One Example Shown, Many Concepts Known! Counterexample-Driven Conceptual
  Reasoning in Mathematical LLMs
Authors: name: Yinghui Li, name: Jiayi Kuang, name: Haojing Huang, name: Zhikun Xu, name: Xinnian Liang, name: Yi Yu, name: Wenlian Lu, name: Yangning Li, name: Xiaoyu Tan, name: Chao Qu, name: Ying Shen, name: Hai-Tao Zheng, name: Philip S. Yu

Leveraging mathematical Large Language Models (LLMs) for proof generation is
a fundamental topic in LLMs research. We argue that the ability of current LLMs
to prove statements largely depends on whether they have encountered the
relevant proof process during training. This reliance limits their deeper
understanding of mathematical theorems and related concepts. Inspired by the
pedagogical method of "proof by counterexamples" commonly used in human
mathematics education, our work aims to enhance LLMs' ability to conduct
mathematical reasoning and proof through counterexamples. Specifically, we
manually create a high-quality, university-level mathematical benchmark,
CounterMATH, which requires LLMs to prove mathematical statements by providing
counterexamples, thereby assessing their grasp of mathematical concepts.
Additionally, we develop a data engineering framework to automatically obtain
training data for further model improvement. Extensive experiments and detailed
analyses demonstrate that CounterMATH is challenging, indicating that LLMs,
such as OpenAI o1, have insufficient counterexample-driven proof capabilities.
Moreover, our exploration into model training reveals that strengthening LLMs'
counterexample-driven conceptual reasoning abilities is crucial for improving
their overall mathematical capabilities. We believe that our work offers new
perspectives on the community of mathematical LLMs.