛。我认为如果这个自动形式化的想法能够成功，将会带来非常令人兴奋的连锁效应。
▶ 英文原文 ⏱
And just imagine, right? Like if you couldn't trust that prior work, what would that mean? Well, it would mean you would spend hours, sometimes days, redoing work. That's not correct. : Yeah. : And so I think there's a second thread of what's going on, which will genuinely lead to a transformation in mathematics. And that is through a connection to what's known as formal verification programming languages. And the potential innovation here is that what if we could automatically translate from natural language math into this formal language where things could be verified. And then we could bootstrap off of the enormous mathematics literature that exists in this sort of chaotic state of papers and textbooks and so forth. We could encode all of that into the computer language and verify it all. And then there's a bunch of downstream consequences. One being just confirming things are correct. The other being making clear what the connections are across papers. A third would be lowering the bar of entry to math, right? There's just like all kinds of knock-on effects that I think are very exciting. If this idea of auto-formalization is successful.
好的。那么请您谈谈DARPA目前在证明验证和自动形式化方面的工作。我们已经讨论了这些工作的重要性，但您是如何开始思考这些问题的？您又是如何开始构建一个能将所有这些庞杂的信息形式化并处理的系统呢？
其理念是：如果我们能将现有的数学翻译成一种形式化语言，就能确认其真实性。此外，这种做法还可能极大加速数学进步，因为思想被以一种有力的方式连接在一起。您还可以拥有AI代理，与数学家共同担任作者。
▶ 英文原文 ⏱
Right. So tell us a little bit about kind of what DARPA is doing now on proof verification and auto-formalization. And, you know, we've talked about all the reasons about why that's important, but how do you start thinking about that even? And how do you start kind of building something that can formalize all of this and just ingest this like massive hot mess, as you say? : The idea is like, if we could translate existing mathematics into a formal language, you get, you know, confirmation that it's true, but also you can potentially radically accelerate mathematical progress because the idea is linked together in a powerful way. And you have AI agents that could be coauthors of mathematicians.
如果我们想要达到这样的世界，关键的瓶颈在于自动形式化的问题。目前手动将每篇论文逐一形式化的工作量太大。因此，如果我们真的希望生活在一个形式化成为数学发展基础的世界，就需要开发能够为我们进行自动形式化的人工智能系统。这样，数学家可以像往常一样进行数学研究，而他们的成果会自动以计算机语言进行验证。一个激动人心的进展是，宣布了球体堆积项目的完成。这个项目是通过一个自动形式化系统完成的，它将八维空间中的最佳球体堆积形式化。这个名为Gauss的人工智能系统由Math Inc.公司推出。球体堆积问题是一个相对直观的问题。
▶ 英文原文 ⏱
And so if we want to get to that world, the key bottleneck is this auto-formalization question. It's too much work to actually manually formalize papers one by one. So if we really want to, you know, live in this world where formalization is really a backbone for mathematics going forward, we need to be able to develop these AI systems that can do auto-formalization for us. Then the mathematicians do math as they always have, and it automatically gets verified in computer languages. One of the really exciting things that's happened is it was announced that the sphere packing project has been completed. And so this is a formalization of optimal sphere packing in eight dimensions by an auto-formalization agent. So an AI system called Gauss from a company called Math Inc. Sphere packing is a pretty intuitive problem.
这段话可以翻译成中文如下：
就像真的在打包球体、球一样。你知道，我们实际上已经很接近数学的前沿，我觉得这真的很令人兴奋。所以我们正接近这样一个地方，那里的事情真的会发生重大变化。我认为这一切变得非常接近现实，而且发展得非常迅速。我们刚刚听说，自动形式化，一种让计算机和人工智能帮助验证证明的方法，已经开始实现。我们稍后会再回到球体堆积问题的形式化概念。这是我们在节目一开始就谈论的一个问题：在一个立方体里能放多少个球？在一个箱子里能装多少个橙子？在此之前，我们对这一切的愿景是什么？
▶ 英文原文 ⏱
Like it's literally like packing spheres, balls. And, you know, we're really like getting close to the frontier of mathematics, which I think is really exciting. And so we're getting close to this place where like things are really going to shift. And so, you know, I think this is becoming very close to real and very rapidly. So we've just heard there, Anjini, that auto-formalization, a way for computers and AI to help to check through proofs is starting to happen. And we'll come back to the idea of formalization of this sphere packing problem again a bit later. This is a problem we talked about right at the start of the show. How many spheres can you get into a cube? How many oranges can you get into a crate? Just before we do that, what's the vision for all this?
我是说，数学文献是否可以变得更加一致，并且成为一个更统一的知识库，以汇集多年来人们所积累的知识？从某种意义上说，是可以的。你知道，不同的公司对这些工作有不同的愿景。比如有一个叫做Harmonic的公司，他们开发了一个名为Aristotle的机器人，目标是实现数学超智能。每个公司似乎都想要自己的超智能，不是吗？没错，这就是一切努力的目标。我也希望我能拥有数学超智能。今年早些时候，它解决了一个长期存在的错误问题，这让我们感到非常兴奋。Harmonic希望创建一个类似GitHub的模型，让研究人员可以公开发布、下载和即时验证数学证明。
▶ 英文原文 ⏱
I mean, is it that the mathematical literature can be made much more consistent and a more uniform repository of the knowledge that's been gathered by people for all these years? In some sense, yes. You know, different companies have different visions for all of this work. You know, there's a company called Harmonic, which has built a bot called Aristotle, which is aiming for mathematical superintelligence. Everyone has to have their own version of superintelligence, don't they? Exactly. That's the goal for everything. I wish I had mathematical superintelligence. And so earlier this year, it solved a long-standing error problem, which we were speaking about earlier. And that was all very exciting. And so they want to build a GitHub-style model where researchers can publish, download, and instantly verify proofs in public.
现在，这对传统出版物意味着什么呢？目前谁也说不准，因为其他公司也在开展类似的工作。而谷歌的“定义”也在进行自己的版本。他们有一个名为“alpha evolved”的工具，旨在为特定类型的问题生成证明，这些问题称为优化问题，其目标基本上是找到一个数学对象，最好地满足某些给定标准。同时，数学家们对人工智能的意义也有自己的看法。这并不意味着人工智能会取代他们。但是，对于持乐观态度的人来说，这些有用的模型，如果适量使用，能够提高生产力，帮助他们探索更多想法，并更好地将想法联系起来。
▶ 英文原文 ⏱
Now, what does all of that mean for traditional publications? Who knows at this point? Because other companies are doing similar work. And Google's defined is doing its own version of all of this. And they've got a tool that's alpha evolved that's designed to generate proofs for specific types of problems called optimization problems, where, you know, the goal is to essentially find kind of a mathematical object that best meets some given criteria. You know, and then mathematicians have their own vision of what AI will mean. And that doesn't mean that it's going to displace them. But it also for the optimistic variety, it means that these kind of helpful models, when taken in small doses, will be able to make them more productive, will help them explore more ideas, will help them connect the dots more.
我认为这是他们所追求的愿景，他们认为数学这个职业和数学家可能在做一些伟大的事情，或者与我们现在所做的不一样的事情，但他们可能会做得更好。我指的是当你与来自DARPA的Patrick Shafto交谈时，他描述了八维球堆积证明的形式化非常接近现实。他这是什么意思呢？AI模型对数学家Marina Vyazovska已经完成的证明进行了形式化，而这项工作为她赢得了菲尔兹奖。现在，她用了两年的时间来完成这项工作，并不是用一种正式的计算机代码完成的，就像我们之前讨论的那样。
▶ 英文原文 ⏱
And I think that's the vision that they're going for, where they think that the profession of mathematics, a mathematician may be doing something great, may be doing something different from what we do now, but they could be doing it better. I mean, Patrick Shafto from DARPA, when you talked to him, described that the formalization of the eight dimensional sphere packing proof as being very close to real. What did he mean by that? So, the AI model formalized proof that a mathematician Marina Vyazovska has already done, and that work won her the Fields Medal. Now, it took her two years, and it wasn't done in a kind of formal computer code, as we were discussing earlier.
一个名为Math Inc.的公司看到了这个机会，并计划将其纳入一个新兴的数学数字图书馆。此外，Vyazovska和一位研究生也在努力推动这项工作并将其规范化。总而言之，目前的AI并没有凭一己之力完成超人级的复杂证明，虽然它已经超越了解决国际数学奥林匹克题目的水平。这些大型语言模型（LLM）并不一定在探索新的数学思想，但它们能够在不同的数学分支之间建立联系。我认为，这在总体上是一个巨大的进步。
▶ 英文原文 ⏱
And so, a company called, you know, Math Inc., they basically saw this as an opportunity to add to an up and coming digital library of mathematics. And Vyazovska and a graduate student were also working on building this out and formalizing this. And so, all of this is to say that AI isn't doing superhuman levels of complex proofs on its own. And it has moved beyond international math Olympiad problems. And LLMs aren't necessarily exploring new ideas. They're able to establish connections between various branches and so on. And I think, ultimately, this is a big step forward.
这段文字翻译成中文是：这也为我们提供了新的机会来理解Vyazovska使用的复杂方法。为了了解她的工作被人工智能形式化是什么感觉，我采访了Marina。她告诉我Gauss AI模型效果很好。这个证明，最初的证明，大概有8万行代码。如果有人给你8万行代码的话，即使代码写得非常好，你可能也需要一些时间来理解。即使它在技术上是完美的，系统目前还是有些有趣的特性。
▶ 英文原文 ⏱
And it can offer new opportunities to understand complex methods that Vyazovska used. And to get a sense of what it was like to have her work formalized by artificial intelligence, I spoke to Marina. She told me how well the Gauss AI model worked. The proof, initial proof, I think it was like 80,000 lines of code. If somebody gives you 80,000 lines of code, it's like, even if it's very well-written, it still would probably take you some time to wrap your head around it. Even if it's a technically perfect code, the system at the moment has some interesting idiosyncrasies.
例如，代码中可能会证明许多简单的陈述。一个有趣的例子是，我在某段代码中发现它有一个声明的定理：2加2等于4。而且从技术上讲，这个定理实际上在证明过程中被用到了，所以不能简单地把它当作无用代码移除。它并不是无用代码，而是一个有用的定理，而且是正确的，所以没有问题。但在最终的代码库中，我们可能不希望将这作为一个单独声明的定理。也许可以把它隐藏在某个地方。
▶ 英文原文 ⏱
For example, it likes proving like a lot of simple statements. So, like one funny place I found in a code is that at some point, it declared the theorem that two plus two equals four. And like, technically, it was actually used in the proof. So, it was not easy to remove it as a dead code. It was not a dead code. It was actually a useful theorem. And it's correct. So, there is nothing wrong with it. But the final repository, maybe we don't want having this as like a separate declared theorem. Maybe it could be hidden somewhere.
你认为人工智能系统最终能够产生那些真正重塑了数学和更广泛领域的证明和结果吗？你认为AI能够做到这一点吗？从技术上讲，我看不出为什么不行。但或许问题也在于，如果想要重塑数学，那也需要有接受这个证明的人。假设有一天你打开新版的、你最喜欢的AI聊天机器人，输入“为我证明黎曼猜想”，它可能会给你生成一百万行的代码链接，甚至几百页的非正式证明。
▶ 英文原文 ⏱
Do you think that AI systems will eventually be able to produce some of these proofs and results that actually have reshaped mathematics and have reshaped this subject more broadly? Do you think AI will be able to do that? No, technically, I don't see why not. But maybe also the question is, if you want to reshape mathematics, there should be also somebody on receiving end of that proof. And suppose that you maybe one day you will open your new version of, I don't know, like your favorite AI chatbot and type like proof. I don't know, Riemann hypothesis for me, and it will give you million lines of link code and maybe even hundreds of pages of informal proof.
那么，我们该如何处理它呢？如果没有人愿意坐下来分析和阅读它，它还能算是为我们做了些什么吗？这并不是一个关于技术能力的问题，而是一个关于我们社会的问题。我可以想象一个幸福的乌托邦社会，人们有很多空闲时间，并对数学、音乐、创意写作非常感兴趣，他们有时间去欣赏这些AI创造的精彩作品，花时间去研究、重写和讨论。这将丰富我们的人类文化，并可能带给我们新的想法，帮助我们将其应用于其他领域的知识。
▶ 英文原文 ⏱
Then what to do with it? If no person would sit down and analyze it and read it, does it count like it has done something for us or not? It's not a question about technical capability. It's more a question of our society. I could imagine a happy utopian society where people have a lot of free time and are very interested in mathematics, music, creative writing, and they have time to take these wonderful artifacts that AI produced and spend time and actually study it and rewrite it and discuss it. And this will enrich and then our human culture. And maybe we will give us new ideas and help us to transfer it to other fields of knowledge.
现实中会发生什么？我不知道。我的意思是，你怎么看待在小范围内运用AI，或者用其它方式，来真正解决其他科学领域的问题，并将不同领域连接起来，从科学发现的角度看其实际应用？是的。我们必须经历这个历史时期，真正与它一起工作、尝试、实验，看看什么有效，什么无效。
▶ 英文原文 ⏱
What will happen in reality? I don't know. I mean, how do you think about, you know, leveraging AI in small doses or whatever it is, to actually kind of solve these problems in other sciences and connect different domains and, you know, the practical uses of it from a scientific discovery perspective? Yes. So I think we have to live through this moment in history and really work with it, play with it, experiment, see what works, what does not work.
所以，我真的希望能让形式化变得不那么困难。要是这个过程能更轻松、更快速就好了。希望在未来的某个版本中，我可以一边用LaTeX写论文，一边同时为同一篇论文写形式化内容，而且大概花费差不多的时间。那将是非常好的。我认为这对这个领域也有益。如果到了那个时候，我们能够得到很多很多证明，可能我们问的数学问题也会有所不同。
▶ 英文原文 ⏱
So I would really like making formalization less difficult. It would be nice to make this process less painful and faster. So in some like version of the future where I write a LaTeX paper and in parallel with it, I also write a formalization of the same paper and it takes maybe around as much time. That would be very nice. And I think it would be good for the field. Also, if we are in this moment when we actually can get many, many proofs of many, many statements, maybe then the questions we are asking about mathematics would be different.
也许我们应该开始提出不同的问题，而这个过程本身可能会为我们提供一些见解，使我们现在能够提出之前可能不觉得有趣或自然的问题。安简，听起来对玛丽娜的工作的规范化过程让你对八维球体堆积问题有了更好的理解。但我在想，如果你有一个由人工智能构建的数学知识库，其中的代码和不同证明的过程是透明的，并且人工智能处理这些问题的不同方法都是开放给大家查询的，这将会带来怎样的变化。
▶ 英文原文 ⏱
Maybe we should start asking different questions and maybe this process itself will provide us with some insights that we now can ask questions we would not find interesting or natural before. So Anjan, it sounds like the process of formalizing Marina's work sort of gives you a better understanding of this eight dimensional sphere packing problem. But I'm just thinking if you had this kind of repository of mathematical knowledge built by AI and you had this code and it was all transparent how different proofs were made and the different ways that AIs approach these things and anyone can interrogate these things.
这对数学以及使用数学的人来说意味着什么？这种特性有什么用处？想象一下，这就像拥有一个庞大的已知真理的图书馆，所有人都能理解，并且在语言上听起来、看起来都一样。当数学研究者探索新想法时，他们可以依赖这些已有的研究并理解它们。因为正如我们之前讨论的，这是数学领域中一个重要但未被广泛认识的瓶颈，对吧？
▶ 英文原文 ⏱
What does it mean for mathematics and for people who use mathematics? What is useful about that? You know, it's like having a vast library of established truths that everyone can understand and that sounds the same, looks the same in terms of the language. And as a mathematics researcher explores new ideas, they can even rely on this work and understand it. Because as we were discussing earlier, you know, this is a fundamental bottleneck in the field of mathematics that is not widely appreciated, right?
因此，如果你有这样一个被称为格式化的东西，即使它是一个统一的研究成果，它为物理学家带来了更多科学发现，因为他们可以依赖它。这对试图突破某个想法界限的数学研究生或理论计算机科学家同样有帮助。因此，我认为，实际上将这些内容集中在一个地方具有巨大的价值。
▶ 英文原文 ⏱
And so if you have this, you know, we call it formalized, but if it is a uniform body of work, it opens up more scientific discovery for the physicist who can also rely on it. It opens up for a graduate mathematician who's trying to push the bounds on an idea, for a theoretical computer scientist. And so I think there is a huge amount of value in essentially putting this in one place.
到目前为止，工程师，我们已经讨论了各种大型语言模型及其不同版本，它们能够证明和重新验证人类已经完成的事情，有时甚至能够自行解决长期存在的难题。不过我在想，人工智能模型是否最终能够更经常地做到后者？换句话说，它们不仅能够解决问题，还能自主提出理论、问题和有趣的想法，以便处理研究问题或为人类创造新的研究课题进行探索？
▶ 英文原文 ⏱
So, so far, engineer, we've talked about various large language models and versions of them proving and reproving things that humans have already done, and sometimes even striking out on their own to solve puzzles that have been longstanding for a long time. But I wonder, will AI models eventually be able to do the latter more regularly? In other words, not only solving problems, but coming up with theories and problems and interesting ideas of their own to tackle research problems or create new ones for humans to sort of investigate?
这就是关键问题，对吧？到目前为止，他们还没完全达到那种程度。但是，这需要一种完全自主的数学思维能力，对吧？这就像我们谈论的通用人工智能（AGI）等内容。我认为在数学领域，实际上比人们想象的要复杂得多。你可能会认为数学像基本的算术那样简单，可以轻松地进行计算。但是，数学有很多不同的领域，对吧？
▶ 英文原文 ⏱
That's the big question, right? You know, so far, they haven't quite gotten there. But it takes another level of this completely autonomous mathematical thinker, right? It's kind of what we talk about AGI and artificial general intelligence and so on. And I think with math, it's actually a lot trickier than people pursue. You know, you would think that something like mathematics, which we typically think of as basic arithmetic is quite easy to just chug and churn along. But there's so many different fields of mathematics, right?
他们在统计学和组合数学方面可能更擅长，但在提出理论或自主证明某些数学命题及其他数学领域上可能并不如意。而且还有许多挑战需要应对。其中一项挑战是“幻觉”，这可能会严重干扰研究人员。正如一位数学家告诉我的那样，如果你，比如说，使用一个大型语言模型（LLM）并给它提示，它往往会倾向于同意你的观点。
▶ 英文原文 ⏱
They may be better at statistics and combinatorics, but it might not be as good at coming up with theories or even autonomously proving certain statements and other areas of mathematics. And there are still a lot of challenges to grapple with. There's hallucinations, which can really throw researchers off. As one mathematician was telling me, if you are, let's say, working with an LLM and you kind of prompt it along, it tends to agree with you.
所以问题在于，它并不是真正自己思考。它是在试图预测数学家接下来会做什么，这本身就削弱了提出新奇想法的过程。因此，我认为还有很多路要走。但问题是，为了什么目的？正如Marina Vyazovskam对我说的，你可以去想出很多不同的证明，但谁会用这些证明呢？我们为什么要这样做？你必须知道目的以及这些证明的用户是谁。如果它就那样放在一边，对任何人都没有吸引力，那也很好，它在那儿，但是增加的价值可能就不那么明显了。
▶ 英文原文 ⏱
And so the problem is that it's not really thinking for itself. It's trying to predict what the mathematician would do next, and that in itself undermines the process of, say, coming up with something novel and new. So I think there's ways to go. But the question is, to what purpose? As Marina Vyazovskam noted to me, she said, look, like, that's fine. You can go and come up with many different proofs, but who's going to use it? And for what are we doing that? And you have to know the purpose and who is the consumer of this proof. And if it's just sitting out there and it's not interesting to anyone, then great, it's there. But the value add part is perhaps less so.
那么，这不正是潜力所在吗？数学家喜欢解决有趣的问题，对他们来说，有趣的意味着所有创造性的东西。他们最感兴趣的是数学中的人性化部分。这常常会引导他们在未来取得最有趣和最有用的成就。我只是在想，数学家们对人工智能的进展有何感受，特别是从更直接的角度来看。他们对机器人摄取他们所有已发表的研究成果，并可能以某种方式与他们竞争有何感想？正如各家公司努力构建数学超智，致力于开发自动形式化工具的过程中，他们也在大量利用数学家们完成的工作。
▶ 英文原文 ⏱
Well, isn't that the potential though? Mathematicians like to solve interesting problems and interesting to them means all the creative stuff. It's the human stuff about mathematics that they're interested in the most. And often that can lead to the most interesting and useful things in the future. I just wonder how mathematicians are feeling about the progress of AI in a more literal sense. How do they feel about bots ingesting all of their published work and going on to perhaps even compete with them in some way? You know, as there's all this development, as companies are trying to build mathematical super intelligence and work to develop auto formalizers, they are also using a lot of work that has been done by mathematicians.
我认为，需要在不干扰他们正在努力进行的工作和给予所有这些数学家应得的荣誉之间取得一个微妙的平衡。因为这些数学家花费了无数个小时和几天的时间专注于自己的研究。Marina和她的一位研究生告诉我们，他们发现自己处于类似的境地。他们对人工智能非常开放。在构建他们的资料库并希望将这个证明形式化的过程中，他们向我们之前讨论过的一家公司Mass Inc敞开了大门。几个月的沉寂之后，他们惊讶地发现，这家公司突然间完成了这个证明的形式化过程。
▶ 英文原文 ⏱
And I think there is a fine balance of not trampling on their work and the work that they are trying to do. And also the credit that is due to all these mathematicians that have spent hours and days of their lives doing this. Marina and one of her graduate students, she told us, found themselves in a similar situation. They had been very open to kind of artificial intelligence. And as they were kind of building their library and wanted to work towards formalizing this proof, they'd opened themselves up to Mass Inc, which is one of the companies we discussed earlier. And a few months after they'd worked them after a period of silence, they found themselves surprised that suddenly this company had formalized the proof.
他们不仅感到惊讶和震惊，还因为事实证明并不像表面上看起来那样而感到失望。我认为在那之后，他们讨论了归属和荣誉等问题，可能也更紧密地合作了。不过，我认为这种情况可能会让数学家们感到警惕，因为它抹杀了所取得的进展。我在想，是否可以利用人工智能来解决数学的技术性问题和形式化问题，以确保知识能够以一种有用的方式共享。也许这能够让真正的数学家——人类数学家，腾出精力去做那些最初看起来有点疯狂但实际上推动数学向前发展的有趣创造性工作。
▶ 英文原文 ⏱
And they were not just surprised and shocked, but upset by the fact that the proof wasn't exactly how it looked. And I think after that they discussed attribution and credit and so on, and perhaps worked more closely. But I think that is somewhere where those kinds of situations will make mathematicians wary and, you know, it negates the progress that's being made. I wonder if AI can be used to solve the sort of mechanics of mathematics and the formalization of things to make sure that the knowledge is shared in a useful way. Perhaps it frees real mathematicians, human ones, to do all the interesting creative stuff that seems a bit bonkers at start, but, you know, is actually the one that pushes mathematics forward.
也许我们并不需要人工智能来推动数学的发展，而是利用它来解放人类数学家，让他们去推动发展。这绝对是一个重大的好处。而且我认为数学家们可能会欢迎这样的变化。但同样值得注意的是，数学是一种实践。当你学习数论、逻辑学、离散数学和应用数学等基础知识，并逐步深入到某个专业领域时，这种学习会以某种特定的方式锻炼你的大脑和思维。
▶ 英文原文 ⏱
Maybe we don't need the AIs to push the maths forward, but we use it to free the human mathematicians to do so. And that's absolutely one of the big prizes. And I think mathematicians would potentially welcome that. But I think it's also worth noting that mathematics is a practice. And when you learn kind of the basics of number theory and logic and the field of discrete math and applied math and kind of go through these various fields and you build your way up into a specialty and so on, it trains the brain and it trains your mind in a certain way.
通过完成所有这些基础工作，我们达到了这一点。如果可以这么说，这就是思考的基础。所以，我认为你需要两者兼备。在这方面，大型语言模型（LLM）可以有所帮助，并扩展我们的思维。也许它们会让数学家拥有更广阔的视野，而不是在某些数学领域视野受限。这本身就能帮助他们更好地连结不同的点。因此，我认为这对数学家和科学家都会有很多价值。
▶ 英文原文 ⏱
And so we get to that point by having done all that grunt work, if you will. And that's what allows the thinking. And so I think you'll need a bit of both. I do think there is a part of it that LLMs can help with and kind of expand the thinking. And perhaps what they will do is allow mathematicians to have a wider aperture, right? Not have their blinders on on certain areas of mathematics. And that in itself will help them connect the dots more. And so I think there's a variety of things that will be of value to mathematicians and scientists in general.
安贾莉，你的分享让人着迷。我相信，随着数学和人工智能的发展，这个领域将会有更多的进展。因此，我们期待在将来能和你继续探讨这个话题。非常感谢你今天抽出时间参与我们的节目。也感谢特伦斯·陶、帕特里克·沙夫托和玛丽娜·维亚佐夫斯卡，以及许多其他在报道中接受安贾莉采访的数学家们。您在节目开始部分已经听到了其中的一些观点。
▶ 英文原文 ⏱
Anjali, that's been fascinating. And I'm sure that as maths and AI develop together, there'll be more on this story. So we'll look forward to talking to you more about it in the future. Thank you very much for your time today. Thank you so much for having me. Our thanks to Terence Tao, Patrick Shafto and Marina Vyazovska, and to the many, many other mathematicians Anjali spoke to in her reporting. You heard some of them in the introduction to the show.
感谢您的收听。如果您喜欢本期内容，并且认识其他可能也会感兴趣的人，您可以在《经济学人》应用程序上免费分享。只需在播客标签中找到这一集，点击分享按钮，然后选择“作为礼物赠送”。本周的Babbage节目就到这里。我们的编辑是Jason Hoskin，他与Peter Granitz和Hannah Fisher一同制作了本期节目。混音和音效设计由Nico Rofast完成，执行制片人是Hannah Mourinho。我是Alan Cha，这里是伦敦的《经济学人》。
▶ 英文原文 ⏱
And thank you for listening. If you enjoyed this episode and know someone else who might do as well, you can share it on The Economist app for free. Just find the episode on the podcast tab, tap the share button, and then select give as a gift. That's all from Babbage this week. Our editor is Jason Hoskin. He produced this episode with assistance from Peter Granitz and Hannah Fisher. Mixing and sound design was my Nico Rofast and the executive producer is Hannah Mourinho. I'm Alan Cha and in London, this is The Economist.