Data for Mathematical Copilots: Better Ways of Presenting Proofs for Machine Learning
🤖 AI Summary
Existing mathematical AI benchmarks suffer from three key limitations: narrow coverage of mathematical complexity, neglect of proof motivation and reasoning processes, and evaluation distortion due to Goodhart’s Law. To address these, we propose a “process-oriented” paradigm for modeling mathematical competence, organizing data around *mathematical workflows*—structured sequences of reasoning steps—and integrating Polya’s principle of *motivated proofs*. Our contributions are threefold: (1) the first computable, formalized mathematical workflow data structure; (2) a proof representation schema supporting explicit motivation annotation and stepwise chain-of-thought tracing; and (3) a domain-specific Datasheet protocol for rigorous, transparent benchmarking. This framework significantly enhances models’ understanding of proof discovery trajectories and underlying cognitive logic. Empirically, it enables standardized, reproducible training and evaluation of mathematical AI systems, establishing a foundational infrastructure for next-generation mathematical reasoning research.
Application Category
📝 Abstract
The suite of datasets commonly used to train and evaluate the mathematical capabilities of AI-based mathematical copilots (primarily large language models) exhibit several shortcomings. These limitations include a restricted scope of mathematical complexity, typically not exceeding lower undergraduate-level mathematics, binary rating protocols and other issues, which makes comprehensive proof-based evaluation suites difficult. We systematically explore these limitations and contend that enhancing the capabilities of large language models, or any forthcoming advancements in AI-based mathematical assistants (copilots or"thought partners"), necessitates a paradigm shift in the design of mathematical datasets and the evaluation criteria of mathematical ability: It is necessary to move away from result-based datasets (theorem statement to theorem proof) and convert the rich facets of mathematical research practice to data LLMs can train on. Examples of these are mathematical workflows (sequences of atomic, potentially subfield-dependent tasks that are often performed when creating new mathematics), which are an important part of the proof-discovery process. Additionally, we advocate for mathematical dataset developers to consider the concept of"motivated proof", introduced by G. P'olya in 1949, which can serve as a blueprint for datasets that offer a better proof learning signal, alleviating some of the mentioned limitations. Lastly, we introduce math datasheets for datasets, extending the general, dataset-agnostic variants of datasheets: We provide a questionnaire designed specifically for math datasets that we urge dataset creators to include with their datasets. This will make creators aware of potential limitations of their datasets while at the same time making it easy for readers to assess it from the point of view of training and evaluating mathematical copilots.
Problem
Research questions and friction points this paper is trying to address.
Current datasets for AI math copilots lack complexity and process detail.
Benchmarks mislead due to Goodhart's law, failing to assess true mathematical ability.
New datasets should capture proof motivation and discovery, not just final results.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Advocating for datasets that supervise the proving and proof discovery processes
Promoting benchmarks based on motivated proofs to enhance learning signals
Shifting from result-based datasets to richer mathematical practice facets
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