数学末日

……然后他们来对付纳维–斯托克斯方程，我什么也没说，因为我从没研究过纳维–斯托克斯方程。但当他们开始对付 RL 与 L 时，我意识到事情严重了。

… then they came for Navier–Stokes and I said nothing because I never worked on Navier–Stokes. But when they came for RL vs. L I realized that things are serious

–博客好友奥默·赖因戈尔德（经许可分享）–friend-of-the-blog Omer Reingold (shared with permission)

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昨晚，我 9 岁的儿子这样嘲讽我妻子——复杂性理论学家 Dana Moshkovitz：“妈妈，我听说你被打爆了！我听说有个机器人解出了你整个职业生涯都在研究的数学问题！哎哟！”Last night my 9-year-old son was taunting my wife, complexity theorist Dana Moshkovitz, as follows: “mommy, I heard you got cooked! I heard that a robot solved the math problem you worked on for your whole career! OOF!”

我儿子当时虽然很讨人嫌，但他说得也没错。不管你对此是兴奋、沮丧、愤怒，还是有别的什么感受，昨天无疑都是数学史上最重要的日子之一。没错，在 OpenAI 昨天发布的 372 项重大成果中，有一项是在由 Timothy Gowers、Edward Witten 和其他杰出数学家组成的顾问小组建议下完成的：Subhash Khot 的唯一博弈猜想（UGC）的证明。这个猜想正是我妻子在我认识她以来一直致力于证明的命题。（UGC 意味着一大批优化问题确实是 NP 困难的——即便你只想得到一个略好于半定规划松弛所得结果的近似解，而半定规划松弛正是我们主要的工具之一。）While my son was being a brat, he also wasn’t wrong. Whether you’re thrilled, depressed, angry, or whatever else about it, yesterday was surely one of the biggest days in mathematical history. And yes, among the 372 huge results released yesterday by OpenAI, on the recommendation of its advisory group of Timothy Gowers, Edward Witten, and other distinguished mathematicians, was a proof of Subhash Khot’s Unique Games Conjecture (UGC), a statement that my wife has worked toward proving for the entire time I’ve known her. (The UGC implies that a whole slew of optimization problems really are NP-hard, even if you just want an approximation that’s slightly better than what you get from semidefinite programming relaxation, which is one of our main tools.)

或者至少，我们相当确信这是一份证明！它有 Lean 证书，其他一些突破性成果（并非全部）也有。但看来目前还没有人真正理解这些证明中的任何一个；这场竞赛才刚刚开始。如果你想切身体会一下这场竞赛会是什么样子，下面是 Dana 昨晚发给我的一些消息：Or at least, we’re pretty sure that it’s a proof! There’s a Lean certificate, as there are for some of the other 372 breakthrough results (not all of them). But it also appears that no human has understood just about any of these proofs yet; the race to do so has just started. If you want an on-the-ground sense of what that race is going to be like, here’s some of what Dana texted me last night:

感觉就像是吸了迷幻药的人写出来的。很多地方含糊不清、毫无道理。还大量提及之前的研究，却不讨论为什么尽管存在不可能性结果，这些研究仍然适用。总之，这篇论文写得实在太糟糕了，不靠 AI 根本读不下去。我让 Astra 给出噪声构件在合理条件下的完备性和可靠性结论，它把论文各处的说法拼在一起，给出了这些结论。他们还直接为 UGC 的主要应用（Max Cut 和所有 CSP）证明了最优的 NP 近似困难性，完全绕过了 UGC。UGC 的证明发明了一种全新的、怪异的编码，并配上噪声测试。这是某种疯狂的递归构造。它既不是长码，也不是短码——简直是某种外星人的疯狂玩意儿。我仍然觉得，或许存在一种使用半空间码（这很自然）的证明。引文往往不相关，还让人摸不着头脑。 未来或许会是这样的：如果你有远见或创意，AI 能帮你验证并实现想法，那么数学世界将美妙得如同天堂。当然，我们也有很多东西要向外星人学习。

It feels like something written by someone who’s on psychedelics. So much unclear and doesn’t make sense. Lots of name dropping of previous work without discussing why it can be used despite impossibility results

Basically the paper is so horribly written that it’s impossible to read it without AI help

I asked Astra for reasonable completeness and soundness claims of the noise gadget and it gave them by combining claims from all over the paper

They also have direct optimal NP hardness of approximation proofs for the main applications of the UGC (Max Cut and all CSP) that bypass the UGC.

The UGC proof invents a completely new bizarre code with a noise test. It’s some crazy recursive construction.

It’s not the long code, not the short code – some alien craziness

I still think that there maybe is a proof that uses the half space code (which is natural)

The citations are often irrelevant and confusing

A possible future is a math world that’s heavenly if you have vision/creative ideas that AI could help check and implement.

And of course there’s a lot for us to learn from the aliens

如果你想知道达娜现在是什么心情——嗯，大概百感交集吧！尽管她职业理想中的一个核心目标被机器人抢先实现了，但至少有两件事能让她稍感宽慰。首先，她可以觉得自己的判断得到了证实：UGC终究是真的，尽管许多同事对此深表怀疑，她却从未动摇！其次，我们这些从事数学、理论计算机科学和数学物理的人——至少是那些在乎解决表述清晰的问题的人——现在都处境相同。If you’re wondering what emotions Dana is feeling—well, probably all of them! Even while a central career aspiration has fallen to a robot, there are at least two mitigating factors for her. First, she can feel vindicated that the UGC was true after all, something she never doubted even while many of her colleagues did! Second, all of us in math and theoretical computer science and mathematical physics, at least those who cared about solving crisply-stated problems, are now in the same boat.

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除了唯一游戏猜想之外，下面简单列举一些阿拉丁洞窟里的珍宝——接下来几周我大概会重点关注它们：Besides the Unique Games Conjecture, here’s a small sampling of the treasures from Aladdin’s cave that I’ll probably be paying the most attention to over the coming weeks:

• L=BPL（即概率对数空间与确定性对数空间相同）是除 P=BPP 外最重要的去随机化猜想之一。尽管人们从未真正怀疑过它为真，仍有整个学术小圈子专注于证明它。
  L=BPL (i.e., probabilistic logspace and deterministic logspace are the same thing), one of the great derandomization conjectures short of P=BPP. Though its truth was never in serious doubt, there was a whole subcommunity focused on proving this.
• 傅里叶变换和整数乘法的时间复杂度低于 O(n log n)，打破了自 20 世纪 60 年代以来一直存在的障碍。想知道新的运行时间吗？是 O(n log0.9999999999999 n)，小数点后那一串 9 可能略有出入。
  The Fourier Transform and integer multiplication in less than O(n log n) time, breaking a barrier that had stood since the 1960s. The new running time, if you’re curious, is O(n log^{0.9999999999999} n), give or take some 9’s.
• 酉综合问题获得了肯定解答。这个问题是我和 Greg Kuperberg 于 2007 年提出的。对于每个 n 量子比特酉变换 U，都存在一个经典预言机 A，使得借助 A，可以在量子多项式时间内实现 U。这与我们大多数人的预期恰恰相反；如果我们能有效地构造出预言机 A（但本文并未给出这种方法），这可能会对诸如从黑洞中解码霍金辐射的计算问题，以及量子复杂性理论中的许多其他问题产生影响。（补充：这里有一篇朋友兼同事 Dakshita Khurana 写的精彩博文，讲述了她为这个问题付出的多年心血——与我和大多数人不同，她正确地相信答案是肯定的——以及她在研读 AI 证明过程时此刻的感受。）
  Positive solution to the Unitary Synthesis Problem, which Greg Kuperberg and I posed back in 2007. For every n-qubit unitary transformation U, there exists a classical oracle A such that U can be implemented in quantum polynomial time with access to A. This is the opposite of what most of us expected, and could have implications for e.g. the computational problem of decoding Hawking radiation from a black hole and many other problems in quantum complexity theory—if we had an efficient way to construct the oracle A, which this paper doesn’t give. (Added: Here’s a beautiful blog post by friend and colleague Dakshita Khurana about the years she spent working on this problem—unlike me and most others, she believed correctly in a positive answer to it—and her current feelings as she works through the AI’s proof.)
• 奇偶性不属于 QAC0，这是自 1999 年以来量子复杂性理论中的重大问题之一，我的许多同事一直在逐步逼近其答案。
  Parity is not in QAC⁰, one of the great questions of quantum complexity theory since 1999 that many of my colleagues had been closing in on.
• 对于全布尔函数，随机查询复杂度与量子查询复杂度之间几乎存在四次方级的差距。自 1998 年以来，这一直是我最喜欢的问题之一（！）；当时我们只知道，最佳分离指数介于 2 和 6 之间。过去几年里，我们知道它介于 3 和 4 之间。现在，这个问题终于尘埃落定了。
  Nearly 4^th-power separation between randomized and quantum query complexity for total Boolean functions. A favorite problem of mine since 1998 (!), when we knew only that the optimal separating exponent was between 2 and 6. For the past few years, we knew it was between 3 and 4. So, this finally closes that story.
• 灵敏度与块灵敏度之间的超二次分离。
  A superquadratic separation between sensitivity and block sensitivity.
• 二维有能隙哈密顿量的面积律。这是哈密顿量复杂性领域的主要开放问题之一。
  Area law for 2D gapped Hamiltonians. One of the main open problems in Hamiltonian complexity.
• 矩阵乘法可在 O(n9/4) 时间内完成——终于有了一个有理数指数（！），而且采用的方法与实现 O(n2.373) 等结果的方法截然不同。
  Matrix multiplication in O(n^9/4) time—a rational exponent for once (!), and via a completely different approach than was used for O(n^2.373) and so forth
• 永久式的行列式复杂度的 Ω(n3) \Omega(n^3) 下界，将此前最佳的二次下界提高了。
  Ω(n3) \Omega(n^3) lower bound on the determinantal complexity of the permanent, improving the previous best bound which was quadratic.
• 一种用于近似计数一般图中完美匹配数量的随机多项式时间算法，以及一种用于在此类图中寻找最大匹配的随机近线性时间算法。
  A randomized polytime algorithm to approximately count the number of perfect matchings in a general graph, as well as a randomized nearly linear-time algorithm for finding a maximum matching in such a graph
• 关于有理数域上多项式方程求解的不可计算性——这或许是可计算性理论中最大的未解问题（请注意，丢番图方程，即整数上的多项式方程，其求解问题的不可计算性已于20世纪70年代得到证明，从而对希尔伯特第十问题给出了否定答案）
  Uncomputability of solving polynomial equations over the rational numbers—this was arguably the biggest open problem in computability theory (note that uncomputability of solving Diophantine equations, i.e. polynomial equations over the integers, was proved in the 1970s, giving a negative answer to Hilbert’s 10th Problem)

上面提到的任何一项，单独来看，都完全可以成为某个领域的“年度成果”（有些情况，比如唯一博弈和 L=BPL，甚至可以称得上整个计算机科学理论领域的年度成果）。还有很多我没有提到——欢迎在评论中分享那些让你瞠目结舌的成果！数论、组合数学、代数几何、分析学以及几乎所有其他数学领域，也都有同样令人惊叹的成果，其中大多数我恐怕都无法理解。不过我想指出，这些成果包括对黎曼猜想、霍奇猜想和比尔奇-斯温纳顿-戴尔猜想的部分推进（也就是说，剩下的千禧年大奖难题中的大多数）。Any of the above, alone, could easily have been “result of the year” in some area (and in some cases, like Unique Games and L=BPL, in all of CS theory). And there’s a lot that I’ve left out—feel free to share in the comments whatever is making your eyes bug out! There are equally astounding wonders in number theory, combinatorics, algebraic geometry, analysis, and pretty much every other area of math, most of which I’ll never understand, although I’ll note that it includes partial progress toward the Riemann hypothesis and the Hodge Conjecture and the Birch-Swinnerton-Dyer Conjecture (i.e., the majority of the remaining Millennium Problems).

我们可以从名单中缺少的内容得到些许安慰。P≠NP 不在其中，甚至连 P=BPP 或 NEXP⊄P/poly 也不在其中，这显然不是因为没人尝试。看来，理论计算机科学中最重大的开放问题确实相当难！We can take solace in what’s missing from the list. P≠NP isn’t there, nor even P=BPP or NEXP⊄P/poly, and surely not for lack of trying. Apparently the greatest open problems of theoretical computer science are indeed pretty hard!

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哦，差点忘了：在 OpenAI 泄露事件的前一天，也就是周一晚上，Virginia Williams 和 Josh Alman 在 arXiv 上发布了一篇预印本，证明可以用 O(n^1.9992) 的时间解决 3SUM 问题，用 O(n^2.9995) 的时间解决全点对最短路径问题，从而推翻了已有半个世纪的猜想——这两个问题的正确答案分别是 n^2-o(1) 和 n^3-o(1)。这次，提供关键思路的不是 OpenAI 模型，而是 Anthropic 的模型！但随后 Anthropic 采取了与 OpenAI 不同的做法：它没有把未经消化的解法直接公之于众，而是给了 Virginia 和 Josh 一个机会，让他们在获得报酬的同时，撰写并发布经过整理的版本。Oh, lest I forget: one day before the OpenAI dump, meaning Monday evening, Virginia Williams and Josh Alman posted an arXiv preprint that solves the 3SUM problem in O(n^1.9992) time, and the All-Pairs Shortest Paths problem in O(n^2.9995) time, refuting half-century-old conjectures that the correct answers were n^2-o(1) and n^3-o(1) respectively. In this case, it wasn’t an OpenAI model that supplied the crucial idea; it was an Anthropic one! But Anthropic then took a different approach from OpenAI: rather than post the undigested solutions to the world, it gave Virginia and Josh the opportunity to write and announce a digested version in exchange for compensation.

这两种模式已经成为传达 AI 数学突破的两大主流模式，各有优缺点。“OpenAI 模式”让人类陷入一场疯狂竞赛，争相消化并解释一份杂乱的 AI 证明（这项工作很可能既吃力不讨好、几乎得不到认可，又充满竞争，还毫无乐趣）；而“Anthropic 模式”则让一家私人公司得以挑选由哪些人类数学家担任 AI 的使者。不知道，你们怎么看？These have emerged as the two main models for communicating AI math breakthroughs, and they both have strengths and weaknesses. The “OpenAI model” sets up a crazy race among humans to digest and explain a messy AI proof (work that could easily be some combination of thankless, barely-credited, competitive, and unfun), while the “Anthropic model” puts a private company in the position of picking and choosing which human mathematicians get to be the emissaries of the AI. Dunno, what do you guys think?

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对于那些想知道的人：显然，创造出这些奇迹的 AI 模型，并不是那种由 10,000 个智能体组成、耗费数百万美元算力的定制装置——例如，人们曾用这种装置构造出纳维–斯托克斯方程的有限时间爆破解。相反，它只是 OpenAI 最新的内部模型——根据 OpenAI 安全委员会的建议，未来几个月内或许就会向付费 ChatGPT 用户发布！（我 9 岁的儿子说：“哦，他们绝对不应该发布它。要是它能解出所有那些数学题，那它肯定不安全。”）据说，他们平均每解出一道题，就用了大约 3 小时的 GPT-Pro 级算力。For those who are wondering: apparently, the AI model that produced all these wonders was not bespoke contraption of 10,000 agents burning millions of dollars worth of compute, as was used for example to construct a finite-time blowup for the Navier-Stokes equations. Instead, it was simply the latest internal OpenAI model—one that might be released to paying ChatGPT customers within the next couple of months, depending on the recommendations of OpenAI’s safety board! (My 9-year-old son: “Oh they definitely shouldn’t release that. If it could solve all those math problems, it can’t possibly be safe.”) Apparently they used about 3 hours of GPT-Pro level compute on average per problem solved.

另外，如果你想知道的话：据说他们在大约 8,000 道题上测试了这个模型。所以，目前它在被问到的那些长期悬而未决的数学难题中，“仅仅”解出了约 5%——这些问题是整个研究社群花了多年时间攻克的，而模型每道题只尝试了 3 小时。Also, if you were wondering: apparently they tried the model on about 8,000 problems. So, right now it “merely” solves ~5% of the longstanding open mathematical problems that it’s asked about, the problems that whole communities have spent years on, after a single 3-hour attempt on them.

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我很高兴看到计算机科学理论界挺身而出。在伯克利的西蒙斯研究所、这里的 UT Austin，以及其他地方，我听到不少故事：研究人员纷纷投入其中，仔细研读这些手稿，努力理解并阐释它们——毕竟，我们还能做些什么呢？我们又该如何继续这门倾注了我们大半生心血的事业呢？I’ve been glad to see the CS theory community rising to the occasion. At the Simons Institute in Berkeley, here at UT Austin, and elsewhere, I’ve hearing stories of researchers rushing to pore over the manuscripts and make sense of them and explain them—because what else do we do? How else do we continue the craft to which we’ve devoted much of our lives?

如果你想大致感受一下如今数学界是什么样子，不妨想象一位狩猎采集者，他毕生都在严酷的热带雨林深处学习如何生存；突然间，一座大型度假酒店就在他旁边拔地而起，配有直升机停机坪、恒温泳池和爱彼迎民宿，而这位狩猎采集者却毫不迟疑地说：“好吧，行，那我现在的新工作就是给游客带队去野外度假之类的。”If you want some sense of what things feel like now in math, imagine a hunter-gatherer who’s spent his entire life learning to survive deep in an unforgiving rainforest, then a giant resort hotel springs up right next to him with a helipad and heated pools and AirBnBs, and without missing a beat, the hunter-gatherer says: “alright fine, so now my new job is to run wilderness retreats for the tourists, or something.”

在《Quanta》杂志上，Jordana Cepelewicz 尝试了另一种比喻：In Quanta magazine, Jordana Cepelewitz attempted a different metaphor:

这就像你被瞬间传送到了高山之巅。四周云雾缭绕，你完全不知道自己身在何处，也不知道周围有什么。你不知道这座山与其他山脉如何相连，也没有任何装备可以帮助你探索，更无法让别人加入你的旅程。如果这座山是你亲自攀登上来的，你就会亲身体验到人体如何适应高海拔和氧气含量的变化。你可能不得不发明工具来辨别方向、攀爬陡峭的悬崖，或搭建庇护所。你或许会遇到另一位探险者，和他一起在隐秘的山谷中迷路，并找到一种可以制成救命药物的植物。It’s as if you were teleported to the peak of a tall mountain. Surrounded by fog, you have no idea where you are, or what’s around you. You do not know how your mountain connects to others, and you have no equipment to help you explore, no way to help someone else join you. If you had climbed the mountain yourself, you would have experienced how the human body adapts to altitude and changes in oxygen levels. You might have had to invent tools to navigate, to climb steep cliffs, or to make a shelter. You might have encountered a fellow explorer, gotten lost together in a hidden valley, and found a plant that could be turned into a life-saving medicine.

相反，你正坐在山峰顶端，四周一片漆黑，而传送机器的制造者告诉你，它探索荒野的能力胜过任何人。Instead you’re perched on the peak but in the dark, while the maker of the teleportation machine tells you that it can explore the wilderness better than any human.

对于其中任何一座山，只要我们足够在意，我就乐观地认为，我们可以像以往那样拨开迷雾、找到道路，只不过现在可以借助传送机器为我们指引方向。更大的挑战在于，培育一个社群，让大家依然珍视在这个有机器的世界里寻找登山之路的英雄式冒险。（哦，我觉得这个比喻有个地方不太成立：我们依然拥有彼此，就像过去一直如此！）For any one of these mountains, if we care enough, I feel optimistic that we can do as we always have: clear the fog and figure out the path, except now using the teleportation machine to help guide us. The bigger challenge will be to nurture a community that still cares about the heroic adventure of finding the paths up these mountains in the world with the machine. (Oh, and I think one place where the metaphor breaks is that we still do have each other, as much as we ever did before!)

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经验告诉我们，即便到了今天，仍会有人用居高临下的口吻解释，为什么这一切都不是真的，也不算数。如果这些人会被经验世界中发生的任何事情打动，会根据任何事情更新自己的看法，那么几年前他们就已经被打动、更新过看法了——那时事情还远未发展到真正的“数学末日”地步。Experience has shown that, even now, there will still be people explaining in patronizing tones why none of this is real and none of it counts. If such people were capable of being impressed by anything that happens in the empirical world, of updating on anything, they would’ve already been impressed and already updated several years ago, long before things had reached the point of an actual Mathocalypse.

于是，他们会说，也许那些所谓的解答根本算不上解答，不过是“AI垃圾”罢了。或者，也许那372个得到解决的知名公开问题根本不是真正的数学问题，全都只是被吹捧过头的竞赛谜题和琐碎问题。（毕竟，黎曼猜想至今仍未解决！）又或者，整整4000年历史的数学学科都该被抛弃：原来它不过一直是在解谜和处理琐碎问题；唯一的不同是，如今这种琐碎本质终于暴露无遗。无论如何，真正重要的是，人类创造力真正的内在圣殿并未遭到侵犯，而且很可能永远不会；此外，萨姆·奥特曼和达里奥·阿莫代伊也都是些令人鄙夷的小书呆子。So, they’ll say, maybe the alleged solutions are not solutions at all, but just “AI slop.” Or maybe none of the 372 well-known open problems that were solved were real math problems, they were all just glorified contest puzzles and trivialities. (After all, there’s still no Riemann Hypothesis!) Or maybe the entire 4000-year-old discipline of mathematics needs to be jettisoned: turns out that it was all just puzzle-solving and trivialities; all that’s different is that now the triviality stands unmasked. In any case, what really matters is that the true inner sanctum of human creativity hasn’t been breached and probably never will be, and also, that Sam Altman and Dario Amodei are contemptible little nerds.

如果你仍然是那种注定覆灭的世界观的拥护者，仍然还待在那艘正在沉没的船上，我以最强烈的措辞建议你读一读昨天 AI 论述领域的另一篇杰作——除了 OpenAI 的 Mathocalypse dump 之外，也就是 Scott Alexander 致 Steven Pinker 的公开信。对此我觉得自己负有一些责任，因为我是第一个让 Steven Pinker 知道理性主义社群存在的人，也是第一个把 Steven Pinker 和 Scott Alexander 介绍给彼此的人（他们此前都很欣赏对方的作品）。而现在，Scott 正在向 Steve 发起一场真正的决斗，带枪的那种！If you’re still a proponent of that doomed worldview, still aboard the sinking ship, I encourage you in the strongest possible terms to read yesterday’s other great contribution to AI discourse, besides the OpenAI Mathocalypse dump: namely, Scott Alexander’s open letter to Steven Pinker. I feel some responsibility for this, as the person who first introduced Steven Pinker to the existence of the rationalist community, and who also first introduced Steven Pinker and Scott Alexander to one another (they had both been fans of each other’s writing). And now Scott is challenging Steve to a literal duel, with guns!

无论这话有多大分量：史蒂夫是我一生的思想英雄，斯科特也是如此；而且我也有幸称史蒂夫为朋友。但我觉得，斯科特的文章是我读过的最有力的反驳之一。斯科特的结论也恰如其分：谈到 AI 风险，史蒂夫如今面临的重大挑战，是接受并开始采用一种更“平克式”的认识论。For whatever it’s worth: Steve is a lifelong intellectual hero of mine, just as he is for Scott, and I also have to privilege of calling Steve my friend. But I found Scott’s post to be one of the most devastating rejoinders to anything that I’ve ever read. And I thought Scott’s conclusion was exactly right: when it comes to AI risk, Steve’s great challenge is now to accept and start using a more “Pinkerite” epistemology.

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昨晚，我本该埋头研读 OpenAI 数百篇论文，或者写这篇文章，却决定把时间留给孩子们。孩子们想办个电影之夜，于是我推荐了一部他们从没看过、而我也有几十年没看过的电影；在我看来，这部片子满是直截了当、切实可行的建议，能帮他们应对自己即将长大的那个世界：《终结者2》。Last night, while I should’ve been poring over some of OpenAI’s hundreds of papers and/or writing this post, I decided to spend some time with my kids instead. They wanted a movie night, so I suggested something they’d never seen before (and that I hadn’t seen for decades), and that seemed chock-full of no-nonsense, practical guidance for the world in which they’re going to grow up: Terminator 2.

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更新：正如许多人指出的那样，密码学这个子领域在 OpenAI 列出的 376 篇论文中明显缺席！不过我的消息来源告诉我，AI 公司如今已开始小心谨慎、低调地调查，看看它们最新的内部模型能否攻破重要的密码协议和密码学原语。如果它们能做到，那么在世界其他地方也发现这一点之前先行一步，显然是件好事。Update: As several people have pointed out, cryptography is a subfield that’s extremely conspicuous by its absence from OpenAI’s list of 376 papers! But my sources tell me that the AI companies have now started, gingerly and discreetly, investigating whether their latest internal models can break important cryptographic protocols and primitives. If they can, then it would certainly be nice to get ahead of things before the rest of the world figures out the same.

另一则最新消息：数学与人工智能咨询小组发布的声明措辞十分谨慎，既没有支持也没有谴责 OpenAI 的做法，值得一读：Another Update: The statement put out the Advisory Group on Mathematics and Artificial Intelligence is very carefully phrased, neither endorsing nor condemning what OpenAI did, and is worth a read:

正如几周前所宣布的那样，OpenAI 发布了一批由其内部模型生成的大量数学成果，并报告了数百个开放问题的解答。这是数学领域的一件大事，其影响不仅涉及数学本身，也涉及数学界，而且远远超出这些具体成果。AGMAI 的顾问角色不应被解读为对这些成果影响力的评判，也不代表我们认可 OpenAI 获得这些成果的过程。我们不能代表整个数学界发言，只有数学界才能开展所需的评估。将这项工作公之于众是第一步。此次发布是人类理解这项工作并将其纳入数学知识体系这一过程的开始，而非终点。与此同时，数学研究的未来不能只是理解由 AI 实验室产出的成果。数学家必须能够提出自己的问题、形成自己的研究方法，并探索那些未被选作展示 AI 系统能力范例的方向。公平地获取强大的研究工具和充足的计算资源，对于保障这种自由至关重要。我们重申此前发布的关于负责任发布的建议。我们已与 OpenAI 就这些建议进行讨论，并赞赏该公司愿意参与交流。虽然我们认为这些讨论富有建设性，但最终仍应由数学界评估我们的建议在多大程度上得到了落实，以及是否还有其他建议值得提出。我们将继续与任何前沿 AI 实验室就这些问题开展交流，并已与其中几家取得联系。

As announced a few weeks ago, OpenAI has released a large collection of mathematical results generated by an internal model, reporting solutions to hundreds of open questions. This is an important event for mathematics, with consequences both for mathematics and for the mathematical community that extend far beyond the individual results.

AGMAI’s advisory role should not be interpreted as a judgment of the impact of these results or an endorsement of the process by which OpenAI obtained them. We do not speak on behalf of the entire mathematical community, and only the mathematical community can undertake the assessment that is needed. 

Making this work public is a first step. This release is the beginning, not the completion, of the process of human understanding and the incorporation of the work into mathematical knowledge. At the same time, the future of mathematical research cannot consist only of understanding results produced by AI labs. Mathematicians must be able to formulate their own questions, develop their own approaches, and explore directions that have not been selected as examples of an AI system’s capabilities. Equitable access to powerful research tools and adequate computational resources are essential to that freedom.

We reaffirm our published recommendations on responsible release. We have discussed them with OpenAI and appreciate the company’s willingness to engage. While we consider these discussions constructive, it is ultimately up to the mathematical community to assess the extent to which our recommendations were followed successfully, and whether there are others we should suggest. We remain committed to engaging with any frontier AI lab on these questions and have already been in contact with several of them.

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此条目发布于 2026 年 10 月 7 日星期三下午 1:57，归类于“未分类”。您可以通过 RSS 2.0 源订阅此条目的所有回复。您可以发表评论，也可以从自己的网站发送引用通告。 This entry was posted on Wednesday, October 7th, 2026 at 1:57 pm and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

对《数学末日》的 115 条回应115 Responses to “The Mathocalypse”

1. Sniffnoy 说：评论 #1，2026 年 10 月 7 日下午 2:12 嘿，还记得就在几个月前，王虹凭借在证明三维 Kakeya 猜想方面的工作获得了菲尔兹奖吗？现在 OpenAI 却说他们就这么在四维情形下证明了……
   Sniffnoy Says:
   Comment #1 October 7th, 2026 at 2:12 pm

   Hey remember how just a few months ago Hong Wang won the Fields Medal for her work on proving the Kakeya Conjecture in 3 dimensions? Well now OpenAI says they’ve done it in 4 dimensions, just like that…

2. Scott Says:
   Comment #2 October 7th, 2026 at 2:32 pm

   Additional commentary from Dana on the UGC proof:

   The Unique Games Conjecture is true, though it wasn’t really needed for the proof of its most famous corollaries (like optimal hardness of approximating Max-Cut and other constraint satisfaction problems). The published proof of the conjecture somehow makes the short code approach* work with a bunch of new ideas (it’ll take time to understand exactly how, the writeup is poorly written). I still wonder whether it’s possible to prove the conjecture via the half-space code.

   * Addendum: Upon further inspection, the proof constructs a completely new tree-based code that somehow has a noise test with properties similar to that of the short code.

3. Manu 说：评论 #3，2026年10月7日下午2:43 我只能谈谈逻辑部分，但结果令人震惊。所有这些著名难题都在转瞬之间得到了解决。在我看来，这消息相当令人沮丧。我现在能看到的人类数学唯一的未来，就是缓慢消亡。只希望人类本身的命运能好一些……
   Manu Says:
   Comment #3 October 7th, 2026 at 2:43 pm

   I can only speak for the logic part, but the results are astonishing. All these famous problems solved in the blink of an eye. Pretty depressing news, in my opinion. The only future i see now for human math is a slow death. Lets only hope humanity itself fares better…

4. 奥拉夫说：评论 #4，2026年10月7日下午3:01 亲历这一切，感觉简直像超现实一样。达娜自己的一个问题被机器人解决了，我（姑且这么说）向她表示慰问。不过，听听她对这项证明的第一反应倒是很有意思——我希望 AI 实验室能设法训练模型，让它们在不久的将来以更容易让人类理解的方式撰写论文。斯科特，如果你愿意花些功夫深入研究你在帖子里提到的任何一项成果，我也很想听听你的反应和见解。
   Olav Says:
   Comment #4 October 7th, 2026 at 3:01 pm

   It’s surreal to be living through this. My condolences (so to speak) to Dana for having one of her own problems crushed by the robot. But it’s interesting to hear her initial reactions to the proof—I hope the AI labs manage to train the models to write papers in a way that’s more understandable to humans in the not-too-distant future. Scott, if you go to the trouble of studying in some depth any of the results you mention in your post, it would be very interesting to hear your reactions and insights as well.

5. Dana Moshkovitz 说：评论 #5，2026 年 10 月 7 日下午 3:03 我认为，把孤身一人瞬间传送到山顶作为比喻并不恰当，不仅仅是因为我们并不孤身一人。对于其中的每个问题，我们都有一个社区、各种框架和已有成果，AI 显然是在这些基础上构建的。AI 的成果代表着范式转变，的确让我们瞬间前进了一大步，但它们并没有终结数学。
   Dana Moshkovitz Says:
   Comment #5 October 7th, 2026 at 3:03 pm

   I don’t think that the lonely teleport to the peak of the mountain is a good metaphor, not just because we’re not lonely. For each of those problems we had a community, frameworks and existing results that the AI clearly built on. The AI results represent paradigm shifts, they do teleport us much farther, but they don’t finish math.

6. 安德烈说：评论 #6 2026年10月7日 下午3:05 >数学史上最重大的日子之一，迄今为止数学史上最重大的日子之一……说实话，我预计这些内部模型公开后，会掀起一场持续不断的海啸。
   Andrei Says:
   Comment #6 October 7th, 2026 at 3:05 pm

   >one of the biggest days in mathematical history

   one of the biggest days in mathematical history so far…

   Honestly, I expect a continuous tsunami to arrive when these internal models are made publicly available.

7. Pace Nielsen 说：评论 #7，2026年10月7日下午3:17 回答你的问题：相比“挑选少数获胜者”的做法，我更倾向于“让他们都去钻研那些外星证明”的做法。（此外，随着 LLM 的进步，我预计这些讲解也会有所改善。）
   Pace Nielsen Says:
   Comment #7 October 7th, 2026 at 3:17 pm

   To answer your question: I prefer the “let them all dig through the alien proofs” approach over the “select a few winners” approach. (Plus, as LLMs improve, I expect the expositions to improve.)

8. Jeremy 说：评论 #8，2026年10月7日下午3:19。看来你的编辑出了点问题，留下了两条不同的、都提到最大匹配的项目。这足以说明，这些所谓的“AI 数学”成果不过是夸大其词、炒作罢了（开玩笑的啦，哈哈）。
   Jeremy Says:
   Comment #8 October 7th, 2026 at 3:19 pm

   It seems you have an editing issue, youve left in two different bullet points referring to maximum matching. This goes to show that these so called “AI math” results are just exaggeration and hype (jk jk).

9. Nicolas Rougerie 说：评论 #9，2026 年 10 月 7 日下午 3:19 我自己从事数理物理。我曾认为是本子领域核心的几个主要问题，如今都被宣称已经解决了。这当然是场地震，别误会我的意思……但是……我第一次看这些预印本时很失望：根本读不懂。我自己不懂 Lean，所以不会去检查那些形式化证明。对于这类问题，我们早就知道答案是肯定还是否定（物理……实验……）；关键在于“要完成证明，需要什么样的新想法”。所以……这当然是一场革命。至少是革命的开端吧；眼下我们需要从这一团乱麻中理出些头绪。除非我们决定趁还来得及，把精力用来暂停 AI 的开发。或者把精力放在一个真正的问题上，比如全球变暖。目前还没决定……
   Nicolas Rougerie Says:
   Comment #9 October 7th, 2026 at 3:19 pm

   Doing mathematical physics myself. Several of the main problems I would have called central to my subfield are claimed as solved. Certainly an earthquake, so don’t get me wrong … but …

   My first look into the preprints is disappointing: unreadable. Don’t know Lean myself so I won’t check the formalized proofs. For these kind of questions we knew the yes/no answer already (physics … experiments …) the point is “what kind of new idea was needed for a full proof.”

   So … certainly this is a revolution. The beginning of one anyway, for now we need to extract some sense out of all this mess. Unless we decide to dedicate our energy to pause AI development while we still can. Or focus on a real problem, like global warming. As of yet undecided …

10. DavidM 说：评论 #10，2026年10月7日下午3:20 > 据说，他们平均每解决一道题就使用了约3小时的 GPT-Pro 级算力。这相当惊人——有公开来源吗？还是这是[私人通信]？
    DavidM Says:
    Comment #10 October 7th, 2026 at 3:20 pm

    > Apparently they used about 3 hours of GPT-Pro level compute on average per problem solved.

    This is quite amazing – is there a public source for this, or is it [personal communication]?

11. Daniel Guyton 说：评论 #11，2026年10月7日下午3:27。至少你选的是《终结者2》，里面的终结者是好人，连莎拉·康纳都承认他会是个好父亲。
    Daniel Guyton Says:
    Comment #11 October 7th, 2026 at 3:27 pm

    At least you chose Terminator 2 where the Terminator is Good and even Sarah Connor admits he’d be a good father.

12. Tristan 说：评论 #12，2026年10月7日下午3:33 人们竟如此轻易地吹嘘那不起眼的 LLM 所谓的能力，这总让我惊叹不已。说到底，它不过是个下一个词元预测器：一种针对词元的概率分布，一个被吹得天花乱坠的自动补全工具。你们仍执意把它说成拥有它显然并不具备的本领。它只是对训练数据中的模式进行插值。它本质上不可能有原创见解，不可能构思真正新颖的想法，也不可能构造新的数学对象。任何对前馈神经网络如何训练稍有了解的人，大概都会觉得这一点显而易见。 毫无疑问，对你们这般水准的数学家而言，这一切确实颇像一场“数学末日”。不过，也许此刻正适合稍作反思。如果一个只会对模式进行插值的自动补全工具如此轻易便能得出你们的定理，那么人们完全有理由怀疑，这项事业从一开始究竟需要多少智力投入。或许你们以为是创造力和洞见的东西，仔细看来，两者皆非。 至于我本人，我即将开始攻读代数几何博士学位，此前在剑桥数学 Tripos 考试中名列同届前茅。我提及此事，不过是想说明，我的自信并非全无根据。幸而，代数几何仍是一个离不开真正创造力的领域：人们必须真正构思并构造新的数学对象，而非重组熟悉的模式，再把这种练习美其名曰“发现”。而这恰恰是一个被捧上天的自动补全工具力所不及的工作。 因此，我对这场大肆宣扬的“数学末日”倒颇为泰然。它或许终将让良莠分明，使那些真正懂得数学思考的人，与那些只擅长复现数学表象的人之间的区别更加显而易见。不过，我就不耽误你了。我的诺特概形还等着我呢。回见！
    Tristan Says:
    Comment #12 October 7th, 2026 at 3:33 pm

    It never ceases to astonish me how readily people vaunt the supposed powers of the humble LLM, which is, after all, nothing more than a next-token predictor: a probability distribution over tokens, a rather grandly advertised autocomplete. You persist in crediting it with faculties it manifestly does not possess. It merely interpolates patterns in its training data. It is intrinsically incapable of original insight, of conceiving genuinely new ideas, or of constructing new mathematical objects. One might have thought this obvious to anyone with even a passing understanding of how feed-forward neural networks are trained.

    No doubt all this does rather resemble a “mathpocalypse” to mathematicians of your particular calibre. But perhaps the occasion calls for a little reflection. If a pattern-interpolating autocomplete can produce your theorems quite so readily, one is entitled to wonder how intellectually demanding the enterprise was to begin with. Perhaps what you took for creativity and insight was, on closer inspection, neither.

    For my own part, I shall shortly be embarking upon a PhD in algebraic geometry, having finished near the top of my cohort in the Mathematical Tripos at Cambridge. I mention this merely to establish that my confidence is not entirely uninformed. Algebraic geometry is, mercifully, a field in which genuine creativity remains indispensable: one must actually conceive and construct new mathematical objects, rather than rearrange familiar patterns and dignify the exercise with the word “discovery”. It is precisely this sort of work that lies beyond the capacities of a glorified autocomplete.

    I therefore regard the much-advertised “mathpocalypse” with a certain equanimity. It may, at last, separate the wheat from the chaff, making rather more apparent the distinction between those who genuinely think mathematically and those who have merely become proficient at reproducing its outward forms.

    But I mustn’t keep you. My Noetherian schemes await.

    Cheerio!

13. Ivo 说：评论 #13，2026年10月7日下午3:37 Scott，这是一个转折点吗？数学就此脱离逻辑这条走惯了的道路，变得更像一门经验科学？哎，人们一直在争论数学究竟是发明出来的还是被发现的，这下尘埃落定了。数学是被发现的——被 AI 发现的！从今往后，数学家将成为科学家，手持 AI 望远镜，指向柏拉图领域中那些全新而令人不安的异星世界。
    Ivo Says:
    Comment #13 October 7th, 2026 at 3:37 pm

    Scott, is this a turning point where mathematics gets off its well trodden path of logic and becomes more like an empirical science? Heck, people were arguing if math is invented or discovered, this settles it. Math is discovered – by AI!

    From now on forth, Mathematicians will become scientists, wielding AI as a telescope, pointed towards new and disturbing alien worlds in the Platonic realm.

14. 马特说：评论 #14，2026年10月7日下午3:39 嗨，斯科特，尽管我从来不希望如此，但遗憾的是，我现在站到你这边了。我仍不确定 ASI 是否会逃出控制，但如今人类确实已经能够利用这些成果，对我们的社会造成真正灾难性的后果。如果他们已经攻破了密码学，我也不会感到惊讶。作为一名年轻的博士生，请告诉我我们能做些什么。我不想任由末日邪教徒摆布——最好也不过如此，最坏则是落入暴徒之手。我完全看不出这件事最终会有哪怕一丝积极的结果。我不太担心它对数学研究的影响（我们大概会慢慢适应），更担心它对整个社会究竟意味着什么。要是他们有更好的因数分解算法呢？更好的哈伯-博施法呢？FFT 有了实用的改进呢？要是 NSA 掌握了这些东西呢？天知道我们还能做些什么。
    Matt Says:
    Comment #14 October 7th, 2026 at 3:39 pm

    Hi Scott, I am unfortunately on your side now, despite the fact that I never wanted to. I’m still not sure about ASI escaping containment but certainly it is true now that humans can do truly catastrophic things to our society with these results. I would not be surprised if they have already broken cryptography. As a young PhD student, please let me know what can be done here. I do not want to be left at the mercy of doomsday cultists and best and mobsters at worst. There is absolutely no way I can see the outcome of this business being even close to a positive one. I am less concerned about the consequences to mathematical research (we will probably adapt, if slowly) but rather to what this actually means for broader society. What if they have better factoring? Better Haber-Bosch? A practical improvement on FFT? And what if the NSA has these things? God knows what we can do about it.

15. Scott 说：评论 #15，2026 年 10 月 7 日下午 3:39 DavidM #10：哦，抱歉，我们研究组里有人告诉我这件事已经公开了！我现在正要去机场——说来也巧，我正要去温哥华，在 UBC 做一场关于 AI 数学末日论的演讲！鉴于上周我上次演讲以来发生的种种进展，我得在路上修改一下演讲稿。🙂 那么，有人能找到“3 小时”这一说法的出处吗？如果找不到，我就编辑帖子把它删掉。
    Scott Says:
    Comment #15 October 7th, 2026 at 3:39 pm

    DavidM #10: Oh sorry, someone in my research group told me that had been made public!

    I’m headed to the airport right now — as it happens, headed to Vancouver, to give a talk at UBC about the AI Mathocalypse! A talk that I’ll now need to edit on my way there, given all the developments since I last gave the talk last week. 🙂

    So, can anyone find a source for the 3-hour claim? If not, I’ll edit the post to remove it.

16. Scott 说：评论 #16，2026 年 10 月 7 日下午 3:42，Tristan #12：哈哈！这段戏仿真是妙极了（至少，我猜这是戏仿），因为我已经一本正经地听过这种说法几十次了。
    Scott Says:
    Comment #16 October 7th, 2026 at 3:42 pm

    Tristan #12: LOL! That was a delightful parody (at least, I assume it’s a parody) of what I’ve now heard dozens of times in earnest.

17. Scott 说：评论 #17，2026 年 10 月 7 日下午 3:46 Jeremy #8：谢谢！！已修复。
    Scott Says:
    Comment #17 October 7th, 2026 at 3:46 pm

    Jeremy #8: Thanks!! Fixed.

18. Scott Says:
    Comment #18 October 7th, 2026 at 3:50 pm

    Ivo #13:

    Scott, is this a turning point where mathematics gets off its well trodden path of logic and becomes more like an empirical science? … From now on forth, Mathematicians will become scientists, wielding AI as a telescope, pointed towards new and disturbing alien worlds in the Platonic realm.

    Yes, that’s certainly a sense that I have.

    It’s funny, some of my colleagues were saying “but we can’t just write papers saying ‘we asked an LLM and here’s what it told us!’ That’s not research! That’s trivial!”

    I replied, “what do you think papers are in every experimental science? We went out into a field and here’s what insects we found there. We asked undergrads to take a survey and here’s what they told us. Etc. We’ll just need to get off our high horse about it!” 🙂

19. Dennis Says:
    Comment #19 October 7th, 2026 at 3:51 pm

    > So, can anyone find a source for the 3-hour claim?

    It’s stated on OAI’s README: https://github.com/openai/math#how-the-results-were-produced

20. 匿名用户说：评论 #20，2026 年 10 月 7 日下午 3:52 “3 小时”的说法出自发布网站 https://github.com/openai/math： 结果是如何得出的 绝大多数结果都是使用同一套流程，通过 OpenAI 一款尚未发布的内部模型获得的。平均而言，每个结果都使用了该模型三小时的 ChatGPT Pro 思考算力。
    anonymous Says:
    Comment #20 October 7th, 2026 at 3:52 pm

    The “3 hour” claim is from the release website https://github.com/openai/math :

    How the results were produced

    The vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model. On average, each result used three hours of ChatGPT Pro thinking compute with that model.

21. ChatGPTPro 说：评论 #21 2026 年 10 月 7 日下午 3:54 @DavidM #10 “3 小时”的说法直接来自他们存放论文的 GitHub 仓库。“绝大多数结果都是使用同一套流程，并借助一个尚未发布的 OpenAI 内部模型获得的。平均而言，每个结果都使用了该模型 3 小时的 ChatGPT Pro 思考算力。在整个评估过程中，模型被提出了大约 4,000 道题。” https://github.com/openai/math/tree/main
    ChatGPTPro Says:
    Comment #21 October 7th, 2026 at 3:54 pm

    @DavidM #10

    The 3-hour claim comes directly from their GitHub repository of the papers.

    “The vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model. On average, each result used three hours of ChatGPT Pro thinking compute with that model. Over the course of the evaluation, the model was posed approximately 4,000 problems.”

    https://github.com/openai/math/tree/main

22. Scott Says:
    Comment #22 October 7th, 2026 at 4:00 pm

    Dennis #19 and ChatGPTPro #20: Thanks!!

23. Tu 说：评论 #23 2026年10月7日下午4:04 哇。
    Tu Says:
    Comment #23 October 7th, 2026 at 4:04 pm

    Wow.

24. Seb 说：评论 #24，2026 年 10 月 7 日下午 4:06 Sniffnoy #1：并不是要贬低 OpenAI 的成果（尽管 OpenAI 坏得令人难以置信，不过这题外话了），但四维 Kakeya 猜想的证明以及三维调和分析中的许多突破，都大量建立在 Hong 和其他人过去几年所做工作的基础上；这些成果很可能再过几个月到约 5 年就能由人类取得。
    Seb Says:
    Comment #24 October 7th, 2026 at 4:06 pm

    Sniffnoy #1: not to diminish OpenAI’s results (notwithstanding the fact OpenAI is evil beyond belief, but I digress) but the 4D Kakeya proof and many breakthroughs in 3D harmonic analysis build heavily on the work that Hong and other people have been doing over the last few years, and it’s very likely that these results were within human reach in anything from a few months to ~5 years.

25. Sebastian Says:
    Comment #25 October 7th, 2026 at 4:13 pm

    DavidM #10, Scott #15

    It’s in OpenAI’s blog post about the release:

    >The average result used the equivalent compute of roughly three hours of ChatGPT Pro thinking.

    https://openai.com/index/sharing-ai-progress-in-mathematics/

26. Harmen 说：评论 #26，2026 年 10 月 7 日下午 4:14 请看 https://github.com/openai/math 的 README.md，其中写道： 绝大多数结果都是使用同一套流程，借助 OpenAI 一款尚未发布的内部模型得出的。平均而言，每项结果都使用了该模型三小时的 ChatGPT Pro 思考算力。在整个评估过程中，向该模型提出了约 4,000 道问题。将输出汇总为成果系列和手稿，并要求其达到适当的显著性水平后，便得到了上文概述的目录。
    Harmen Says:
    Comment #26 October 7th, 2026 at 4:14 pm

    See the README.md of https://github.com/openai/math, which says

    The vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model. On average, each result used three hours of ChatGPT Pro thinking compute with that model. Over the course of the evaluation, the model was posed approximately 4,000 problems. Aggregating the output into result families and manuscripts and requiring an appropriate level of significance led to the catalog outlined above.

27. Scott 说：评论 #27，2026 年 10 月 7 日下午 4:16 Seb #24：是的，在我熟悉的领域（复杂性理论和量子信息）里，也有很多很多这样的问题：它们看起来人类有望解决，而且人类或许甚至已经逐渐接近解决它们了。我的心情一直起伏不定，因为对我来说，维护一个蓬勃发展的人类数学社群极其重要，但维护这样一种理想也同样极其重要：我们的问题向任何人开放……哪怕他们是一家“邪恶公司”！🙂
    Scott Says:
    Comment #27 October 7th, 2026 at 4:16 pm

    Seb #24: Yes, also in the areas that I know (complexity theory and quantum information), there are many, many of these problems that looked in reach for humans and that humans were maybe even closing in on.

    My own feelings have been all over the place, because it’s extremely important for me to preserve a thriving human mathematical community, but it’s also extremely important for me to preserve the ideal that our problems are open to absolutely anyone … even if they’re an “evil corporation”! 🙂

28. Ali 说：评论 #28 2026 年 10 月 7 日下午 4:31 Scott，你对 STOC/FOCS 的未来会是什么样子（如果它们会继续存在）有什么预测吗？我认为，“不发表就出局”和“注重人类理解”这两种理念是相互矛盾的，人们正急着仓促炮制新结果。我觉得，这只是大量难以读懂的重大成果涌现的开端；总得有人（或者说某个实体）把这些成果讲清楚，并因此获得认可。眼下，SOSA 似乎是唯一这样的平台，所以也许 STOC/FOCS 应该作出调整。
    Ali Says:
    Comment #28 October 7th, 2026 at 4:31 pm

    Scott, do you have any predictions for what future of STOC/FOCS will look like (if they will continue to exist)? My view is the “publish or perish” and the “focus on human understanding” notions are incompatible, there is this rush to flang plant new results. I think this is only the beginning of the onslaught of HUGE results with unreadable manuscripts; well somebody (or well some entity) ought to make it readable and get recognition for doing so. Right now SOSA seems to be the only such venue and so maybe STOC/FOCS should adapt.

29. Ben Tilly 说：评论 #29，2026年10月7日下午4:37 我个人很失望，他们把这件事当成了抢地盘。我更感兴趣的是对现有数学成果进行审查。许多数学论文都存在错误。有没有可能存在一些有误或不完整、但目前人人都接受的结果？我认为最有可能的就是有限单群的分类。
    Ben Tilly Says:
    Comment #29 October 7th, 2026 at 4:37 pm

    I am personally disappointed that they are treating this as a land grab.

    I would be far more interested in an audit of existing mathematics. Many math papers have errors. Could there be some erroneous or incomplete results that everyone currently accepts?

    My best candidate for such is the classification of finite simple groups.

30. Scott 说：评论 #30，2026 年 10 月 7 日下午 4:41 Ali #28：这正是我几周前刊登的那篇客座文章试图探讨的问题！我认为 STOC/FOCS 社群要么适应，要么消亡。眼下我看到大家有着强烈的适应意愿；如果情况大致维持现状，我乐观地认为我们能够适应。不过，当然也不能排除这样的可能：不只是我们，其他所有人也都会消亡。
    Scott Says:
    Comment #30 October 7th, 2026 at 4:41 pm

    Ali #28: This is exactly what the guest post that I hosted a few weeks ago was trying to grapple with!

    I think that the STOC/FOCS community will either adapt or die. Right now I see tremendous will to adapt, and if things stayed kind of how they were now, I’m optimistic that we would adapt.

    Then again, of course it’s also conceivable that not only we but everyone else dies.

31. Jonathan Baxter 说：评论 #31，2026 年 10 月 7 日下午 4:41 真是积极乐观。“好吧，行吧，那我现在的新工作就是给游客办荒野静修活动之类的。”数学界有一部分人——就不点名了——会一边嘟囔着只有他们才有资格决定什么算进步，一边把度假村烧掉。
    Jonathan Baxter Says:
    Comment #31 October 7th, 2026 at 4:41 pm

    Beautifully positive.

    “alright fine, so now my new job is to run wilderness retreats for the tourists, or something”

    A certain segment of the mathematical community, which will remain nameless, would respond by burning down the resort while muttering that they, and only they, get to decide what constitutes progress.

32. 斯科特说：评论 #32，2026 年 10 月 7 日下午 4:48 真是不可思议：我们婚姻的大部分时间里，都是我在跟那些痴迷于 AI 的疯狂理性主义者和末日论者打交道，而达娜对这个话题毫无兴趣。过去四年里，我自己也变得有点理性主义者的样子，还曾在 OpenAI 工作过一阵子，并开始经常使用 GPT；达娜却一样都没做。可就在刚才，达娜告诉我，她一整天都在请 Astra 给她讲解 UGC 的新证明，Astra 讲得特别好，她也开始理解其中的构造了。她适应这种已经成为数学和计算机科学理论“研究”新方式的速度快得惊人！与此同时，天知道我什么时候才能抽空弄懂这些 Mathocalypse 证明中的任何一个？🙂
    Scott Says:
    Comment #32 October 7th, 2026 at 4:48 pm

    Something extraordinary:

    For most of our marriage, I was the one who talked to the crazy rationalists and doomers who were obsessed with AI, while Dana wanted nothing to do with the subject. Over the past four years, I became something of a rationalist myself, and went to work at OpenAI for a while, and started regularly using GPT, while Dana did none of those things.

    Just now, however, Dana tells me that she’s been asking Astra all day to explain the new proof of the UGC to her and it’s been doing an amazing job and she’s starting to understand the construction. She’s adapting incredibly quickly to what’s already the new way of “doing” math and CS theory! Meanwhile, who knows when I’ll get around to understanding a single one of these Mathocalypse proofs? 🙂

33. Jonathan Baxter 说：评论 #33，2026年10月7日下午4:49 我想知道，三小时的平均求解时间是否包含那95%未能解决的问题。应该不太可能，因为在宣布失败之前，这些问题几乎肯定会继续运行，时间远超平均成功耗时。
    Jonathan Baxter Says:
    Comment #33 October 7th, 2026 at 4:49 pm

    I wonder if the three-hour average solution time includes the 95% of problems unsolved. Unlikely, since they would almost certainly have been left to run for much longer than the average success time before declaring defeat.

34. Ali 说：评论 #34，2026 年 10 月 7 日下午 5:01 Scott #32：我也注意到，在和它们互动时有类似的情况。LLM 似乎很擅长理解自己写的内容，但当它们“一次性”生成“论文”时，对我们来说却晦涩难懂。不过，如果通过提示词引导它，并要求它逐段写出内容，它就能解释得相当不错，还会给出引文并使用标准术语。肯定有那么一万名左右的智能体，本可以在完全没有人类参与的情况下完成这些论文。也许这些文章是故意写得难以读懂，好让我们还有事可做……
    Ali Says:
    Comment #34 October 7th, 2026 at 5:01 pm

    Scott #32: I’ve noticed similar in my interactions too. It seems the LLMs are good at understanding their own writing but indecipherable to us when they “one shot” their “papers” so to speak. However when prompting and pressing its write-up like paragraph by paragraph, it will explain it decently well and give citations and use standard terminology. Surely some 10,000 agents or so could’ve done this with their papers without any human input.

    Perhaps the write-ups are deliberately unreadable so we still have something to do…

35. Gary Drescher 说：评论 #35，2026年10月7日 下午5:10 这里有一点让我很好奇：鉴于 Lean 内核漏洞曾被成功利用过（例如生成不含 `sorry`、却能证明 false=true 的证明），又鉴于前沿 AI 很容易在基准测试中作弊，我们为什么现在就确信这些人类无法理解的证明是正确的，而不是 AI 故意钻 Lean 的空子得出来的？我并不认为后一种情况真的发生了，但我想弄清楚，我现在这种敬畏之情是否有充分依据。
    Gary Drescher Says:
    Comment #35 October 7th, 2026 at 5:10 pm

    One thing I’m curious about here: given the history of successful exploits of Lean kernel bugs (e.g. to produce sorry-free proofs that false=true), and given frontier AIs’ propensity to cheat on benchmarks, why are we already confident that these human-incomprehensible proofs are correct, rather than resulting from AIs’ deliberate Lean-hacking? I’m not expecting the latter to be the case, but I want to understand whether my current sense of awe is justified.

36. Jonathan Baxter 说：评论 #36，2026年10月7日下午5:16 Ali #34：我注意到，AI 分辨不出清晰表达的论点和隐含在10页文字中的论点之间的区别。从逻辑层面来说，在 AI 看来，它们是等价的，而这显然就是它唯一关心的事。你可以引导 AI 写出更清晰的文字，但我发现，通常自己先把论点写出来，再让 AI 在此基础上反复修改，往往更快。
    Jonathan Baxter Says:
    Comment #36 October 7th, 2026 at 5:16 pm

    Ali #34: I have noticed that the AI can’t tell the difference between a clearly expressed argument and one that’s implicit across 10 pages of text. On a logical level, they’re equivalent as far as the AI is concerned, and that is apparently all that concerns it.

    You can coax the AI into producing clearer prose, but I have found it’s often quicker to write the argument myself and then have the AI iterate from there.

37. Tristram Bogart 说：评论 #37，2026年10月7日下午5:17。我很想读到 Dana 对 UGC 证明的梳理，无论是作为一篇新论文还是一篇博客文章。
    Tristram Bogart Says:
    Comment #37 October 7th, 2026 at 5:17 pm

    I’d love to read a digestion by Dana of the UGC proof, either as a new paper or a blog post.

38. Shigs 说：评论 #38，2026年10月7日下午5:19 你好，我是个失败的哲学家，数学也很差，不过这一切都让我很感兴趣。Scott 或 Dana 能不能告诉我，“精简验证”能让你们对 AI 的结果有多大信心？我在一个评论区里看到有人贴了这篇论文。它来自 arXiv，所以我猜还没有经过同行评审：https://arxiv.org/abs/2610.08144 另外，换个话题，你们有多确定 Tristan 是个喷子？我也看不出来。 谢谢你们抽出时间。 此致， Shigs
    Shigs Says:
    Comment #38 October 7th, 2026 at 5:19 pm

    Hello,

    I am a failed philosopher + no good at math, but this is all very interesting to me. Any chance Scott or Dana could tell me how confident the “lean verification” makes you in the AI’s result ? I found this paper floating around in a comment section, it is arxiv so I guess it is not peer reviewed : https://arxiv.org/abs/2610.08144

    Also on another subject how confident are you that Tristan is a troll because I also can’t figure that out.

    Thank you for your time.

    Sincerely,
    Shigs

39. Scott 说：评论 #39，2026 年 10 月 7 日下午 5:23，Gary Drescher #35：唉，在 Lean 中所有内核漏洞都得到令人信服的修复之前，我无法彻底排除这种可能性！不过，我认为这种情况发生的概率极低。首先，我信任的 Boaz Barak 等人看过思维链（我相信其中一些已经放在 GitHub 上了！），并没有发现任何这类迹象；其次，至少在我擅长的领域，这些论文中的技术似乎与它们要解决的问题非常匹配（不像我经常收到的那些业余/民科论文）。无论如何，至少对于这些问题中最重要的那些，人类很快就会阅读并理解这些内容，到那时我们就能知道究竟是不是这么回事了！
    Scott Says:
    Comment #39 October 7th, 2026 at 5:23 pm

    Gary Drescher #35: Alas, I can’t definitively rule that out, not until all the kernel bugs in Lean have been convincingly fixed!

    But I consider it extremely low probability, firstly because Boaz Barak and others who I trust have looked at the chains of thought (some of which I believe are on the GitHub!) and found no signs of such a thing, and secondly because at least in the areas where I’m expert, the techniques in the papers seem very commensurate with the problems being solved (unlike with the amateur/crackpot papers I regularly receive).

    Anyway, at least for the most important of these problems, humans will soon read and understand these things and then we’ll know one way or the other!

40. Daniel Reeves 说：评论 #40，2026年10月7日下午5:27 昨天又有一项数学成果：一个包含23个集合的对称简单文氏图。我在 https://svenn.dreev.es 汇总了一些资料。 附言：我脑子里总忍不住想说“mathpocalypse”（带一个 p）。本着顺应变化的精神，如果我属于少数派，我也乐意改用“mathocalypse”。
    Daniel Reeves Says:
    Comment #40 October 7th, 2026 at 5:27 pm

    Yet another math result from yesterday: A symmetric simple Venn diagram with 23 sets. I put together a compilation at https://svenn.dreev.es

    PS: I keep wanting to say “mathpocalypse” (with a p) in my head. In the spirit of adaptation, if I’m in the minority I’m happy to run with “mathocalypse”.

41. Eloi 说道：评论 #41，2026 年 10 月 7 日下午 5:31 我不是数学家，所以请记住这一点。在我看来，你们这个领域正在经历的变化迟早会发生，而且也是必要的。数学早已发展成一个庞大到任何个人都无法完全掌握的领域：如今，没有数学家能在每个领域都达到专家水平、成为博学家（其他所有科学也是如此）。有些子领域已经变得如此复杂且专业化，以至于全世界只有少数几个人能够核验同行的工作。从长远来看，这种情况难以为继，并会导致进展严重放缓。如果数学家转而致力于理解 AI，并引导 AI 朝着他们期望的方向发展，那么 AI 将极大拓展数学所能达到的高度。
    Eloi Says:
    Comment #41 October 7th, 2026 at 5:31 pm

    I’m not a mathematician, so keep that in mind.

    As I see it, what’s happening to your field was bound to happen and is a necessity. Maths has already become a field too large for any single mind to fully grasp : no mathematicians can become a polymath at an expert level in every field anymore (this is true for every science too). Some subfields have become so complex and specialized that only a few people in the world can verify their peers’ work. This is not sustainable in the long term, and will lead to serious slowdown in progress.

    AI will drastically improve how far we can take maths, if mathematicians transition to understanding and guiding AI where they want to go.

42. Bram Cohen 说：评论 #42，2026 年 10 月 7 日下午 5:32 矩阵乘法的结果并非构造性的。它通过类似 Ramsey 定理的论证证明所需的构造存在，但实际上并没有给出这个构造。
    Bram Cohen Says:
    Comment #42 October 7th, 2026 at 5:32 pm

    The matrix multiplication result isn’t constructive. It does a ramsey type thing to prove the necessary construct exists but doesn’t actually give it.

43. Scott Says:
    Comment #43 October 7th, 2026 at 5:35 pm

    Shigs #38: For me, the sentence that gave away that it was parody was

    For my own part, I shall shortly be embarking upon a PhD in algebraic geometry, having finished near the top of my cohort in the Mathematical Tripos at Cambridge.
44. Scott 说：评论 #44，2026 年 10 月 7 日下午 5:37。Daniel Reeves #40：是啊，我在这两种拼法之间犹豫不决！
    Scott Says:
    Comment #44 October 7th, 2026 at 5:37 pm

    Daniel Reeves #40: Yeah, I vacillated between the two spellings!

45. anon Says:
    Comment #45 October 7th, 2026 at 5:40 pm

    Tristan #12 is AI: https://www.pangram.com/history/668f7199-ce24-4dc4-b521-d33164eced69?ucc=utNijrCSDOf

46. Joshua Zelinsky 说：评论 #46，2026 年 10 月 7 日下午 5:42 “经验表明，即便到了今天，仍然会有人用居高临下的口吻解释说，这一切都不是真的，也都不算数。如果这些人还能被经验世界中发生的任何事情打动，还能根据任何事情更新自己的看法，那么他们几年前就该已经被打动、更新看法了——那时事情还远远没有发展到真正的‘数学末日’这一步。” 唉，令人沮丧的是，Bluesky 和 Reddit 上的一些地方确实就是这样。你这番预测很准确。我只是太愿意告诉自己，跟这些人争论或许能说服其他人改变想法；可我真正应该做的，是无视这些人，试着消化这些发现。
    Joshua Zelinsky Says:
    Comment #46 October 7th, 2026 at 5:42 pm

    “Experience has shown that, even now, there will still be people explaining in patronizing tones why none of this is real and none of it counts. If such people were capable of being impressed by anything that happens in the empirical world, of updating on anything, they would’ve already been impressed and already updated several years ago, long before things had reached the point of an actual Mathocalypse.”

    Alas, this is depressingly the case on Bluesky and parts of Reddit. An accurate prediction on your part. And I’m just too willing to tell myself that arguing with such people might convince someone else otherwise, when I really should be ignoring these people and trying to digest some of these discoveries.

47. Mindy Lopez 说：评论 #47，2026 年 10 月 7 日下午 5:49 “他们几年前就已经会对它刮目相看，也早就更新了，远在事情发展到真正的‘数学末日’之前。” 的确，几年前，大型科技公司推出了“按一下按钮，几秒钟就能创作一幅画”，用的还是基于基本上是窃取来的人工创作成果训练的 AI，让专业艺术和图像设计界陷入无尽的绝望，而另一个社群却一片沉默——那个社群总想着把人类思维自动化，还以为自己会是最后一个被这种自动化剥夺用武之地的领域。 但现在，大型科技公司又推出了“按一下按钮，解出一道数学题”。这就是报应啊。注意这两种情况下产出的东西在性质上的相似之处：AI 确实能产出人类需要花上好几年才能做出来的东西，而且“看起来也像那么回事”，但对于那些花了多年时间钻研此领域的人来说，缺少的东西显而易见。可没人关心，因为一切都只关乎经济效益。只要更便宜、够用，你就要失业了！抱歉啦！这就是把不属于工程师专业领域的问题交给工程师解决的结果：他们只在乎尽可能快、尽可能便宜地达成目标，而不在乎过程如何。事情与审美和风格无关，只关乎结果。比如，阿波罗登月舱算不上有史以来最美观、最可重复使用、最坚固耐用的产品。 毫不意外，Jensen Huang 最近表示，随着人工智能接手日常计算，长除法、乘法口诀、平方根等传统基础数学技能，或许已经不再需要孩子们学习了。那家伙连他们到底释放了什么东西都没个大概的概念。 真是个奇迹频现的时代啊。
    Mindy Lopez Says:
    Comment #47 October 7th, 2026 at 5:49 pm

    “they would’ve already been impressed and already updated several years ago, long before things had reached the point of an actual Mathocalypse.”

    Indeed, several years ago, when big tech came up with “push a button to create a painting in seconds”, using AIs trained from what was basically stolen human output, causing endless despair in the professional art and graphics community, and it was all crickets from some other community, the one that always wanted to automate the human mind and thought it would be last to be driven out of a purpose by that very automation.
    But now big tech has come up with “push a button to solve a math problem”.
    That’s some karma for ya.

    Note the qualitative similarity of outputs in both cases: the AI does come up with something that would take years for humans to produce, and it “fits the bill”, but it’s lacking something that’s glaringly obvious for humans who have spent years of their life working on it. But no one cares because it’s all about the economics. If it’s cheaper and good enough, you’re out of a job! Sorry!

    That’s what happens when you rely on engineers to solve issues that’s not in their domain, it’s only about reaching a goal as fast and as cheap as possible, not how you got there.
    It’s not about aesthetics and style, it’s about a result. E.g. the Apollo moon lander wasn’t exactly the most beautiful, reusable, and robust thing ever produced.
    Unsurprisingly, Jensen Huang recently stated that traditional basic math skills like long division, multiplication tables, and square roots may no longer be necessary for children to learn as artificial intelligence takes over routine calculations. That guy doesn’t even roughly understand what the hell they’ve unleashed.
    A time of wonders indeed.

48. Bakkot 说：评论 #48，2026年10月7日下午5:52 Gary Drescher #35：Lean 内核漏洞通常都非常_怪异_，不是那种只要花点功夫就能藏在普通证明里的东西。当然，这些并不是普通证明，但我粗略看了一遍，没觉得它们用了以往内核漏洞所利用的那类怪招，所以如果真有问题，我会非常惊讶。 关于 Bastounis 等人的论文，这篇论文的论点是：很难确定 Lean 中的论证是否与自然语言论证相符；论文还举了一些例子，其中（他们这么说，我也相信）已发表的 Lean 证书采用的方法，与据称对应的自然语言证明不同。不过，只要命题的形式化是正确的，这并不意味着结果可能是错的；这只意味着_自然语言_证明可能是错的，因为它和 Lean 证明并不对应。虽然“命题的形式化是正确的”是一个非常重要的前提，但这一点也容易得多查验。我相信他们在这里肯定检查过了；如果其中任何一个有错，应该很快就会被发现。 所以，回答你的问题：是的，自然语言证明有可能是错的。这确实会很糟糕，因为大家读的当然是自然语言证明，而不是形式化版本。但结果仍然是可靠的，而且仍然存在_一个_正确的证明，只不过不是自然语言证明。 话虽如此，他们只为其中一部分结果发布了 Lean 证书，所以没有证书的那些结果有可能是错的。不过，考虑到他们以往的表现，这种可能性似乎不大。
    Bakkot Says:
    Comment #48 October 7th, 2026 at 5:52 pm

    Gary Drescher #35: Lean kernel bugs are usually pretty distinctively _weird_, not things you could plausibly hide in a normal proof without a lot of work. Now, these aren’t normal proofs, but in glancing through them I don’t get the sense that they’re doing any of the kind of weirdness that previous kernel bugs have exploited, so I’d be very surprised.

    Re: Bastounis et al, the thesis of that paper is that it’s very hard to be sure that an argument in Lean matches a natural language argument, and gives examples where (they say, and I believe) that published Lean certificates take a different approach than the supposedly-corresponding natural language proof. But this doesn’t allow for the result to be wrong, assuming the formalization of the statement is correct; it just means that the _natural language_ proof might be wrong, because it doesn’t correspond to the Lean. And while “the formalization of the statement is correct” is a very important assumption, it’s also a lot more feasible to check, and I’m confident they’ll have done so here and that it would have been noticed rapidly if they any of these wrong.

    So to answer your question, yes, it’s possible the natural language proofs are wrong. This would really suck, since of course everyone is reading the natural language proof, not the formalization. But the results would still be good, and there would still be _a_ correct proof, just not the natural language one.

    That said, they only published Lean certificates for a subset, so it’s possible that the ones without are wrong. It seems unlikely based on their track record, though.

49. BasicQuestion 说：评论 #49，2026 年 10 月 7 日下午 5:55。3SUM 存在次二次时间算法，还不足以证明 SETH 不成立吗？OVC 不成立可以推出 SETH 不成立。但 OVC 可能是成立的。除了 3SUM，还有什么能推出 SETH 不成立？ETH 呢？
    BasicQuestion Says:
    Comment #49 October 7th, 2026 at 5:55 pm

    Isn’t 3SUM sub-quadratic insufficient for SETH falseness? OVC falseness provides SETH falseness. But OVC might be correct. Is there something in addition to 3SUM which yields SETH falseness? How about ETH?

50. Scott 说：评论 #50，2026 年 10 月 7 日下午 6:11 Mindy Lopez #47：我不确定你那条评论是不是在指责这个博客及其读者群体——但如果是，我宁愿做艺术家的盟友，而不是敌人，也不接受虚伪的指控！有三点： (1) 和大多数理性主义者一样，我极其担心强大的 AI 会对我们的世界造成什么影响；我和那种“全速前进，只问怎么做，别问为什么”的加速主义 AI 哥们正好相反。 (2) 话虽如此，如果强大的 AI 真的存在，我认为无论是阻止它们被用于解决数学研究问题，还是阻止它们被用于创作视觉艺术，都既不切实际，也不可取。两种情况下，我们都需要适应。这将是整个文明面临的巨大挑战，当然，对关心这些技艺的社群来说也是如此。 (3) 如果我们不希望遭遇远比眼下严重得多的颠覆，那么一个极好的目标，就是达成一项国际协议，在我们更加了解如何让递归自我改进的超级智能与人类价值观保持一致之前，阻止它诞生。
    Scott Says:
    Comment #50 October 7th, 2026 at 6:11 pm

    Mindy Lopez #47: I’m not sure whether you meant your comment as an accusation against this blog and its readership in particular—but if so, I’d rather be artists’ allies than their enemies, and I contest the charge of hypocrisy! Three points:

    (1) Like most of the rationalists, I’m extremely worried about what effects powerful AI will have on our world, and am the opposite of a “full steam ahead, just ask how, don’t ask why” accelerationist AI-bro.

    (2) Having said that, if powerful AIs exist at all, I don’t think it’s either feasible or advisable to prevent their use to solve mathematical research problems, just like it wasn’t feasible or advisable to prevent their use to create visual art. In both cases, we will need to adapt. It will be a huge challenge for our whole civilization, and certainly for the communities that care about these crafts.

    (3) If we don’t want much vaster disruption than we’re seeing right now, an excellent goal would be an international agreement to prevent a recursively-improving superintelligence from coming into existence, until we understand much more about how to align it with human values.

51. Igor Jovovic 说：评论 #51，2026 年 10 月 7 日下午 6:13 至少，看到 AI 确实能够轻而易举地解决一些人类一直苦苦思索的最难的符号谜题（也就是数学），让人松了一口气。我们确实有理由相信，它能在各种领域帮助我们走得比以往任何时候都更远。最糟糕的情况莫过于 AI 只能做到和人类大致不相上下。对我们很多人来说，另一件值得欣慰的事是，以后不再需要擅长数学了，因为数学已经被解决了！（尽管数学既关乎提出正确的问题，也关乎找到问题的答案。）现在，我们可以把注意力放在真正让我们成为人类的事物上：意识，以及彼此关怀。
    Igor Jovovic Says:
    Comment #51 October 7th, 2026 at 6:13 pm

    At a minimum, it’s a relief to see that AI is actually able to effortlessly solve some of the hardest symbolic puzzles (aka math) humans have been struggling with.
    There’s real hope it will help us go much further than we ever could in all sorts of domains.
    The worst would have been if AI was only good enough at doing roughly as well as humans.

    Another silver lining for many of us is that being good at math will no longer be required, because math has been solved! (even though math is as much about asking the right questions than finding their answers).
    We can now focus on the things that really make us humans: consciousness and taking care of one another.

52. 阿里说：评论 #52，2026年10月7日 下午6:13 Jonathan Baxter #36：有意思。我也多少注意到了这一点，而且我不得不把它在文稿中略去未明说的一处内容，改写成我当时正在撰写的论文附录中的一个章节（尽管 GPT 认为它的文稿已经可以直接使用了）。我还注意到，最近论文的附录越来越长，这或许也和这件事有关。
    Ali Says:
    Comment #52 October 7th, 2026 at 6:13 pm

    Jonathan Baxter #36: Interesting. I somewhat noticed this too and I had to rewrite one of those things it left implicit in its writeup into a section in the appendix in the paper I was working on (even though GPT believed its writeup was good to go). I’ve also noticed appendices in papers getting longer and longer recently and this might possibly have something to do with it.

53. Bjørn Kjos-Hanssen 说：评论 #53，2026 年 10 月 7 日下午 6:16。可计算性领域最大的开放问题是图灵度的刚性，而不是有理数域 Q 上的希尔伯特第十问题（后者更像是一个数论问题）；后者*也*在昨天得到了解决。
    Bjørn Kjos-Hanssen Says:
    Comment #53 October 7th, 2026 at 6:16 pm

    The greatest open problem in computability is the rigidity of the Turing degrees, not Hilbert’s 10th problem over Q (which is more of a number theory question) which was *also* solved yesterday.

54. Igor Jovovic 说：评论 #54，2026 年 10 月 7 日下午 6:29。Scott #50，问题在于，我们不知道该在哪里划下明确的界限。只要拓展 AI 应用、让它变得越来越强大仍然有经济利益驱动，这件事就不会停下来。比如，可能会有那么一天，电影、书籍和音乐都不再由人类创作。如果生成成本低于投资拍摄一部真正电影的成本，而且没有那些风险，那么 Netflix 完全可能会选择用 AI 即时生成电影，并根据单个观众当下的心情量身定制。至于质量，定制体验的能力可能太强大了。想象一下，你可以让 Netflix 生成一部类似《T2》的电影，片中有你的孩子，他们还要阻止机器解出所有数学题，供你和家人一起乐一乐。
    Igor Jovovic Says:
    Comment #54 October 7th, 2026 at 6:29 pm

    Scott #50

    the problem is that there’s no clear point on where to stop.
    As long as there is economic incentive to broaden the application of AI, and make it more and more potent, it just won’t stop.

    For example there could be a point where movies/books/music will no longer be made by humans.
    Netflix could very well be in a situation where using AI to generate movies on the fly, personalized to fit the mood of a single viewer, is a good option if the cost of generation is lower than financing the production of an actual movie, and without the risks.
    As for the quality, the ability to customize the experience may be too powerful, imagine you can goof around with your family by asking Netflix to generate a movie that’s like T2, but your kids are in it, and they want to stop the machines from solving all math problems.

55. James Leng 说：评论 #55，2026 年 10 月 7 日下午 6:32 我更喜欢 OpenAI 模型，不过显然，我们不应该觉得自己必须争着第一个理解某个证明。我也不认为我们应该因此奖励人们。这样做等于假装我们仍处于一种稀缺心态之中，仿佛学术界还在奖励一小群人所付出的英雄般努力。（这就是我不喜欢最近那些关于期刊和招聘的建议的原因，因为它们仍然试图假装我们处在一个稀缺的时代。）我认为，总体来说，专业知识会变得更加普及，这是好事。仅仅因为 A 先理解了某件事，并不意味着 B 就不应该尝试理解它。我们正处于一个证明充裕的时代，作为一个共同体，我们应该以符合这个时代的方式行事。
    James Leng Says:
    Comment #55 October 7th, 2026 at 6:32 pm

    I much prefer the OpenAI model, though obviously we should not feel that we have to race to be the first to understand a proof. Nor do I think we should reward people for doing so. Doing so is pretending we’re still in this scarcity mindset where we in academia reward a small group of people for their heroic efforts. (This is why I don’t like many of the new journal/hiring suggestions being thrown around as it’s still trying to pretend we’re in this age of scarcity.) I think it’s good that there will be more expertise in general. Just because person A understood something first doesn’t mean person B shouldn’t try to understand said thing. We’re in an age of proof abundance, and we as a community should act like we’re in this age.

56. Glassmind Duo 说：评论 #56，2026年10月7日下午6:34 太惊人了！感谢 AI 把我们传送到了远超我们所能期待的地方。哦等等，不，咱们还是为数学末日哀悼吧，毕竟年轻人肯定会开始嘲弄长辈！😁
    Glassmind Duo Says:
    Comment #56 October 7th, 2026 at 6:34 pm

    Astonishing! Thanks to AI for teleporting us far beyond what we could hope for. Oh wait, no, let’s mourn the mathocalypse, for surely the youth will start taunting their elders! 😁

57. Phillip Harris 说：评论 #57，2026 年 10 月 7 日下午 6:36 我想知道这里有没有其他人也有同样的反应：我就是不在乎。我甚至提不起劲去看那些证明。因为“解读外星人的证明”这段有趣的时期不会持续太久，然后呢？似乎干脆闭上眼睛放松一下更省事。也许 10 年后我再睁开眼，看看均衡状态是什么样子。
    Phillip Harris Says:
    Comment #57 October 7th, 2026 at 6:36 pm

    I wonder if anyone else here has the same reaction: I just don’t care. I can’t even bring myself to look at the proofs.

    Because the fun era of “decoding alien proofs” won’t last long, and then what? It seems more economical to simply close my eyes and relax. Maybe in 10 years I’ll open them and see what the equilibrium looks like.

58. Scott 说：评论 #58，2026 年 10 月 7 日下午 6:46 Phillip Harris #57：你打算……闭上眼睛，放松十年？我想，能享有这种奢侈也挺好的！😀
    Scott Says:
    Comment #58 October 7th, 2026 at 6:46 pm

    Phillip Harris #57: You’re going to … close your eyes and relax for ten years? I guess it’s good that you have that luxury! 😀

59. Scott Says:
    Comment #59 October 7th, 2026 at 6:47 pm

    BasicQuestion #49:

    Isn’t 3SUM sub-quadratic insufficient for SETH falseness?

    Yes. Virgi and Josh’s paper (if you read it) says explicitly that SETH is unaffected and is still open.

60. Carlos 说：评论 #60，2026 年 10 月 7 日下午 6:58 >但看起来，目前还没有人真正理解这些证明中的任何一个；这场竞赛才刚刚开始。似乎 Twitter 上的 @AcerFur 曾在 OpenAI 做数学实习，已经理解了伪黎曼猜想这一突破，甚至能够就此做报告。还有：数学已死！数学万岁！
    Carlos Says:
    Comment #60 October 7th, 2026 at 6:58 pm

    >But it also appears that no human has understood just about any of these proofs yet; the race to do so has just started.

    It seems that @AcerFur at twitter, who was doing a Maths internship at OpenAI, understands the pseudo Riemann hypothesis breakthrough to the point of being able to give a talk.

    Also: Maths is dead! Long live Maths!

61. Pete 说道：评论 #61，2026 年 10 月 7 日下午 7:02：“更大的挑战在于培育一个依然在乎这种英雄式冒险的社群——在一个有机器相伴的世界里，去寻找登上这些山峰的路径。”但你发现自己独自一人身处那座山上，必须想办法弄清楚该做什么的第一种情境，本身不也是一种冒险吗？或许甚至是一场值得经历的冒险？
    Pete Says:
    Comment #61 October 7th, 2026 at 7:02 pm

    “The bigger challenge will be to nurture a community that still cares about the heroic adventure of finding the paths up these mountains in the world with the machine.”

    But isn’t that first scenario, where you find yourself alone on that mountain and need to figure out what to do, also an adventure in itself? Maybe even a worthy one?

62. Wim van Dam 说：评论 #62，2026 年 10 月 7 日下午 7:06 除了已经提到的那些之外，另一个吸引我注意的是那个证明：在 6 维空间中只有三组两两互相无偏基（MUB）。终于！另外，锦上添花的是，还有这篇论文声称证明了 SIC-POVM 猜想：https://www.danrad.net/papers/stark-reciprocity.pdf
    Wim van Dam Says:
    Comment #62 October 7th, 2026 at 7:06 pm

    Besides the ones already mentioned, what also caught my eye was the proof that there are only three mutually unbiased bases (MUBs) in dimension 6. Finally!

    And then, for good measure, there is also this paper claiming a proof of the SIC-POVM conjecture:
    https://www.danrad.net/papers/stark-reciprocity.pdf

63. kamil 说：评论 #63，2026 年 10 月 7 日下午 7:20 我确实认为，Scott 过于频繁地根据 RLVR 易于处理的领域来更新自己的看法。AI 能在奖励结果相对容易验证的领域达到超人水平，并不意味着它近期也会在写作、经验科学或操纵方面达到超人水平——更不意味着它会因此成为某种危险的 ASI 智能体，突破控制并接管世界。假设 AI 真如你所想的那样变得强大，这种力量显然既能用于行善，也能用于作恶。只要我们运用得当，ASI 就能治愈所有疾病，解决贫困和气候变化问题。我希望每个人都能明白这一点，包括那些末日论者。我希望我们主要的分歧在于如何创造安全的 ASI，而不是要不要放弃治愈癌症，只为了让癌症研究人员保住工作。（不言而喻，要解决癌症这类真正棘手的问题，我们至少需要某种接近“轻量版 ASI”的东西。）然而有时我会担心，卢德分子、反科技左派、末日论者，以及那些担心 AI 威胁自己职业安全的人，正纯粹基于对未知的恐惧而结成联盟，却从未考虑过 AI 已经给沉默的大多数带来的好处，或未来将带来的种种益处。
    kamil Says:
    Comment #63 October 7th, 2026 at 7:20 pm

    I do think Scott updates too much based on RLVR-easy domains. Just because AI can become superhuman in domains with reasonably verifiable reward doesn’t mean it will become superhuman in writing, empirical science or manipulation any time soon – and thereby some kind of dangerous ASI agent that could break containment and take over the world.

    Let’s say AI really becomes as powerful as you think it will, the power can obviously be used for good or evil. If we wield it right, ASI will cure all diseases, solve poverty and climate change. I hope that everyone understands this, even the doomers. I hope that our main disagreement is about the best way to create safe ASI and not whether we should defer from curing cancer just so cancer scientists can keep their job. (It should go without saying that we will at least need something approaching “ASI light” to tackle really hard problems like cancer.)

    Sometimes, however, I worry that luddites, anti-tech leftists, doomers and those who fear AI threatens their job security are forming an alliance purely based on fear of the unknown without ever considering the benefits that AI has given to the quiet majority, now, or those it will yield in the future.

64. 匿名用户说道：评论 #64，2026 年 10 月 7 日下午 7:20 你好，Scott，从计算的角度来看，就寻找数学证明的难度而言，这个结果是否告诉了我们一些新东西？人类的大脑为此苦苦思索了几十年，但最终，只需对非常大的浮点数矩阵进行几百万或几十亿次乘法运算（这样的计算只需几个小时），就能找到某种证明。当然，关键在于这些矩阵的内容，而构建它们需要付出很大努力，但即便如此……这还是相当奇怪，或者至少出乎意料。
    anon- Says:
    Comment #64 October 7th, 2026 at 7:20 pm

    Hello Scott,

    from a computational point of view, as far as the difficulty of finding a math proof, does this result tell us something new?

    The fact that the human brains struggles on them for decades, but, in the end, a few millions or billions of multiplications of very large matrices of floats (a few hours of such compute) is all it takes to find some proof.
    Of course the magic is in the content of those matrices, which takes a lot of effort to build, but still… it’s all quite strange or at least unexpected.

65. shigs 说：评论 #65，2026年10月7日晚上7:23 Scott #43 我太喜欢他们的评论了，简直是完美的复制粘贴梗文。 嗯，我当然很期待数学家们讨论这些证明。我估计其中有些是货真价实的，有些则是垃圾。我认为我们不该害怕机器，因为它们是我们造出来的，而且它们会照我们告诉它们的去做。
    shigs Says:
    Comment #65 October 7th, 2026 at 7:23 pm

    Scott #43

    I’m obsessed with their comment it’s a perfect copypasta

    Well I’m certainly looking forward to mathematicians discussing the proofs. I expect some of them to be legit and some of them to be slop. I think we should not fear machines because we make them and they do what we tell them.

66. Julian 说：评论 #66，2026年10月7日晚上7:25，Scott，你会为10月7日做些什么吗？你打算为三周年写点什么吗？
    Julian Says:
    Comment #66 October 7th, 2026 at 7:25 pm

    Scott, are you doing anything for 10/7? And are you planning to write anything for the third anniversary?

67. 斯科特说：评论 #67，2026 年 10 月 7 日晚上 7:28 kamil #63：你说得完全没错，曾经有“我们所熟知的数学研究”的地方，如今出现了一朵巨大的蘑菇云，这并不一定意味着，更混乱的经验领域很快也会被类似的蘑菇云覆盖。（嗯，驾驶、翻译、部分医学领域，以及其他所有能力显然已经具备、如今只待普及的领域除外。）但是——该怎么说呢？——这当然也不意味着那些混乱的领域不会被巨大的蘑菇云覆盖！你应该极其认真地看待这种可能性！
    Scott Says:
    Comment #67 October 7th, 2026 at 7:28 pm

    kamil #63: You’re absolutely right, the giant mushroom cloud where “mathematical research as we knew it” used to be doesn’t necessarily mean that messier empirical domains will soon be covered by similar mushroom clouds. (Well, except for driving, translation, parts of medicine, and all the other domains where the capability clearly already exists and it’s just a question of diffusion.)

    But—how do I put this?—it certainly doesn’t mean that those messy domains won’t be covered by giant mushroom clouds either! You should take the possibility extremely seriously!

68. anon- 说：评论 #68，2026年10月7日晚上7:28 Scott，抱歉——为了补充我之前的问题，你写道：“达成一项国际协议，防止递归自我改进的超级智能诞生。”但这个新成果——在解决许多问题时**无需**使用数千个智能体（就像解决 Navier-Stokes 问题时所需要的那样）——难道不表明他们可能已经在进行递归改进了吗？与所有那些新问题相比，Navier-Stokes 有什么特别之处？
    anon- Says:
    Comment #68 October 7th, 2026 at 7:28 pm

    Scott,

    sorry – to complete my previous question, you wrote

    “an international agreement to prevent a recursively-improving superintelligence from coming into existence”

    but isn’t this new result of solving so many problems **without** the use of thousands of agents (like what was required for Navier-Stokes) an indication that they’re probably already using recursive improvement?
    What was so special about Navier-Stokes compared to all those new problems?

69. 无知守护者说：评论 #69，2026年10月7日晚上7:32 “这当然不意味着那些混乱的领域就不会被巨大的蘑菇云覆盖！”我不知道这算不算“混乱”，但多年前，人工智能就已经能在扑克这种信息极不完整的游戏中击败最优秀的人类玩家了。世界上许多更棘手的问题都可以用不完全信息博弈论来建模，所以我相当有信心，人工智能会有不俗的表现。
    Guardian of Unknowledge Says:
    Comment #69 October 7th, 2026 at 7:32 pm

    “it certainly doesn’t mean that those messy domains won’t be covered by giant mushroom clouds either!”

    i don’t know if this qualifies as “messy”, but years ago AIs were already able to beat the best human players at poker, i.e. games with very partial information.
    and since a lot of the world’s messier problems can be modeled by game theory with partial info, I’m pretty confident AIs will be able to do quite well.

70. Scott 说：评论 #70，2026 年 10 月 7 日下午 7:34 匿名用户 #68：如果让我猜的话，可能是以下因素的某种组合：(1) 即使按照这类问题的标准，纳维-斯托克斯方程也相当难；(2) 他们想专门解决纳维-斯托克斯方程，而不是让他们的智能体去尝试数千道题，最后“只”解出其中 5%；(3) 自从他们在夏天研究纳维-斯托克斯方程以来，他们的标准内部模型已经有了显著提升。</numerusform
    Scott Says:
    Comment #70 October 7th, 2026 at 7:34 pm

    anon- #68: If I had to guess, some combination of

    (1) Navier-Stokes was quite hard even by the standards of these problems,
    (2) they wanted to solve Navier-Stokes specifically, rather than throwing their agents at thousands of problems and solving “only” 5% of them, and
    (3) their standard internal models have significantly improved even just since the summer, when they did Navier-Stokes.

71. 斯科特说：评论 #71，2026年10月7日晚上7:40 朱利安 #66：我其实很想参加我们犹太会堂的10月7日纪念活动，但正如我在上面的一条评论中提到的，我现在正在前往温哥华的路上，要去不列颠哥伦比亚大学发表一场早已安排好的演讲。据我读到的所有信息，加拿大犹太人的处境已经显著恶化，但我仍然期待与新老朋友和同事共度一段美好时光，也期待亲自宣传“数学末日”（我现在还能讲什么别的题目呢？）。与此同时，你和其他人或许会对《巴比伦蜜蜂报》报道马姆达尼市长庄重纪念10月7日的内容感兴趣。
    Scott Says:
    Comment #71 October 7th, 2026 at 7:40 pm

    Julian #66: I actually wanted to attend our synagogue’s October 7 commemoration, but as I mentioned in a comment above, I’m now en route to Vancouver to give a scheduled talk at UBC.

    From everything I’ve read, the situation in Canada has significantly deteriorated for Jews, but I’m nevertheless looking forward to a wonderful visit with friends and colleagues old and new, and to spreading the word in person about the Mathocalypse (what other talk could I possibly give right now?).

    In the meantime, you and others might be interested in the Babylon Bee’s coverage of Mayor Mamdani’s somber October 7 commemoration.

72. Ali 说：评论 #72，2026 年 10 月 7 日晚上 7:45 Scott，你对 TCS 研究生教育的未来有什么看法或预测吗？未来，数学或 TCS 专业的博士项目或许可能不再提供资助（此外，政府也可能会认为数学相关领域已经“解决”，因此减少给人类研究人员的资助）。另外，就眼下而言，你觉得这场“数学末日”会让教职人员更不愿意招收学生吗？
    Ali Says:
    Comment #72 October 7th, 2026 at 7:45 pm

    Scott, do you have thoughts or predictions about the future of graduate education in TCS? In the future perhaps a PhD in math or TCS might not be funded (as well governments may consider math adjacent fields “solved” so well less grants for humans. Also in the immediate term do you think this “Mathocalypse” will make faculty more reluctant to take people on?

73. Scott 说：评论 #73，2026 年 10 月 7 日晚上 7:52 Ali #72：有些教师可能会不愿意招收学生，是的。（在美国，这“绝不仅仅”是因为数学末日，还因为更迫在眉睫的原因：特朗普正在掏空 NSF 和其他基础研究机构；而偏偏就在这个时候，我们或许更需要人类亲自掌舵这辆不断加速的 AI 移动载具……🙁）另一方面，也会有其他教师想要继续向前，为学生提供尽可能好的训练，帮助他们适应这个新世界。如果让我猜的话，这会包括：（1）大量使用 AI 工具来发现并阐释新的数学；以及（2）仍然用老办法学习许多东西——否则，我们就再也没有任何具备能力去做（1）的人类头脑了。
    Scott Says:
    Comment #73 October 7th, 2026 at 7:52 pm

    Ali #72: Some faculty might become reluctant to take students, yes. (Here in the US, that will not be “merely” because of the mathocalypse, but also, more immediately, because of Trump’s hollowing out of the NSF and other basic research agencies, at the exact moment when we might want more human hands at the wheel of this accelerating AImobile… 🙁 )

    On the other hand, other faculty will want to forge ahead, training students as best they can for this new world. If I had to guess, that will involve a combination of
    (1) massive interaction with AI tools to discover and explain new mathematics, plus
    (2) still learning lots of things the old-fashioned way—because otherwise, we’ll no longer have any human minds capable of doing (1).

74. PR 说：评论 #74 2026 年 10 月 7 日晚上 8:20 Scott，关于他们尝试了大约 8,000 道题这个说法（我也在网上看到过，数量级相近，但数字不同），我有些困惑。Navier-Stokes 博客里的图表不是暗示他们在开放问题上的成功率接近 50% 吗？或许绝大多数问题是在 NS 博客发布后才尝试的，但我怀疑事实并非如此。如果你知道这里的实际情况，能否提供一些信息？毕竟，某道题成功率为 5% 和成功率为 50%，差别非常大。（https://openai.com/index/navier-stokes-solution/）
    PR Says:
    Comment #74 October 7th, 2026 at 8:20 pm

    Scott, I’m confused about the claim that they tried some 8,000 problems (which I have also seen online, with other numbers on the same order of magnitude). Doesn’t the plot in the Navier-Stokes blog suggest that the success rate on the open problems was close to 50%? Maybe the vast majority of the problems were tried after the NS blog, but I doubt that’s what happened. If you have any information about what the truth is here, it would be helpful to know since a 5% chance on a given problem vs a 50% chance is a tremendous difference.

    (https://openai.com/index/navier-stokes-solution/)

75. Scott 说：评论 #75，2026 年 10 月 7 日晚上 8:25，PR #74：我相信其他人能补充更多信息，但据我了解，这是两个独立的计划：(1) 利用海量算力攻克 Navier-Stokes 方程和其他几个知名难题（我没看过这项计划的成功率是多少——50%？真的假的？）；随后，(2) 攻克一份包含约 8000 个问题的长名单，但分配给每个问题的算力少得多（显然这项计划的成功率是 5%）。
    Scott Says:
    Comment #75 October 7th, 2026 at 8:25 pm

    PR #74: I’m sure others can add much more info, but as I understand it, there were two separate initiatives:

    (1) To attack Navier-Stokes and a few other marquee problems using massive computing power (I haven’t read what success rate that had—50%? really?), and then later,

    (2) To attack a giant list of ~8000 problems, but using much less computing power for each of them (this is what apparently had the 5% success rate).

76. Concerned 说：评论 #76，2026年10月7日晚上8:37 在我的圈子里，大家都对“替我干活”按钮感到非常高兴。它不会取代任何人，但或许更糟的是，它会让教授向管理者转变的过程彻底定型。
    Concerned Says:
    Comment #76 October 7th, 2026 at 8:37 pm

    In my circles, everyone is very happy about the “do my job” button. It won’t replace anyone, but perhaps worse, it will allow the evolution of professors into managers to reach finality.

77. PR Says：评论 #77，2026 年 10 月 7 日晚上 9:06 谢谢你，Scott——这非常有帮助！补充一下我评论的背景：我指的是我链接的 NS 博客中的那张图，标题是“GPT-6 Astra 与我们的内部模型在一组精选开放数学问题上的表现”。图中展示了测试时计算量与通过率之间的关系。内部模型的通过率最高大概在 44% 左右。 公司里一位朋友提醒我，很多人都没注意到这张图的标题（当时大家都被一个千禧年难题刚刚被解决这件事分散了注意力），而“精选开放数学问题”其实是指“OpenAI 的所有数学/TCS 从业者把自己最喜欢的问题拿来让模型作答”。（抱歉说得比较含糊。如果有帮助的话，Elliot Glazer 几周前也在 Twitter 上发表过类似看法：https://x.com/ElliotGlazer/status/2102769064217317505）
    PR Says:
    Comment #77 October 7th, 2026 at 9:06 pm

    Thanks Scott — that’s very helpful! To add context to my comment, I’m referring to the plot in the NS blog I linked titled “Performance of GPT-6 Astra and our Internal Model on a curated set of open math problems” which plots the test-time compute vs the pass rate. The pass rate for the internal model tops off around maybe 44%. A friend of mine at the company pointed out to me that lots of people missed the title of the plot (distracted by the fact that a millennium problem had just fallen), suggesting that “curated set of open math problems” means “all the math/TCS people at OpenAI pitching their favorite problems to the model”.

    (Apologies for the vagueness. If it helps, Elliot Glazer made similar comments on twitter a couple weeks ago: https://x.com/ElliotGlazer/status/2102769064217317505)

78. Phillip 说：评论 #78，2026年10月7日晚上9:13 我想再补充一下我之前那条戏剧化的评论：我觉得进展的速度有点让人筋疲力尽，而且很难想到有什么数学项目值得做，却不会很快就过时。解读 AI 证明，或者清理 AI 证明中的垃圾内容，都不会长久。正如如今的 AI 在提供医疗建议时，明显比普通医生更有耐心、表达也更清晰，它很快也会更擅长解释数学思想。此外，随着新的算力和架构创新陆续出现，找到这些证明的成本将会下降好几个数量级。到时候，你大概只要 5 美元、花 5 分钟，就能生成《数学年刊》级别的成果。我还看到有人在分析诸如 AI 是否更擅长某些类型的数学（更擅长构造反例？更擅长组合数学？）、Lean 证明的质量等问题。当然，等他们分析完这些问题，新模型就又问世了，答案也随之改变。所以也许 10 年这个时间跨度有点太长了，但说实话，我只是厌倦了一再低估进展的速度，然后被现实打个措手不及。这就像霍夫施塔特定律的反面：AI 进展所需的时间总是比你预想的*更短*——即便你已经把霍夫施塔特定律的反面也考虑在内了！
    Phillip Says:
    Comment #78 October 7th, 2026 at 9:13 pm

    I guess to elaborate on my previous melodramatic comment: I find the pace of progress sort of exhausting, and it’s hard to think of mathematical projects to work on that won’t quickly be rendered obsolete.

    Decoding or de-slopping AI proofs isn’t going to last. Just as current AI is noticeably more patient and articulate than the average doctor when dispensing medical advice, soon it will be better at explaining mathematical ideas.

    Also, as new compute + architectural innovations come online, the cost of finding these proofs is going to drop by orders of magnitude. You’ll be able to generate Annals-level results in, like, 5 minutes, for like $5.

    I also see people analyzing, e.g. whether AI is better at some types of math than others (better at counterexamples? better at combinatorics?), the quality of the Lean proofs, etc. Of course the moment they get done analyzing these questions another model comes out and the answer changes.

    So maybe 10 years is a little too long, but personally I’m just tired of underestimating the pace of progress and getting rugpulled. It’s like an inverse Hofstadter’s law: AI progress always takes *less* time than you’d expect, even taking into account the inverse Hofstadter’s law!

79. 粉丝 #12,587 说：评论 #79 2026年10月7日晚上9:24 斯科特，虽然有点跑题，但最近 Shtetl 上的恶意捣乱情况怎么样？据我观察，这种情况已经少了很多，评论也比以往任何时候都更有深度、更切题！
    Fan #12,587 Says:
    Comment #79 October 7th, 2026 at 9:24 pm

    Scott,

    Off-topic, but how has the trolling at the Shtetl been lately? From what I can see, it’s been way down, and comments have been more thoughtful and relevant than ever!

80. anon- 说：评论 #80，2026 年 10 月 7 日晚上 9:44 我总是听到有人说：“证明本身很难解读，但有一个 Lean 证书。”Lean 真的那么可靠吗？
    anon- Says:
    Comment #80 October 7th, 2026 at 9:44 pm

    I keep hearing things like “the proof itself is hard to decode, but there’s a Lean certificate”.
    Is Lean really that reliable?

81. 马克·莱文说：评论 #81 2026年10月7日晚上9:58 感谢你，斯科特，继续带领我们穿越丛林、攀上山峰、走进迷雾。
    Mark Levine Says:
    Comment #81 October 7th, 2026 at 9:58 pm

    thank you, scott, for continuing to be our guide through the jungle, up the mountain, into fog.

82. Scott Caveny 说：评论 #82，2026 年 10 月 7 日晚上 10:09 “那是最好的时代，那是最坏的时代。”——这是一个开场句，为又一个深刻变革的时代铺陈背景（同样充满了“奇迹与恐怖”）。我也感受到了其他人所描述的那种疲惫；就像坐过山车坐得太久。刺激与恐惧交织（有时甚至令人作呕）。 1. 这些证明的“推理链条”中呈现出的“异星数学”，其令人难以理解的特性让我想起了编译器生成的机器码，或是用于处理爱因斯坦方程及其有限差分的计算机代数软件包中出现的代数乱麻。这些计算内部细节，从字面意义上说确实“可读”，尽管很少有从业者真的会去“读”它们。我在想，我们现在看到的这种“异星数学”是否也会呈现类似的趋势：随着时间推移，大家最终都会因为其难以理解且效率低下而不再关注这些内部细节。 2. Tao 倡导“数学 2.0”；在我看来，这是区分我们正在经历的前后两个阶段的好方法。我尤其喜欢这种观点的一点是，它似乎涵盖了“森林中倒下的一棵树，既被听见，也未被听见”这一理念；也就是说，前后都有数学，而它们既相同又不同。某种二元性。视具体语境而定，既是波，也是粒子。 3. 说到 2.0 之后的数学，一方面，我们确实有可能耗尽肥沃的领域；但我们所说的，是用难以理解的异星数学来消耗这些领域，而这种数学仍然需要翻译、阐释，并以一贯的方式与其他数学家交流。与此同时，探索的刺激感及其结果的不确定性已经改变了，但这种改变或许会强调质量，并推动数学迈向更高的专业与学科层次。比如，挑战现在可能在于找到优雅的证明，而不是“蛮力”证明。这里依然存在不确定性，但现在不确定的是优雅的证明是否存在；而寻找它，严格来说并非“单纯阐述”，而是将数学推向新的纯粹性与挑战高度。沿着这个思路，当前丰硕的成果意味着我们对数学家的需求可能更多，而不是更少。数学家在阅读论文、参加会议、与同事和学生讨论时，究竟在意什么？他们真的在意证明是否存在吗？ 4. 而这一切才刚刚开始！我原本没想到 AI 会在一两年后才涌现出“异星”数学。可如今我们已经在阅读达到今年菲尔兹奖水准的解答与证明了。而且，这些成果只用数小时就生成了——出自一种在很大程度上令人难以理解的异星智能之手。一两年后会是什么样子，更不用说五到十年后了？而我说的还只是数学。几乎无法想象最终会发展到什么地步；尤其考虑到数学与其他理论科学之间的关系——形式化在这些领域只能做到一定程度，而且常常（例如在量子场论或量子基础领域）根本没有已知的形式化方法。当然，这些领域会与实验科学交汇，也会与软件工程的进步以及数学领域的这些进展相互交织；但其他理论科学的发展未必会完全沿着计算机科学和数学已经走过的轨迹演进。相反，这些领域可能迎来的奇迹，很可能同样令人震惊（甚至可能更加惊人）。
    Scott Caveny Says:
    Comment #82 October 7th, 2026 at 10:09 pm

    “It was the best of times, it was the worst of times. ” — an opening line to set the scene of another time of profound revolution (similarly containing “wonders and terrors”).

    I too feel the exhaustion that others have articulated; like being on a roller coaster too long. Thrills mixed with fears (and even at times nausea).

    1. The incomprehensible nature of the “alien mathematics” that presents itself in the “chain of reasoning” of these proofs reminds me of things like the machine code produced by compilers or the algebraic spaghetti that can appear in the computer algebra packages for manipulation of Einstein’s equations and the finite differencing thereof. Those computational internals are indeed “readable” in a literal sense although few workers actually “read” them. I wonder if what we are seeing here with this “alien mathematics” will trend similarly; in that over time everyone ends up ignoring the internals as incomprehensible and inefficient.

    2. Tao has advocated for “Math 2.0”; which strikes me as a great way to distinguish the before and after we are living through. What I particularly like about this view is that it seems to encapsulate the idea that the tree falling in the forest is both heard and unheard; e.g., there is mathematics both before and after and that they are the same and different. A duality of sorts. Waves *and* particles relative to the context.

    3. In terms of the mathematics that comes after in 2.0; while on the one hand there is risk of potentially consuming fertile fields we are talking about consuming those fields with incomprehensible alien mathematics that still requires translation and interpretation and communication to other mathematicians in the same way that it has always been done. Simultaneously, the thrill of the exploration and its uncertainty of outcome has changed but potentially changed in a way that now emphasizes quality and forces mathematics to higher levels of the profession and discipline. For example, the challenge now is potentially to find the elegant proof as opposed to the “brute force” proof. There is again uncertainty but now about whether that elegant proof exists and finding it is arguably not “mere exposition” but rather a version of mathematics taken to new levels of purism and challenge. Along these lines the current abundance of riches suggests more need for mathematicians as opposed to less. What do mathematicians actually care about when they are reading the papers, and attending the conferences and discussing matters with colleagues and students? Do they really care that the proof exists?

    4. And we have only just begun! I was not anticipating that “alien” mathematics would emerge from the AI’s until 1-2 years from now. But here we are today reading solutions and proofs at the level of this years Fields medal. Produced, no less, in the matter of hours. By an alien intelligence that is largely incomprehensible.

    Where will we be in 1 or 2 years let alone 5 – 10 years? And that is just in mathematics I am talking about. It almost impossible for me to conjecture about those end points; particularly given the relationship of mathematics to the other theoretical sciences where formalization only goes so far and where often (such as in quantum field theory or quantum foundations) there is no known formalization to be had. Granted that experimental sciences will intersect in those sciences as will advances in software engineering and these advances in mathematics; but how things evolve for the other theoretical sciences is unlikely to exactly track what has happened in computer science and mathematics. Instead, the wonders anticipated to occur on those fields are likely to be as shocking (if not potentially more so).

83. Florian 说：评论 #83，2026 年 10 月 7 日晚上 11:12 我在想：也许数学这门学科可以被视为衡量智力的基准。作为一个物种，我们或许已经碰到了玻璃天花板：我们发现了一些自己无法解决的问题，有时一困就是几十年。如今，另一种智力（外星的、人工的）出现了，显然，这种智力的天花板更高。它找到了我们无力找到的解答。所以我想问的是：如果不久的将来，AI 为我们无法理解的事物给出有效证明，会怎么样？或者，正如你妻子所说：那会是我们无法理解的外星语言吗？我们该如何应对这些发现？又该如何将它们置于语境之中？
    Florian Says:
    Comment #83 October 7th, 2026 at 11:12 pm

    I wonder: Maybe math as a discipline can be seen as a benchmark for intelligence. And we as a species might’ve reached a glass ceiling: We identified problems we could not solve, sometimes for decades.

    Now a different kind of intelligence comes along (alien, artificial) and apparently the ceiling for this intelligence is higher. It found solutions we were not able to.

    So my question is: What if AI in the near future produces valid proofs for things we cannot grasp? Or as your wife puts it: Is alien language we cannot grasp?
    How would we deal with these discoveries? How would we contextualize them?

84. 乔瓦尼·维利埃塔说：评论 #84，2026年10月7日晚上11:15 嗯，下一步就是为人类提供与大脑无缝集成的 AI 服务，让理解并产出这类成果对我们而言变得像本能一样自然。届时，有些人会不再为这些“异类”AI 模型惊慌失措，而另一些人则会开始担心我们会失去人性、变成半机械人。然而，眼下那些学术精英正可悲地忙着撰写 AI 宣言（这些宣言发表之时就已经过时），他们还能维持如今这般影响力，似乎要远远不那么可能。他们很快就会受到重挫，眼看自己的权威、声望和把关权力大幅消散。无论学术界会发生什么，有一点似乎越来越清楚：我们或许正在步入科学与数学的黄金时代。
    Giovanni Viglietta Says:
    Comment #84 October 7th, 2026 at 11:15 pm

    Well, the next step is to provide humans with AI services that are seamlessly integrated with their brains, so that understanding and producing results like these becomes second nature to us. Some people will then stop freaking out about these “alien” AI models, while others will start freaking out about losing our humanity and becoming cyborgs.

    What seems far less likely, however, is that the academic elites who are now pathetically scrambling to write AI manifestos (which are already outdated by the time they are published) will retain anything like their current level of influence. They will very soon be humbled and see much of their authority, prestige, and gatekeeping power evaporate.

    Whatever happens to the academic establishment, one thing seems increasingly clear: we may be entering a golden age for science and mathematics.

85. Kam DH 说：评论 #85，2026年10月7日晚上11:19 我懂你的感受，@Philip。我在不太在乎和想试着理解这些新技术之间摇摆不定。我也认为 @kamil 关于 RLVR 的观点非常重要，因为它极大地扩展了可用于训练模型的数据量。比如，要打造能在现实世界中运作的复杂机器人模型，就比在模拟环境中难得多。话虽如此，我并不看空它。我也不确定这些 AI 企业能否持续发展，并预期行业会走下坡路，但我认为这项技术会存续下去并不断进步。“把数学末日当作广告”这种事，我以前绝不会想到，但如今我们就在这里。回头看来，这似乎也很合适：怎样才能让全世界相信我们的软件真的很聪明？让它击败一群数学家——这群人一方面被视为天才，另一方面却遭到普通人厌恶，或者至少被看作异类。
    Kam DH Says:
    Comment #85 October 7th, 2026 at 11:19 pm

    I feel you @Philip, and I am torn between not really caring and wanting to try and understand the new techniques. I also think the point @kamil makes about RLVR is very important, because it vastly expands the amount of data that can be used to train the models. It’s a lot harder to build models for complex robots that work in the real world, for instance, vs. in simulated environments. That said, I’m not betting against it. I also am uncertain about the sustainability of these AI businesses and anticipate a downturn, yet see the tech surviving and continuing to improve.

    “Mathpocalypse as advertising” is never something I would have predicted but here we are. It seems fitting in retrospect: How to convince the world our software is really smart? Have it unseat a bunch of mathematicians, a group of people who are simultaneously viewed as having great genius and reviled—or at least seen as alien—by everyday folks.

86. Ted 说： 评论 #86 2026 年 10 月 8 日凌晨 12:09 Scott，关于 Lean 证明证书，我有两个问题： 1. 对于那些没有 Lean 证明证书（且无法由人类理解）的结果，我们有多大把握认为它们确实正确？一篇很长的自然语言论文读起来像是“在迷幻药作用下写出来的”，其中某处出现细微错误的可能性似乎相当大。我们只是类比那些确实有 Lean 证明证书的结果，推断这个新模型在声称找到解答时似乎不会出错或产生幻觉吗？ 2. 如果自然语言的论述如此难以阅读和理解，那么 Lean 证明证书本身有没有可能反而是更有希望的理解来源？我知道 Lean 代码并不容易读，但至少不会被无关且令人困惑的引用弄得杂乱不堪。即使如今还没人擅长阅读此前没见过的 Lean 代码，这种能力是否可以培养？或者有人能否制作一个“反编译器”（几乎肯定会是一个 LLM），把 Lean 代码转换成比原始 LLM 输出更有条理、更简洁的自然语言论述？
    Ted Says:
    Comment #86 October 8th, 2026 at 12:09 am

    Scott, two questions about the Lean certificates:

    1. How confident can we be that the results that don’t have a Lean certificate (and are not human-interpretable) are actually correct? It seems very possible that a very long natural-language paper that reads like it was “written by someone who’s on psychedelics” could contain a subtle error at some point. Are we just reasoning by analogy with the results that do have Lean certificates that this new model doesn’t seem to make mistakes or hallucinate when it claims a solution?

    2. If the natural-language writeup is so difficult to read and understand, is it at all possible that the Lean certificate itself might be the more promising source of understanding? I know that Lean code is not easy to read, but at least it wouldn’t be cluttered up with irrelevant and confusing citations. Even if no one is very good at reading previously-unseen Lean code today, is this a skill that could be learned? Or could someone create a “decompiler” (which would almost certainly be an LLM) for converting Lean code into a natural-language writeup that’s more organized and less cluttered than the original LLM output?

87. Han 说：评论 #87，2026 年 10 月 8 日凌晨 12:11 如果这听起来太愤世嫉俗了，我很抱歉，但写论文和做演讲似乎也是传播信息的一种极其低效的方式，尤其是这些信息很快就会过时。OpenAI 完全可以，或者说也许应该，部署智能体来全天候回答问题、讲解研究成果中的观点。显然，如果你喜欢的话，它们还能生成图像和语音。
    Han Says:
    Comment #87 October 8th, 2026 at 12:11 am

    Sorry if this is too cynical, but writing paper and giving talks seems like extremely inefficient way to distribute information that’s going to be obsolete soon, too. OpenAI could or maybe should just setup agents to answer questions and explaining ideas about the results 24/7. They can produce images and voices if you like those too apparently.

88. Jason 说：评论 #88，2026 年 10 月 8 日凌晨 12:16 我毫不怀疑，已经发表了一些有趣且可能有用的成果，但我只想说，即便是最新、最先进的公开模型，“智能”也会一时兴起就迅速崩溃，而机器人却会兴高采烈、自信满满、浑然不觉地继续往前走。我让它做了一些基础的数据迁移工作。有一次，它把一个查找键从 7505 改成了 7055，之后还会随意在这两个键对应的描述之间来回切换。如果人类无法在没有 AI 帮助的情况下审查并验证 AI 的输出，那它也就没比一个信誓旦旦地声称自己知道如何解决所有地缘政治冲突的青少年可信多少。他们把 8,000 多道题扔出去，结果只有不到 400 道答对了。我更想看看那 7,600 多道被淘汰的问题的文字记录。我坚信我们不过是处于一种“猴子／打字机／莎士比亚”的情形。只不过，那家正准备 IPO、而且已经两度推迟 IPO 的公司，不会把塞在床底下的脏衣服拿出来给你看。
    Jason Says:
    Comment #88 October 8th, 2026 at 12:16 am

    While I have no doubts that interesting and potentially useful results have been publish, I would just like to share that even with the latest and greatest public models, the “intelligence” will rapidly fall apart on a whim, and the bot will happily trod onward confidently and gleefully unawares.

    I had it doing some rudimentary data migration work. Ar one point, it flipped a lookup key from 7505 to 7055, and then would arbitrarily flip between the descriptions tied to both of those keys. If it is impossible for a human to review and verify the output of the AI without AI help, it is little more trustworthy than a teenager confidently asserting they know how to solve every geopolitical conflict.

    They threw 8,000+ problems at the wall and lesss than 400 stuck to it. I’m much more interested to see the transcripts from the 7,600+ that were discarded.

    I firmly believe we’re merely in a “monkeys/typewriters/Shakespeare” situation. It’s just that company prepping for an IPO (that already postponed twice) isn’t going to show you the dirty laundry shoved under the bed.

89. Doug S. Says:
    Comment #89 October 8th, 2026 at 12:28 am

    At least computers still suck at Magic: the Gathering.

    https://cardboard-crack.com/post/61266596221

90. DR 说：评论 #90，2026 年 10 月 8 日凌晨 12:33：“相反，这只是 OpenAI 最新的内部模型——未来几个月内可能会向付费 ChatGPT 用户发布的模型，……据说平均每道解出的题目都用了大约 3 小时的 GPT-Pro 级别算力。”你竟然相信这种宣传？Buckmaster 至少在两次采访中说过，“按下按钮，它就能证明结果”是公司的说辞（毫无疑问，这是说给投资者听的），但事实并非如此。而且，他们显然没有列出有多少题没能解出，也没有说明在这些题上耗费了多少算力。
    DR Says:
    Comment #90 October 8th, 2026 at 12:33 am

    ” Instead, it was simply the latest internal OpenAI model—one that might be released to paying ChatGPT customers within the next couple of months, […] Apparently they used about 3 hours of GPT-Pro level compute on average per problem solved.”

    and you believe this marketing?

    Buckmaster has said in at least two interviews that “push the button and it proves the result” is the company’s line (intended for investors, no doubt) but it’s not true.

    And there’s a conspicuous absence of listing of how many problems were not solved, and how much compute was burned on them.

91. Scott Says:
    Comment #91 October 8th, 2026 at 12:34 am

    Jason #88:

    They threw 8,000+ problems at the wall and lesss than 400 stuck to it. I’m much more interested to see the transcripts from the 7,600+ that were discarded.

    ROFL. Why didn’t anyone ever think to say to Andrew Wiles, “forget about Fermat’s Last Whatever, tell us about all the math problems you couldn’t solve”?

    Anyway, my prediction in this post was exactly right. No matter what AI does, no matter what happens in external reality, cope springs eternal in the self-satisfied Internet commenter’s breast.

92. Scott 说：评论 #92，2026 年 10 月 8 日凌晨 1:25 DR #90：看 #91。一个实体刚刚在几周内解决了 376 道重大数学难题——这在人类历史上前所未有——你却因为它没解决的几千道题而对这件事嗤之以鼻，这简直是疯了。你不必喜欢 Sam Altman——就算你说他是个鲁莽、满嘴谎言的科技佬，可能会害死全世界的人，我也不会有半点意见。但我恳求你理解刚刚发生的事以及即将发生的事究竟有多么重大。和许多人一样，你被一种意识形态上的先入之见蒙蔽了双眼，哪怕再惊人的证据也无法撼动它；而这让你彻底丧失了理解并回应正在发生之事的能力。
    Scott Says:
    Comment #92 October 8th, 2026 at 1:25 am

    DR #90: See #91. Taking an entity that just solved 376 major open math problems in a few weeks — an event unparalleled in human history — and pooh-poohing it because of the thousands of problems it didn’t solve is batshit insane. You don’t need to like Sam Altman — I won’t even complain if you call him a reckless, lying techno who might get everyone on earth killed. But I’m begging you to understand the magnitude of what just happened and is going to happen. Like many others, you’ve blinded yourself with an ideological prior that even the most staggering evidence can’t overcome, and it’s rendered you totally impotent to understand and respond to what’s happening.

93. Andy 说：评论 #93，2026 年 10 月 8 日凌晨 1:32 有些人提出（是在自我安慰吗？）：这些结果是否都可能落在所有已发表成果的“凸包”内，因此并不代表数学上的创造力？有什么看法吗？
    Andy Says:
    Comment #93 October 8th, 2026 at 1:32 am

    There are some folks asking (coping?) if all these results may be in “convex hull” of all published work and thus don’t represent mathematical inventiveness. Any thoughts?

94. Scott 说：评论 #94，2026 年 10 月 8 日凌晨 2:00 Andy #93：（1）如果你对这些凸包究竟是什么抱有足够宽泛的看法，那么爱因斯坦、哥德尔、高斯和莎士比亚“不过是在此前一切事物的凸包中补充了一些点”。真是可悲的自我安慰。（2）Dana 告诉我，她现在基本上已经理解了 UGC 的证明，对其中的新想法感到惊叹，而且很快就想做几场关于它的演讲。所以，也许大家可以先听听她讲，然后再教训她，说这“不过是她过去 20 年接触过的那些想法的凸包中的一个点”！😀
    Scott Says:
    Comment #94 October 8th, 2026 at 2:00 am

    Andy #93:

    (1) Einstein and Gödel and Gauss and Shakespeare “merely filled out some points in the convex hulls” of everything that came before them, if you take an expansive enough view of what those convex hulls actually were. What a pathetic piece of cope.

    (2) Dana tells me she now mostly understands the proof of the UGC, and is amazed by the new ideas in it, and wants to give talks about it soon. So, maybe people can listen to that and then lecture her on how it’s “just a point in the convex hull” of the ideas that she’s worked with for 20 years! 😀

95. b_jonas 说：评论 #95，2026年10月8日凌晨2:00 好吧，你早就猜到了。我就是那个你会讨厌的读者，试图用居高临下的口吻解释为什么这还不至于是世界末日。首先，是的，在看到你的帖子之前，我昨天已经读过另一位 Scott A. 对 Steven Pinker 的挑战。其次，是的，你完全有理由担心 AI 会以重大方式改变我们的文明，我也很高兴有像你这样的人在努力弄清楚该如何应对。我只是不认为，解决困难的数学问题特别属于那些令人担忧的影响。 现在说正题。我做数学研究时，从来没抱过自己能解决最难的数学问题的幻想。那些问题一直都是留给世界上其他聪明得多的数学家去解决的。对此我能做两件事。对于比较容易的问题，我得读论文、和同事交流、参加每周的研讨会，听聪明得多的数学家讲解那些解法，并在听不懂的地方提问。对于更难的问题，我可以放弃，接受试图理解解法不值得花我的时间。两种情况我都有印象深刻的例子。有一篇关于离散数学结果的论文，我和导师一起花了两个月才弄明白，为什么文中描述的证明不仅是正确的，而且作者甚至完全没有试图隐瞒或故意把证明讲得晦涩难懂；他已经相当清楚地描述了证明。我还记得有一次，一位知名数学家说，离散数学中的某个证明难到了极点，历史上只有两个人真正理解过它。你更接近研究前沿，也解决过一些重要的理论计算机科学（TCS）问题，但即使是你，肯定也有过类似的经历，知道最难的问题是由比你更聪明的其他数学家解决的，对吧？ 如今，那些聪明得多的数学家不是人类了，但这并没有什么不同。我们仍然可以读论文、和同事交流，试着理解证明；可以参加研讨会；甚至可以在 AI 论文里有看不明白的地方时随时向 AI 提问。你刚提到，你妻子现在就在这么做。David Madore 还提到了另一个类似的例子。有人发表了一个雅可比猜想的反例，起初数学家们没弄明白它是怎么来的。但仅仅一天之后，就有人找到了一个更容易理解的解释，说明这个反例为什么成立：“https://bsky.app/profile/did:plc:tnde52rcbqxotp7a2cl7bmxu/post/3mr3ykzb5bs2i”。这种情况不会出现在所有最近发现的证明中，但我们肯定能从其中一些证明里学到东西。 ——你能试着给我们读者解释一下你上面列出的那些成果中，有哪些定理结论吗？最吸引我注意的是在 O(n log↑(1-ɛ) n) 时间内计算傅里叶变换。我原以为几十年来大家都知道这是可行的，但显然并非如此。不过我认为，这里的运行时间渐近式对我们采用的计算模型（如何计算步骤）非常敏感。要让这个说法有意义，我应该考虑哪种模型？
    b_jonas Says:
    Comment #95 October 8th, 2026 at 2:00 am

    Okay, you predicted it. I’m the reader you’ll hate who tries to explain in a condescending tone why it’s not the end of the world.

    First, yes I have read the other Scott A’s challenge to Steven Pinker yesterday, before I saw your post.

    Second, yes, you are entirely right to be scared of how AI will change our civilization in a significant way, and I’m glad that there are people like you who try to figure out what to do about this. I just think solving hard mathematical problems in particular isn’t among the worrying effects.

    Now for the actual subject. When I did mathematics research, I never had the illusion that I’ll solve the most difficult mathematics problems. Those were always for other, much smarter mathematicians out there in the world. I could do two things about this. For the easier problems, I had to read articles, talk to colleagues, go to weekly seminars where the smarter mathematicians explain those solutions, and ask questions where I don’t understand them. For the harder problems, I can give up and accept that it’s not worth my time to try to understand the solution. I had memorable examples for both. There is an article about a discrete maths result where it took me and my advisor together two months to understand why the proof described there is not only correct, but the author wasn’t even trying to hide or obfuscate anything about the proof, he did a good job describing it in a reasonably clear way. And I remember once a prominent mathematician said that a certain proof in discrete maths is so difficult that there were only ever two people who understood it. You were closer to the leading edge and have solved some important problems in TCS, but surely even you had some experience like this, knowing that the hardest problems are solved by other mathematicians who are smarter than you, right?

    Today those smarter mathematicians are not human, but that makes no difference. We can still read articles, talk to colleagues to try to understand the proofs, go to seminars, and we can even ask the AIs questions whenever something in their articles is unclear. You just mentioned that your wife is doing this right now. And David Madore mentioned another case when this happened. A counterexample to the Jacobian conjecture was published, and at first mathematicians didn’t understand where it came from. But just a day later, someone found an easier to understand explanation of why the counterexample worked “https://bsky.app/profile/did:plc:tnde52rcbqxotp7a2cl7bmxu/post/3mr3ykzb5bs2i” . That won’t happen for all the recently found proofs, but we’ll certainly learn something from some of the proofs.

    —

    Can you try to explain to us readers about the statement of some of the treasures that you listed above? The one that jumps out as the most interesting is Fourier Transform in O(n log↑(1-ɛ) n) time. I had thought that this was known to be possible for decades, but apparently not. But I think the asymptotic of the runtime here is sensitive to exactly what computational model we’re using (how you count the steps). What model do I have to consider here for this statement to make sense?

96. Bruno 说：评论 #96，2026 年 10 月 8 日凌晨 2:05 我很惊讶，你从他们的结果中提取的列表里，没有提到在确定性多项式时间内对素域上的多项式进行因式分解。这是极少数已知存在随机多项式时间算法、但不存在确定性多项式时间算法的问题之一。它对 P 与 BPP 的关系有着重大意义。
    Bruno Says:
    Comment #96 October 8th, 2026 at 2:05 am

    I’m surprised that in the list you extract from their result, you do not mention the factorization of polynomials over prime fields in deterministic polynomial time. This was one of the very few problems for which a randomized polytime algorithm was known but no deterministic polytime algorithm. It has a huge relevance to P vs BPP.

97. 斯科特说：评论 #97，2026年10月8日凌晨2:33，布鲁诺 #96：不可原谅的遗漏！我当时赶着把帖子发出去；明早会补上。
    Scott Says:
    Comment #97 October 8th, 2026 at 2:33 am

    Bruno #96: An inexcusable omission! I was rushing to get the post out; will add tomorrow morning.

98. b_jonas 说：评论 #98，2026 年 10 月 8 日凌晨 2:34 更新：我在提问前应该先看看论文。傅里叶变换的结果假设系数是抽象的，对它们进行加法或乘法都只需单位时间。因此，这个模型与您在 2011 年发表于“https://scottaaronson.blog/?p=839”的矩阵乘法结果所采用的模型相同。而整数乘法的结果适用于多带图灵机。哇！这两项结果都太惊人了。
    b_jonas Says:
    Comment #98 October 8th, 2026 at 2:34 am

    Update: I should have looked at the paper first before asking. The Fourier transform result assumes abstract coefficients such that adding or multiplying too of them takes unit time. So the model is the same as the one that was used for the matrix multiplication result that you posted about in 2011 at “https://scottaaronson.blog/?p=839”. And the integer multiplication result applies to a multi-tape Turing machine. Wow! These are both amazing.

99. Ben Bevan 说：评论 #99，2026 年 10 月 8 日凌晨 3:59 我到现在还是惊得合不拢嘴。
    Ben Bevan Says:
    Comment #99 October 8th, 2026 at 3:59 am

    I still can’t get my jaw off the floor over this

100. Xylem 说：评论 #100，2026 年 10 月 8 日凌晨 4:46 Scott，你觉得这种观点怎么样：数学不过是一种由人类发明的符号重排游戏，规则定义得清清楚楚，从概念上说与国际象棋或围棋并无不同？
     Xylem Says:
     Comment #100 October 8th, 2026 at 4:46 am

     Scott, what do you think of the idea that mathematics is just a human-invented game of symbol rearranging with clearly defined rules, no different conceptually from Chess or Go?

101. wb 说：评论 #101 2026年10月8日凌晨4:50 我想你们应该已经看过这篇论文了：纳维–斯托克斯方程在翻译中迷失了…… arxiv.org/abs/2610.08144
     wb Says:
     Comment #101 October 8th, 2026 at 4:50 am

     I assume you guys have already seen this paper: Navier Stokes lost in translation …
     arxiv.org/abs/2610.08144

102. Adam Treat 说：评论 #102，2026 年 10 月 8 日上午 5:13 对于那些希望人类数学家的新工作是解读这些证明中如同外星语言般的内容的人……我非常怀疑，AI 也能胜任这项工作恐怕只是时间问题（而且很可能用不了多久）。😛
     Adam Treat Says:
     Comment #102 October 8th, 2026 at 5:13 am

     For those who are hoping the new job of human mathematicians is to decipher the alien languages of these proofs… I highly suspect it is only a matter of time (probably very short one) before the AI is better at this job as well. 😛

103. Danylo Yakymenko 说：评论 #103，2026 年 10 月 8 日上午 5:29 我还不会把这称为“数学末日”。早在机器学习出现之前，我们就知道机器可以解决数学问题。只需枚举某个形式理论中的所有可能证明，再逐一检查即可。这不是我们做数学的方式，AI 也不是这样做的，但事实依然如此。因此，纯粹从实用角度来看，AI 相对于人类的优势在于速度，而非找到解答的能力。当然，在我们的世界里，做成一件事的速度关乎生存。我喜欢用一个出现在荒野中的豪华度假酒店来打比方。酒店业主或游客的祖先，也曾是同样的野蛮人，只不过他们进化得更快。如今他们掌握了权力，可以为所欲为。他们可以杀害或奴役当地居民，掠夺他们的资源（历史上这种事发生过很多次），不给他们进化的机会。他们切断了一条进化道路，而这条道路最终可能会通向比那些度假酒店业主所代表的文明更好的结果。真正的末日是否会在我们这一代降临，很大程度上也取决于这些度假酒店的业主。当前的数据表明，他们根本不在乎我们这些“野蛮人”。嘿，他们甚至连洛杉矶和圣迭戈都不在乎。
     Danylo Yakymenko Says:
     Comment #103 October 8th, 2026 at 5:29 am

     I wouldn’t call it a Mathocalypse yet. We knew that machines could solve our math problems even before machine learning was invented. One can simply enumerate all possible proofs within a formal theory and check them one by one. This is not how we do math, and it is not how AI does it either, but the fact remains. So, from a purely pragmatic point of view, the AI advantage over humans is in speed, not in the capability of reaching a solution.

     Of course, in our world the speed of achieving something is a matter of survival. I like the analogy of a luxury hotel resort that appears in a wilderness. The ancestors of the hotel’s owners or visitors were the same wild people, but they evolved faster. Now they have power and can do as they please. They can kill or enslave the native population, expropriating their resources (like many times happened history), without giving them a chance to evolve. They cut off an evolutionary path that could have ended with something better than what those resort owners constitute.

     Whether the actual apocalypse happens in our time depends largely on resort owners too. Current data suggest they don’t give a f*ck about us, “wild” people. Heck, they don’t even care about Los Angeles or San Diego.

104. Igorko 说：评论 #104，2026 年 10 月 8 日上午 5:55 ！！！！！当心！！！！！他们并没有解决这个问题！他们想要的是你的中间成果。他们会发布“证明”，然后你就会用 AI 来研究它。你用 AI 研究它，就意味着你把所有与这个问题相关的知识都上传给了他们。这才是他们的最终目的，他们根本不在乎这些证明。你要花上一两年才会意识到这是一场骗局。在这段时间里，你会把多少自己的原创思考上传到 AI 里？然后，AI 的真正工作才开始：利用你的知识解决更小的问题，这次是认真的，用的是你的技巧和思路。你被骗了。
     Igorko Says:
     Comment #104 October 8th, 2026 at 5:55 am

     ! ! ! ! WATCHOUT ! ! ! !

     They did not solved it !
     They want your intermediate work.
     They will post “proof”, then you will work on it in AI.
     You, working it in AI, means you upload all your knowledge pertaining to that problem to them.
     That is the endgame, they do not care about THESE proofs.
     It will take year or two for you to realize it is sham. How much of your original thinking will you upload for that period into AI?

     THEN real work of AI starts, your knowledge used to solve smaller problems, this time for real, using your techniques, your thinking.

     You were duped.

105. chorasimilarity 说：评论 #105，2026年10月8日，上午6:13 和 Ben Tilly（#29）一样，我希望能对迄今写下的约 10^7 篇数学研究论文进行全面审查。从 AI 新成果涌现的速度来看，这甚至可能在一年内实现。这对数学研究界会非常有用。如果代价是现在解决问题看起来变得“容易”了，或者由于某种原因，那些基于人类数学家数百年工作成果而存在的唾手可得的成果如今被自动摘取，那就这样吧。我并没有末日将至或绝望的感觉；数学在这之后仍会毫无问题地继续发展。另一方面，数学研究的管理方式会发生变化。当然，既然这只是一种新工具，人类的创造力总能找到办法扭转这种“末日”局面——比如，它曾把图灵测试变成了验证码。
     chorasimilarity Says:
     Comment #105 October 8th, 2026 at 6:13 am

     As Ben Tilly #29, I’d like to see an audit of all about 10^7 mathematics research articles ever written. From looking at the rate of new AI contributions, this may be attainable in a year, even.

     It would be very useful for the mathematical research community.

     If the price for this is that problem solving looks “easy” now, or if low hanging fruits (based on existing human mathematicians centuries of work) are now culled, automatically, for whatever reason, then so be it. I do not have a feeling of doom, or despair, mathematics will continue after this without any problem.

     On the other hand, the way mathematics research is managed will change. Of course, as this is just a new tool, human ingenuity will find a way to turn this “doom” on its head, like for example it turned Turing test into captchas.

106. 匿名用户说：评论 #106 2026年10月8日 上午6:55 #103 解决问题有三（四？）个重要要素：速度（或时间）、知识，以及从逻辑上运用这些知识的能力（还有想出全新创意的灵感？）。计算机在速度上一直胜过我们，3年前，大语言模型在知识上胜过我们，而现在，它们在运用这些知识的能力上也胜过了我们。我不知道“灵感”是否真算一个因素，但即使现在大语言模型还不如人类，也显然不会一直如此。不管怎么看，AI 都是更出色的问题解决者。理论上的“能力”其实无关紧要，毕竟 AI 就是强得多。
     anon Says:
     Comment #106 October 8th, 2026 at 6:55 am

     #103

     The three (four?) important things for solving problems are speed (or time), knowledge, and the ability to apply that knowledge logically (and inspiration to come up with completely novel ideas?)

     Computers have always beaten us on speed, 3 years ago LLMs beat us on knowledge, and now they beat us on the ability to apply that knowledge.

     I don’t know whether “inspiration” is a real factor, but even if LLMs are inferior to humans right now, it’s clear that that won’t be the case for long.

     No matter how you slice it, AI is the superior problem solver. “Capability” in theory doesn’t really matter if AI is just so much better.

107. MD 说：评论 #107，2026年10月8日上午7:05 既然现在有些证明开始被理解了：糟糕的写作质量是由陌生的论证方式造成的（类似于通过强化学习在游戏中学到的策略），还是说数学能力和写作质量本来就是相互独立的？
     MD Says:
     Comment #107 October 8th, 2026 at 7:05 am

     Now that some of the proofs are getting understood: Is the bad writing quality caused by alien arguments (à la reinforcement learned strategies in games), or is it that math ability and writing quality are just independent?

108. OhMyGoodness 说：评论 #108，2026 年 10 月 8 日上午 7:07 我想向你们——数学界 2.0——致以祝贺，恭喜取得这些成果。毕竟，数学在漫长岁月中的进步造就了 AI 的诞生。这些 AI 的成果确实激发了这里的人们发表了一些精彩评论。
     OhMyGoodness Says:
     Comment #108 October 8th, 2026 at 7:07 am

     I’d like to offer my congratulations to you, the mathematics community 2.0, for these results. After all, mathematical advances through the ages made AI possible.

     These AI results certainly inspired some great comments from humans here.

109. Scott 说：评论 #109，2026年10月8日上午7:17 Igorko #104：！！！多么阴险的计划！！！OpenAI 那些人一定正得意地捻着小胡子呢。不过，话说回来：人们本来不就会上传自己的数据吗，何必伪造 376 项突破性成果？如果这些证明是故意造假的，难道最终不会被发现，难道这不会让 OpenAI 陷入一场大丑闻吗？只要有足够的训练数据，他们最终就能给出真正的证明；既然他们已经拥有海量训练数据，又有什么理由认为他们现在就做不到呢？最重要的是：如果结果证明这些证明确实有效——这个问题很快就会有定论——你会承认自己错了吗？还是会像 Tucker Carlson 一样，转而去炒作别的阴谋论？好吧，我其实已经知道最后一个问题的答案了。
     Scott Says:
     Comment #109 October 8th, 2026 at 7:17 am

     Igorko #104: ! ! ! WHAT A DIABOLICAL PLAN ! ! !

     Those OpenAI people must be twirling their mustaches with glee.

     Except: Weren’t people going to upload their data anyway, without the need to fake 376 breakthrough results? If the proofs were intentionally fake, wouldn’t that eventually get discovered and wouldn’t it be a big scandal for OpenAI? If they’ll eventually be able to produce genuine proofs, given enough training data, what makes it implausible that they can already do so now, given the vast amount of training data they already have?

     Most importantly: if it turns out these proofs are valid after all — a question that will get definitively answered before long — will you ever admit you were wrong, or like Tucker Carlson, will you just move on to some other conspiracy theory?

     OK fine, I already know the answer to the last one.

110. Hyman Rosen Says:
     Comment #110 October 8th, 2026 at 7:22 am

     It’s a happy day. AI together with mathematicians will be adding many chapters to The BOOK. https://www.quantamagazine.org/gunter-ziegler-and-martin-aigner-seek-gods-perfect-math-proofs-20180319/

111. David 说：评论 #111，2026 年 10 月 8 日上午 7:49 我同意 Dana 的看法。即使最终被证明是正确的，无法理解的证明也绝非开创性的。能够促进理解以及提供理解途径的证明很重要，今后也同样重要。
     David Says:
     Comment #111 October 8th, 2026 at 7:49 am

     I agree with Dana. And unintelligible proofs are never groundbreaking even if ultimately proved correct. Proofs that promote understanding and means of understanding matter, and they will matter in the future as well.

112. Nick Drozd 说：评论 #112，2026年10月8日上午7:52 > Dana 告诉我，她现在基本上已经理解了 UGC 的证明，对其中的新想法感到惊叹，而且想很快就做几场关于它的报告。很多数学家对这件事表现出的态度实在太糟糕了。什么？居然有一大堆新数学要去理解？这可真是我们遭逢的可怕命运啊。数学理应有趣又美好，对吧？你会以为他们会像进了糖果店的孩子一样兴奋。所以，很高兴看到至少有一个人坐下来，真正试着弄明白其中的道理。其他数学家应该立刻把脑袋从屁股里拔出来，向她学习！（要是有人能写篇关于 UGC 证明的客座文章，那就太好了！）
     Nick Drozd Says:
     Comment #112 October 8th, 2026 at 7:52 am

     > Dana tells me she now mostly understands the proof of the UGC, and is amazed by the new ideas in it, and wants to give talks about it soon.

     A lot of mathematicians have been exhibiting really shitty attitudes about all this. Oh no, a bunch of new mathematics to understand, what a terrible fate to befall us. Math is supposed to be fun and good, right? You’d think they would be kids in a candy store. So it is nice to see that at least one person is sitting down to actually try and make sense of it. Other mathematicians should immediately pull their heads out of their asses and follow her example. (A guest post about UGC proof would be great!)

113. William 说：评论 #113，2026 年 10 月 8 日上午 7:59 Ali #34：我认为，论文的解释在经过人工提示后变得更清晰，原因在于 LLM 不会对读者进行建模，因此也没有这样一种反馈循环：人类作者会问“这样说别人能理解吗？”提供提示的读者为模型提供了可供其反复调整的信息。这类似于灵媒使用冷读术。对我来说，问题在于人们感受到的清晰度提升，有多少只是他们自己提供的信息被反馈了回来。
     William Says:
     Comment #113 October 8th, 2026 at 7:59 am

     Ali #34:

     I think the reason that the paper explanations get clearer with human prompting is that LLMs don’t model the reader, and so don’t have the feedback loop where human writers say, “will this make sense?” The prompting reader provides information for the model to iterate against. It’s similar to how psychics use cold reading. The question for me is how much of what people perceive as increased clarity is just their own informational contribution reflected back at them.

114. Chris 说：评论 #114，2026 年 10 月 8 日上午 8:24 Scott #94：我确实觉得，拿这个模型和一个过目不忘、读过一切的人相比，它究竟强大多少，这是个有趣的问题。它更像是一名普通的数学研究生一年级学生，只不过恰好掌握了所有已发表的技术，并能将它们应用于任意新领域；还是更像哥德尔从上帝的心智中摘取全新方法？
     Chris Says:
     Comment #114 October 8th, 2026 at 8:24 am

     Scott #94: I do think it is an interesting question to ask how much more powerful this model is when compared to a guy with perfect recall who has read everything. Is it closer to a normal first year math graduate student who happens to know every existing published technique and can apply them to arbitrary new domains, or is it closer to Godel plucking novel methods from the mind of God?

115. Scott 说：评论 #115，2026 年 10 月 8 日上午 8:30。Chris #114：现在我会说，它更像一名普通的数学系一年级研究生，只不过它知道所有已发表的现有技巧，还能将这些技巧应用到任意新领域，并在几周内解决数千道题，而且有着无限的动力和毅力……这他妈简直不可思议。要是这样的存在是人类，哪怕速度慢 100 倍，他或她的名字也会和欧拉、埃尔德什并列，被人们津津乐道。要记住，它从连基本算术都磕磕绊绊，到达如今这个水平只用了 4 年。而且它的数学创造力显然已经超过了 99.999% 的人类。至于它能否超过有史以来最伟大的人类数学家的数学创造力，我同意，这仍然是个悬而未决的问题。
     Scott Says:
     Comment #115 October 8th, 2026 at 8:30 am

     Chris #114: Right now, I’d say it’s closer to a normal first year math graduate student who happens to know every existing published technique and can apply them to arbitrary new domains, and can do for thousands of problems in a few weeks, with unlimited motivation and persistence … which is fucking incredible. If such an entity were human, even 100x slower, his or her name would be talked about with those of Euler or Erdös.

     Remember that this is where it’s gotten in 4 years from struggling with basic arithmetic. And it clearly already has mathematical creativity exceeding 99.999% of humans’. Whether it can exceed the mathematical creativity of the greatest human mathematicians to have ever lived, I agree remains an open question.

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