内容来源：https://www.quantamagazine.org/live-from-icm-2026-what-is-math-for-in-the-age-of-ai-20260903/

内容总结：

ICM 2026特别报道：AI时代，数学的意义何在？

顶尖数学家热议人工智能对数学领域的深刻影响

费城讯——在2026年7月26日于费城举行的国际数学家大会（ICM）上，一场关于人工智能与数学未来的特别讨论吸引了广泛关注。《The Joy of Why》播客首次进行现场录制，邀请了三位世界顶尖数学家——普林斯顿高等研究院的Akshay Venkatesh、斯坦福大学及美国数学会主席Ravi Vakil、罗格斯大学的Alex Kontorovich——共同探讨AI正在如何重塑数学这门古老的学科。

AI解题能力引发思考

讨论从近期AI在数学竞赛中的突破性表现切入。2024年，AlphaProof在国际数学奥林匹克竞赛中获得银牌；2025年，配备DeepThink的进阶模型更斩获金牌。曾参与过奥赛的Vakil教授对此持辩证态度："模型首次答对题目确实令人兴奋，但这是因为我们达到了一个里程碑；而学生首次做出题目则不同，因为他们学会了思考。"

Kontorovich教授则更为直接地指出，AI公司所谓"金牌"的自我宣称存在炒作成分。他认为AI的真正价值不在于解决多少问题，而在于为数学家提供新的研究思路。他以近期被AI攻克的Erdős单位距离猜想为例，强调更珍贵的是随后人类数学家借鉴AI思路解决其他未决问题所产生的连锁效应。

数学的本质：证明还是叙事？

当话题转向数学的核心价值时，三位数学家不约而同地挑战了"证明是数学中心活动"的传统观念。Venkatesh教授提出一个发人深省的观点："我做数学时，更倾向于将其视为讲故事的过程。证明或许只是其中的语法部分。"

Vakil教授对此深表赞同，并以费马大定理为例说明："一个伟大的数学成果之所以引人入胜，是因为它属于一个绵长而丰富的故事脉络。每解决一个问题，就像完成一部书的一章，却总留下让人期待下一部的悬念。"他认为，费马大定理真正伟大之处，不在于结论本身，而在于探索过程中所建立的、连接不同数学分支的"戏剧性桥梁"。

对年轻一代的担忧

当话题转向AI对数学教育的冲击时，Vakil教授坦言真正令他忧虑的不是技术本身，而是它可能成为反智主义的"特洛伊木马"。"有人以AI为借口削减研究生项目和科研经费，声称学生不再需要学习如何思考，因为机器可以代劳——这种论调毫无道理且正在发生。"

三位教授都观察到学生群体中出现的"两极分化"——善用AI辅助理解的学生进步加速，而直接让AI代做作业的学生则在考试中暴露短板。Kontorovich用生动的比喻劝诫学生："做数学练习就像在健身房举重，你若开着叉车去举杠铃，练出的肌肉永远不会是自己的。"

未来展望

在回答现场观众提问时，几位数学家对数学界的未来持谨慎乐观态度。他们认为，AI的冲击或许会促使数学界重新重视长期以来被忽视的教学与阐释工作，使数学回归其作为"人文学科"的一面。正如Venkatesh所言："人类的理解始终是属人的、主观的。数学兼有科学与人文的双重基因，或许现在是时候在AI的推动下重新拥抱它的人文特质了。"

中文翻译：

2026年国际数学家大会现场直播：在人工智能时代，数学的用途是什么？
引言
数学家们正在目睹自己领域的深刻变革。人工智能系统现在能够生成证明、发现相距遥远领域之间的联系，并且在某些情况下，解决了困扰数学家数十年的难题。过去几个月的进展速度引发了一系列具有挑战性的问题：人工智能系统究竟能做什么？一旦机器在解决问题方面能够匹敌甚至超越人类，数学领域中那些更人性化、更具创造性的方面又会发生什么变化？

在本期《为什么的乐趣》特别现场节目中，我们录制于2026年7月26日在费城举行的国际数学家大会。主持人Janna Levin和Steven Strogatz邀请了三位数学家一同探讨人工智能在数学领域的崛起：来自高等研究院的Akshay Venkatesh、来自斯坦福大学并担任美国数学学会主席的Ravi Vakil，以及来自罗格斯大学的Alex Kontorovich。他们共同讨论了近期人工智能生成的证明究竟证明了什么、随着数学在前沿领域越来越依赖机器辅助将会得到什么又失去什么，以及这对下一代数学家意味着什么。对话最终转向了一个更深刻的问题：数学家们最初从事数学研究时，究竟珍视的是什么？答案涉及对证明和理解真正含义的深入思考，以及讲故事这一出人意料的的重要性。

请在Apple Podcasts、Spotify、TuneIn或你最喜欢的播客应用上收听，也可以从Quanta网站在线收听。

【观众掌声】

JANNA LEVIN：非常感谢大家来到这里。我是Janna Levin。
STEVE STROGATZ：我是Steve Strogatz。
LEVIN：这里是《为什么的乐趣》。欢迎来到我们的第一期现场节目。

【观众掌声】

STROGATZ：是的，《为什么的乐趣》，一档我们一起探讨当今数学和科学中一些最大未解问题的播客。我们非常兴奋能来到2026年国际数学家大会现场为大家播出。
这确实感觉像是一个历史性的时刻。数学的未来面临着相当高的风险。我们人类靠自己在这条路上已经走了三四千年，一直走得不错。但城里来了个新玩家——人工智能。它开始参与到我们的游戏中来，参与到我们的领域中。
所以我们很期待听听我们的嘉宾们对数学与人工智能之间互动关系的看法，无论是现在、短期内，也许还有更长远来看。

那么让我来介绍一下我们杰出的嘉宾阵容。首先，我看到了来自高等研究院的Akshay Venkatesh。坐在他旁边的是来自斯坦福大学的Ravi Vakil。最后一位是来自罗格斯大学的Alex Kontorovich。好的，Janna，你来开启我们的讨论吧？

LEVIN：好的，我想也许我们可以从一些让部分数学家感到不安、而另一些人则不以为然的近期进展开始。我要回溯到2024年国际数学奥林匹克竞赛，当时AlphaProof获得了银牌，然后在2025年，一个更先进的模型搭配DeepThink获得了金牌。
在我们深入探讨它是如何做到的之前，我想先问一下，你们几位当年有没有参加过国际数学奥林匹克竞赛？你参加过？你也参加过？你们能给大家讲讲吗？因为有些听众可能不太了解，那是六道题、九个小时、为期两天、参赛者是高中生的比赛。给我们讲讲那是怎样的体验。

STROGATZ：我听说Ravi当年在这项比赛中可是个令人畏惧的角色。Jordan Ellenberg还在说你当年是怎么在比赛中碾压他的。

LEVIN：是啊，来参加会议的人都在说你当年是怎么把他们打服的。

RAVI VAKIL：其实我要说，我就是在那个赛场上第一次见到Akshay的，虽然他可能不记得了。

AKSHAY VENKATESH：我确实不记得了。抱歉。

VAKIL：没错。因为他当时还很小。呃，但这是通往数学的众多道路之一。通往数学没有捷径，但这是众多吸引人们进入数学领域的方式之一，也是一种很好的数学宣传方式，而且有助于培养一种特定的数学思维方式。
关于这个比赛，重要的不是那六个人或者那些题目，而是它如何激励了成百上千万的孩子磨炼他们的心智。所以，当模型表现出色的时候，那真是太棒了。这确实是一个概念的验证。我认为这极其有趣，同时我认为也许有人对它产生了误解。
当一个模型第一次答对一道问题时，那很令人兴奋。一个里程碑达成了。当一个学生第一次答对一道问题时，兴奋的原因不同，因为他们学会了如何思考。而当一个模型第七次答对一道问题时，就没那么有意思了；同样地，当一个学生第七次答同一道题时，也没那么有意思了——原因类似。
所以，这确实令人兴奋，在我看来这完全是件好事，但这并不代表超人智能或任何那种疯狂的东西。

STROGATZ：是的，当然。

LEVIN：而且即使是它出现的那一年，它也没有拿到最高分。

VAKIL：没错，但第二年它就到了，之后的每一年它都会到。对此我很高兴，而且我完全不感到惊慌。

LEVIN：那么Akshay，你的经验是什么？

VENKATESH：我觉得Ravi说得很到位。我觉得，你知道，这件事的一个不幸之处在于，它加剧了一种印象——即使在数学家中间——即数学是一场竞赛，但这不是我对数学的看法。而且我认为这在当下尤其没有益处。确实存在某种竞争驱动力。你知道，当你年轻的时候，如果人们认为你在某方面很出色，你就会得到认可。我对这种事情是否健康有复杂的感受。

VAKIL：正如Akshay所说，这可能是一种危险。凡事都有两面性。有人因为竞赛没考好就觉得自己不行，从而被赶出了数学领域，而这完全是没道理的。

STROGATZ：我对你的观点非常感兴趣。有句老话说“重要的是旅程，而不是目的地”，你似乎就是在倡导这种哲学。但我觉得这也是某种雏形，也许我们今天讨论结束时就能触及这个话题，因为我们都可能需要开始思考：数学究竟关乎旅程，而非目的地？

LEVIN：我正要说，对Alex来说，除此之外，那还是2025年的事。最近我们又看到了一些重大的进展被发表。其中一个Erdős问题被解决了。我只是好奇，第一，你是否感到印象深刻、惊讶，你对这件事的反应是什么；第二，你能否为大家介绍一下被解决的那个问题是什么。

ALEX KONTOROVICH：好的。关于IMO，我只想说一句，你知道，两年前我们有了AlphaProof，它离金牌就差一分。顺便说一句，没有任何人给这些公司颁发奖牌。是他们自己宣布——

LEVIN：对，自封的金牌得主。

VAKIL：他们给自己发奖牌。

KONTOROVICH：没错。他们——是的，就是这样。嗯，我当时预测今年夏天不会有AI模型获得金牌，因为它们都不屑于再参赛了。因为这对它们来说已经没意思了。我可以按一下GPT 5.6上的按钮就在IMO拿到满分。所以说，好吧，下一个。还有什么更有趣的事？那个已经不再有趣了，正如你们刚才说的那样。
但让我回到你问的关于Erdős问题的话题。好的，我们有正在研究的问题，正如Akshay所说，解决问题只是数学家工作的一小部分。我们寻找定义，发现我们感兴趣的新结构，我们一直在创造新的问题和新的难题。
有一个人特别喜欢提问，而且不介意不管对错都把问题写下来，那就是Erdős，他写下了数千个问题。幸运的是，Tom Bloom——不管出于什么原因——几年前决定把这些整理成一个数据库。
在数学中有时会发生这样的情况，实际上相当频繁，那就是一些我们以为很难的问题，最终被发现并没有我们想的那么难，用现有的技术就能解决——只要你肯去研究那个问题，并且具备必要的背景知识来应用那些最终能攻克它并让它变得简单的技术。
而AI最擅长的事情是什么：你给它1000个问题，也许它有1%的成功率，那一年就有10篇好论文，这在数学界已经是非常出色的职业生涯了。所以，现在我们每天都能看到越来越多的这类Erdős风格的问题被宣布解决。其中有多少是真正的解决方案，而不是部分进展或者完全是胡说八道——因为那只是LLM在胡言乱语——这是另一回事。
我想你指的应该是Erdős单位距离猜想，这个猜想很多人都知道，也有很多人研究过。在试图把这个常数逼近到猜想值方面，已经取得了非常重要的进展。结果发现，猜想的值是错误的。
而且，GPT自主地找到了一个反例，你知道，构建了一个不满足该指数的解。对我来说最有趣的是，我的意思是，首先，这很精彩。太棒了。如果是一个人类提出了这个反例，这将是一个非常伟大的贡献。嗯，最有趣的地方在于，一周之后，一个完全不相关的问题，叫做和积问题，也被人类用否定的方式解决了——他们应用了从这个AI解决方案中展现出来的想法和技术。

LEVIN：是的，太有意思了。

KONTOROVICH：所以对我来说，这大概就是我们希望进入的黄金时代。我们将通过这些技术学到各种惊人的数学。但话说回来，“如果一个定理倒在森林里，而没人在乎，它还有价值吗？”之类的问题。
所以我不认为定理在宇宙中有内在价值。数学家珍视它们，而现在AI公司也珍视它们，因为它们非常适合做营销。它们是这些公司说“嘿，我的工具比以前那个工具以及市面上其他所有工具都好”的绝佳方式。从这个意义上说确实非常有效。我认为很快AI公司就需要其他真正能给世界带来价值的东西，而定理除了对数学家之外并不带来价值。但在它们走过之后，会给我们留下一件奇妙的工具，用来学习大量惊人的事实。

LEVIN：我确实担心，生成如此多可能是错误的证明的能力可能会成为数学家的一个黑洞——现在有一个产业就是专门核查这些证明。
你们几位有没有发现自己被核查某些证明的吸引力所诱惑？我是说，我们一些非常著名的同事就这么做了，对吧，主动请缨提供服务。Akshay，你会去做这种事吗？像Terry Tao那样去做审稿人。

VAKIL：100个AI槽位证明。你会报名做那个吗？

KONTOROVICH：得付你多少钱你才肯干？

STROGATZ：这里面有很多光环啊。来吧，你怎么能抵挡得住？

VENKATESH：但我觉得这跟我们的学术共同体中一个非常现实的问题密切相关，那就是期刊的运作方式以及它们将如何应对这一切。所以我认为作为一个共同体，有一大堆系统我们需要关注，这些系统将承受巨大的压力。

STROGATZ：嗯。

VAKIL：对。部分问题在于人们不会排着队来做这件事。它们之所以承受压力，是因为我们被淹没了，而洪水才刚刚开始——大量材料涌入，其中确实有一些真正了不起的事情在发生。而且，我在这两个小时里收到了邮件，我可不期待看到人们会给我发来什么样的AI垃圾——他们认为自己证明了什么惊人的东西。说实话，我不会去核查。那不好玩。

STROGATZ：这让我想到Terry Tao一直提到的三件事：证明生成、证明验证——这就是我们现在在说的——还有证明消化，也就是我们开始用人的语言来理解这些证明，并评估它们是否有趣、有多有趣。但我在想，也许还有另一个类别，那就是证明营销。

VENKATESH：嗯，我想质疑一下证明是核心活动这个假设。

STROGATZ：好，继续说。

VENKATESH：嗯，如果我想想，你知道，当我在做一项数学工作的时候，我可能更倾向于把它看作是在讲故事。你知道，我在试图讲述一个故事。嗯嗯。也许证明只是那个故事的语法的一部分。
这是我们应该思考的事情。比如，我们是否应该用证明来定义这门学科？当然，我们一直都是这么做的，但我不确定这是否真的与我们珍视的东西和我们正在做的事情相符。

VAKIL：我想强力支持Akshay说的话，因为我觉得……我也认为这才是我们一直珍视的东西。这不是什么变化。这是我们从始至终一直珍视的，我们有过各种各样的替代指标，而现在因为技术的变化，我们被迫去审视这些指标。
但人类的理解和讲故事——我的研究生们，他们已经很出色了。我教他们的是讲故事。学习如何做数学，就是当你写一篇论文的时候，你必须理解人类的理解方式以及如何传达这种理解。我认为这才是我们一直想要的。
有些才华横溢的数学家提出了想法，但因为无法说服别人而不了了之。还有一些人真正改变了领域，因为他们非常擅长表达。Erdős就是一个很好的例子，他改变了数学；或者在一个完全不同的层面上，Martin Gardner可能比大多数人都更多地推动了数学的发展，因为他通过讲故事做到了这一点。

STROGATZ：我觉得讲故事这个词不会出现在大多数人的脑海里。所以你们两位现在有责任给我们解释一下，你们到底在说什么？

VENKATESH：我想那是我能给这个过程的找到的最好的词了。你知道，我在研究某个东西。那里有一片我正在探索的景观，里面有各种各样的东西，我在思考：“我要如何讲述一个故事，让我的同事们觉得有趣和吸引人呢？”

LEVIN：这是你被问题吸引的方式吗？我的意思是，数学是少数几个人们可以板着脸谈论美、优雅，还有在这个情况下，讲故事的地方之一。你能给那些不熟悉“数学是美的”或“数学具有叙事力量”这种想法的人举一个例子吗？

STROGATZ：Ravi，你看起来已经准备好了。

VAKIL：我要说的是，我想不出数学中有什么重大进展不符合这个范式。就拿费马大定理来说吧。是什么让费马有趣，而哥德巴赫猜想则无趣得多？那是因为费马是一个漫长而丰富的故事的一部分。人们开始深入挖掘它，在19世纪，它开启了许多丰富的问题。需要说明的是，它很美，但同时也很有力量。数学的美妙之处在于，当你拥有一个在经验上很美的惊人故事时——我不知道为什么宇宙会这样运作——但当你理解了某件事，故事变得越来越简单时，你会得到越来越强大的东西。
毫无疑问，19世纪的这些发展对技术、科学和我们的生活水平带来了什么。但那是因为人们被好奇心驱动，所以费马带来了很大一部分发展。嗯，不是单独地。它是一个更大故事的一部分。
然后最终的证明，它最终的样子要好得多，因为它需要构建一个更有趣的故事，在不同思维方式之间架起戏剧性的桥梁——这些桥梁本不该存在。它们完全遥不可及，直到它们变得可以触及。直到出现了戏剧性的思维飞跃。有扣人心弦的悬念。有看起来我们已经要完成了的例子，然后——哎呀——其实没有。然后故事还没有结束。
当你读完一本书，你把它放下，最好的定理是那些让你期待下一卷的定理。全世界的人都在讨论它，试图思考故事如何继续。很难找到一个不符合这个模式的例子。

STROGATZ：嗯。我想试着把这个“故事”的概念再精炼一下。你说的是数学在时间维度和学科维度上的互联性。数学是这种惊人的、往往是地下的东西，偶尔冒出地面，我们能看到山峰，但我们没有意识到有这些地下的分支连接着那些看不见的部分，直到……好吧，我的比喻有点混乱了。
但是，嗯，就在那方面——当你说故事的时候，你指的是这个学科的内在连贯性，我想。但是，我不知道。Alex，你想试着比我说得更清楚一些吗？

KONTOROVICH：也许有一种说法是，费马大定理这个问题，你知道，这个X的N次方加Y的N次方等于Z的N次方，在宇宙中没有任何实际后果。没有人真的在乎它是否正确。你知道，如果有一个反例，它会摧毁宇宙中的什么东西吗？嗯，事实证明它会摧毁宇宙中的某些东西。它会摧毁朗兰兹纲领。呃，但是，嗯，那是一个必须被证明的定理。
但我认为，那个问题引发了一系列绝对惊人的突破性发现。从那个问题的开端开始，它导致了对数域中因式分解非唯一性的理解，以及我们必须把数域理解为其自身实体的事实。所以那一个问题——如果你只想着“我想知道那是不是真的”，没有任何理由——你只是沿着那个故事的线索从一个突破到另一个突破，你就会一路走到这个巨大的谷山-志村猜想，然后是Taylor-Wiles的突破。所以那是一个美丽的、贯穿始终的宏大故事。我们在数学中有很多这样的故事。
再举一个例子，平行公设的问题，欧几里得《几何原本》中的第五公设，能否从其他四条公设中推出。看起来是最深奥、最无用的东西。谁在乎第五公设能不能从其他四条推出啊？我们甚至不是在争论第五公设本身是否真实。我们说的是，它应该是一个定理还是理论中的一个公理？你沿着那个问题两千年的线索一路走到爱因斯坦的广义相对论。

LEVIN：而这件事对宇宙来说确实非常重要。

KONTOROVICH：没错。

STROGATZ：那个例子也是如此。Akshay，我觉得你确实挖掘到了一个丰富的矿脉，把我们的注意力引向了数学故事这个概念。我还想知道，因为你似乎在说，证明并不是我们追求的最高目标——它们可以为不断展开的故事或更丰富的故事做出贡献，是的。但数学中我们有时会说某事“在道德上是真的”，这又是什么呢？比如在我们有证明之前，甚至在我们有精确的陈述之前，我们就隐约知道是怎么回事了。你能谈谈那种感觉是什么样的吗？我们说的“在道德上为真”是什么意思？

VENKATESH：哦，呃，是的，当然。你知道吗，就拿我的经验——大概也是在座很多人的经验——举个例子吧。当你写一篇论文时，你在填补所有细节之前很久就知道什么是对的。你抓住了一种直觉。获得那种直觉实际上是最令人享受的部分。有时候这种直觉如何运作、以及它有时如何失效，是很神秘的。但是的，在某种意义上，知道某件事却不知道你是怎么知道的，要有趣得多得多。

LEVIN：那么，选择一个问题难道不存在某种意义上的直觉调用吗？

VENKATESH：是的，当然，对吧？你不知道那里有什么。但你有一种感觉。你感觉到那里有某种丰富的东西。

LEVIN：我想，Akshay，我同样想问，你是否在乎一个证明来自机器、来自一位前辈还是来自一位年轻新秀？来源对你来说重要吗？

VENKATESH：你知道，我当然看不出为什么机器不能产生我觉得有趣的东西。但我也认为，随着我们使用机器，这些概念会非常迅速地改变。所以现在去看我们珍视什么、然后问机器能否做到，可能没有太大意义。更有趣的可能是思考它将如何改变我们所珍视的东西。
我的意思是，我们这么多人正在以不同方式把AI作为日常研究的一部分，这本身就向你展示了数学实践正在发生变化。

STROGATZ：那么在这一点上，我想问一个我觉得有点敏感、但也可能非常有趣的问题。在这次会议上，以及我来之前和人们交谈时，似乎存在某些禁忌话题——有些事我们作为一个共同体应该讨论却没有讨论，还有些事我觉得我们很多人都在担心但不公开说出来。

VENKATESH：好吧，Ravi必须回答这个问题。

STROGATZ：有谁愿意——作为美国数学学会的主席，Ravi。不，你不必说什么，你是主席……

VAKIL：不不不不，我完全不会委婉。没有禁忌。呃，所以我认为人们在公开场合或在社交媒体上担心的事情，并不是我们应该担心的事情。那些我不太担心。
我确实认为有些事情我们应该非常担心。我担心年轻数学家们。而好的方面是，这个共同体——至少是一致地——似乎意识到这个职业的危险来自于对年轻数学家的威胁。但威胁不是那个显而易见的威胁。
首先，我需要说的是，在实践中，到目前为止的证据表明，在历史上技术总是推动了科学的加速发展。我想不出历史上哪一次技术对科学是灾难性的。目前证据表明它是积极的。我们处在一个大变革时期，误差范围很大，我对未来不做任何保证。所以有不确定性和焦虑，也有机遇。
但那不是我所担心的。我所担心的是，这被用作一个借口，一匹攻破知识分子精神的特洛伊木马——有人拿着干草叉和火把要来烧掉图书馆。我们担心也许几年后，如果这导致灾难性的削减怎么办。你可以想象五年内，研究生项目可能被大幅削减。科学经费可能被削减。但这种事实际上现在就在发生。而那些借口根本说不通。
所以当人们说因为是AI我们必须削减研究生资助、减少工作岗位、取消REU、我们不再需要教孩子们数学思维因为他们可以谷歌、他们不再需要理解科学因为他们可以问模型——危险在于这是一匹攻击智识精神的特洛伊木马，而且它已经在发生了。我们现在正在对抗这些削减。所以这是我面临的第一个真正的危险，而且，是的，这完全没有道理。
我担心的第二件事是学生的学习。我们之前谈到过，为一道题苦战数周，也许永远解不出来。你不能把所有答案都放在书后是有原因的，而现在你有了这样的诱惑。危险在于某些工具——那些需要思考一周的习题集——现在我们必须真正相信学生会独立完成作业，而它的目的不是为了得到答案。从来就不是为了在考试中得到答案。那是为了训练你的思维。
最好的学生——我不是指最聪明的，而是最明智的学生——会被加速。他们会学得更快，因为AI会给他们线索，告诉他们应该如何进行更长时间的思考挣扎。但我担心大量的人会以为自己理解了某些东西。我们会失去大量非常聪明的学生，他们会因为……的存在而思维变得不那么清晰——

LEVIN：作业成绩和考试成绩之间存在负相关。

VAKIL：正是如此。

LEVIN：是的。这已经变得非常明显了。学生们所有作业都拿100分，因为他们用AI。所以他们去参加考试——那是限时的——他们什么都不懂。

STROGATZ：哇。

LEVIN：现实就是这样运作的。

KONTOROVICH：成绩呈双峰分布。有一类人用AI来帮助自己理解那些他们感到困惑、不知道去哪里学的东西……

STROGATZ：是的。

KONTOROVICH：……还有一类人用AI来做作业。然后你能在考试中看到——你知道——不再是一个单一的高斯分布了。而是双峰。一个峰集中在那些真正用AI加速理解的学生周围。

STROGATZ：嗯。

KONTOROVICH：他们训练大脑的方式和我们当年完全一样，只是——你知道——当我们卡住了，甚至不知道某个词是什么意思，还得等五天在答疑时间去问教授的时候，他们可以立刻问AI。所以这就是他们加速教育的方式。反观那些直接把作业扔给AI的学生。他们有了答案，好极了，然后他们在考试中不及格。
但你知道，有趣的是，我们数学家应该拥有最长的时间视野来理解教育如何与技术互动，因为袖珍计算器已经存在了大约70年，我们都知道有教育者告诉自己的学生：“你不需要背乘法表，因为计算器会帮你算。”

VAKIL：这似乎是个不言而喻的道理。我们经历过计算器时代，有过很多愚蠢的争论，但有一件事似乎是不可否认的：用计算器最好的人，也是不用计算器最好的人。

STROGATZ：【笑】这是个好格言，我觉得没错。

VAKIL：所以，我们不怕计算器。高斯做了那些超长的表格，我们不再需要做这些事了，这是好事。但我们仍然培养数感。我们改变了我们做的事情。我们不完全做同样的事。同样，正如Akshay所说，我们会改变我们珍视的东西。我们的长期目标是一样的，但我们会改变这些东西。

KONTOROVICH：而不改变可能也不好，因为如果高斯没有制作那些表格，他可能就不会注意到某个范围内素数的增长速度与对数函数之间的关系。所以有各种各样的洞见，我认为我们会——

VAKIL：我们失去了。

KONTOROVICH：我们可能会……

VAKIL：这是一种权衡。

KONTOROVICH：这确实是一种权衡，是的。而且我认为这种权衡我愿意接受——是的……有了技术。但是，你知道，没有解决方案。这里一切都是权衡。

STROGATZ：好吧，让我们稍微聚焦在人类理解这个问题上。这是你们几位演讲中的一个主题。Akshay，我看得出来你对此很关注。随着数学在前沿领域、甚至只是在日常生活中与AI的互动越来越多，会失去什么，又会得到什么？

VENKATESH：是的。我认为，我所希望的是它会引导我们实际上更加重视人类理解。

STROGATZ：嗯。

VENKATESH：人类理解是人类的，对吧？它是主观的。我认为数学有双重传统——既是科学也是人文学科。

STROGATZ：是的。

VENKATESH：而我们已经——至少在我有生之年，肯定自二战以来——我认为强调了科学性。我们把自己看作科学，因为那样更容易获得资助。资源在那里。不，我是认真的。但是，嗯——我所希望看到的是，真正拥抱数学的人文学科属性。这是对AI的一个可能的回应，我认为那会非常好。

STROGATZ：嗯。

LEVIN：Steve和我有时会谈到这个。在超级计算机问世时，也有类似的歇斯底里，认为超级计算机将取代物理学家或科学家。当然，产生一串数字并不是人类的理解。它对我们来说毫无意义。
所以我们不认为问题被超级计算机解决了。是那个以某种方式——正如你所说的——让作为人类的我们能够欣赏、理解并将其整合到更大的知识领域中的那个人。
而且我们当然不是计算机科学家。你不是计算机科学家。但只是想知道，我们是否认为AI在某个时刻会理解？那种理解是否会与人类理解的方式完全不同？

KONTOROVICH：你们为什么都看着我？

LEVIN：【笑】

KONTOROVICH：回答这个问题有几种方式。你知道，AI会有意识吗？我不知道。谁知道呢？我喜欢这样想……我记得是Richard Feynman很久以前说过，在谈到技术和人类驾驭技术的方式时说到，我们看到自然界中的事物并试图为己所用。他指出，鸟拍打翅膀的方式完全不是我们发明飞行时所采用的方式。所以，我们有一套完全不同的系统让我们在天空中从A点到B点。
大概我们观察了鸟类，然后幻想出了这个东西。嗯，也许同样的道理，你知道，取决于我们的目标是什么，我们会发明技术，它看起来不完全像自然解决问题的方式，但用稍微不同的方式解决了它。这就是LLM已经变成的东西——一种替代机制，通过它我们可以完成数学实践所追求的一小部分。
这是否意味着它理解了？我不知道“理解”这个词在这个语境下是什么意思。

STROGATZ：确实。

KONTOROVICH：这一切都……你知道，图灵从未说过理解语言意味着什么。他只是说：“它看起来像语言吗？那东西跟我说话的方式……我分辨不出它是人还是机器。”
所以同理，也许我对此更加实用主义。AI是否有助于我实现我的目标？我的目标是理解数学。我想每天学习新的数学。使用AI是否帮助了这个目标？在它有帮助的范围内，我会使用它；在它没有帮助的范围内，我会停下来去散个步，或者跟孩子们打球什么的。

STROGATZ：我喜欢你的操作性提问——它们能帮到我们吗？我认为答案已经有了：是的，当然它们有用。
但我感觉——回到我的禁忌问题清单——的问题是，当它们对我们没有用时怎么办，因为它们能做我们能做的一切，而且做得更好得多。所以我会承认——也许你不同意——但我认为很明显，它们不仅能比我们更好地解决问题，它们还能提出我们会觉得有趣的问题。它们能做比Martin Gardner更好的阐述。那时候我们做数学还会觉得好玩吗？

VAKIL：嗯，我认为你说了两种层次的东西，一个非常合理，另一个则完全不同。

LEVIN：【笑】

STROGATZ：说吧。哪些是？【笑】

VAKIL：所以，一方面你可以说，好吧，一旦你假设到达了奇点，那我觉得一切都无法预测了，那就没什么可讨论的了。

STROGATZ：因为太难知道接下来会发生什么了，你是说？

VAKIL：或者说，我只是觉得奇点之后的辩论往往没什么价值。你没有用“超级智能”这个词。每次用到那个词，我个人觉得讨论就没那么有用了，因为缺乏定义。
但你说了一些非常合理的话，因为它们很精确，是关于证据的陈述：它们将能写出有趣的东西。而那些东西已经在这里了。那个时代已经到来，我认为它们已经在告诉我们了。

STROGATZ：我没听到说它们已经到了。我听到人们说的话——再说一遍，因为我觉得这是个社会学问题——我们私下里在谈什么？我听到的是我们仍然可以有品味。我们更擅长知道什么是有趣的。

VAKIL：啊，好吧，那么……

STROGATZ：而我要说的是那只是短期内的。

VAKIL：所以，很明显可能会有新的技术突破，但按照目前的工作方式，任何你能爬山优化的东西，它也能爬山优化。如果有任何东西你有足够的样本，它就能改进。我们目前训练学生去应对未知的情况——他们从未见过类似的情况——而这按定义就是无法训练的东西。
所以，有人会预测三年内会有一本普利策奖获奖小说由AI写出来吗？我跟你赌20美元不会。过去有人打过这种赌，他们每一次都输了。如果你听某些非常富有的人十年前说的话，我们今天本该在火星上了。
所以，大量可接受的平庸材料我预期会非常有用。我不是贬低它。这会改变一切，会发生。但我认为按照目前的方式，杰出的事情将会发生，AI参与其中，人类也参与其中。而且，我猜人们在声称终有一天不会再有人类了，但他们需要一些证据。

VENKATESH：我非常同意我们必须非常谨慎地对待那些前提。但让我说——就假设我们在这样一个世界里，AI能证明定理，能提出我们也同样觉得有趣的问题。
你知道，当我刚到研究生院的时候，对吧？我遇到了所有这些数学比我好得多的人。那时候我在想：“我永远不会像这些人一样擅长数学。”但这很好。我能研究这个领域。这是个美丽的领域。我不会成为最优秀的那个。我很乐意这样度过一生。所以我认为，你知道，是的，我们仍然会做数学，因为它对我们作为人类有一些基本的价值。

LEVIN：我觉得Akshay的观点至关重要。还有关于你说的学生问题——对年轻学生的真正威胁在于态度上的。如果他们觉得这里没有愉悦、没有快乐、没有发现——不仅仅是一份职业或成为最优秀的人——而是成为一名数学家这件事本身不再值得追求，而且还意味着必须付出艰苦努力、必须抵制翻书后答案的诱惑——也就是说去问那些模型。那么你们打算如何应对这个问题，让下一代数学家能挺过来？

KONTOROVICH：我一直在试着向我的学生解释三种情景，三个小故事，作为对这个问题的类比。
故事一：你有一些沉重的托盘需要搬到卡车上。你开了一辆叉车过来，把托盘叉起来，放到卡车上，故事结束。明白吗？
故事二：你在学习怎么开叉车。你坐进叉车，移动一些托盘——不是因为它们需要从A点搬到B点，而是因为你在学习如何操作叉车。
好了，故事三：你决定想锻炼一下身体。你去健身房，往卧推上加了两百磅。你坐进你的叉车，开过去，用叉车做了十次卧推。其中有一个是对技术的错误使用。

【群笑】

好了吧？所以如果你只是向学生解释，你为什么在这里？你是在……你知道，如果我不需要知道这道实分析作业的答案。我知道怎么做。答案在书后面。在AI之前，你可以用谷歌搜答案。好吗？那么，这次操作的意义是什么？是为了让你学会使用那个技术，学会提示AI解决实分析问题？是我想要这个问题的解，还是你应该坐在那里感到沮丧，培养那种挫败耐力——做任何人类想做的困难事情都需要的那种耐力？
如果这就是这次练习的目标，那么要么自愿地按照设计的方式去做这个练习，要么退掉这门课，因为这就是我要教的东西。

LEVIN：是的，我觉得健身房那个类比很好。我确实觉得这非常有意思。我曾经不得不试着解爱因斯坦方程寻找一个黑洞解，不是因为那还没被解出来，也不是因为我找不到答案。那是我生命中一个伟大的时刻，对吧？经历那个过程。但我认为要说服这代在AI陪伴下长大的人，还需要一些功夫。

STROGATZ：你的例子很有趣，因为这是我自己一直以来的一个小纠结。有人——不喜欢数学的人——说高中几何毫无价值，但几何会教你思考。它会帮你培养逻辑和耐心，等等等等。
我不喜欢这种为几何辩护的方式。几何是人类的一项辉煌创造。这就像，你知道，你可以踢足球是为了锻炼你的腿，或者因为它是“美丽的运动”，你知道吗？
所以对我来说，数学的内容是美的。它不只是让我变得更强壮的体操训练，以便为一家金融公司解决问题——因为那才是真正重要的。我想为了人类的繁荣而做它，就像Francis Su会说的那样。就像我觉得你一直在提倡的，Ravi。不只是为了让自己变强壮，而是因为内容本身就很丰富，让我在死之前成为一个更快乐的人。生命的意义是什么？我活着的时候想做数学。

VAKIL：我喜欢我们可以两者兼得。我喜欢足球在美国中学里大规模开展的方式，你玩得开心。你和其他人在一起，你——而你——他们当然在享受乐趣和培养习惯的同时，偷偷地变得更健康更强壮。你知道，这似乎是类似的模式——同时让几何在相同的年龄段支持类似的东西。

LEVIN：这可能非常不受欢迎，我先向数学家们道歉。

STROGATZ：好吧，别问……

LEVIN：但有这样一种信念：如果明天所有人类知识都被抹去——不是人类——而是人类知识。最终我们都会重新发现它们。一加一等于二。如果明天人类知识被抹去了，AI就停止了。

STROGATZ：但人类会发生什么——

LEVIN：它得从头开始。

KONTOROVICH：不，但这个问题有一个你现在就能问的版本，那就是：好吧，有这些……我们没来得及深入形式化或交互式定理证明器，但如果你取任何一个交互式定理证明器——它有逻辑公理系统、从一个状态到下一个状态的逻辑推理——然后你什么都不给它，就像AlphaZero一样。你只给它国际象棋的规则，然后说：“下一万亿盘棋”，它就会产生自己的理论和自己的状态评估等等。
如果你用一个交互式定理证明器，不给它任何数学库呢？

LEVIN：正是。

KONTOROVICH：然后你放一个AI自由发挥，它能发现费马大定理的证明吗？

LEVIN：没错。没有那些强化和——

KONTOROVICH：是的，不给它任何人类知识，什么都不给。所以这——我想Peter Sarnak正在推广这个问题。你知道吗，有没有一个“数学零号”？有人能训练AI真正做到那样吗？这才是我们目前还没有答案的真正问题。

LEVIN：但我们知道人类能做到。我的意思是，是的，我们做到过一次，我们还能再做一次，对吧？但这也有点回到了——我认为这就是我担心数学家不喜欢的那部分——“发现”还是“发明”的问题。这些是可被发现的真理吗？一个思维方式与人类完全不同实体能发现它们吗？

VAKIL：当然，你可以想象自己是第一个看到某个新发现岛屿森林中湖泊的人类。两种说法都可以为真。人类发现中有乐趣，也许一架无人机也会飞过岛屿发现那个湖。

STROGATZ：好吧，我想现在是时候把时间交给观众了。你们一直很耐心，我也感谢我们的嘉宾们接受了一些相当大胆的问题。接下来的问题可能会更大胆。我们准备好了，请提问。

观众提问：到目前为止，AI被私营公司控制着，最好的模型越来越多地被金钱所门槛化。我担心那些能用上这些资源的研究者和不能用的研究者之间的鸿沟。历史上，我们的领域以只需要纸笔而闻名。这在多大程度上将不再成立？我们如何确保数学仍然对所有人开放？

KONTOROVICH：有很多团队在做开源替代方案。当然，它们落后了，但正在追赶。我在一个团队里做开源模型，专门解决数学问题，或者帮助数学家形式化数学等等。所以人们对此非常担忧。他们在努力解决这个问题。这很难。我们需要更多资源。我们应该总是说我们需要更多资源。

STROGATZ：我们总是可以这么说。

KONTOROVICH：是的。

观众提问：我的问题是，我认为有一些缺失的环节。现在，随着AI开始大量产出证明，我认为我们开始意识到这不是数学家想要的——只是一堆证明的数量。我觉得缺失的是美学。而我们还从未真正定义过它。
比如，为什么一个问题有趣？为什么一个证明有趣？美在哪里？深度在哪里？互联性在哪里？我认为因为这些还没被定义，当你面对AI生成的几百个证明时，你完全不知道该看哪个。所以，有人在研究测量这些东西吗？

VAKIL：我会从另一个方向说——你可能是在问：“我们能机械化美学，从而衡量什么是最美的证明吗？”我认为这不是该走的方向。
我们已经有了很好的人类直觉来判断什么在审美上令人愉悦。我们知道什么是有趣的。我们知道AI产生的大多数证明并不有趣。我们知道其中一些非常有趣和美丽。
所以我不认为那是可以机械化的，而这让我有点欣慰——这是一个人的学科，重要的是人类的理解。我意识到这不是一个令人满意的答案，不是一个很结论性的回答，所以也许我应该把话筒给别人。

KONTOROVICH：我只想说，其他科学试图通过引用指数、H指数这类东西来机械化衡量质量的各种方式，在数学中基本上几乎是零相关性。
我很高兴我们的领域不关注指标。有菲尔兹奖得主的H指数只有四之类的。这些东西完全没有意义。呃，当我们不得不去找院长说“不不，这个人真的很优秀。请看他们的论文”的时候，这就是个问题——院长看不懂。

STROGATZ：嗯。哦，请继续。

VENKATESH：不，我只是想说，作为数学家，我认为我们应该非常警惕用数字来替代另一个概念的做法，我们要记住我们真正感兴趣的是那个另个概念。

VAKIL：而且我们在数学界有一些有趣的辩论——我很享受这些辩论——关于“这到底有没有趣”？

VENKATESH：是的。

VAKIL：不是说有一个单一的指标说这个定理在十分制上值3.5分。相反，你知道，我们会争论，会在喝咖啡时进行有趣的讨论：“哦，这个确实了不起”和“其实我并没有被打动”。这是一个非常人性化的衡量体系。有人会说那它就容易受操纵、容易被滥用——确实完全可能。但另一方面，大多数人在善意地执行这个体系，而且数学中确实存在美感。
很难……我不认为我想——我认为不可能量化。但这是那种你在ICM上遇到一个来自另一个国家、在完全不同的教育体系中长大的人，你们见面，听说一个新定理，你们俩都点点头，心想：“那太了不起了。”

KONTOROVICH：而不同的人会对价值有不同的评价，甚至同一个人在不同时间对同一个结果也会有不同评价。我知道有些结果，我当初想：“哦，那个嘛，我不觉得那有什么意思。”然后我开始更多了解它，我发现：“不，那是个惊人的突破。”

STROGATZ：【笑】太真实了。好的，请讲。

观众提问：嗯，感谢这场精彩的对话。我想，不幸的是，从历史上看，数学界并不像重视解决难题那样重视阐述和教学。这可以追溯到伯特兰·罗素把那些“描述数学”的人看作比“解决数学问题”的人低一等。你认为在AI时代，这种情况会转向对教学法和阐述的重视吗？

VENKATESH：我觉得会的，肯定。

VAKIL：我觉得我们三个在这里的人，在AI出现之前就会往那个方向推进。这不是新东西，但我认为是这样，我希望如此，而且我认为我们早就应该推进了，而且……

英文来源：

Live from ICM 2026: What Is Math For in the Age of AI?
Introduction
Mathematicians are witnessing a profound shift in their field. AI systems are now producing proofs, spotting connections between distant fields, and, in a few cases, solving problems that had stumped mathematicians for decades. The pace of progress over the past several months has raised challenging questions: What can AI systems actually do? And what happens to the more human, creative aspects of the field once machines can match – or exceed – people at problem-solving?
In this special live episode of The Joy of Why, recorded at the International Congress of Mathematicians in Philadelphia on July 26, 2026, hosts Janna Levin and Steven Strogatz are joined by three mathematicians responding to the rise of AI in math: Akshay Venkatesh at the Institute for Advanced Study; Ravi Vakil at Stanford University and president of the American Mathematical Society; and Alex Kontorovich at Rutgers University. Together they discuss what recent AI-generated proofs really demonstrate, what will be gained and lost as mathematics becomes more machine-assisted at the frontier, and what this means for the next generation of mathematicians. The conversation turns to a deeper question: What do mathematicians value about doing mathematics in the first place? The answer involves grappling with what proof and understanding really mean, and the surprising importance of storytelling.
Listen on Apple Podcasts, Spotify, TuneIn or your favorite podcasting app, or you can stream it from Quanta.
[Audience applause]
JANNA LEVIN: Thank you so much for being here. I’m Janna Levin.
STEVE STROGATZ: And I’m Steve Strogatz.
LEVIN: And this is The Joy of Why. Welcome to our first live.
[Audience applause]
STROGATZ: Yes, The Joy of Why, a podcast where we consider some of the biggest unanswered questions in math and science today. And we’re very excited to be coming to you from the International Congress of Mathematicians 2026.
It really does feel like a historic moment. The stakes are pretty high for the future of math. And we’ve had a good run for three, four thousand years doing this on our own. But there’s a new kid in town — artificial intelligence. And it’s starting to play with us, and our field.
And so we’re excited to hear what our panelists have to say about the interaction between math and artificial intelligence, now and in the short term and maybe in the longer term.
So let me introduce our distinguished panel here. First up, I see Akshay Venkatesh from the Institute for Advanced Study. And next to him, we’ve got Ravi Vakil from Stanford University. And rounding out the panel, there’s Alex Kontorovich from Rutgers. All right, Janna, do you wanna get us launched in our discussion?
LEVIN: Yeah, I thought maybe we’d start with some of the recent advances that gave some mathematicians pause, while others were very blasé and unimpressed. I’m going back to the 2024 International Mathematical Olympiads when AlphaProof achieved a silver medal, and then later in 2025, a more advanced model with DeepThink achieved a gold medal.
I first want to ask before we get into exactly how it was done, how many of you participated in the International Mathematical Olympiads back in your day? You did? You did? And will you tell people about it? ’Cause some of our audience might not know it’s, six problems, nine hours, two days, and high-school-aged competitors. Tell us what it was like.
STROGATZ: I’ve heard that Ravi was a terror at this. Jordan Ellenberg was talking about how you used to kick his you-know-what at that competition.
LEVIN: Yeah, people at the conference have been talking about how you terrorize them.
RAVI VAKIL: So, so I first, I first met Akshay at the event, I should say, although he may not remember.
AKSHAY VENKATESH: I don’t. Sorry.
VAKIL: Exactly. Because he was quite young. Uh, but it’s one of the many roads into mathematics. There’s no royal road to mathematics, but this is one of the many ways by which we attract people into mathematics, and it’s also a great way of advertising mathematics, and it helps to develop a certain way of mathematical thinking.
Uh, the important thing about the competition isn’t the six people or the problems, it’s how it encourages millions and millions of kids to hone their minds. So, it was wonderful when the models did well. It was really a proof of concept. I think it was extremely interesting, and I think it can both be an amazing milestone reached, while also I think perhaps people might have misinterpreted it.
So, the first time a model answers a question, it’s exciting. A milestone is reached. The first time a student answers a question, it’s exciting for a different reason, because they’ve learned how to think. And the seventh time a model answers a question, it’s far less interesting, and the seventh time a student answers the same question is far less interesting, for similar reasons.
So, it was exciting, and I, I see it as only a good thing, but it did not represent, uh, superhuman intelligence or anything crazy like that.
STROGATZ: No, sure.
LEVIN: And it wasn’t the highest score even in the year that it, it appeared.
VAKIL: Sure, but the next year it was, and next year after it will be, and I’m happy about that, and I’m absolutely not alarmed.
LEVIN: And Akshay, what’s your experience?
VENKATESH: I think Ravi said it well. I think, you know, one unfortunate thing about this is it has contributed to the image, even among mathematicians, of mathematics as a competition, which is not really how I see it. and I think it’s particularly unhelpful at the moment. There’s some competitive drive. You know, when you’re young, you get validation if people think you’re good at something. I have mixed feelings about how healthy this thing is.
VAKIL: As Akshay said, it could be a danger. Everything’s a double-edged sword. People are driven out of mathematics because they think they’re not good because they didn’t do well in a competition, whereas that just makes no sense.
STROGATZ: I’m very taken with your view of this. There’s that old saying about “It’s the journey, not the destination,” and you seem to be espousing that kind of philosophy, but it feels to me like that is also in embryonic form, something we might get to by the end of our discussion today because all of us may have to start wondering, is mathematics about the journey and not the destination?
LEVIN: I, I was going to say for Alex, beyond that, that was 2025. We’ve recently had some major advances recently published. One of the Erdős Problems was solved, and I’m just curious, one, if you were impressed, surprised, and sort of your reaction to that, and if you could set the stage of what the problem was that was solved.
ALEX KONTOROVICH: Sure. Just to make one comment on the IMO, you know, two years ago we had, uh, AlphaProof, that you know, it was one point off of a gold medal. And by the way, no one’s handing medals to these companies. They’re declaring themselves. As, as having, uh —
LEVIN: Right, self-declared gold medalist.
VAKIL: They’re handing medals to themselves.
KONTOROVICH: Yeah. They- yeah, that’s right. Um, and my prediction was that this summer, no AI models would get a gold medal ‘cause none of them would bother competing anymore. So it’s not interesting for them. The fact that I can push a button on GPT 5.6 and get perfect scores on the IMO. So, okay, next. What’s the next interesting thing? That’s not interesting anymore, exactly as, as you were saying.
But let me go back to your question about these Erdős Problems. Okay, so we have problems that we’re working on, and as Akshay said, solving problems is just one small component of what mathematicians do. We find definitions, we find new structures we’re interested in, we create new questions and new problems all the time.
One person who just loved to ask questions and didn’t mind writing them down whether he was right or wrong, uh, was Erdős, and he wrote down thousands of them. And, thankfully, Tom Bloom, for whatever reason, several years ago decided to collect these into a database.
And what happens sometimes in mathematics, in fact, it happens with some frequency, is that problems that we thought were difficult end up being not nearly as difficult as we thought and doable by existing techniques if you only bothered to look at that problem and, uh, had the background necessary to apply the techniques that end up cracking it and making it easy.
And so what AI is amazing at is: you give it 1,000 problems, and maybe it has a 1% hit rate, and that’s 10 good papers a year, which is a fantastic, uh, career in mathematics. So, right now we’re seeing every day more and more of these Erdős, uh, style problems being announced. How many of them are actual solutions as opposed to maybe partial progress or complete abject nonsense ’cause it’s just LLMs saying things, is one thing.
I think what you’re referring to is the Erdős unit distance conjecture, which is something that quite a lot of people knew, quite a lot of people worked on. There was very significant, progress trying to, you know, get this constant to the conjectured value. And, it turns out that the conjectured value is just wrong.
And, autonomously, uh, GPT found a counterexample, you know, constructed a solution where this exponent is not satisfied. What to me is most interesting about that, I mean, first of all, it’s brilliant. It’s fantastic. This would be a really great contribution if it was a human being that came up with this counterexample. Um, the most fun thing about this is, a week later, a completely unrelated problem called the sum-product problem was also solved in the negative by human beings who were applying the ideas, the techniques that were exposed to us from this AI solution.
LEVIN: Yeah, so interesting.
KONTOROVICH: So this, to me, is sort of the hopefully golden age that we’re going to enter. We’re gonna learn all kinds of amazing mathematics thanks to these techniques. But what is it? “If a theorem falls in the woods and no one cares, you know, does it have any value?” kind of thing.
So I don’t think theorems have intrinsic value in the universe. Mathematicians value them, and right now, AI companies value them because they’re great at marketing. They’re a great way for them to say, “Hey, my tool is better than the previous tool and all the other tools out there.” And it is very effective in that sense, and I think pretty soon AI companies will need other, you know, things that actually bring value to the world, which theorems do not, except for to mathematicians. So but in their wake, they’ll leave us a fantastic tool to learn lots of amazing facts.
LEVIN: I do worry that the ability to generate so many proofs that could be wrong could be a sinkhole for mathematicians, that there is now an industry of just checking those proofs.
Do any of you find yourself lured by the attraction of checking some of these proofs? I mean, some of our very prestigious colleagues have done that, right, have offered their services. Would you ever do it, Akshay? To referee like Terry Tao did.
VAKIL: 100 AI slot proofs. Would you, would you, would you sign up for that?
KONTOROVICH: How much would someone have to pay you?
STROGATZ: There’s a lot of glamour. Come on, how can you resist?
VENKATESH: But this is, I think, adjacent to a very real problem in our community, which is about the functioning of journals and how they will manage in this. So I think as a community, there are a lot of systems that we have to be looking at which are gonna be placed under immense strain.
STROGATZ: Hmm.
VAKIL: Right. Part of the issue is that people are not gonna be lining up to do this. The reason they’re under strain is we’re being flooded, where the flood is only beginning of a large quantity of material, some of which, really amazing things are happening. And also, I have in my inbox during these two hours, I’m not looking forward to seeing what kind of AI slop people are gonna send me, what amazing things they think they’ve proved. And to be honest, I’m not going to check them. It’s not fun.
STROGATZ: It makes me think a little of this, um, trifecta that Terry Tao has been mentioning about concepts like proof generation, proof verification, which is what we’re talking about now, and also proof digestion, where we start to make sense of these proofs in human terms and can assess whether, how interesting they are. But I’m wondering if there might be another category of proof marketing.
VENKATESH: Well, I think I wanna also even question the assumption that proof is the central activity.
STROGATZ: Okay, go on.
VENKATESH: Well, if I think about, you know, if I’m doing a piece of math, I probably think of it more akin to storytelling. You know, I’m trying to tell a story. Uh-huh. You know, perhaps the proofs are part of the grammar of that.
That’s, you know, something that we should be thinking about. Like, should we define this discipline by proofs? And of course, we have done so, but I’m not sure it really matches with what we value and what we’re doing.
VAKIL: I’d also like to strongly endorse what Akshay’s saying, ’cause I feel like… I also think this is what we have always valued. This is not a change. This is something we’ve always valued, and we’ve had various proxies, and we’re forced to examine these proxies because of the change in technology.
But human understanding and the storytelling, My graduate students, they’re brilliant already. What I teach them, it’s the storytelling. Learning how to do mathematics, is when you write a paper, you have to understand the human understanding and the conveying of the understanding. I claim this is what we’ve always wanted.
There have been brilliant mathematicians who’ve had ideas that went nowhere because they could not convince people of them. And there are other people who’ve really changed fields who’ve just been excellent. Erdős is a good example of someone who’s changed mathematics, or there even, on a very different level, Martin Gardner is someone who probably has advanced mathematics much more than most people because of what he’s managed to do by storytelling.
STROGATZ: I think storytelling is not a term that would occur to most people, so I think you’re on the hook now, both of you, to explain to us what are you talking about?
VENKATESH: I think that that was the best, uh, word I could use for the process of You know, I’m working on something. There’s some landscape which I’m exploring, and there are all these things in that landscape, and I’m sort of thinking, “How will I tell a story that will be interesting and engaging for my colleagues?”
LEVIN: Is that how you’re drawn to problems? I mean, math is one of the last places people with a straight face talk about beauty and elegance and, in this case, storytelling. Can you give an example of that for somebody who’s not familiar with the idea that this is beautiful or narratively powerful.
STROGATZ: Ravi, you, you look like you’re ready go.
VAKIL: I would say I cannot think of a major advance in mathematics that does not fit this paradigm. Let me just take Fermat’s Last Theorem as an example. What makes Fermat interesting and Goldbach far less interesting. Uh, It’s because Fermat is part of a long and rich story. People started to dig into it and, in the 1800s, it led to the opening of lots of rich questions. And to be clear, it was beautiful, but it also was powerful. The thing about mathematics is when you have an amazing story that is beautiful empirically. I don’t know why the universe works this way, but when you understand something and the story becomes simpler and simpler, you get something which is more and more powerful.
And there’s no question what the developments from the 19th century have led to in technology and science and in our standards of living. But it’s because people were curiosity-driven and so Fermat led to the development of a good chunk. Well, not alone. It’s part of a grander story.
And then the final proof, it was way better the way it turned out because it required the building of a far more interesting story with dramatic bridges that had no right to be there between one way of thinking and completely different way of thinking. And they were completely out of reach until they weren’t. Until you had dramatic leaps of thought. There were cliffhangers. There were examples where it looked like we were all done, and then, uh-oh, we weren’t. And then the story is not over.
Once you finish the book, you put the book down, and the best theorems are the ones where you’re waiting for the next volume. And everyone is talking about it around the world to try to think of how the story continues. It’s hard to think of an example where things do not fit this.
STROGATZ: Uh-huh. I would like to just try to sharpen up this idea of story. You’re talking about the interconnectivity across time and across disciplines of math. That math is this amazing, often sort of subterranean thing that occasionally pops up above the surface and we can see peaks, but we d-didn’t realize that there are these subterranean branches connecting parts that are invisible until… Okay, I’m my metaphor’s getting mixed up here.
But, uh, it’s something about that, that when you talk about a story, you mean something about the coherence of the subject, I think. Uh, but I don’t know. You wanna try to unpack that better than I just just did, Alex?
KONTOROVICH: Maybe one way of saying it is that the question of Fermat’s Last Theorem, you know, this X to the N plus Y to the N equals Z to the N, has no consequences whatsoever in the universe. No one really cares if it’s true or not. You know, if there was some counterexample, would it destroy something about the universe? Well, it turns out it would destroy something about the universe. It would destroy the Langlands program. Uh, but, um, but that, that was a, a theorem that had to be proved.
But I, I think that question led to a series of other absolutely amazing breakthrough discoveries. From the very beginning of that question, it led to the understanding of non-uniqueness of factorization in number fields, and the fact that we have to understand number fields as their own entities and so that one question, if all you think about is, “I wonder if that’s true or not,” for no reason whatsoever, and you just follow the thread of that story from one breakthrough to another, you get all the way down to, you know, this, uh, massive Taniyama-Shimura and, and so on, and Taylor-Wiles breakthrough. So that’s a beautiful arching, overarching story, and we have many of these in mathematics.
Just to give one more, you know, the question of whether the parallel postulate, this fifth postulate in Euclid’s elements, can or can’t be proved from the other four. Seems like the most esoteric, useless thing. Who the hell cares, uh, if the fifth postulate can or can’t be proved from the other four? It’s not like, we’re not even arguing about whether the fifth postulate is true or not. We’re saying, you know, should it be a theorem or should it be an axiom in the theory? And you follow the thread of that question 2,000 years all the way to Einstein’s general theory of relativity.
LEVIN: And it very much matters to the universe.
KONTOROVICH: Exactly.
STROGATZ: In that case too. So Akshay, I feel like you really tapped into a rich vein with putting our attention on this concept of mathematical stories. I’m also wondering, since you seem to be saying that proofs are not the zenith of what we’re after, insofar as they can contribute to an unfolding story or a richer story, yes. But what about this thing that we say in math when we say something is morally true? Like before we have a proof, maybe even before we have a precise statement, we sort of know what’s going on. Can you talk about that, what that feels like, and what, do we mean by that?
VENKATESH: Oh, uh, yeah, absolutely. You know, like, just to give an example of my experience and probably of many other people in the room, right? When you write a paper, you know what’s right long before you fill in all the details, right? You grasp an intuition. Gaining that intuition is really the most enjoyable part. And it’s sometimes mysterious how that intuition operates and, and how it fails sometimes. But yeah, it is in a way, it’s, it’s much more fun to know something without knowing how you know it.
LEVIN: Well, isn’t there a sense where choosing a problem invokes that intuition?
VENKATESH: Yeah, absolutely, right? you don’t know what’s there. But you have a sense. You have a sense that there’s some richness there.
LEVIN: I guess, Akshay, I would also ask similarly, are, do you care either way if a proof if it’s coming from a machine or if it’s coming from an elder or if it’s coming from a young upstart? Does it matter to you where it came from?
VENKATESH: You know, I certainly don’t see why a machine couldn’t produce, something I found interesting. But I also think as we use machines, it will change those notions very rapidly. So it probably doesn’t make a lot of sense to look at what we value now and ask, can machines do them? It may be more interesting to think how it’ll change what we value.
I mean, the fact that so many of us are using AI in different ways as part of our regular research, I I think it shows you the practice of mathematics is in flux.
STROGATZ: So at this point, I’d like to ask a question that I feel is sensitive, but I think it could also be very interesting, which is here at this meeting and in conversations I’ve been having with people before I came to the meeting, there seem to be certain taboo subjects, things that we probably should be talking about that we’re not, as a community, and also things that I feel like a lot of us are worrying about, but we don’t say out loud.
VENKATESH: Okay, Ravi has to answer this.
STROGATZ: Anyone like to– As president of the American Math Society, Ravi. No, you don’t have to say anything, you’re the president…
VAKIL: No, no, no, no, I won’t be circumspect at all. But there are no taboos. Uh, So I think things people worry about in public or, or on social media are not the things that we should be worried about. I’m less worried about those.
I do think there are things we should be very worried about. I’m worried about younger mathematicians, I think. And the good thing is the community, at least uniformly, seems to realize the danger to the profession comes from the threat to younger mathematicians. But the threat isn’t the obvious one.
First, I need to say in practice, the evidence so far is that technology has always accelerated science in history. I can’t think a single time in history when it’s been disastrous for science. And currently the evidence is that it’s been positive. We’re in a period of great transition, and the error bars are huge, and I make no guarantees for the future. And so there’s uncertainty and anxiety, and also opportunity.
But that’s not what I’m worried about. What I’m worried about is, is this is being used as a pretext, as a Trojan horse for anti-intellectualism, which is that people are coming with pitchforks and torches to burn down the library. And we worry that maybe in a few years, what if this leads to disastrous cuts. You could imagine in five years, graduate programs might be cut by huge amounts. Funding for science might be cut. But that’s actually happening right now. And the pretexts for it don’t make sense.
So when people say it’s because of AI that we have to cut funding for graduate students, reduce jobs, cut REUs, that we don’t no longer need to teach kids how to think mathematically because they can just Google it, they no longer need to understand science because they could just ask a model. So the danger is that it’s a Trojan horse for anti-intellectualism, and it’s already happening. And we’re fighting the cuts right now. So that’s, that’s my first real danger and so, yeah, I mean, it makes no sense.
So the second thing I’m worried about is students learning. We talked about earlier, just fighting with a problem over weeks and maybe never solving it. And there’s a reason why you don’t want all the answers to be the back of the book, and now you have the temptation. The danger is that some of the tools, the problem sets to think over a week, now we have to really trust students to work on their own, and the purpose of it is not to get the answer. It was never to get the answer on a test. It was to train your mind.
The best students, and I don’t mean the smartest, but the wisest students are gonna be accelerated. They will learn faster because it will give them clues as to how they should struggle with problems for longer. But I worry that a large number of people will think they understand something. That will lose a large number of very bright students who will think less clearly because of the existence of…
LEVIN: There’s an anti-correlation between homework scores and test scores.
VAKIL: Exactly.
LEVIN: Yeah. And that’s become very clear. Students are getting 100% on all of their homeworks ’cause they’re using AI. So they go into a test, yeah, that’s timed. They understand none of it.
STROGATZ: Wow.
LEVIN: That’s the way it’s working.
KONTOROVICH: Scores coming in bimodally. So there are the people who use AI to help themselves understand when there’s something that they’re confused about and they don’t really know where to go to learn it…
STROGATZ: Right.
KONTOROVICH: … and then there are the ones who use AI to do their homework. And you can see then on the tests, you know, it’s no longer a single Gaussian. It’s sort of a double peak. One peak is concentrated around the students who are really using that AI to accelerate their understanding.
STROGATZ: Uh-huh.
KONTOROVICH: And train their brains in exactly the same way that we trained our brains, except, you know, when we would get stuck and I just even don’t even know what this word means and have to wait five days to ask the professor in office hours, they can ask the AI right now. And so that’s the way in which they’re accelerating their education. As opposed to the students who just put the homework into the AI. They have the answers, great, and then they’re failing the test.
But you know, it’s interesting, we mathematicians should have the longest time horizon over which to understand how education interacts with technology, because pocket calculators have been around for 70 years or something, and we all know educators who tell their own students, “You don’t have to know your times tables ‘cause a calculator will do it.”
VAKIL: I mean, it seems a truism. We’ve gone through this with calculators and there’ve been a lot of silly debates, but one thing which seems undeniable is the best person with a calculator is the best person without a calculator.
STROGATZ: [laughs] That’s a good aphorism, I think that’s right.
VAKIL: So, uh, we’re not afraid of calculators. Gauss made these huge long, uh, tables and we no longer have to do these things, uh, and that’s good. Uh, but we still develop number sense. We change what we do. We don’t do exactly the same things. And similarly, as Akshay had said, we’re gonna change what we value. Our long-term goals are the same, but we’ll change these things.
KONTOROVICH: And it may not be good because if Gauss hadn’t made those tables, he may not have noticed the relationship between the growth rate of the primes in a range versus the logarithm function. So there, there are all kinds of insights that I think we stand to-
VAKIL: We lost.
KONTOROVICH: We potentially…
VAKIL: It’s a trade-off.
KONTOROVICH: It’s definitely a trade-off, yeah. And I think it’s the trade-off that I will take- Yes … uh, to have the technology. But, you know, there’s no solutions. Everything is a trade-off here.
STROGATZ: Well, let’s focus a little bit on this question of human understanding. It’s a theme in some of your talks. Akshay, I’ve seen that you’re concerned about this. What will be lost and, and what could maybe be gained as math interacts more and more with AI at the frontier and even just in everyday life?
VENKATESH: Right. I, I think, something I would hope is it would lead to us, in fact, putting much more value on human understanding.
STROGATZ: Mm-hmm.
VENKATESH: Human understanding is human, right? It’s subjective. And I think math has this dual heritage of being a science and a humanity.
STROGATZ: Yes.
VENKATESH: And we have, uh, certainly in my lifetime, probably certainly since World War II, I think emphasized the scientific, right? We think of ourself as a science because it’s easier to get funding. That’s where the resources are. Uh, no, I mean, right. But, um- What I would like to see is, is really a, kind-of embrace the humanistic aspect of math. That’s a possible response to AI, which I think would be very nice.
STROGATZ: Mm-hmm.
LEVIN: Steve and I sometimes talk about this, at the advent of supercomputers, there was a similar kind of hysteria that the supercomputer would replace the physicist or the scientist. And of course, producing a string of numbers is not human understanding. It’s meaningless to us.
So we don’t consider the problem to be solved by the supercomputer. It’s the human who adapts it in a way that, as you said, as human beings, we’re able to appreciate it, understand it, and integrate it into the larger sphere of knowledge.
And we’re not computer scientists, of course. You’re not a computer scientist. But just wonder if at any point, do we think the AI will understand? And will that understanding be completely different than the way in which human beings understand?
KONTOROVICH: Why are you guys looking at me?
LEVIN: [Laughs]
KONTOROVICH: There’s a, There’s a number of ways to try to answer that question. You know, will the AI be conscious? I have, I have no idea, Uh, who knows? I like to think of, um… I think it was Richard Feynman who said a long time ago when speaking about technology and the way human beings harness technology that we see in nature and try to make it our own. And he pointed out that the way a bird flaps its wings is not at all how we figured out flight. And so, we have a completely different system that gets us from point A to point B in the air.
Presumably we looked at birds and dreamed up such a thing. And, um, maybe in the same way, you know, uh, depending on what our goals are, we’ll, we invent technology that doesn’t look exactly the way nature figured out to solve that problem, but solves it slightly differently. And so that’s kind of what LLMs have become, is this alternative mechanism by which we can do a small part of what mathematical practice strives for.
Does it mean that it understands? I don’t know what the word understands means in this context.
STROGATZ: No.
KONTOROVICH: And it’s all… you know, Turing never said, uh, what it means to understand language. He just said, “Does it look like language? Does the thing talk to me the way… and I can’t tell if it’s a human or, or a machine.”
So in the same way, maybe I’m even more pragmatic about it. Is the AI useful to me in what my goals are? My goals are to understand mathematics. I wanna learn new math every single day. Is using AI helping that goal or not? To the extent that it’s helping, I’m gonna use it, and to the extent that it’s not, I’m gonna stop and go for a walk and, you know, play ball with my kids, or something.
STROGATZ: I like your operational question, can they be helpful to us? And I think the answer’s already in, that yes, of course, they’re helpful.
But the part that I feel is, back to my list of taboo questions, is what happens when they’re not useful to us because they could do everything that we could do except much, much better. So I, I will stipulate, but maybe you won’t agree. But I think it’s obvious that not only will they be able to solve problems better than us, they’ll be able to ask questions that we will find interesting. They’ll be able to do exposition better than Martin Gardner and the whole thing. Will it still be fun for us to do math?
VAKIL: Well, well, I think you said two flavors of things, one of which was very reasonable, the other of which was very different.
LEVIN: [Laughs]
STROGATZ: Go ahead. What were those? [Laughs]
VAKIL: So yeah, so on one hand, you could say, okay, o- o- once you postulate reaching the singularity, then I feel like all bets are off, and then it’s not really worth having a discussion.
STROGATZ: Because it’s so hard to know what will happen next, you mean?
VAKIL: Or, I just feel like the debates post-singularity tend not to have much value. You didn’t use the phrase superintelligence. Anytime that gets used, I personally feel those are less useful discussions because of the lack of definitions.
But then you said things that were very reasonable because they were precise, and they were statements about evidence of, they will be able to write things that will be interesting. And those things were already here. That time has already come, and I think they are already telling us.
STROGATZ: I’m not hearing that they’re already here. I hear people saying things, you know, again, ’cause this is a, I feel like a sociological question. What are we talking about privately? And I, I hear that we can still have taste. We’re better at knowing what’s interesting.
VAKIL: Ah, okay, so…
STROGATZ: And I’m claiming that that’s only for the short term.
VAKIL: So, so obviously, there might be a new technological advance, but with the way things currently work, anything you can hill climb, it can hill climb. If there’s anything that you can have enough examples of it can improve. We train currently students to try to deal with the unknown, the situations that they have never seen anything like before, which is by definition the thing you can’t sort-of train for.
So, is someone gonna predict that a Pulitzer Prize-winning novel will be written by an AI within three years? I’ll put $20 against you on this. And people have made bets like this in the past, and they’ve lost those bets every single time. We were supposed to be on Mars by today if you listen to certain very wealthy people 10 years ago.
So, absolutely large quantities of acceptable material I expect to be very useful. And I’m not belittling it. This will change everything, uh, will happen. But I think the way things are currently happening is outstanding things are going to happen, and AI is involved, and humans are involved. And, uh, I guess people are claiming that a time will come when there’ll be no more humans, but they need some evidence.
VENKATESH: I very much agree that we have to be very careful about the premises. But let me give a, just let’s suppose we’re in this world where AIs can prove things, they can write questions that we find just as interesting.
You know, like when I arrived at grad school, right? I met all these people who were just much better at math than I was. And at that time, I was like, “I’m never gonna be as good at math as these people.” But this is great. I get to study this field. It’s a beautiful field. I’m not gonna be the best at it. I’m happy to spend my life this way.” So I, I think you know, yes, we will still do math because it has some basic value to us as human beings.
LEVIN: I mean, I think Akshay’s point is crucial. And also to your point about students and the, the threat really to the young student is attitudinal. If they think there is no pleasure here, no joy here, no discovery. Not just a career or being the best, but becoming a mathematician is no longer on offer and the hard work that it’s going to require and having to resist looking up the answers in the back of the book, which is to say to ask one of the models. And so how are you imagining addressing this to have the next generation of mathematicians survive?
KONTOROVICH: So I’ve been trying to explain to my students sort of three scenarios, three vignettes, as an analogy to exactly this question.
Vignette number one, uh, you have some heavy pallets and you need to move them onto a truck. So you get a forklift, you drive it over, you lift the pallets, you put them on the truck, end of story. Okay?
Scenario number two, you’re trying to learn how to use a forklift. So you get in the forklift and you move some pallets, not because they need to be moved from point A to point B, but because you’re learning how to operate a forklift.
Okay, scenario three, you decide you wanna get a, a little workout in. You go to the gym, you put 200 pounds on the bench press. You get in your forklift, you drive over, and you do 10 reps with the forklift. One of these is the wrong use of technology.
[Group laughter]
Okay? So if you just explain to students, why are you here? What are you… You know, if I don’t need to know the answer to this real analysis homework. I know how to do this. It’s in the back of the book. Before AI, you could Google the answers. Okay? So, uh, what is the purpose of this operation? Is it for you to learn how to use that technology, to learn how to prompt AI to solve real analysis problem? Is it that I want the solution to the problem, or is it that you’re supposed to sit there being frustrated and get the stamina for frustration that it takes to do any kind of difficult thing that human beings want to do?
If that’s what the goal of this exercise is, then either sign up voluntarily to do the exercise the way it’s designed, or drop this course ’cause that’s, that’s what I’m trying to teach.
LEVIN: Yeah, I think the gym analogy is pretty good. I do think it’s very interesting, I once had to try to solve Einstein’s equations for a black hole solution, not because it wasn’t solved already or because I couldn’t find the answer somewhere. And it was a great moment in my life, right? Going through that process. But I think there’s gonna take some convincing for this generation that’s growing up on AI.
STROGATZ: It’s interesting your example because this has always been a little bugaboo of my own, that there are people who talk about high school geometry is worthless, say the people who don’t like math, but geometry will teach you to think. It will help you develop logic and patience and blah, blah, blah.
I don’t like that defense of geometry. Geometry is a magnificent creation of humanity, and this is like, you know, you could play soccer because you wanna exercise your leg, or because it’s the beautiful game, you know?
And so math to me, the content is beautiful. It’s not just the calisthenics that I wanna make myself stronger for solving a problem for a finance company, ’cause that’s what really matters. I wanna do it for human flourishing, like Francis Su would say. Like what I think you’ve been pushing, Ravi. Not just for my own bulking up because the content is inherently rich and makes me a happier person before I’m dead. What’s the meaning of life? I wanna do math while I’m alive.
VAKIL: And I like that we can have both. I like the fact that soccer in the United States is played in middle schools en masse, and you have fun. You’re with people, and you s- and they’re secretly, of course, getting healthier and stronger by enjoying it and building habits. You know, so this seems a great model for similarly doing geometry to support similar things at the same ages.
LEVIN: This might be very unpopular, and I apologize to the mathematicians ahead of time.
STROGATZ: OK, don’t ask…
LEVIN: But there’s a belief that if all of human knowledge is wiped out tomorrow, not humanity, but human knowledge. That eventually we’ll all be rediscovered. One plus one is two. If human knowledge was wiped out tomorrow, the AI stops.
STROGATZ: But what happens to the human…
LEVIN: It’s got to start over.
KONTOROVICH: No, but there’s a version of this question which you can ask right now, which is, okay, so there are these… we didn’t get into formalization in interactive theorem provers, but if you take any interactive theorem prover which has the logical axioms of inference, logical inference from one state to the next, and you seed it with nothing, like AlphaZero. You’ve just given it the rules of chess, and you say, “Play a trillion games, and it comes up with, you know, its own theory and its own evaluation of states and so on.
If you seed an interactive theorem prover with no library of what mathematics is.
LEVIN: Exactly.
KONTOROVICH: and you cut an AI loose, will it discover a proof of Fermat’s Last Theorem?
LEVIN: Right. Without the reinforcements and…
KONTOROVICH: Yeah, without giving it any human knowledge, without, without anything. So that is, uh, I think Peter Sarnak is popularizing this question. You know, is there, like, a MathZero? Could someone train an AI to actually do that? That’s the real question that we have an answer to at the moment.
LEVIN: But we know humanity could do it. I mean, yes we’ve done it once, we could do it again, right? But it, it also sort of harkens, and I think this was the part I was concerned about mathematicians wouldn’t like, to the “discovered” or “invented” question. Are these discoverable truths that an entity that thinks entirely differently from human beings could discover?
VAKIL: Sure, and you can imagine being the first human being to see a lake wandering in the woods in some new island that’s discovered. Both can be true. That there’s joy in human discovery and also maybe a drone would fly over the island and discover the lake.
STROGATZ: Well, I think this is the place where we turn it over to the audience. You’ve been patient, and I appreciate, our panel taking some pretty wild questions from us. They may get even wilder here. We’re ready when you are please.
Audience Question: So far, AI is controlled by private companies, and access to the best models is increasingly being gated by financial resources. And I’m concerned about this divide between researchers who are going to have access and those who don’t. Historically, our field has been known for all you need is pen and paper. To what extent will that no longer be true, and how can we make sure math is still accessible to everyone?
KONTOROVICH: There are a lot of groups that are working on open source analogs. Of course, they’re behind, but they’re catching up. I’m on a team that’s working on open source models that solve math problems in particular, or help mathematicians formalize mathematics and so on. So people are, uh, are very concerned about this. They’re working on it. It’s hard. We need more resources. We should always say we need more resources.
STROGATZ: We can always say that.
KONTOROVICH: Yeah.
Audience Question: My question is, I think there are some missing pieces. Now, as AI begins churning out proofs, I think we are realizing that this is not the- what mathematicians want, just like the number of proofs. I think the missing piece I feel is aesthetics. And we haven’t really defined that.
Like, why is a problem interesting or why is a proof interesting? Where is the beauty? Where is depth? Where is interconnectedness? I think because these haven’t been defined, when you’re faced with hundreds of proof generated by AI, you have no idea what to look at. So, is somebody working on, like, measuring these?
VAKIL: I will go the other direction and say that y- y- y- you might be saying, “Can we mechanize aesthetics so we can measure what is the most aesthetic proof?” And I, I don’t think that’s the direction to go.
We already have a good human sense as what’s aesthetically pleasing. We know what’s interesting. We know most of the proofs that are produced by AIs are not interesting. We know some of them are fantastically interesting and beautiful.
So I don’t think it’s mechanizable, and that pleases me a little bit that, it’s a human subject, that it’s a human understanding that matters. I realize that’s not a very happy, it’s not like a very conclusive answer, so maybe I should pass it to someone else.
KONTOROVICH: I’m just gonna say the, the various ways that other sciences try to mechanize quality by, you know, citation inde- you know, H-index and these kinds of things, uh, which in mathematics are almost, you know, have zero correlation basically.
I’m really happy that our field is not looking at metrics and there are Fields Medalists with an H-index of four or something. These, these things are completely meaningless. Uh, it’s a problem when we have to go to the dean and say, “No, no, this person really is very good. Please look at their papers,” which the dean will not understand.
STROGATZ: Uh-huh. Oh, go ahead please.
VENKATESH: No, I just say as mathematicians, I think we should be very conscious of the use of numbers to stand in for another concept, and we remember it’s really this other concept we’re interested in.
VAKIL: And we have interesting debates in the mathematical community, which I enjoy, about is this interesting or not?
VENKATESH: Right.
VAKIL: It’s not like there’s a single metric, this theorem deserves a 3.5 on the scale. But instead, you know, we’ll argue, we’ll have interesting discussions over coffee that, “Oh, this was really something,” and “Actually, I wasn’t that impressed.” It’s a very human measurement system. Some would argue then that it’s open, it’s ripe for abuse, which it absolutely is. But on the other hand, it also is most people are, doing it in good faith, and there’s the sense of beauty in mathematics.
It is hard to… I don’t think I would want… I don’t think it’s possible to quantify, but it’s something that you meet someone at the ICM from another country, brought up in a completely different school system, and you meet and you hear about a new theorem, and you both nod your head and you think, “That was amazing.”
KONTOROVICH: And different people will have different evaluations of the value, and even the same person at different times will have different evaluations of the same result. I know that there are results that, you know, I, I thought, “Oh, that, I don’t think that’s that interesting,” and then I started learning more about it, and I realized, “No, that’s an amazing breakthrough.”
STROGATZ: [Laughs] So true. Okay, please.
Audience Question: Um, thanks for the great conversation. think unfortunately, historically, the mathematical community has not appreciated exposition and teaching in the same way that it has solving really hard problems. I mean, this goes back to, like, Bertrand Russell describing people that describe math as being less than the people that solve math problems. Do you think this will shift in the age of AI towards an appreciation of pedagogy and of exposition?
VENKATESH: I think yes, for sure.
VAKIL: I think the three of us here would have pushed before AI in exactly that direction. This is not new, but I think so, and I hope so, and I think we should have pushed before, and…
VENKATESH: Yeah, I think this is an example of where, y-you know, the, stress that AI puts on the system will lead to something healthy, something that was healthy quite without the issue of AI.
STROGATZ: That’s an optimistic reading that I like hearing. Actually, that it, it may push us in a direction that we should have gone in before.
KONTOROVICH: I mean, [G. H.] Hardy explicitly wrote, you know, “Here I am in my old age looking back and sort of surveying things,” and surveying is the work of lesser men that, uh… I mean, that’s I think a literal quote.
STROGATZ: Yeah. He seemed to have had a lot of self-loathing. Some of it deserved, I think, maybe, but not all of it.
[Group laughter]
STROGATZ: No, that essay did a lot to damage our values.
VAKIL: That is true.
STROGATZ: We’ll save that for another time.
Audience Question: Thank you for this conversation. I had a question for the entire panel. So the panelists laid out a vision of what AI use in education ideally looks like, right? You ask students not to use AI to train their minds.
But this vision is sort of incompatible with the current infrastructure for like grad school applications or postdoc applications or tenure track applications or what have you, where, you know, it seems like the only thing that matters is how much you publish, how hard the problems you’re solving are, and so on, so on.
How do you see this infrastructure changing? And what is your advice to a current graduate student worried about the fact that not wanting to use AI, say, in order to train their minds, would leave them behind, especially given how slow the timelines of infrastructure changes tend to be?
VAKIL: I’ll begin, but I think what the other two have to say is gonna be very important. I do think part of the system is currently quite robust, and I feel happy that we can deal with it. Parts are going to have to change. I will say, for example, for applications for graduate school, our department now interviews in a way we didn’t, because we want to really see what people know and see the human being. And we ask questions not to test, but see their actual understanding. So, I feel like that part of the system is, is robust. And we’re looking for at every single stage is can you do the best mathematics?
And I do think the best mathematics, some people will do very well augmented by AI, other people will not. I will say to graduate students, if they wonder, “Do I need to use AI?” I would say, “No, currently.” But I would also say don’t be afraid of it if it helps. If you need to look for references, use it in the way you feel comfortable with.
But the evidence currently out, don’t go to Twitter, don’t follow what other people are saying. Don’t be afraid. Uh, what really is happening on the ground is that the people who are using it effectively are using it in a very specific way, and I think the people who are best at it, who will give you the best advice are fellow graduate students who are similarly struggling with it at the same level, where they will say use it to perhaps… track down the thing to do. How do I think about this? But then you go away, and learn who among your peers is learning the fastest, and maybe they’re using an AI to guide them on what to think, or maybe not. And so I don’t think you need to use AI. I don’t think you should be afraid of it either.
VENKATESH: I, I was gonna say the same thing about interviews. You know, I’m also considering a-adding them for postdocs for exactly this reason. I think we, you know, we wanna understand who the person is and language models are gonna make that different, difficult if we just go on unwritten things.
I think the broader question of how systems have to change is, is important. And I don’t know, but I, I think the community is seriously thinking about it.
KONTOROVICH: And just to add, even before AI, it wasn’t a number of publications or your GRE score, it was the letters. We wanna know who the professors are that you’ve interacted with and what is their opinion. So which, you know, is an interpersonal communication and that’s what a lot of the measure was. You know, GRE scores are nice to look at, but I go straight to the letters.
VENKATESH: I think, you know, mathematics has always, I think it’s made space for people who are independent-minded and wanted to go and do their own thing. And I think we will continue to do that. And maybe doing your own thing means saying I refuse to use AI. But, I think as a culture, we respect that kind of independence of mind.
STROGATZ: Nice. Okay.
Audience Question: Um, So one thing that we’ve heard a lot about is how AI is changing the way that we’re attacking these problems, the way we’re thinking really. And I get a little pessimistic and wonder, you know, is it gonna change the way we think in a bad way? I don’t think students necessarily start using AI as a tool thinking, “Okay, I’m going to hurt my knowledge and the way I go about these problems,” but slowly over time it does, and I worry that the same thing might happen to mathematicians.
So my question is, are we really just like frogs boiling in a pot of water? Are we gonna know when we’re supposed to jump out?
[Audience giggles]
VENKATESH: Who knows?
STROGATZ: Who knows!
VAKIL: I think some people will make bad decisions, and the question is if too many do, I, I think that is a real danger, and it’s hard to know how warm the water is, and people will think their homework is due the next day and just this one time, they’ll do it.
So I, I think that’s a fair question, and it’s tough when you’re the person. It’s easy to see it from the outside. It’s tougher when you’re the person deciding.
I feel it a little bit when I learn a new subject with AI that sometimes I get the link to the paper and I get, then maybe I’ll get the summary. But in the old days, I would’ve read the paper and I would’ve fought with it for a few days.
So, there’s a trade-off, and as I s-said earlier, we need to fight with things for a long period of time and be stuck. And I don’t have much time right now, and so I sometimes make this trade-off. But I’m making potentially the same mistake the undergraduate is the day before the homework is due.
KONTOROVICH: I still can’t understand anything without printing it out. So I’m, I’m very old-fashioned in that way, you know. If I’m looking at something on a screen, it’s just going by. I tried annotating PDFs. It’s just not the same as printing it out, laying seven pages across my desk, circling things and, you know. I don’t know. I’m a Luddite… who loves AI.
[Group laughs]
STROGATZ: Yes, please.
Audience Question: Um, As a grad student, I feel, uh, low-hanging fruit is very important for, uh, grad students for, like, getting started, getting confidence in math. And my concern is that AI will swoop, and take all the low-hanging fruit, and then maybe, uh, it’ll be hard for a lot of people to get started. And I’m concerned about like, the Average Joe mathematician who doesn’t work on things that are, like, safe from AI. But, yeah. I was wondering what your thoughts are on that.
STROGATZ: I wonder about that, though. we didn’t get a chance to talk about applied math versus pure math, and maybe you’re in pure math, so this isn’t relevant. But I think even within pure math, isn’t it possible to start a new area sort of? Like, there’s this question of interpolating within math and extrapolating to do something brand new, and I realize it’s hard to start a new subject or even a new tiny piece of a subject, but I’m just thinking that might increase your chances, that it won’t be low-hanging fruit if it doesn’t exist yet. You know what I mean?
Like, there may be tractable graduate student beginning level questions that you could get by talking to your friend who’s a psychologist. I am gonna go applied on you for a minute. We want you in applied math. There’s a lot to do, and science may be more resistant than math.
LEVIN: It’s happened in particle physics. It was a victim of its own success. I mean, this is a different kind of a success. It’s a technological success, but where the standard model seemed done.
STROGATZ: Mmm, okay.
LEVIN: And it was no longer attracting tons of students who had lots of work to do. It was all kind of solved. And then, you know, luckily, we realized we only know 5% of what’s out in the universe, so we’re back. But, you know, dark energy and dark matter kind of saved the fields… the unknowns. So it, it may be that there’s just, as you’re saying, there’s a way of, sort of, pivoting. But it’s very curious for mathematics. I think it is a curious question.
VAKIL: So I wanna express worry as well. I wanna share your concern that this, I think uh, there are certain kinds of problems that would be excellent for training graduate student minds, such as you take a solution, a theorem that worked in a certain situation, and I as an advisor know it’s going to work. It’s got to work in a slightly different situation. And that’s perfect ’cause you would have to really understand it to make it work. But now that really feels something that is accessible to models, and rightly so that now this should be handed over to models.
So, the good thing is advisors are concretely thinking about how next to think about the right way, the right sources of problems. I don’t feel like graduate students are the ones that are gonna invent new fields. Unless, except for maybe three or four.
But, I do worry. That’s the, one of the few things that I think are real concerns. Uh, but at least I share the concern, and we’re thinking hard about it. So don’t give up, uh, because it’s important to get the graduate students through this stage and continue to train because they’re the ones who are going to drive the field forward four years later.
KONTOROVICH: But I think even your example of I know this theorem works in, uh, situation X, I wanna, you know, make a slightly, slight modification X prime. The act of what we’re asking the student to do to transfer from X to X prime, we’re not saying you have to come up necessarily with any new ideas.
The goal is for you to learn actually what theorem X is, learn the innards of how it works, because when you shake it just a little bit, you, you learn some more about, oh, actually the reason this is happening is because of this and not that and you thought it would work, but it actually doesn’t work and here’s where I’m stuck and so on.
There’s still so much, even with the AI assisting you in that path, there’s still so much for you to learn. The goal isn’t necessarily just to arrive at X prime. Now, if you’re saying, “Okay, but AI already knows X prime,” who’s pressing the button on the AI to get X prime, right? At the end of the day, someone has to ask the AI the question, and of course it can ask its own questions, but even then someone’s prompting it to say, “Go ask questions.”
So, it’s like it’s not turtles all the way down. Someone is paying the electric bill of what’s being asked and, uh, that person’s doing it for a reason. At the end of the day, what is being done has to be valued by human beings, otherwise it’s not going to get done.
STROGATZ: And isn’t it possible that we’re in a peculiarly anomalous time right now where the big AI companies are playing around with our field for their own purposes…
KONTOROVICH: Yes.
STROGATZ: … but they may lose interest very quickly.
KONTOROVICH: That’s exactly what I expect.
STROGATZ: Right? So, so in fact, what we’re experiencing now may not last much longer, and maybe we’ll have it back to ourselves again for a while.
KONTOROVICH: And we’ll have this great tool that they…
STROGATZ: And we’ll have this great tool while they’re solving something else.
Well… okay, please, go ahead.
Audience QA: So like many people in this room, I of course agree that math does amazing things to sharpen minds and to enrich lives. But at the same time, there’s a lot of other disciplines that can lay similar claims, and much has been said even today about, you know, humanities and that maybe math is going to start looking a little more like those.
So my question is do you think that as a profession, we need to be thinking about some sort of a value-added statement that would actually work to, in real life, nurture an ecosystem of people who would be engaged in this brave new math that’s being shaped up? Or do you think that such a value-added statement will just kind of happen by itself?
VAKIL: Okay, that was like softly lobbed across home plate. So this is something which is, i-i-is, so this is part of the mistakes that were made with the Hardy style: I only wanna talk to those students who are future PhD mathematicians. That’s decades ago.
At many schools, math is one of the biggest majors. At our school, math is one of the biggest majors. Why? Not because people are going to become PhD mathematicians. It’s because this, these habits of mind, this, way of thinking are empirically, more and more today, even more than ever, are incredibly useful.
Our students, if you want a job where you know what you’re gonna be doing in 50 years, you should not be a math major. If you want to be a job in a career that doesn’t even exist yet, that where you’re gonna make the future, our students rule the world.
In fact, I was just given from the Italian Mathematical Society, they did a study on the value added of mathematics to the economy. This is, incredibly safe, and I think by hiding ourselves and pretending we’re not useful, that’s been really dangerous. I think we need to have an informed public. They want to know math. They want to understand math. We can easily convince them how useful it is if we would just make the case for it. So thank you for that that very easy question.
[Laughter]
STROGATZ: Well, you have been a fantastic audience, and so we thank you. We also thank our panel here. So we’ve been speaking today with Alex Kontorovich, Ravi Vakil, Akshay Venkatesh, and as always, Janna Levin is here co-hosting. We’re signing off from The Joy of Why. Thanks again, Janna.
LEVIN: Thank you, Steve, and thanks to our guests. Till next time.
[Audience applause]
STROGATZ: You turned on the recording?
[Laughter]
KONTOROVICH: Run it again!
[Music plays]
STROGATZ: If you’re enjoying The Joy of Why and you’re not already subscribed, hit the subscribe or follow button where you’re listening. You can also leave a review for the show. It helps people find this podcast. Find articles, newsletters, videos, and more at quantamagazine.org. [email protected].
LEVIN: The Joy of Why is a podcast from Quanta Magazine, an editorially independent publication supported by the Simons Foundation. Funding decisions by the Simons Foundation have no influence on the selection of topics, guests, or other editorial decisions in this podcast or in Quanta Magazine.
The Joy of Why is produced by PRX Productions. The production team is Caitlin Faulds, Jade Abdul-Malik, Genevieve Sponsler, and Merritt Jacob. The executive producer of PRX Productions is Jocelyn Gonzales. Edwin Ochoa is our project manager.
From Quanta Magazine, Simon Frantz and Samir Patel provided editorial guidance with support from Samuel Velasco, Kit Sudol, Simone Barr, and Michael Kanyongolo. Samir Patel is Quanta’s editor-in-chief.
The episode art is by Chanelle Nibbelink, and our logo is by Jacky King and Kristina Armitage.
STROGATZ: Special thanks to Garth Avery at the Cornell Broadcast Studio and Andrew Stelzer for PRX.
LEVIN: I’m your host, Janna Levin. If you have any questions or comments, please email us at [email protected]. Thanks for listening.
[ends]
Note: Alex Kontorovich is a member of Quanta Magazine’s advisory board.