Conjectures have historically played an important role in the development of pure mathematics. We propose a systematic approach to finding abstract patterns in mathematical data, in order to generate conjectures about mathematical inequalities, using machine intelligence. We focus on strict inequalities of type f < g and associate them with a vector space. By geometerising this space, which we refer to as a conjecture space, we prove that this space is isomorphic to a Banach manifold. We develop a structural understanding of this conjecture space by studying linear automorphisms of this manifold and show that this space admits several free group actions. Based on these insights, we propose an algorithmic pipeline to generate novel conjectures using geometric gradient descent, where the metric is informed by the invariances of the conjecture space. As proof of concept, we give a toy algorithm to generate novel conjectures about the prime counting function and diameters of Cayley graphs of non-abelian simple groups. We also report private communications with colleagues in which some conjectures were proved, and highlight that some conjectures generated using this procedure are still unproven. Finally, we propose a pipeline of mathematical discovery in this space and highlight the importance of domain expertise in this pipeline.

翻译：猜想在纯数学的发展历史上一直扮演着重要角色。我们提出了一种系统性的方法，用于在数学数据中发现抽象模式，从而利用机器智能生成关于数学不等式的猜想。我们聚焦于形如 f < g 的严格不等式，并将其与一个向量空间关联起来。通过对这个空间进行几何化处理——我们将其称为猜想空间——我们证明了该空间同构于一个巴拿赫流形。通过研究该流形的线性自同构，我们发展了对猜想空间的结构性理解，并表明该空间容许若干自由群作用。基于这些洞见，我们提出了一种利用几何梯度下降生成新猜想的算法流程，其中度量由猜想空间的不变性所决定。作为概念验证，我们给出一个示例算法，用于生成关于素数计数函数和非阿贝尔单群凯莱图直径的新猜想。我们还报告了与同事的私下交流，其中部分猜想已被证明，并强调使用此流程生成的某些猜想仍未得到证明。最后，我们提出了该空间中的数学发现流程，并强调了领域专业知识在这一流程中的重要性。