导读
人眼能够实现在380至720纳米的可见光波段，人类的视觉分辨率约为1角分（1/60度）。当宇宙还处于萌芽阶段，只有现在年龄的3%时，伽利略并不知道恒星会从如此遥远的地方发出光芒，也不知道太阳是宇宙历史最后三分之一才形成的，算是姗姗来迟。
Ancient astronomy was practiced for thousands of years. For example, the Mayans used their eyes to track changes in the positions and relative brightness of the Sun, Moon, planets and stars for over 2,000 years. Mayan astronomers acquired the highest status in their society as priests who advised their rulers on major decisions about state events, marriages, and wars.
The human eye achieves an angular resolution of about an arcminute (1/60th of a degree) in the visible wavelength band between 380 and 720 nanometers. Given the limited collecting area of the pupil, 3–50 square millimeters, humans were able to see only relatively bright sources of visible light, as needed for their survival. Mayan astronomers with sharper eyes had the privilege of identifying fainter sources in the sky. Given the interest of Mayan rulers in their findings, they must have been celebrated and rewarded based on the quality of their eyesight.
Starting with Galileo Galilei in 1610, astronomers employed telescopes to observe the sky. 380 years after Galileo’s first observations, the Hubble Space Telescope (HST) was launched beyond the Earth’s atmosphere, to alleviate the compromising atmospheric glow and blur. HST features a primary mirror with a collecting area of 4.52 square meters, about 100,000 times bigger than a fully dilated human eye. Its angular resolution is 1,200 times sharper and it can detect sources that are 10 billion times fainter than what an unaided human eye can see.
Whereas early telescopes probed electromagnetic waves at the frequency range visible to the human eyes, their advanced successors probed other frequencies including the radio, infrared, ultraviolet, X-ray and gamma-ray bands. In addition, the detection of cosmic-rays, neutrinos, and gravitational waves offered new insights on otherwise invisible processes throughout the cosmos.
Are we witnessing an accelerated repeat of this history in the context of mathematics in the age of artificial superintelligence?
Traditionally, mathematicians used their biological brains to prove or disprove research conjectures. Over the past few months, advanced reasoning models from OpenAI, Anthropic, and DeepMind have generated thousands of pages of manuscripts, found counterexamples to long-standing math problems, and computerized historic human proofs. These include discovering a partial proof of the Quasi-Riemann Hypothesis and a major special case of the Hodge Conjecture, disproving long-standing conjectures such as the Köthe Conjecture, the Ibragimov–Iosifescu Conjecture, the Discrete Geometry Conjecture, the Crouzeix Conjecture, and the Erdős Unit Distance Conjecture. Superintelligence (SI) agents successfully resolved several open Erdős problems and verified dozens of integer sequence conjectures. SI labs now pair Large Language Models with interactive proof assistants like Lean. The SI acts as the creative engine, guessing lines of a proof, and Lean acts as a fact-checker, telling the SI if a step violates mathematical rules. This feedback loop allows SI agents to autonomously debug their logic until a bulletproof code certificate is produced
The use of instruments with super-human abilities to explore mathematical truths resembles the use of astronomical instruments to explore invisible truths in the sky. In both cases, the limited biological abilities of humans are assisted by hardware to extend the human reach.
There was no way for Galileo Galilei to imagine the discovery of the galaxy GN-z11 by HST. This galaxy existed during the cosmic dawn, 400 million years after the Big Bang, when the infant Universe was only 3% of its current age. Galileo did not know that stars emit light from so far away, nor that the Sun is late to the cosmic party since it formed in the last third of cosmic history. His concept of the world included the solar system and some bright stars near it. That was it.
Similarly, traditional mathematicians used the limited neural network at their disposal in their biological brain to explore mathematical truths in the domain accessible to their brain. As of now, SI helps us resolve open questions that were defined by these same human brains. But in the future, SI could reveal new mathematical truths that the human brain did not even imagine.
Whereas the revolution in our understanding of the cosmos was brought about by centuries of improving astronomical instrumentation, the SI revolution in mathematics takes place rapidly over a period of only months. It took the Vatican 350 years after the death of Galileo to admit that he was right. But it takes Lean at most hours to check math derivations by SI.
What are humans to do under these circumstances?
Mathematicians can learn from the historical precedence of astronomy in how to adapt to emerging technologies. The proper approach is to take advantage of technological assists and change the skills necessary for making breakthroughs. The expertise needed to take advantage of HST data is very different from that employed by Mayan astronomers.
The Physics Nobel Prize was awarded in recent decades to astronomers who did not necessarily have exceptional eyesight like the Mayan astronomers, but instead had the vision to employ the latest technologies to discover the faint radio glow from the Big Bang (here and here), the existence of black holes and neutron stars in the infrared (here), X-rays (here), gravitational waves (here) and neutrinos (here and in 2026 here).
Similarly, the Fields Medal could be awarded in the future to mathematicians who discovered, with the help of SI, mathematical territories that extend well beyond the imagination of traditional mathematicians.
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古代天文学实践已有数千年历史。例如，玛雅人用肉眼观测太阳、月亮、行星和恒星的位置和相对亮度变化长达两千多年。玛雅天文学家在社会中地位崇高，他们作为祭司为统治者提供建议，指导国家大事、婚姻和战争等重大决策。
人眼能够实现在380至720纳米的可见光波段，人类的视觉分辨率约为1角分（1/60度）。由于瞳孔的收集面积有限（3-50平方毫米），人类只能看到相对明亮的可见光源，以满足生存所需。而视力更敏锐的玛雅天文学家则有幸能够识别天空中更暗的光源。玛雅统治者对他们的发现非常感兴趣，他们一定会根据视力的好坏来表彰和奖励他们。
自1610年伽利略·伽利莱开始，天文学家们就利用望远镜观测天空。在伽利略首次观测380年后，哈勃太空望远镜（HST）被发射到地球大气层之外，以减轻大气辉光造成的干扰。模糊。哈勃太空望远镜的主镜集光面积达4.52平方米，约为人眼完全睁开时的10万倍。它的角分辨率比人眼高1200倍，能够探测到比肉眼可见光暗100亿倍的天体。
早期的望远镜探测的是人眼可见频率范围内的电磁波，而它们的广告更先进的后续探测器探测了其他频率，包括射电、红外线、紫外线、X射线和伽马射线波段。此外，宇宙射线、中微子和引力波的探测也为了解宇宙中原本不可见的物理过程提供了新的视角。
在人工智能时代，我们是否正在目睹数学领域中这段历史的加速重演？传统上，数学家们依靠生物大脑来证明或反驳研究猜想。而在过去几个月里，来自 OpenAI、Anthropic 和 DeepMind 的先进推理模型已经生成了数千页手稿，找到了长期存在的数学难题的反例，并将历史上人类的证明用计算机化。其中包括发现了准黎曼 H 定理的部分证明。假设以及霍奇猜想的一个重要特例，推翻了诸如科特猜想、伊布拉吉莫夫-约西费斯库猜想、离散几何猜想、克鲁泽猜想和埃尔德什单位距离猜想等长期存在的猜想。超级智能（SI）代理成功解决了几个未解决的埃尔德什问题，并验证了数十个整数序列猜想。现在，bs 将大型语言模型与 Lean 等交互式证明助手结合使用。系统集成 (SI) 充当创意引擎，猜测证明的步骤，而 Lean 则充当事实检查器，告知 SI 某个步骤是否违反数学规则。这种反馈循环使 SI 代理能够自主调试其逻辑，直至生成万无一失的代码证书。
使用具有超人能力的仪器探索数学真理类似于利用天文仪器探索天空中不可见的真理。在这两种情况下，人类有限的生物能力都需要借助硬件来扩展人类的认知范围。
伽利略·伽利莱绝对无法想象哈勃太空望远镜会发现GN-z11星系。这个星系存在于宇宙黎明时期，也就是大爆炸后4亿年。当宇宙还处于萌芽阶段，只有现在年龄的3%时，伽利略并不知道恒星会从如此遥远的地方发出光芒，也不知道太阳是宇宙历史最后三分之一才形成的，算是姗姗来迟。他眼中的世界只有太阳系和附近的一些明亮恒星，仅此而已。
同样，传统数学家也只能利用当时有限的神经网络资源。人类大脑利用其生物特性来探索其大脑可及领域的数学真理。目前，SI 帮助我们解答了由这些人类大脑提出的未解之谜。但未来，SI 或许能够揭示人类大脑甚至无法想象的全新数学真理。
我们对宇宙的理解发生了革命性的变化，而这种变化是由几个世纪以来天文技术的不断进步带来的。在计量仪器领域，数学领域的国际单位制（SI）革命在短短几个月内就迅速发生。梵蒂冈在伽利略去世350年后才承认他的正确性。但Lean最多只需几个小时就能用国际单位制验证数学推导。
在这种情况下，人类应该怎么办？
数学家可以从天文学的历史先例中学习如何适应环境变化新兴技术。正确的方法是利用技术辅助，并提升自身技能以取得突破。利用哈勃太空望远镜数据所需的专业知识与玛雅天文学家所使用的专业知识截然不同。
近几十年来，诺贝尔物理学奖颁给了那些未必像玛雅天文学家那样拥有超凡视力的天文学家。但他却有远见地运用最新技术发现了宇宙大爆炸产生的微弱射电辐射（此处和此处），发现了红外线（此处）、X射线（此处）、引力波（此处）和中微子（此处和2026年此处）中黑洞和中子星的存在。
同样，菲尔兹奖未来可能会颁发给在国际单位制（SI）的帮助下发现数学奇迹的数学家。主题领域远远超出了传统数学家的想象。
专业网站：
https://lweb.cfa.harvard.edu/~loeb/
https://avi-loeb.medium.com/https://www ... sorAviLoeb
https://open.spotify.com/show/1zhndXkvS ... rofAviLoeb
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原文链接：https://avi-loeb.medium.com/will-mathem ... 529f21cbb3
作者：Avi Loeb（哈佛大学天文系教授）
