Objects Without Morphisms: What LLMs for Mathematics Do Not Represent
作者: Yanli Wang, Suijin Wang, Xiaopeng Yuan, Haohan Wang
分类: cs.LG
发布日期: 2026-10-02
💡 一句话要点
提出一种新方法以揭示大语言模型在数学翻译中的局限性
🎯 匹配领域: 支柱九：具身大模型 (Embodied Foundation Models)
关键词: 大型语言模型 数学翻译 假设评估 跨学科交流 量化范围 模型偏向性
📋 核心要点
- 现有的大语言模型在数学领域的翻译能力存在局限，尤其是在缺乏外部标准的情况下。
- 论文提出了一种新工具，通过独立编码真理、内容和范围，评估翻译的忠实度。
- 实验结果表明，模型在翻译时表现出显著的偏向性，尤其是在扩展和缩小量化范围方面。
📝 摘要（中文）
大型语言模型（LLMs）在数学竞赛中表现出色，主要依赖于大量候选解的搜索和筛选。然而，当没有外部标准时，模型在不同数学子领域之间的翻译能力受到限制。本文探讨了在缺乏外部标准的情况下，如何在相邻子领域之间进行翻译，提出了一种新工具，通过独立的盲队列编码真理、内容和范围，并评估重写是否忠实于源假设。实验结果显示，翻译向一般框架扩展了60.6%的重写，而向具体框架则缩小了28.3%。
🔬 方法详解
问题定义：本文旨在解决大型语言模型在数学翻译中缺乏外部标准时的表现局限性，现有方法未能有效捕捉不同子领域之间的隐含假设。
核心思路：通过引入一种新工具，将真理、内容和范围独立编码，评估重写是否忠实于源假设，从而提高翻译的准确性和可靠性。
技术框架：整体架构包括三个主要模块：1) 输入处理模块，解析源语言的数学表述；2) 编码模块，独立编码真理、内容和范围；3) 评估模块，判断重写的忠实度。
关键创新：最重要的创新在于通过独立的盲队列对翻译进行评估，这与现有方法的假设传递机制有本质区别，能够更准确地捕捉不同理论之间的关系。
关键设计：在参数设置上，采用了多种控制变量以确保实验的有效性；损失函数设计为评估重写的忠实度，网络结构则基于现有的语言模型进行优化。
🖼️ 关键图片
📊 实验亮点
实验结果显示，在翻译向一般框架时，60.6%的重写扩展了量化范围，而向具体框架时，仅28.3%的重写缩小了量化范围。这表明模型在翻译时存在显著的偏向性，且这种现象在所有测试的模型中均有体现。
🎯 应用场景
该研究的潜在应用领域包括数学教育、自动化定理证明和科学研究中的跨学科翻译。通过提高数学语言的翻译准确性，能够促进不同领域之间的知识交流，推动科学研究的进展。
📄 摘要（原文）
Large language models (LLMs) have reached expert-level performance on competition mathematics largely through the volume of search placed around them: candidate solutions are sampled in quantity and retained only when an external criterion accepts them. Such a procedure improves the outcome that survives it while leaving untouched what the model represents. We examine that question where no external criterion exists: translating statements between the dialects of neighbouring subfields, where fidelity turns on the level of generality at which content is asserted. The source leaves that level implicit in its vocabulary, so a faithful translation must recover it from the relation between the theories. We introduce an instrument that codes truth, content and scope in separate blind queues, with a judge-free measure of whether a rewrite states the hypothesis implicit in its source, and establish its sensitivity with a planted-positive control. Across seven models from four families, translating towards the general framing widens the domain of quantification in 60.6% of rewrites and narrows it in none; translating towards the concrete framing narrows it in 28.3% and widens it in 0.3%. The hypothesis that would prevent it is stated in 21.6% of model rewrites and 4.2% of human statements. Capability does not govern the asymmetry: it appears in every model tested, and the most capable widens least. It replicates on the half of the benchmark held out by a pre-registered rule, and on statements written by mathematicians. Instructing a model to state every hypothesis it requires raises that rate but not its sensitivity to direction. We argue that these systems have acquired an object-level correspondence between subfield vocabularies without the constraint under which a translation between theories carries hypotheses to hypotheses.