Why I didn’t sign the Fields medallists’ letter

为什么我没有签署那封菲尔兹奖得主联名信

When I was around 11 I heard for the first time about Fermat’s Last Theorem. I was immediately captivated by the problem statement, as well as by the accompanying story, and made a fairly serious attempt to prove it. And while, unsurprisingly, I failed, I learned a lot from the attempt. 大约 11 岁时，我第一次听说费马大定理。我立刻被这个问题的陈述及其背后的故事所吸引，并相当认真地尝试去证明它。虽然不出所料，我失败了，但我从这次尝试中学到了很多。

Blissfully ignorant of the fact that the case had been proved by Euler over 200 years earlier, I decided that that would be a good place to start: once I had sorted that out, I was optimistic that I would be ready to tackle the general case. Since I still couldn’t really see where to start, I decided to simplify the problem further and concentrate on successive differences of cubes, with a view to showing that such a difference could not itself be a cube. 当时我并不知道欧拉早在 200 多年前就证明了该情况，我天真地认为这是一个很好的切入点：一旦我搞定了那个，我就乐观地认为自己可以去挑战一般情况了。由于我仍然不知道从何下手，我决定进一步简化问题，专注于立方数的连续差，试图证明这种差值本身不可能是立方数。

At the time I did not know how to express what I was doing in algebraic language, so I did not explicitly try to prove that the Diophantine equation had no solution. Rather, I just worked out some successive differences and stared at them, trying to get some idea of why none of them was a perfect cube. (I should be clear that this story is a reconstruction of what I think probably happened given the few memory traces that remain half a century later rather than a completely reliable account.) 当时我不知道如何用代数语言表达我的做法，所以我并没有明确尝试去证明该丢番图方程无解。相反，我只是计算了一些连续差值并盯着它们看，试图弄明白为什么它们都不是完全立方数。（我需要说明的是，这个故事是我根据半个世纪后仅存的少量记忆碎片重构的，而非完全可靠的记录。）

At some point, I had the idea of taking the difference sequence of the difference sequence, and discovered that it formed an arithmetic progression. That felt like progress, so I investigated difference sequences a bit more and discovered, purely empirically, the rule that if you start with $n$th powers and keep taking successive differences, then eventually you get to the constant sequence $n!$. 在某个时刻，我萌生了对差分序列再求差分的想法，并发现它构成了一个等差数列。这感觉像是取得了进展，于是我进一步研究了差分序列，并纯粹通过经验发现了一条规律：如果你从 $n$ 次幂开始不断求连续差分，最终会得到常数序列 $n!$。

Somehow I never managed to turn this observation into a proof of Fermat’s Last Theorem, and later on my dream of solving it got replaced by other mathematical dreams. However, when I reached the point in my mathematical education where I was taught about taking difference sequences and about what happened to polynomials, I understood those topics much better than I would have if I had not discovered difference sequences for myself and spent happy hours playing around with them. 不知何故，我始终没能将这一观察转化为费马大定理的证明，后来我解决它的梦想也被其他数学梦想所取代。然而，当我的数学教育进展到学习差分序列和多项式性质时，我比那些从未自己发现差分序列并花时间钻研的人，对这些主题的理解要深刻得多。

I mention this story just as an illustration of the phenomenon that was strongly emphasized in this letter signed by 25 Fields medallists, that one learns a lot from thinking about a problem, regardless of whether one solves it. In the end, however, I felt that I could not sign the letter, despite agreeing with much of what it said. 我提到这个故事只是为了说明那封由 25 位菲尔兹奖得主签署的联名信中所强调的现象：无论是否解决问题，思考过程本身就能让人学到很多。然而最终，尽管我同意信中大部分内容，我还是觉得无法签署它。

Instead, it seemed better to do what I did with the Leiden Declaration and set out my own position in a blog post. But it should be understood that by doing that I am not setting myself up as a member of some opposing camp: indeed one of my worries at the moment is that the mathematical community might become bitterly divided, something I would very much like to avoid. 相反，我觉得像对待《莱顿宣言》那样，在博客文章中阐述自己的立场会更好。但需要明确的是，我这样做并不是为了将自己置于某个对立阵营：事实上，我目前担忧的一点是数学界可能会陷入激烈的分裂，这是我非常希望避免的。

Also, I agree on the fundamental point that we are facing a crisis: I just want to offer a slightly different analysis of what that crisis is. I don’t claim full originality for this analysis, as I know that several other mathematicians have already put forward thoughts that are similar to the ones I have, though (for what it’s worth) I have largely come to these conclusions independently. 此外，我同意我们正面临危机这一基本观点：我只是想对这场危机的本质提供一种略有不同的分析。我并不声称这种分析完全原创，因为我知道其他几位数学家已经提出了与我类似的观点，尽管（无论是否有价值）我基本上是独立得出这些结论的。

On the subject of independence, it will perhaps help if I clarify that while I have contacts in the mathematics group at OpenAI, and have also been given early access to some of their models (typically only a few days before they have been released), and have been given free access to their Pro models once released, I have never been paid by OpenAI. I mention this in the hope, perhaps naive, that what I write will not be dismissed for ad hominem reasons. 关于独立性，如果我澄清一下可能会有帮助：虽然我在 OpenAI 的数学团队中有联系人，也曾获得过他们部分模型的早期访问权限（通常只比发布时间早几天），并在模型发布后获得了免费使用 Pro 版本的权限，但我从未从 OpenAI 领取过报酬。我提到这一点是希望（或许有些天真）我所写的内容不会因为人身攻击的原因而被驳回。

Another potential reason for my being regarded as “pro-AI” is that, as I have stated publicly several times, I have a group in Cambridge devoted to automatic theorem proving. However, that is actually more of a reason to be anti-AI, since our group has been trying to attack the problem of getting computers to prove interesting theorems by understanding as well as possible how humans prove interesting theorems, so now that LLMs can clearly do it without the help of such insights as we have had, one of the main motivations for our work has disappeared. 我被视为“亲 AI”的另一个潜在原因是，正如我多次公开声明的那样，我在剑桥有一个致力于自动定理证明的研究小组。然而，这实际上更是一个反 AI 的理由，因为我们小组一直试图通过尽可能深入地理解人类如何证明有趣的定理，来攻克让计算机证明定理的难题。现在，既然大语言模型（LLM）显然可以在没有我们这些见解帮助的情况下做到这一点，我们工作的主要动机之一已经消失了。

To put it another way, we have had to swallow the bitter lesson (which of course we were always aware was a distinct possibility, even if the speed at which it happened has taken us by surprise). I do in fact think that it is still a very interesting and valuable intellectual exercise to try to gain this understanding, even if we can use LLMs as black boxes, but that’s a topic for another blog post. 换句话说，我们不得不吞下这个苦果（当然，我们一直意识到这是一种明显的可能性，尽管它发生的速度让我们感到惊讶）。事实上，我确实认为，即使我们可以将 LLM 作为黑盒使用，尝试获得这种理解仍然是一项非常有趣且有价值的智力练习，但这将是另一篇博客文章的主题。

So why didn’t I sign the letter? Let me extract a couple of sentences from it that express what I see as the principal argument being put forward. 那么，我为什么没有签署这封信呢？让我从中摘录几句，它们表达了我所认为的该信提出的主要论点。

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. “但解决问题只是实现概念理解和洞察力这一主要目标的工具和代理。在 AI 世界中忘记这一点，可能会使工具背离主要目标。事实上，以越来越快的速度大规模生产‘真/假’陈述，可能会摧毁肥沃的土壤，而不是为新思想注入活力。”

Perhaps the main reason I didn’t sign is that I don’t fully subscribe to this view. Instead, I have a more complicated view, which I actually expressed in my essay The Two Cultures of Mathematics a quarter of a century ago, and which can be summarized by saying that there is a spectrum of attitudes in mathematics to the relationship between problem-solving and conceptual understanding. At one end of the spectrum you have mathematicians who are primarily motivated by the wish to solve problems, who see conceptual understanding as a very important means to that end. At the other you have mathematicians who are primarily motivated by the wish to attain conceptual understanding. 我没有签署的主要原因可能是我并不完全认同这一观点。相反，我持有一种更复杂的看法，这实际上我在四分之一个世纪前的文章《数学的两种文化》中已经表达过。可以总结为：在数学界，对于解决问题与概念理解之间的关系，存在着一系列不同的态度。在这一谱系的一端，是那些主要受解决问题愿望驱动的数学家，他们将概念理解视为实现该目标的重要手段；而在另一端，则是那些主要受获得概念理解愿望驱动的数学家。