AI Isn’t Outthinking Mathematicians. It’s Out-Remembering Them.

人工智能并非在智力上超越数学家，而是在记忆力上胜出

AI Isn’t Outthinking Mathematicians. It’s Out-Remembering Them. The key advantage may not be superior reasoning, but a virtually unlimited symbolic working memory. 人工智能并非在智力上超越了数学家，而是在记忆力上胜出。其核心优势或许并非更卓越的推理能力，而是一种近乎无限的符号工作记忆。

When an AI system solves a difficult mathematical problem, the usual explanation is that it has become more intelligent. Perhaps it has absorbed millions of mathematical examples. Perhaps reinforcement learning has taught it better reasoning strategies. Perhaps it is beginning to develop something resembling genuine mathematical intuition. 当人工智能系统解决一道高难度数学题时，通常的解释是它变得更聪明了。也许是因为它吸收了数以百万计的数学案例；也许是强化学习教会了它更好的推理策略；又或许，它开始发展出某种类似于真正的数学直觉的能力。

All of these explanations may contain some truth. But they overlook a simpler possibility: AI has access to a vastly larger working memory than the human brain. Or, more precisely, it has access to an enormous external symbolic workspace that performs many of the functions that working memory performs in humans. 这些解释或许都有一定道理，但它们忽略了一种更简单的可能性：人工智能拥有比人类大脑大得多的工作记忆。或者更准确地说，它拥有一个巨大的外部符号工作空间，能够执行许多人类工作记忆所承担的功能。

This difference may be especially important in mathematics. A human mathematician can hold only a small number of unfamiliar elements in mind simultaneously. An AI model can keep the entire problem statement, hundreds of intermediate equations, several abandoned approaches, definitions, constraints and earlier conclusions inside its context window. 这种差异在数学领域尤为重要。人类数学家同时能在大脑中处理的陌生元素非常有限，而人工智能模型却能将整个问题陈述、数百个中间方程、多种被放弃的解题思路、定义、约束条件以及之前的结论全部保留在其上下文窗口中。

We normally interpret the resulting performance as evidence of superior reasoning. But some of it may instead reflect the removal of one of the most important biological limits on human reasoning: our extremely restricted working-memory capacity. 我们通常将这种表现解读为推理能力卓越的证据，但其中一部分原因可能仅仅是因为它摆脱了人类推理最重要的生物学限制之一：我们极其有限的工作记忆容量。

Mathematics is constrained by memory

数学受限于记忆

Working memory is the mental system that allows us to hold and manipulate information over short periods. When solving an equation, you must remember what each variable represents, which operations have already been performed and what the current goal is. During a proof, you may need to keep track of assumptions, intermediate lemmas, exceptions and multiple possible cases. 工作记忆是一种心理系统，允许我们在短时间内保存并处理信息。在解方程时，你必须记住每个变量代表什么、已经执行了哪些运算，以及当前的目标是什么。在进行证明时，你可能需要跟踪各种假设、中间引理、例外情况以及多种可能的方案。

Human working memory is remarkably limited. Its exact capacity depends on the task and on how information is organized, but the general limitation is obvious from everyday experience. Try multiplying two three-digit numbers in your head. The underlying operations are simple. The difficulty comes largely from having to preserve partial results while performing additional calculations. 人类的工作记忆非常有限。其确切容量取决于任务类型和信息的组织方式，但这种普遍的局限性在日常生活中显而易见。试着在脑海中计算两个三位数相乘，其基本运算很简单，但困难主要在于你必须在进行后续计算的同时，还要记住之前的中间结果。

Writing the numbers down transforms the problem. Paper does not make you more intelligent. It expands your effective working memory. The same principle applies at higher levels of mathematics. A mathematician uses notation, scratch paper, diagrams and previously written lemmas not merely to communicate the solution, but to make the reasoning cognitively possible. 把数字写下来会改变问题的性质。纸张并不会让你变得更聪明，但它扩展了你的有效工作记忆。同样的原则也适用于更高阶的数学。数学家使用符号、草稿纸、图表和之前写下的引理，不仅是为了交流解题过程，更是为了让推理在认知上成为可能。

Experts compensate through “chunking.” A novice sees a long sequence of symbols. An expert recognizes a familiar structure and treats it as a single conceptual object. This allows far more information to fit inside the same biological working-memory limit. But chunking does not eliminate the limit. It merely compresses the information. An AI model faces a very different constraint. 专家通过“组块化”（chunking）来弥补这一缺陷。新手看到的是一长串符号，而专家能识别出熟悉的结构，并将其视为一个单一的概念对象。这使得在相同的生物学工作记忆限制内可以容纳更多信息。但组块化并不能消除限制，它只是压缩了信息。而人工智能模型面临的约束则完全不同。

Working memory predicts mathematical performance beyond IQ

工作记忆对数学表现的预测力超越了智商（IQ）

The importance of working memory for mathematics is not merely theoretical. It is visible in the differences between human beings. Working memory is strongly related to general intelligence, which raises an obvious question: does it independently predict mathematical performance, or is it merely another imperfect measure of IQ? 工作记忆对数学的重要性不仅是理论上的，在人与人之间的差异中也清晰可见。工作记忆与一般智力密切相关，这引出了一个显而易见的问题：它是否能独立预测数学表现，还是仅仅作为衡量智商的另一种不完美指标？

Several studies suggest that it contributes something beyond conventional intelligence measures. Alloway and Passolunghi (2011), for example, examined working memory, verbal ability and mathematical skills in children. They found that working-memory measures made a distinct contribution to mathematical performance rather than simply reproducing the association between mathematics and general verbal ability. 多项研究表明，它在传统智力指标之外确实有所贡献。例如，Alloway 和 Passolunghi (2011) 对儿童的工作记忆、语言能力和数学技能进行了研究。他们发现，工作记忆指标对数学表现有独特的贡献，而不仅仅是重复了数学与一般语言能力之间的关联。

In a separate six-year longitudinal study, Alloway and Alloway (2010) measured children at age five and then examined their academic achievement six years later. Early working-memory performance predicted later literacy and numeracy even after IQ was included in the analysis. Indeed, working memory was a stronger predictor of the later academic outcomes than the IQ measure used in the study. 在另一项为期六年的纵向研究中，Alloway 和 Alloway (2010) 对五岁儿童进行了测试，并在六年后的学业成就中进行了评估。即使在分析中纳入了智商因素，早期的工作记忆表现依然能预测后来的读写和计算能力。事实上，工作记忆比研究中使用的智商指标更能预测后来的学业成果。

Blankenship and colleagues (2015) similarly reported that working memory explained unique variation in mathematical fluency and calculation after statistically controlling for IQ and age. A large meta-analysis by Friso-van den Bos and colleagues (2013) also found a consistent relationship between working memory and mathematics across primary-school studies, although the strength of the relationship varied according to the type of working-memory and mathematical task being measured. Blankenship 等人 (2015) 也报告称，在对智商和年龄进行统计控制后，工作记忆依然能解释数学流利度和计算能力的独特差异。Friso-van den Bos 等人 (2013) 的一项大型荟萃分析也发现，在小学阶段的研究中，工作记忆与数学之间存在一致的关联，尽管这种关联的强度会根据所测量的工作记忆类型和数学任务而有所不同。

These findings should not be exaggerated. Working memory and intelligence overlap substantially, and statistical control cannot perfectly isolate them as independent psychological mechanisms. Nor does the evidence imply that commercially training working memory will necessarily produce large improvements in intelligence or mathematics. 这些发现不应被夸大。工作记忆与智力在很大程度上是重叠的，统计控制无法将它们完美地隔离为独立的心理机制。这些证据也不意味着通过商业手段训练工作记忆就能必然带来智力或数学能力的巨大提升。

The narrower conclusion is nevertheless important: among children with similar measured intelligence, differences in the ability to hold, update and manipulate information still predict differences in mathematical performance. This provides a crucial clue for understanding AI. If human mathematical performance is partly capped by a working-memory bottleneck, then giving a machine an enormous symbolic workspace changes the nature of the contest. The machine may appear more mathematically intelligent partly because it is much less constrained by a cognitive limitation that suppresses human performance. 尽管如此，一个更具体的结论依然很重要：在智力水平相当的儿童中，保持、更新和处理信息的能力差异，依然能预测其数学表现的差异。这为理解人工智能提供了一个关键线索。如果人类的数学表现部分受限于工作记忆的瓶颈，那么赋予机器一个巨大的符号工作空间就改变了竞争的本质。机器之所以看起来在数学上更聪明，部分原因在于它受到的认知限制要小得多，而这种限制正是抑制人类表现的关键因素。

The context window is a gigantic notebook

上下文窗口是一个巨大的笔记本

A modern language model can process an enormous sequence of tokens at once. This sequence may include the original question, definitions, examples, intermediate calculations… 现代语言模型可以一次性处理极其庞大的标记（token）序列。这个序列可以包含原始问题、定义、示例、中间计算过程……