[Paper Review] Mathematical Capabilities of ChatGPT

Simon Frieder, Luca Pinchetti+6 more|arXiv (Cornell University)|Jan 31, 2023

Artificial Intelligence in Healthcare and EducationMedicine296 citations

TL;DR

The paper introduces GHOSTS and miniGHOSTS, natural-language datasets to benchmark graduate-level mathematical reasoning in ChatGPT versions (Jan 2023) and GPT-4, showing limited graduate-level proficiency but strong use as a mathematical search/knowledge assistant. It provides a comprehensive evaluation framework and discusses model weaknesses, improvements over time, and practical integration insights for mathematicians.

ABSTRACT

We investigate the mathematical capabilities of two iterations of ChatGPT (released 9-January-2023 and 30-January-2023) and of GPT-4 by testing them on publicly available datasets, as well as hand-crafted ones, using a novel methodology. In contrast to formal mathematics, where large databases of formal proofs are available (e.g., the Lean Mathematical Library), current datasets of natural-language mathematics, used to benchmark language models, either cover only elementary mathematics or are very small. We address this by publicly releasing two new datasets: GHOSTS and miniGHOSTS. These are the first natural-language datasets curated by working researchers in mathematics that (1) aim to cover graduate-level mathematics, (2) provide a holistic overview of the mathematical capabilities of language models, and (3) distinguish multiple dimensions of mathematical reasoning. These datasets also test whether ChatGPT and GPT-4 can be helpful assistants to professional mathematicians by emulating use cases that arise in the daily professional activities of mathematicians. We benchmark the models on a range of fine-grained performance metrics. For advanced mathematics, this is the most detailed evaluation effort to date. We find that ChatGPT can be used most successfully as a mathematical assistant for querying facts, acting as a mathematical search engine and knowledge base interface. GPT-4 can additionally be used for undergraduate-level mathematics but fails on graduate-level difficulty. Contrary to many positive reports in the media about GPT-4 and ChatGPT's exam-solving abilities (a potential case of selection bias), their overall mathematical performance is well below the level of a graduate student. Hence, if your goal is to use ChatGPT to pass a graduate-level math exam, you would be better off copying from your average peer!

Motivation & Objective

• Introduce GHOSTS and miniGHOSTS datasets to evaluate advanced mathematical reasoning in LLMs.
• Benchmark two ChatGPT versions (Jan 9 and Jan 30, 2023) and GPT-4 on diverse, graduate-level problems.
• Identify strengths, failure modes, and practical uses of ChatGPT as a mathematical assistant for professionals.
• Provide a framework to track mathematical progress across model iterations and guide future improvements.

Proposed method

• Create six subdatasets (Grad-Text, Holes-in-Proofs, Olympiad-Problem-Solving, Symbolic-Integration, MATH, Search-Engine-Aspects) to test various mathematical skills.
• Annotate outputs with rating, errorcodes, warnings, and confidence; manually label 1636 expert-rated evaluations.
• Use JSON-formatted data points with prompts and model outputs to analyze capabilities and failure modes.
• Compare two ChatGPT versions (9-Jan-2023 and 30-Jan-2023) and GPT-4 on miniGHOSTS and GHOSTS datasets.
• Employ a thorough testing methodology including warning and error codes to classify failure modes.
• Provide qualitative and quantitative analyses across subdatasets, including cross-domain performance and prompt-engineering effects.

Experimental results

Research questions

• RQ1 How capable are ChatGPT versions and GPT-4 at graduate-level mathematics across diverse tasks?
• RQ2 What are the specific strengths and failure modes of ChatGPT as a mathematical assistant?
• RQ3 Can GPT-4 extend undergraduate math proficiency while ChatGPT struggles at graduate level?
• RQ4 How does model performance evolve over time between Jan 2023 releases and GPT-4?
• RQ5 In what ways can these models best assist professional mathematicians in practice?

Key findings

• ChatGPT versions show limited success on graduate-level tasks, with average ratings around 3.2 and strong weakness in proofs and complex symbolic calculations.
• GPT-4 achieves higher performance on miniGHOSTS, with many perfect ratings, but still falls short of graduate-level mastery on full GHOSTS.
• GPT-4 substantially outperforms ChatGPT, though both remain below the level of a graduate student on many tasks.
• ChatGPT excels as a mathematical search engine and knowledge-base interface for quick fact retrieval and context understanding.
• Prompt engineering yields only marginal improvements for complex tasks, and GPT-4 often provides longer, more rambling answers that can aid or hinder readability.
• Overall, ChatGPT is more suitable as an assistant for lookup and organization rather than as a sole solver for advanced mathematical problems.