A Severe Misalignment of AI in Mathematics

数学领域中人工智能的严重错位

I am proud to be among the list of 25 initial signatories — all Fields Medallists — to the declaration below, which grew out of discussions between ourselves over the last week. We have also posted our declaration on this web page, and (similarly to the Leiden declaration) invite further signatures. (It is unfortunate that we did not have the time to have a more consultative process, as with Leiden; but we decided that the urgency of the situation was such that we needed to release a statement sooner rather than later.) See also this recent article in the Economist regarding our declaration.

我很荣幸能成为下方声明的首批 25 位签署人之一——他们全都是菲尔兹奖得主。这份声明源于我们过去一周的讨论。我们已将声明发布在此网页上，并（参照莱顿宣言）邀请更多人签署。（遗憾的是，我们没有时间像莱顿宣言那样进行更广泛的磋商；但我们认为形势紧迫，必须尽快发表声明。）另请参阅《经济学人》近期关于我们这份声明的文章。

Over the last few months, the mathematical capabilities of LLMs have improved dramatically, to the point that they can solve major outstanding problems in many fields of mathematics. However, the push by AI companies to solve mathematical problems as a benchmark is detrimental to the science of mathematics, and to the mathematical community. The goals of the AI companies and the goals of the mathematical community are severely misaligned. We see these as part of broader alignment issues impacting other scientific and creative professions, as well as the whole of society.

在过去几个月里，大语言模型（LLM）的数学能力得到了显著提升，甚至能够解决数学领域中许多重大的未决问题。然而，人工智能公司将解决数学难题作为基准的推动方式，对数学科学和数学共同体是有害的。人工智能公司的目标与数学共同体的目标存在严重的错位。我们认为，这属于影响其他科学和创意行业乃至整个社会的更广泛的对齐问题的一部分。

Research mathematics deals with understanding basic structures of shapes, numbers, and natural phenomena. Over the course of generations, it has built a large corpus of sophisticated ideas, methods, abstractions, and other tools to comprehend the mathematical landscape. In turn, modern technologies and sciences are based on mathematical tools. Famous problems have often served as landmarks and lighthouses against which one can measure an improved understanding of this landscape. Solving one of these problems has been a certain sign of new insights and interesting methods, which would then be studied by a community of mathematicians, through a long and arduous process of talks, discussions, simplifications. At the end of this process, one will ideally find a textbook presentation of the results suitable for any graduate or even undergraduate student to study. Some of the mathematical ideas pursue their journey even further to become, decades or centuries after, tools that are understood and used by the whole population.

数学研究旨在理解形状、数字和自然现象的基本结构。经过几代人的努力，数学界构建了庞大的复杂思想、方法、抽象概念和其他工具体系，以理解数学图景。反过来，现代技术和科学也建立在这些数学工具之上。著名的数学难题往往充当着地标和灯塔，人们以此衡量对这一图景理解的深入程度。解决其中一个难题，通常标志着获得了新的见解和有趣的方法，随后数学共同体会通过漫长而艰苦的讲座、讨论和简化过程对其进行研究。在这一过程的最后，理想情况下，人们会得到一份适合任何研究生甚至本科生学习的教科书式的成果呈现。其中一些数学思想会走得更远，在几十年或几个世纪后，成为被全人类理解和使用的工具。

The mathematical community functions, in many ways, as a miniature version of humanity. It consists of individuals using a wide variety of different approaches, joined by core values. The most precious resources of our profession are students and ideas, and these we nurture with great care. We feel responsible to let them grow to their full potential, until they can live a life of their own in the mathematical world. For students we often suggest problems with the core intention of developing skills making them well-positioned for advances in research and elsewhere. Our ideas we disseminate in talks, private discussions and careful writeups, connecting them to the previous ideas of others. These processes invariably take time and are based on human interaction.

数学共同体在许多方面就像是人类社会的缩影。它由使用各种不同方法的个体组成，并由核心价值观凝聚在一起。我们行业最宝贵的资源是学生和思想，我们对此悉心培育。我们有责任让他们充分发挥潜力，直到他们能在数学世界中独立发展。对于学生，我们通常建议一些问题，其核心意图是培养他们的技能，使他们能够为研究及其他领域的进步做好准备。我们通过讲座、私人讨论和严谨的论文来传播思想，并将它们与他人的既有思想联系起来。这些过程总是需要时间，并基于人与人之间的互动。

In recent months, the success of AI in solving major mathematical problems has made headlines even outside mathematical circles. But solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight. Forgetting this in the world of AI may turn the tool against the primary goal. Indeed, the mass production at faster and faster pace of “true/false” statements could destroy fertile ground instead of breathing life into new ideas. Often these solutions are announced in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work of others. As in all creative professions, this raises severe attribution and plagiarism questions. Moreover, without the willing mathematicians who must take care of their development and integration into the mathematical canon, AI-conceived ideas would never become fully alive and the crucial human transmission chain between mathematicians would be lost.

近几个月来，人工智能在解决重大数学问题上的成功甚至在数学界之外也成为了头条新闻。但解决问题仅仅是实现“概念理解和洞察”这一主要目标的工具和代理。在人工智能世界中忘记这一点，可能会使工具背离其初衷。事实上，以越来越快的速度大规模生产“真/假”陈述，可能会破坏肥沃的土壤，而不是为新思想注入活力。这些解决方案往往被匆忙宣布，没有时间进行适当的撰写、提炼新方法和新思想，也没有引用他人的相关前人工作。正如所有创意行业一样，这引发了严重的归属权和剽窃问题。此外，如果没有愿意负责其发展并将其整合进数学经典体系的数学家，人工智能构思的思想将永远无法真正“活”起来，数学家之间至关重要的人类传承链条也将随之断裂。

We are witnessing a general threat to intellectual work, with misalignment between the outcome of the use of AI and its initial purpose. In many fields and activities, years of training have traditionally served not only to produce a final answer or product, but also to develop understanding and the ability to formulate new questions and ideas. However, building on a vast body of previous human work, AI systems are becoming increasingly capable of producing the results of such work directly, and these goals cease to align. The issues the mathematical community faces now are similar to issues that other scientific and creative professions are facing, and indicate issues that all of humanity might face: how to make sure that, as AI changes the way work is done, we do not lose sight of what that work was meant to achieve in the first place.

我们正在目睹对智力工作的普遍威胁，即人工智能的使用结果与其初衷之间存在错位。在许多领域和活动中，多年的训练传统上不仅是为了产生最终答案或产品，更是为了培养理解力以及提出新问题和新思想的能力。然而，基于人类以往庞大的工作成果，人工智能系统正变得越来越有能力直接产出这些工作的结果，而这些目标不再对齐。数学共同体现在面临的问题，与其他科学和创意行业面临的问题类似，这也预示着全人类可能面临的问题：当人工智能改变工作方式时，我们如何确保不偏离工作最初旨在实现的目标。

AI offers the potential of enhancing and accelerating genuine mathematical study and understanding. Mathematics as a profession will need to adapt to these changes in several ways. However, whether these changes ultimately benefit the field or have a destructive effect will in large part be determined by the decisions of the humans in control of this new technology. These issues must be addressed urgently, in the mathematical community, by the companies developing these technologies and, more broadly, by a society that will confront similar problems in many other forms of intellectual work.

人工智能提供了增强和加速真正数学研究与理解的潜力。数学作为一门职业，将需要在多个方面适应这些变化。然而，这些变化最终是有益于该领域还是具有破坏性，很大程度上将取决于掌控这项新技术的决策者。这些问题必须在数学共同体内部、在开发这些技术的公司中，以及更广泛地在将面临许多其他形式智力工作类似问题的社会中，得到紧急解决。