A Severe Misalignment of AI in Mathematics
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.
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.
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.
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.
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.
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.
Artur Avila (Fields Medal 2014)
Manjul Bhargava (Fields Medal 2014)
Caucher Birkar (Fields Medal 2018)
Pierre Deligne (Fields Medal 1978)
Yu Deng (Fields Medal 2026)
Simon Donaldson (Fields Medal 1986)
Hugo Duminil-Copin (Fields Medal 2022)
Alessio Figalli (Fields Medal 2018)
Martin Hairer (Fields Medal 2014)
June Huh (Fields Medal 2022)
Maxim Kontsevich (Fields Medal 1998)
Elon Lindenstrauss (Fields Medal 2010)
Pierre-Louis Lions (Fields Medal 1994)
James Maynard (Fields Medal 2022)
Curtis McMullen (Fields Medal 1998)
Shigefumi Mori (Fields Medal 1990)
Ngô Bảo Châu (Fields Medal 2010)
Andrei Okounkov (Fields Medal 2006)
Peter Scholze (Fields Medal 2018)
Stanislav Smirnov (Fields Medal 2010)
Terence Tao (Fields Medal 2006)
Maryna Viazovska (Fields Medal 2022)
Cédric Villani (Fields Medal 2010)
Wendelin Werner (Fields Medal 2006)
Efim Zelmanov (Fields Medal 1994)
Add your name
If you support this declaration, you are invited to endorse it.
Your endorsement appears right after you sign in with ORCID, or after you confirm your academic email address.
Math and AI
AI 與數學的嚴重失配
在過去幾個月裡，大型語言模型（LLM）的數學能力取得了極其顯著的進步，以至於它們如今已經能夠解決數學許多領域中的重大未解問題。然而，AI 公司將解決數學問題作為能力基準而不斷推進的做法，正在損害數學這門科學，也正在損害數學共同體。AI 公司的目標與數學共同體的目標之間存在著嚴重的失配。我們認為，這也是更廣泛的 AI 對齊問題的一部分；類似問題正在影響其他科學與創造性職業，也正在影響整個社會。
數學研究致力於理解形狀、數以及自然現象中的基本結構。經過一代又一代人的積累，數學建立起了一個龐大的知識體系，其中包含高度精密的思想、方法、抽象概念以及其他用來理解數學世界的工具。反過來，現代技術與科學也建立在這些數學工具之上。
著名的數學問題往往充當地標與燈塔，人們可以藉由它們衡量自己對這片數學世界的理解取得了多大的進展。解決其中一個問題，過去一直是出現新洞見和有趣新方法的可靠標誌。隨後，一個數學家共同體會通過漫長而艱苦的報告、討論與簡化過程，研究這些成果。在這一過程的終點，理想的結果是形成教科書式的表述，使任何研究生，甚至本科生，都可以學習這些成果。有些數學思想還會繼續它們的旅程，在幾十年甚至幾百年之後，成為全體人類都能理解和使用的工具。
數學共同體在許多方面都像是一個微縮的人類社會。它由採取各種不同方法的個體組成，而所有人又由一些共同的核心價值聯結在一起。我們這一職業最寶貴的資源是學生與思想，我們以極大的慎重培育二者。我們認為自己負有責任，讓他們充分成長，直到他們能夠在數學世界中擁有自己的生命。
對於學生，我們經常為他們提出問題，其主要目的在於培養他們的能力，使他們處於有利的位置，能夠在數學研究以及其他領域取得進展。對於我們的思想，我們通過學術報告、私人討論和審慎撰寫的文章加以傳播，並將它們與前人的思想聯結起來。這些過程無一例外都需要時間，而且建立在人與人之間的互動之上。
最近幾個月，AI 成功解決重大數學問題的消息甚至登上了數學圈之外的新聞頭條。然而，解決問題本身只是一種工具，也只是用來衡量我們是否接近真正目標的一項代理指標（proxy）；真正的首要目標，是概念上的理解與洞見。
在 AI 的世界裡忘記這一點，就可能使原本的工具反過來損害真正的目標。事實上，以越來越快的速度大規模生產一個又一個「真／假」陳述，可能摧毀孕育新思想的沃土，而非為新的思想注入生命。
這些解答往往被匆忙公布，沒有留下足夠的時間來完成妥善的書面整理，從中分離、提煉出新的方法與思想，也沒有足夠的時間恰當引用他人的相關先行工作。與所有創造性職業一樣，這也帶來了嚴重的成果歸屬與抄襲問題。
此外，如果沒有願意接手的數學家去照料這些思想的發展，並將它們整合進數學的經典知識體系（mathematical canon），由 AI 構想出的思想就永遠無法真正獲得完整的生命，而數學家與數學家之間至關重要的**人類傳承鏈條（human transmission chain）**也將因此斷裂。
我們正在目睹一種對智識工作的普遍威脅，其表現就是 AI 的使用結果與最初使用它所要達成的目的之間出現失配。
在許多領域與活動中，多年的訓練傳統上並不只是為了產出一個最終答案或產品。這些訓練同時也是為了形成理解力，以及提出新問題、新思想的能力。然而，AI 系統建立在此前人類積累的龐大成果之上，現在已經越來越能夠直接產出這類工作的結果，於是，產出這些結果與從事這些工作的原有目的開始不再對齊。
數學共同體現在面臨的問題，與其他科學和創造性職業正在面臨的問題相似，同時也預示著整個人類可能面臨的問題：
當 AI 改變工作的完成方式時，我們如何確保自己不會忘記，這些工作最初究竟是為了實現什麼？
AI 有潛力增進並加速真正的數學研究與數學理解。數學作為一個專業共同體，也需要以多種方式適應這些變化。然而，這些變化最終究竟會使這個領域受益，還是產生破壞性的效果，在很大程度上將由掌握這項新技術的人類所作出的決定來決定。
這些問題必須得到緊迫的處理。數學共同體需要處理它們，開發這些技術的公司需要處理它們，而更廣泛的社會也需要處理它們，因為整個社會將在其他許多形式的智識工作中遭遇類似的問題。
Artur Avila（2014 年菲爾茲獎）
Manjul Bhargava（2014 年菲爾茲獎）
Caucher Birkar（2018 年菲爾茲獎）
Pierre Deligne（1978 年菲爾茲獎）
Yu Deng（2026 年菲爾茲獎）
Simon Donaldson（1986 年菲爾茲獎）
Hugo Duminil-Copin（2022 年菲爾茲獎）
Alessio Figalli（2018 年菲爾茲獎）
Martin Hairer（2014 年菲爾茲獎）
June Huh（2022 年菲爾茲獎）
Maxim Kontsevich（1998 年菲爾茲獎）
Elon Lindenstrauss（2010 年菲爾茲獎）
Pierre-Louis Lions（1994 年菲爾茲獎）
James Maynard（2022 年菲爾茲獎）
Curtis McMullen（1998 年菲爾茲獎）
Shigefumi Mori（1990 年菲爾茲獎）
Ngô Bảo Châu（2010 年菲爾茲獎）
Andrei Okounkov（2006 年菲爾茲獎）
Peter Scholze（2018 年菲爾茲獎）
Stanislav Smirnov（2010 年菲爾茲獎）
Terence Tao（2006 年菲爾茲獎）
Maryna Viazovska（2022 年菲爾茲獎）
Cédric Villani（2010 年菲爾茲獎）
Wendelin Werner（2006 年菲爾茲獎）
Efim Zelmanov（1994 年菲爾茲獎）
添加你的名字
如果你支持這份宣言，歡迎參與聯署。
聯署本宣言
使用 ORCID 登入後，你的聯署將立即顯示；你也可以使用學術電子郵件地址，在完成確認後顯示聯署。
Math and AI
