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数学，路在何方？

AI将不可逆地终结纯数学的“手工英雄主义”时代，迫使学术评价体系从“解题难度崇拜”转向“问题定义与AI协同能力”，但作者对AI构建深层理论的能力存在严重过度外推。
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2026-10-05 原文链接 ↗
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核心观点

  • 价值锚点转移 纯数学的声望分配体系必然瓦解，未来核心竞争力将从“徒手攀登高难度猜想”降级为“精准定义问题、调度AI算力与跨学科翻译”。
  • 技术演进非线性 将AI在特定猜想上的突破等同于理论构建能力是范畴错误，当前AI仅能完成概率优化与模式匹配，尚未跨越概念发明与严格逻辑演绎的鸿沟。
  • 身份认同陷阱 将职业价值绑定于“AI无法完成某项任务”是高风险策略，抗拒技术迭代的学者将面临学术边缘化与目标感丧失的双重危机。
  • 工业化不可逆 数学研究正从精英探险转向基础设施产业，AI将大幅降低“地形测绘”成本，但会伴随学术审美流失与商业化侵蚀的阵痛。

跟我们的关联

  • 对 ATou 意味着个人技术护城河必须从“执行深度”转向“问题嗅觉与AI编排”，下一步应停止打磨单一解题技巧，转而构建多模型协同工作流并主导高模糊性项目的定义。
  • 对 Neta 意味着组织激励体系需彻底重构，下一步必须废除以单点突破为核心的KPI，建立以“地形选择（战略眼光）”、“机队调度（流程效率）”和“价值翻译（业务对齐）”为权重的评价矩阵。
  • 对 Uota 意味着需警惕技术决定论的叙事包装，下一步应在引入AI工具时强制保留人类可解释性验证环节，并建立“静态思维谬误”检测清单以对冲过度外推风险。

讨论引子

  • 当AI将证明猜想的边际成本降至趋近于零时，学术界应如何重新定义“原创性贡献”以避免陷入跑分竞赛？
  • 在“手工攀岩”与“飞机测绘”的范式切换期，研究者应如何平衡对数学审美与直觉的坚守与对AI工具链的妥协？
  • 如果下一代模型在形式化证明上遭遇架构瓶颈，当前押注“AI全面接管理论构建”的学术投资是否会引发系统性资源错配？

报告不当行为详情

毕业

十几岁时，我在匈牙利最负盛名的学校就读数学特长班。我们每周有八个小时的数学课，外加为国内和国际竞赛做准备的特别辅导。我甚至在巴拉顿湖畔为期一周的夏令营中遇见了我的前妻，在那里我们唯一的任务就是做国际数学奥林匹克竞赛（IMO）的题目。我青春的大部分时光都花在了解决竞赛题上。

高中毕业考试结束后的头两个小时，是我一生中最感自由的时刻。我们复习了一周，涵盖了五门学科的一长串知识点，当最后一门考试结束时，我感到了巨大的如释重负：终于结束了！紧接着，空虚感袭来。我突然意识到：再也没有竞赛了。并不是说我特别喜欢竞争，而是它已经成了我生命中不可或缺的一部分。我们对此非常认真，并深深敬佩那些表现出色的人。在匈牙利全国竞赛中跻身前十能带来真正的地位，更不用说进入国家 IMO 代表队了。我从未接近过那支队伍，但归属于那个围绕竞赛建立的社区，是我生命中最重要的部分之一。

那两个小时过后，一种令人悲伤的空虚感油然而生。不再有竞赛，不再有备赛，不再有激动人心。再也没有人指导我，也没有人关心我。多年来，我第一次失去了训练的目标，没有了它，我甚至不知道自己是谁。这感觉很糟，仿佛我失去了自身至关重要的一部分。我难过了好几天。

丛林

想象一片无边无际、未经测绘的荒野：丛林、沙漠、崎岖的山脉、宽阔到足以吞噬村庄的河流。现在想象你用一生的时间去探索它。这大致就是做纯数学（Pure Mathematics）的感觉。让我解释一下这个类比。

这片地形危险而美丽，有点像达连隘口（Darién Gap），但探索起来更加困难。初学者会在几天内迷路，染上疟疾并死在丛林里，或者被野兽吃掉。探险者的工作有三重。首先，他们测绘地形，这有助于他们快速找到最美丽的地方并探索正确的区域。其次，他们寻找自然资源，因为探险需要资金。他们几乎不关心经济回报本身，但必须找到为旅行买单的方法。第三，他们发明并制造小工具来一路辅助自己。

明确一下这个类比：终身教职数学家（Tenured Mathematicians）可以自由探索他们喜欢的任何东西，但社区仍然根据他们的贡献来评价他们。最高的荣誉属于那些通过巨大努力和艰辛攀上一座高山，并从山顶勾勒出粗略地图的人。人们最钦佩的不仅是成果（地图），还有其背后的英雄壮举。如果这座山极其困难，以至于以前的探险者都失败了，你就会受到真正的敬畏。朗兰兹纲领（Langlands program）就可以被视为这样一项成就。发现一条连接两个遥远地区并让人们可以在其间穿行的河流，也会受到高度评价。

工具也受到重视，尽管它们不那么算得上是神来之笔。如果你创造了一种被广泛使用的新工具，比如矩阵的伪逆（pseudo-inverse of a matrix）或 3-SAT 的 NP 完全性（NP-completeness），你会很有名，但你不会获得像征服一块看似无用、许多人尝试攀爬却失败的岩石的人那样的光环。事实上，在你用工具爬上过一块无人攀爬过的岩石之前，没人会相信你的工具有用。

最后，通过发现宝贵的矿藏——那些真正让非探险者的普通人受益的东西——你也能赢得一些尊重。如果你发现椭圆曲线（Elliptic Curves）可用于密码学，大多数数论学家并不怎么关心，但他们会在科研经费申请书中提到它。如果出资者相信附近可能埋藏更多矿藏，这当然有助于为他们的探险提供资金。尽管如此，探险者（数学家）主要关心的还是美丽的风景、探索的刺激，以及最重要的，同侪对征服难爬岩石的敬畏。那些岩石（猜想）本身通常没那么有趣，但它们是新攀岩工具的试验场。不过偶尔，一块岩石会被厚雾笼罩，而更罕见的是，它最终被证明是一座宏伟的山峰。从它的顶端你可以看得更远，探索地形中更有趣的部分。归根结底，这就是人们留下的原因：不是为了地图或金钱，而是为了那个罕见的时刻——一块无人在意的岩石变成了一座山峰，整个地形在你脚下豁然开朗。

飞机

每一场革命起初看起来都像个玩具。莱特兄弟的首次飞行持续了十二秒，飞过的距离还不及大型喷气式客机（jumbo jet）的翼展。这大致就是当今 AI 在数学领域的处境。直到今年，我们的飞机还极其简陋：它们只能飞几百米（英尺），高度只有几米（英尺）。它们也依然昂贵得离谱，且仅在少数狭窄的场景下有用。但它们进步神速，在过去几个月里，其航程和可靠性已达到可以飞入达连隘口而不坠毁的程度。我们甚至降落在了几块以前无人攀爬过的岩石上（我们解决了几个猜想）。这些飞机仍然非常原始，以低速和低空飞行不到半小时，而且这些飞行的成本远超探险者手工打造工具的耗费。

航空公司对在一些大岩石上插上旗帜感到无比自豪，全然不顾探险者们正忙着攀爬这些岩石且已接近山顶。他们将此视为进步的伟大展示，并期望探险者们会为探索丛林中有趣部分的速度变快而感到兴奋。相反，探险者们生气了：“请不要在我们的岩石上插旗。这会阻止其他探险者尝试攀爬。而且，我们已经爬上过其中几块了。说实话，这些岩石没那么有趣。我们攀爬它们是为了打磨和测试用于真正山峰的工具，而你们的飞机飞得还不够高，够不到那些山峰。不过，你们能送我们几架飞机当礼物吗？我们想测试一下，看看能派上什么用场。”

当然，探险者们很不满。这一切来得太突然，早期的飞机看起来又原始又没用。他们中少数人试飞了飞机，结果在丛林边缘就坠毁了，看起来十分滑稽。谁能想到那些飞机突然就能在他们的岩石上插旗呢？很可笑，对吧？

但是，认为飞机无法到达真正山峰的顶端（解决最难的问题）或绘制大面积区域的近似地图（建立理论）的论点，是一种静态思维的谬误。与几个月后甚至几年后的飞机相比，今天的飞机还很原始。

很自然地，仅仅为了打磨攀岩工具而去爬岩石将失去意义。直到今天，这些工具可能还是必不可少的，但不久后就不会了：私人飞机将无处不在，任何负担得起的人都可以包机降落在任何一块岩石甚至一座山峰上。

另一方面，在随机的岩石上插旗在现阶段只是一种技术演示或公关。这很快就会过时，尽管航空公司会继续利用这片地形来测试和改进他们的飞机，就像数学家利用岩石来打磨他们的手工工具一样。

一旦飞机变得廉价，我们也会发现卫星。我们将为整个地形绘制地图以寻找矿藏和其他有价值的资源，然后提取它们。这将变得杂乱且肮脏，今天的探险者将对他们美丽丛林所遭受的破坏感到深感震惊。他们会怀念过去的好时光。他们会说这不再是探索，而是一项商业活动，大公司正在破坏他们曾如此热衷于探索的区域。他们是对的，但这不会改变什么，大公司将在那里发现难以想象的财富。

当然，攀岩不会过时。许多新的、更休闲的探险者甚至可能会加入：包机飞到你最喜欢的岩石，在工具店买合适的工具（而不是手工打磨），然后和朋友一起爬上去。体能较差的人会乘飞机去寻找更宏伟的山峰，或者从飞得比任何山峰都高的飞机上俯瞰整个山脉。旧日的氛围和刺激将永远消失。数学将比英雄主义的旧探险时代更加辉煌、有用，同时也更加平凡。

路在何方？

不要和飞机作对。那些说 AI 不能猜想、不能建立理论、不能解释的怀疑论者，描述的只是去年春天的模型。几个月内，这种描述将不再成立。证据是明确的，有目共睹。任何将自己的身份认同押在“AI 做不了任务 X”上的人，都是在沙上建塔。

许多数学家，包括我的兄弟 Balázs，早早地拥抱了这些变化。其他人，如 Jared Duker Lichtman 和 Michael R. Douglas 以及许多其他人，也非常直言不讳，在过去几年里一直朝着同样的愿景努力。数学家们充满激情且客观地工作，数学仍然是最唯才是举的学术领域。学术界不会迅速改变。大学、期刊和终身教职评审委员会在多年内看起来还会是老样子。但他们奖励的内容将发生根本性的转变。数学以其唯才是举而自豪，对探险者——那些能攀上别人爬不上去的地方的人——怀有深深的敬意。这种特定的英雄主义将变得不那么重要，而其他技能将变得更加重要：选择值得探索的地形，驾驶飞机编队，以及向世界其他地方解释他们的发现。对于系统中的许多人来说，这会感觉像是一种降级，有些人会为此抗争。其他领域也经历过类似的动荡并幸存了下来。改变的是什么才算数，以及谁因什么而受到尊重。

许多数学家会感受到我高中毕业、竞赛生涯结束时所感受到的：迷茫，以及突然失去目标。这种悲伤是真实的，我不会假装它不存在。但它不会持续太久。数学即将毕业。它将成长为一个全面的产业，成为所有科技进步的真正基础设施，成为所有其他领域所依赖的基石，而不是学术界一个安静的角落。孤独攀登者的时代正在落幕，测绘整个大陆的时代正在开启。至于我，我已迫不及待地想从我的超音速飞机上俯瞰那些广袤的山脉了。

Quo Vadis, Mathematics?

不正行為を報告する詳細

Quo Vadis, Mathematics?

Is Mathematics Over, or Just Graduating?

报告不当行为详情

Graduation

As a teenager, I attended a math-specialized class at the most prestigious school in Hungary. We had eight hours of mathematics every week, plus special sessions to prepare for national and international competitions. I even met my ex-wife at a week-long camp at Lake Balaton, where our only job was to work on IMO problems. Much of my youth went into solving competition problems.

The first two hours after my high school finals were the freest of my life. We had studied for a week, covering a long list of topics in five subjects, and when the last exam ended, I felt enormous relief: finally, it was over! Then came the emptiness. It hit me: there would be no more competitions. Its not that I particularly enjoyed competing, but it had become an organic part of my life. We took it very seriously and deeply respected those who did well. Placing in the top ten of a Hungarian national contest brought real status, never mind making the national IMO team. I never came close to that team, yet belonging to the community around it was one of the most important parts of my life.

After those two hours, a sad emptiness set in. No more competitions, no more preparing for them, no more thrill. Nobody coached me anymore, and nobody cared. For the first time in years, I had nothing to train for, and I didnt know who I was without it. It felt terrible, as if I had lost a vital part of myself. I stayed sad for days.

毕业

十几岁时，我在匈牙利最负盛名的学校就读数学特长班。我们每周有八个小时的数学课，外加为国内和国际竞赛做准备的特别辅导。我甚至在巴拉顿湖畔为期一周的夏令营中遇见了我的前妻，在那里我们唯一的任务就是做国际数学奥林匹克竞赛（IMO）的题目。我青春的大部分时光都花在了解决竞赛题上。

高中毕业考试结束后的头两个小时，是我一生中最感自由的时刻。我们复习了一周，涵盖了五门学科的一长串知识点，当最后一门考试结束时，我感到了巨大的如释重负：终于结束了！紧接着，空虚感袭来。我突然意识到：再也没有竞赛了。并不是说我特别喜欢竞争，而是它已经成了我生命中不可或缺的一部分。我们对此非常认真，并深深敬佩那些表现出色的人。在匈牙利全国竞赛中跻身前十能带来真正的地位，更不用说进入国家 IMO 代表队了。我从未接近过那支队伍，但归属于那个围绕竞赛建立的社区，是我生命中最重要的部分之一。

那两个小时过后，一种令人悲伤的空虚感油然而生。不再有竞赛，不再有备赛，不再有激动人心。再也没有人指导我，也没有人关心我。多年来，我第一次失去了训练的目标，没有了它，我甚至不知道自己是谁。这感觉很糟，仿佛我失去了自身至关重要的一部分。我难过了好几天。

The Jungle

Picture an endless, unmapped wilderness: jungles, deserts, jagged mountains, rivers wide enough to swallow a village. Now imagine spending your life exploring it. That, roughly, is what doing pure mathematics is like. Let me explain the analogy.

The terrain is perilous but beautiful, something like the Darién Gap, but even harder to explore. A beginner would get lost within days, catch malaria and die in the jungle, or be eaten by wild animals. The explorers work is threefold. First, they map the terrain, which helps them find the most beautiful places quickly and explore the right areas. Second, they look for natural resources, because expeditions need funding. They hardly care about the financial rewards for their own sake, but they must find a way to pay for their travels. Third, they invent and build gadgets to help them along the way.

To make the analogy explicit: tenured mathematicians are free to explore whatever they like, but the community still judges them by their contributions. The most recognition goes to those who climb a tall mountain through great effort and hardship and sketch a rough map from the summit. What people admire most is not just the outcome (the map) but the heroic effort behind it. If the mountain is so hard that earlier explorers failed, you will be truly revered. The Langlands program can be seen as such an accomplishment. Finding a river that connects two distant regions and lets people travel between them is highly valued too.

Tools are valued as well, though they are less of a tour de force. If you create a new tool that becomes widely used, like the pseudo-inverse of a matrix or the NP-completeness of 3-SAT, you will be well known, but you wont earn the same halo as someone who conquers a seemingly useless rock that many had tried and failed to climb. In fact, nobody will believe your tool is useful until you use it to climb a rock that nobody has climbed before.

Finally, you earn some respect by finding a valuable mineral deposit: something that actually benefits ordinary people, the non-explorers. If you discover that elliptic curves can be used for cryptography, most number theorists wont care much, but they will mention it in their grant proposals. It certainly helps fund their expeditions if their financiers believe more deposits may lie nearby. Still, the explorers (mathematicians) mostly care about the beautiful views, the thrill of exploration and, above all, the veneration of their peers for conquering a hard-to-climb rock. Those rocks (conjectures) are usually not that interesting in themselves, but they serve as testing grounds for new climbing tools. Occasionally, though, a rock is shrouded in thick haze, and, even more rarely, it turns out to be a majestic mountain. From its top you can see much farther and explore far more interesting parts of the terrain. That, in the end, is why people stay: not for the maps or the money, but for the rare moment when a rock nobody cared about turns out to be a mountain, and the whole terrain opens up below you.

丛林

想象一片无边无际、未经测绘的荒野：丛林、沙漠、崎岖的山脉、宽阔到足以吞噬村庄的河流。现在想象你用一生的时间去探索它。这大致就是做纯数学（Pure Mathematics）的感觉。让我解释一下这个类比。

这片地形危险而美丽，有点像达连隘口（Darién Gap），但探索起来更加困难。初学者会在几天内迷路，染上疟疾并死在丛林里，或者被野兽吃掉。探险者的工作有三重。首先，他们测绘地形，这有助于他们快速找到最美丽的地方并探索正确的区域。其次，他们寻找自然资源，因为探险需要资金。他们几乎不关心经济回报本身，但必须找到为旅行买单的方法。第三，他们发明并制造小工具来一路辅助自己。

明确一下这个类比：终身教职数学家（Tenured Mathematicians）可以自由探索他们喜欢的任何东西，但社区仍然根据他们的贡献来评价他们。最高的荣誉属于那些通过巨大努力和艰辛攀上一座高山，并从山顶勾勒出粗略地图的人。人们最钦佩的不仅是成果（地图），还有其背后的英雄壮举。如果这座山极其困难，以至于以前的探险者都失败了，你就会受到真正的敬畏。朗兰兹纲领（Langlands program）就可以被视为这样一项成就。发现一条连接两个遥远地区并让人们可以在其间穿行的河流，也会受到高度评价。

工具也受到重视，尽管它们不那么算得上是神来之笔。如果你创造了一种被广泛使用的新工具，比如矩阵的伪逆（pseudo-inverse of a matrix）或 3-SAT 的 NP 完全性（NP-completeness），你会很有名，但你不会获得像征服一块看似无用、许多人尝试攀爬却失败的岩石的人那样的光环。事实上，在你用工具爬上过一块无人攀爬过的岩石之前，没人会相信你的工具有用。

最后，通过发现宝贵的矿藏——那些真正让非探险者的普通人受益的东西——你也能赢得一些尊重。如果你发现椭圆曲线（Elliptic Curves）可用于密码学，大多数数论学家并不怎么关心，但他们会在科研经费申请书中提到它。如果出资者相信附近可能埋藏更多矿藏，这当然有助于为他们的探险提供资金。尽管如此，探险者（数学家）主要关心的还是美丽的风景、探索的刺激，以及最重要的，同侪对征服难爬岩石的敬畏。那些岩石（猜想）本身通常没那么有趣，但它们是新攀岩工具的试验场。不过偶尔，一块岩石会被厚雾笼罩，而更罕见的是，它最终被证明是一座宏伟的山峰。从它的顶端你可以看得更远，探索地形中更有趣的部分。归根结底，这就是人们留下的原因：不是为了地图或金钱，而是为了那个罕见的时刻——一块无人在意的岩石变成了一座山峰，整个地形在你脚下豁然开朗。

The Airplane

Every revolution looks like a toy at first. The Wright brothers first flight lasted twelve seconds and covered less distance than a jumbo jets wingspan. That is roughly where AI stands in mathematics today. Until this year, our planes were extremely rudimentary: they flew a few hundred meters (feet), at an altitude of a few meters (feet). They are also still obscenely expensive, and useful only in a few narrow cases. But they have improved quickly, and in the past few months they have reached the range and reliability to fly into the Darién Gap without crashing. We even landed on several rocks that nobody had climbed before (we solved a few conjectures). The planes are still very primitive, flying for less than half an hour at low speed and low altitude, and those flights cost far more than the explorers hand-crafted tools ever did.

The aviation companies were extremely proud of planting flags on some of the large rocks, never mind that explorers were busy climbing them and were close to the top. They considered it a great demonstration of progress and expected the explorers to be thrilled at how much faster exploring the interesting parts of the jungle could become. Instead, the explorers got upset: Please dont put flags on our rocks. It stops other explorers from trying to climb them. Also, we already climbed a few of them. And honestly, these rocks arent that interesting. We climb them to hone and test our tools for the real mountains, and your planes cant fly high enough to reach those. Still, could you give us some airplanes as a gift? Wed like to test them and find some use for them.

Of course, the explorers are upset. It all arrived so suddenly, and the early airplanes looked so primitive and useless. A few of them tried the planes and crashed right at the edge of the jungle, which looked ridiculous. Who would have thought that those planes could suddenly plant flags on their rocks? Laughable, right?

But the argument that airplanes cant reach the top of the real mountains (solve the hardest problems) or draw approximate maps of large areas (build theories) is a static-thinking fallacy. Todays airplanes are primitive compared to what they will be in a few months, let alone years.

Naturally, climbing rocks just to hone climbing tools will lose its point. Those tools may have been essential until today, but they wont be for long: private airplanes will be everywhere, and anyone who can afford it will charter one and land on any rock, or even a mountain.

On the other hand, planting flags on random rocks is just a technical demonstration, or PR, at this point. It will get old quickly, though the aviation companies will keep using the terrain to test and improve their planes, just as mathematicians used the rocks to hone their artisan tools.

Once airplanes get cheap, we will also discover satellites. We will map the whole terrain for mineral deposits and other valuable resources, and then extract them. It will be messy and dirty, and todays explorers will be deeply appalled at what it does to their beautiful jungle. They will long for the good old days. They will say this is no longer exploration but a commercial undertaking, large corporations wrecking the area they were so keen to explore. They will be right, but it wont change much, and large companies will find unimaginable riches there.

Rock climbing will not go out of fashion, of course. Many new, more casual explorers might even join: charter a plane to your favorite rock, buy the right tools at the tool shop (instead of honing them by hand), and climb it with friends. Less athletic people will fly in search of even more majestic mountains, or look at entire ranges from airplanes that fly far higher than any peak. The old vibe and thrill will be gone forever. Mathematics will be far more glorious, useful and mundane at the same time than it was in the heroic old days of exploration.

飞机

每一场革命起初看起来都像个玩具。莱特兄弟的首次飞行持续了十二秒，飞过的距离还不及大型喷气式客机（jumbo jet）的翼展。这大致就是当今 AI 在数学领域的处境。直到今年，我们的飞机还极其简陋：它们只能飞几百米（英尺），高度只有几米（英尺）。它们也依然昂贵得离谱，且仅在少数狭窄的场景下有用。但它们进步神速，在过去几个月里，其航程和可靠性已达到可以飞入达连隘口而不坠毁的程度。我们甚至降落在了几块以前无人攀爬过的岩石上（我们解决了几个猜想）。这些飞机仍然非常原始，以低速和低空飞行不到半小时，而且这些飞行的成本远超探险者手工打造工具的耗费。

航空公司对在一些大岩石上插上旗帜感到无比自豪，全然不顾探险者们正忙着攀爬这些岩石且已接近山顶。他们将此视为进步的伟大展示，并期望探险者们会为探索丛林中有趣部分的速度变快而感到兴奋。相反，探险者们生气了：“请不要在我们的岩石上插旗。这会阻止其他探险者尝试攀爬。而且，我们已经爬上过其中几块了。说实话，这些岩石没那么有趣。我们攀爬它们是为了打磨和测试用于真正山峰的工具，而你们的飞机飞得还不够高，够不到那些山峰。不过，你们能送我们几架飞机当礼物吗？我们想测试一下，看看能派上什么用场。”

当然，探险者们很不满。这一切来得太突然，早期的飞机看起来又原始又没用。他们中少数人试飞了飞机，结果在丛林边缘就坠毁了，看起来十分滑稽。谁能想到那些飞机突然就能在他们的岩石上插旗呢？很可笑，对吧？

但是，认为飞机无法到达真正山峰的顶端（解决最难的问题）或绘制大面积区域的近似地图（建立理论）的论点，是一种静态思维的谬误。与几个月后甚至几年后的飞机相比，今天的飞机还很原始。

很自然地，仅仅为了打磨攀岩工具而去爬岩石将失去意义。直到今天，这些工具可能还是必不可少的，但不久后就不会了：私人飞机将无处不在，任何负担得起的人都可以包机降落在任何一块岩石甚至一座山峰上。

另一方面，在随机的岩石上插旗在现阶段只是一种技术演示或公关。这很快就会过时，尽管航空公司会继续利用这片地形来测试和改进他们的飞机，就像数学家利用岩石来打磨他们的手工工具一样。

一旦飞机变得廉价，我们也会发现卫星。我们将为整个地形绘制地图以寻找矿藏和其他有价值的资源，然后提取它们。这将变得杂乱且肮脏，今天的探险者将对他们美丽丛林所遭受的破坏感到深感震惊。他们会怀念过去的好时光。他们会说这不再是探索，而是一项商业活动，大公司正在破坏他们曾如此热衷于探索的区域。他们是对的，但这不会改变什么，大公司将在那里发现难以想象的财富。

当然，攀岩不会过时。许多新的、更休闲的探险者甚至可能会加入：包机飞到你最喜欢的岩石，在工具店买合适的工具（而不是手工打磨），然后和朋友一起爬上去。体能较差的人会乘飞机去寻找更宏伟的山峰，或者从飞得比任何山峰都高的飞机上俯瞰整个山脉。旧日的氛围和刺激将永远消失。数学将比英雄主义的旧探险时代更加辉煌、有用，同时也更加平凡。

Quo Vadis?

Do not bet against the airplane. The skeptics who say AI cant conjecture, cant theorize, and cant explain are describing last springs models. Within months, that description will no longer hold. The evidence is clear, and those with eyes can see it. Anyone who stakes their identity on AI cant do task X is building on sand.

Many mathematicians, including my brother Balázs, embraced the changes early on. Others, like Jared Duker Lichtman and Michael R. Douglas and many others have been very outspoken, working toward the same vision over the past years. Mathematicians work passionately and objectively, and mathematics is still the most meritocratic academic field. Academia will not change quickly. Universities, journals and tenure committees will look much the same for years. But what they reward will shift fundamentally. Mathematics prided itself on being meritocratic, with deep reverence for the explorers: the people who could climb what nobody else could. That specific kind of heroism will matter less, and other skills will matter more: choosing which terrain is worth exploring, steering fleets of airplanes, and explaining what they find to the rest of the world. For many people in the system, this will feel like a demotion, and some will fight it. Other fields have been through similar upheavals and survived. What changes is what counts and who gets respected for what.

Many mathematicians will feel what I felt when I finished high school and my competing days ended: disorientation, and a sudden loss of purpose. That grief is real, and I wont pretend otherwise. But it will not last. Mathematics is about to graduate. It will grow into a full-scale industry and become the true infrastructure of all scientific and technological progress, a foundation that every other field stands on rather than a quiet corner of academia. The age of lone climbers is closing, and the age of mapping the whole continent is opening. As for me, I cant wait to see those vast mountain ranges from my supersonic airplane.

路在何方？

不要和飞机作对。那些说 AI 不能猜想、不能建立理论、不能解释的怀疑论者，描述的只是去年春天的模型。几个月内，这种描述将不再成立。证据是明确的，有目共睹。任何将自己的身份认同押在“AI 做不了任务 X”上的人，都是在沙上建塔。

许多数学家，包括我的兄弟 Balázs，早早地拥抱了这些变化。其他人，如 Jared Duker Lichtman 和 Michael R. Douglas 以及许多其他人，也非常直言不讳，在过去几年里一直朝着同样的愿景努力。数学家们充满激情且客观地工作，数学仍然是最唯才是举的学术领域。学术界不会迅速改变。大学、期刊和终身教职评审委员会在多年内看起来还会是老样子。但他们奖励的内容将发生根本性的转变。数学以其唯才是举而自豪，对探险者——那些能攀上别人爬不上去的地方的人——怀有深深的敬意。这种特定的英雄主义将变得不那么重要，而其他技能将变得更加重要：选择值得探索的地形，驾驶飞机编队，以及向世界其他地方解释他们的发现。对于系统中的许多人来说，这会感觉像是一种降级，有些人会为此抗争。其他领域也经历过类似的动荡并幸存了下来。改变的是什么才算数，以及谁因什么而受到尊重。

许多数学家会感受到我高中毕业、竞赛生涯结束时所感受到的：迷茫，以及突然失去目标。这种悲伤是真实的，我不会假装它不存在。但它不会持续太久。数学即将毕业。它将成长为一个全面的产业，成为所有科技进步的真正基础设施，成为所有其他领域所依赖的基石，而不是学术界一个安静的角落。孤独攀登者的时代正在落幕，测绘整个大陆的时代正在开启。至于我，我已迫不及待地想从我的超音速飞机上俯瞰那些广袤的山脉了。

Quo Vadis, Mathematics?

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Quo Vadis, Mathematics?

Is Mathematics Over, or Just Graduating?

Graduation

As a teenager, I attended a math-specialized class at the most prestigious school in Hungary. We had eight hours of mathematics every week, plus special sessions to prepare for national and international competitions. I even met my ex-wife at a week-long camp at Lake Balaton, where our only job was to work on IMO problems. Much of my youth went into solving competition problems.

The first two hours after my high school finals were the freest of my life. We had studied for a week, covering a long list of topics in five subjects, and when the last exam ended, I felt enormous relief: finally, it was over! Then came the emptiness. It hit me: there would be no more competitions. Its not that I particularly enjoyed competing, but it had become an organic part of my life. We took it very seriously and deeply respected those who did well. Placing in the top ten of a Hungarian national contest brought real status, never mind making the national IMO team. I never came close to that team, yet belonging to the community around it was one of the most important parts of my life.

After those two hours, a sad emptiness set in. No more competitions, no more preparing for them, no more thrill. Nobody coached me anymore, and nobody cared. For the first time in years, I had nothing to train for, and I didnt know who I was without it. It felt terrible, as if I had lost a vital part of myself. I stayed sad for days.

The Jungle

Picture an endless, unmapped wilderness: jungles, deserts, jagged mountains, rivers wide enough to swallow a village. Now imagine spending your life exploring it. That, roughly, is what doing pure mathematics is like. Let me explain the analogy.

The terrain is perilous but beautiful, something like the Darién Gap, but even harder to explore. A beginner would get lost within days, catch malaria and die in the jungle, or be eaten by wild animals. The explorers work is threefold. First, they map the terrain, which helps them find the most beautiful places quickly and explore the right areas. Second, they look for natural resources, because expeditions need funding. They hardly care about the financial rewards for their own sake, but they must find a way to pay for their travels. Third, they invent and build gadgets to help them along the way.

To make the analogy explicit: tenured mathematicians are free to explore whatever they like, but the community still judges them by their contributions. The most recognition goes to those who climb a tall mountain through great effort and hardship and sketch a rough map from the summit. What people admire most is not just the outcome (the map) but the heroic effort behind it. If the mountain is so hard that earlier explorers failed, you will be truly revered. The Langlands program can be seen as such an accomplishment. Finding a river that connects two distant regions and lets people travel between them is highly valued too.

Tools are valued as well, though they are less of a tour de force. If you create a new tool that becomes widely used, like the pseudo-inverse of a matrix or the NP-completeness of 3-SAT, you will be well known, but you wont earn the same halo as someone who conquers a seemingly useless rock that many had tried and failed to climb. In fact, nobody will believe your tool is useful until you use it to climb a rock that nobody has climbed before.

Finally, you earn some respect by finding a valuable mineral deposit: something that actually benefits ordinary people, the non-explorers. If you discover that elliptic curves can be used for cryptography, most number theorists wont care much, but they will mention it in their grant proposals. It certainly helps fund their expeditions if their financiers believe more deposits may lie nearby. Still, the explorers (mathematicians) mostly care about the beautiful views, the thrill of exploration and, above all, the veneration of their peers for conquering a hard-to-climb rock. Those rocks (conjectures) are usually not that interesting in themselves, but they serve as testing grounds for new climbing tools. Occasionally, though, a rock is shrouded in thick haze, and, even more rarely, it turns out to be a majestic mountain. From its top you can see much farther and explore far more interesting parts of the terrain. That, in the end, is why people stay: not for the maps or the money, but for the rare moment when a rock nobody cared about turns out to be a mountain, and the whole terrain opens up below you.

The Airplane

Every revolution looks like a toy at first. The Wright brothers first flight lasted twelve seconds and covered less distance than a jumbo jets wingspan. That is roughly where AI stands in mathematics today. Until this year, our planes were extremely rudimentary: they flew a few hundred meters (feet), at an altitude of a few meters (feet). They are also still obscenely expensive, and useful only in a few narrow cases. But they have improved quickly, and in the past few months they have reached the range and reliability to fly into the Darién Gap without crashing. We even landed on several rocks that nobody had climbed before (we solved a few conjectures). The planes are still very primitive, flying for less than half an hour at low speed and low altitude, and those flights cost far more than the explorers hand-crafted tools ever did.

The aviation companies were extremely proud of planting flags on some of the large rocks, never mind that explorers were busy climbing them and were close to the top. They considered it a great demonstration of progress and expected the explorers to be thrilled at how much faster exploring the interesting parts of the jungle could become. Instead, the explorers got upset: Please dont put flags on our rocks. It stops other explorers from trying to climb them. Also, we already climbed a few of them. And honestly, these rocks arent that interesting. We climb them to hone and test our tools for the real mountains, and your planes cant fly high enough to reach those. Still, could you give us some airplanes as a gift? Wed like to test them and find some use for them.

Of course, the explorers are upset. It all arrived so suddenly, and the early airplanes looked so primitive and useless. A few of them tried the planes and crashed right at the edge of the jungle, which looked ridiculous. Who would have thought that those planes could suddenly plant flags on their rocks? Laughable, right?

But the argument that airplanes cant reach the top of the real mountains (solve the hardest problems) or draw approximate maps of large areas (build theories) is a static-thinking fallacy. Todays airplanes are primitive compared to what they will be in a few months, let alone years.

Naturally, climbing rocks just to hone climbing tools will lose its point. Those tools may have been essential until today, but they wont be for long: private airplanes will be everywhere, and anyone who can afford it will charter one and land on any rock, or even a mountain.

On the other hand, planting flags on random rocks is just a technical demonstration, or PR, at this point. It will get old quickly, though the aviation companies will keep using the terrain to test and improve their planes, just as mathematicians used the rocks to hone their artisan tools.

Once airplanes get cheap, we will also discover satellites. We will map the whole terrain for mineral deposits and other valuable resources, and then extract them. It will be messy and dirty, and todays explorers will be deeply appalled at what it does to their beautiful jungle. They will long for the good old days. They will say this is no longer exploration but a commercial undertaking, large corporations wrecking the area they were so keen to explore. They will be right, but it wont change much, and large companies will find unimaginable riches there.

Rock climbing will not go out of fashion, of course. Many new, more casual explorers might even join: charter a plane to your favorite rock, buy the right tools at the tool shop (instead of honing them by hand), and climb it with friends. Less athletic people will fly in search of even more majestic mountains, or look at entire ranges from airplanes that fly far higher than any peak. The old vibe and thrill will be gone forever. Mathematics will be far more glorious, useful and mundane at the same time than it was in the heroic old days of exploration.

Quo Vadis?

Do not bet against the airplane. The skeptics who say AI cant conjecture, cant theorize, and cant explain are describing last springs models. Within months, that description will no longer hold. The evidence is clear, and those with eyes can see it. Anyone who stakes their identity on AI cant do task X is building on sand.

Many mathematicians, including my brother Balázs, embraced the changes early on. Others, like Jared Duker Lichtman and Michael R. Douglas and many others have been very outspoken, working toward the same vision over the past years. Mathematicians work passionately and objectively, and mathematics is still the most meritocratic academic field. Academia will not change quickly. Universities, journals and tenure committees will look much the same for years. But what they reward will shift fundamentally. Mathematics prided itself on being meritocratic, with deep reverence for the explorers: the people who could climb what nobody else could. That specific kind of heroism will matter less, and other skills will matter more: choosing which terrain is worth exploring, steering fleets of airplanes, and explaining what they find to the rest of the world. For many people in the system, this will feel like a demotion, and some will fight it. Other fields have been through similar upheavals and survived. What changes is what counts and who gets respected for what.

Many mathematicians will feel what I felt when I finished high school and my competing days ended: disorientation, and a sudden loss of purpose. That grief is real, and I wont pretend otherwise. But it will not last. Mathematics is about to graduate. It will grow into a full-scale industry and become the true infrastructure of all scientific and technological progress, a foundation that every other field stands on rather than a quiet corner of academia. The age of lone climbers is closing, and the age of mapping the whole continent is opening. As for me, I cant wait to see those vast mountain ranges from my supersonic airplane.

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