Thinking Machines: Mathematical Reasoning in the Age of LLMs

Abstract

Large Language Models (LLMs) have demonstrated impressive capabilities in structured reasoning and symbolic tasks, with coding emerging as a particularly successful application. This progress has naturally motivated efforts to extend these models to mathematics, both in its traditional form, expressed through natural-style mathematical language, and in its formalized counterpart, expressed in a symbolic syntax suitable for automatic verification. Yet, despite apparent parallels between programming and proof construction, advances in formalized mathematics have proven significantly more challenging. This gap raises fundamental questions about the nature of reasoning in current LLM architectures, the role of supervision and feedback, and the extent to which such models maintain an internal notion of computational or deductive state. In this article, we review the current state-of-the-art in mathematical reasoning with LLMs, focusing on recent models and benchmarks. We explore three central issues at the intersection of machine learning and mathematical cognition: (i) the trade-offs between traditional and formalized mathematics as training and evaluation domains; (ii) the structural and methodological reasons why proof synthesis remains more brittle than code generation; and (iii) whether LLMs genuinely represent or merely emulate a notion of evolving logical state. Our goal is not to draw rigid distinctions but to clarify the present boundaries of these systems and outline promising directions for their extension.