Data for Mathematical Copilots: Better Ways of Presenting Proofs for Machine Learning

Abstract: The suite of datasets commonly used to train and evaluate the mathematical capabilities of AI-based mathematical copilots (primarily LLMs) 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 LLMs, 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.

• The paper demonstrates that existing datasets overlook advanced mathematical complexities by relying on binary evaluations of final results.
• It introduces a methodology incorporating mathematical workflows and motivated proofs to better mirror the iterative nature of research.
• The study proposes math datasheets as a tool for dataset creators, aiming to elevate AI systems' ability to assist in mathematical discovery.

Essay: Data for Mathematical Copilots: Better Ways of Presenting Proofs for Machine Learning

In the paper titled "Data for Mathematical Copilots: Better Ways of Presenting Proofs for Machine Learning," the authors address significant limitations in the datasets used for training AI-based mathematical assistants, particularly LLMs. The authors argue that the current datasets do not adequately capture the complexity and richness of mathematical research practices, which limits the capabilities of these AI systems.

Key Issues with Existing Datasets

The authors systematically examine the shortcomings of the existing datasets used for evaluating the mathematical capabilities of AI systems. These datasets typically focus on undergraduate-level mathematics and employ binary rating protocols for evaluation. The authors identify several critical issues:

1. Limited Scope: The datasets often cover only elementary mathematics up to lower undergraduate levels, neglecting more complex and advanced mathematical concepts.
2. Binary Evaluation: Current benchmarks predominantly use binary evaluation metrics, which are not sufficient for capturing the nuances of mathematical problem-solving and proof generation.
3. Result-based Datasets: Most datasets are designed around final results, such as theorem statements and proofs, rather than the processes leading to these results.

Proposed Improvements

The authors propose a shift in the design of mathematical datasets and evaluation criteria:

• Inclusion of Mathematical Workflows: They advocate for datasets that incorporate mathematical workflows — sequences of atomic, often subfield-dependent tasks involved in mathematical research. This aims to reflect the iterative and exploratory nature of mathematical practice.
• Motivated Proofs: The concept of "motivated proof," introduced by George Pólya, is recommended as a foundation for new datasets. Motivated proofs provide a learning signal that helps alleviate some of the limitations of current datasets by making the reasoning process explicit.
• Math Datasheets: The authors introduce math datasheets (extensions of the general datasheet concept) that include a questionnaire specifically for math datasets. This tool is intended to help dataset creators identify potential limitations and improve the quality of the datasets.

Implications and Future Developments

The implications of this research are profound both practically and theoretically. Practically, better datasets can enhance the training of AI systems, enabling them to function as more effective mathematical copilots. Theoretically, this shift can lead to a deeper understanding of how machines can emulate human reasoning and creativity in mathematics.

The authors suggest that future AI developments could benefit significantly from datasets that better emulate the rich facets of mathematical practice. They envision a future where AI not only provides correct proofs but also enhances mathematicians' understanding and assists in mathematical discoveries. This evolution could result in systems that not only solve problems but also generate new mathematical theories and theorems autonomously.

The paper highlights the need for collaboration between AI practitioners and mathematicians to align AI tools with the practical needs of mathematicians. The authors underscore that without proper datasets, the goal of creating AI mathematicians remains out of reach. They conclude by encouraging the community to create more diverse and comprehensive datasets that better capture the full scope of mathematical research and practice.

In sum, this paper presents a compelling case for rethinking how datasets are constructed and evaluated for AI systems in mathematics. By moving beyond result-based datasets and focusing on the processes of mathematical discovery, we can significantly advance the capabilities of AI as a tool for mathematicians.