内容来源：https://www.sciencenews.org/article/ai-guardrails-erdos-math-problem

内容总结：

数学难题八十载悬而未决，OpenAI模型意外破解引发学界热议

1946年，著名数学家保罗·埃尔德什提出了一个看似简单却困扰学界近80年的“单位距离问题”：在平面上放置尽可能多的点，使它们两两之间的距离恰好等于某个固定值，最多能有多少对这样的点？这一问题长期未能得到证实或证伪，直到近日，OpenAI的研究人员用人工智能模型给出了出人意料的答案。

据报道，研究人员将埃尔德什的猜想输入AI模型后便不再干预。当他们返回时，发现模型已在5月20日于OpenAI官网发布了一份数学证明，成功推翻了这一猜想。哈佛大学数学家梅兰妮·马切特·伍德评论称：“这是一项美丽的数学发现。”她指出，AI将代数和数论这两个古老的基础数学工具应用于几何问题，展示了跨领域应用的巨大潜力。

然而，这一突破在学界引发了复杂反响。一方面，AI通过“笨办法”——穷举大量看似不可能的路径——而非创造性灵感完成了证明。OpenAI研究员塞巴斯蒂安·布贝克坦言，“这份证明并非数学中常见的天才火花。”另一方面，AI的不可靠性令人担忧。英国曼彻斯特大学的数学家托马斯·布鲁姆警告，已有网友用AI生成数百页无法验证的数学内容，“可能正确，也可能是胡说八道，谁来检查？”

更引发争议的是，6月2日，一批专家联合发表声明，呼吁为数学研究中的AI应用设置“严密护栏”。截至6月5日，该声明已获得1590人签名。学者们还担忧，AI无法像人类一样注明灵感来源，且若最强大的工具被私有化，数学研究可能失去开放与民主的特质。

尽管存在诸多疑虑，伍德仍持谨慎乐观态度：“我认为AI终将成为数学领域不可或缺的工具。”但当前，学界显然需要在拥抱新技术与维护学术严谨性之间寻找平衡。

中文翻译：

试想在一个平面上放置若干点，要使任意两点之间的距离尽可能多地相等。对于任意数量的点，最多能有多少对点之间的距离恰好等于同一个数值？
这个被数学家称为"单位距离问题"的难题看似简单，答案却极为复杂。早在1946年，著名数学家保罗·埃尔多斯就提出了他心目中的答案，但八十年来始终无人能证实或推翻这一猜想——至少到最近才出现转机。
OpenAI的研究人员将埃尔多斯猜想输入人工智能模型后便不再干预。当他们再度查看时，发现模型已在5月20日发表于OpenAI官网的数学证明中推翻了该猜想。
"这堪称一项优美的数学发现。"哈佛大学的梅兰妮·马切特·伍德评论道。她为同行专家评审AI成果后附带的论文撰写了评议。这项发现进一步证实了人工智能推动科学认知的可能性。
但这场突破仰赖的是AI的顽强毅力而非创意洞见，并引发了对未来数学研究方式的担忧。6月2日，专家组联合发布声明，呼吁为数学研究中的AI应用设置严格护栏。截至6月5日，该声明已获1590人联署。

数学突破，未必是AI突破
产生此成果的AI模型尚未公开，但OpenAI表示这是为推理训练设计的通用大型语言模型，未使用任何数学专用工具或软件。"我们没有以任何特定方式引导模型。"OpenAI研究员塞巴斯蒂安·布贝克说。
由AI撰写的原始指令描述了猜想，并明确要求解决方案必须证实或推翻该猜想。数学家们原以为猜想为真，但模型却选择尝试推翻它。
伍德将这一成果视为数学领域的突破。AI运用代数与数论这两个最古老、最基础数学分支的工具提出了反例。"代数与数论看似与几何问题无关，但这项成果证明不同数学分支的工具能有效应用于交叉领域。"她认为这将启发数学家探索这些工具的新应用方式。
但她并不认为这是人工智能领域的突破。审阅解决方案时她发现，当前公开的AI模型也能得出相同结论。（事实上，已有研究员在社交平台上表示用公开模型复现了该证明。）
英国曼彻斯特大学的数学家托马斯·布鲁姆也有同感。他在相关论文中指出，若AI能成功证实猜想才堪称"真正的奇迹"，因为这类解决方案需要创造性洞见。
布贝克承认："这份证明算不上数学中偶尔闪现的天才火花。"人工智能仍难以实现发现性飞跃，但能耐心排查海量无效策略。

谁来核查AI成果？
正如呼吁护栏的声明所言，AI技术也威胁着数学研究的可靠性、可验证性和伦理性。
布鲁姆表示，该AI模型的证明对人类专家而言相对容易验证。但他发现网络上有自称解决开放问题的人，用AI生成数百页自己都无法理解的数学内容。"这些内容可能正确，也可能纯属胡诌——谁能来核查？"
若数学家能了解AI生成证明的正确概率会大有助益。但伍德指出，OpenAI并未公开内部模型在解决数学难题时失败，甚至用错误推理生成错误解决方案的全部案例。
OpenAI的布贝克称，团队用同一模型对埃尔多斯猜想进行多次测试，其中50%的尝试生成了正确解。同事陈立杰表示，新模型在遇到困难时更擅长生成"无法解决"的反馈。但相关数据尚未公开发布或经同行评审，OpenAI也未透露模型解题耗时。
伍德、布鲁姆及联署声明者还有其他担忧。当前AI生成数学推理时不显示灵感来源，这与数学家标明突破性工作源头的惯例相悖。
"大型语言模型已阅读所有论文、评注和网络内容……人工智能难以合理标注思想来源。"伍德说。
布鲁姆指出，可获取性也是问题。若最强大的分析工具昂贵且私有，数学可能失去开放与民主性，甚至让人质疑学习数学的意义。
尽管如此，伍德、布鲁姆等数学家仍持谨慎乐观态度。"我坚信人工智能将成为数学领域不可或缺的工具。"伍德表示。

英文来源：

Think about placing dots on a flat surface. You want as many pairs as possible to be separated by the same distance. For any amount of dots, what is the greatest possible number of pairs that can be exactly that far apart?
The question, what mathematicians call the unit distance problem, seems simple. The answer is tricky. Eighty years ago, in 1946, the famous mathematician Paul Erdős proposed what he thought was the answer, but no one had been able to prove or disprove his conjecture. At least, not until now.
Researchers at OpenAI gave an AI model Erdős’ conjecture and walked away. When they returned, they discovered a breakthrough: The model had disproved the conjecture in a mathematical proof posted May 20 on OpenAI.com.
“It’s a beautiful piece of mathematics that has been discovered,” says Melanie Matchett Wood of Harvard University, who contributed remarks to an accompanying paper in which outside experts reviewed the AI’s result. The discovery bolsters hopes that AI can contribute to scientific understanding.
But the AI proof relied on perseverance rather than creative insight and has raised concerns about how mathematics will be done going forward. On June 2 a group of experts published a declaration calling for tight guardrails around AI in mathematical research. As of June 5, the declaration has 1,590 signatures.
A breakthrough for math, but maybe not for AI
The AI model that produced this result isn’t publicly available yet, but Open AI says it is a general-purpose large language model trained for reasoning. It did not use any math-specific tools or software. And “we didn’t guide the model in any particular way,” says OpenAI researcher Sébastien Bubeck.
The original prompt, composed by AI, described the conjecture and instructed the model that a complete solution must either prove or disprove it. Mathematicians had believed the conjecture was true. Yet the model tried to disprove it instead.
Wood sees the result as a breakthrough for mathematics. The AI came up with a counterexample using tools from two of the oldest and most foundational mathematical fields: algebra and number theory. It seems that these areas shouldn’t have anything to do with this geometry question, Wood says. But the result “shows that tools from one part of mathematics can be applied really fruitfully in this other area of mathematics.” She thinks this result will inspire mathematicians to think of new ways to apply those same tools.
She’s not convinced, however, that this is a breakthrough in artificial intelligence. When she read the solution, it seemed to her that the latest, publicly available AI models could have come up with it. (In fact,one researcher posted on X that he had reproduced the proof using a publicly available model.)
Mathematician Thomas Bloom of the University of Manchester in England had a similar reaction. He noted in the paper from outside experts on the achievement that it would have been “truly incredible” if the AI had managed to prove the conjecture, as that kind of solution would require creative insight.
Bubeck concedes that “this proof isn’t exactly the spark of genius that we see sometimes in mathematics.” AI still struggles to make leaps of discovery. But the tech can patiently slog through a huge number of unlikely strategies.
Who will check AI’s work?
Still, as the declaration calling for guardrails notes, AI technology also threatens our ability to produce responsible, verifiable and ethical mathematics.
For one thing, AI’s reasoning can be unreliable. In this case, the AI model’s proof happened to be relatively easy for a human expert to verify, Bloom says. But he has seen people on the internet who claim they have a solution to some open problem. These people have used AI to generate hundreds of pages of math that they can’t understand or even read. “It could be right. It could be nonsense. Who’s going to be able to check this?” Bloom says.
If mathematicians knew the probability that an AI-generated proof was correct, that would help. But as Wood notes, OpenAI does not share all the times their internal model failed to solve an open problem in math or, even worse, produced an incorrect solution with flawed reasoning.
OpenAI’s Bubeck says that the team ran their prompt on the Erdős conjecture through the same model multiple times, and it produced the correct solution in 50 percent of those trials. His colleague Lijie Chen says that the new model is better than current models at generating an “I cannot solve it” response when it runs into difficulty on a problem. But data to support these claims have not been released or peer-reviewed. And OpenAI will not reveal how much time the model spent working on its solution.
Wood, Bloom and those who signed the declaration have other concerns too. Right now, AI generates mathematical reasoning without showing what work inspired the ideas. That clashes with mathematicians’ standard practice of giving credit to the work that inspired a breakthrough.
“LLMs have read ALL the papers. They have read all the commentary and notes, and everything that’s online…. It’s not clear that there’s a way for [AI] to reasonably attribute the source of the ideas,” Wood says.
Access is another concern, Bloom says. If the most powerful tools are expensive and private, mathematics could become less open and democratic, and some people may question why they should learn math at all, he says.
Wood, Bloom, and some other mathematicians are cautiously optimistic, however. “I do think [AI] is going to become an indispensable tool in mathematics,” Wood says.