Large Language Models for Mathematicians

ChatGPT and similar models can aid professional mathematicians by improving work speed and quality.

Summary of “Large Language Models for Mathematicians”

Major Takeaways

• Large language models (LLMs), such as ChatGPT and GPT-4, have demonstrated the potential to aid professional mathematicians in various tasks, including theorem proving, filling gaps in proofs, acting as a mathematical search engine, and performing simple computations.
• LLMs have shown proficiency in tasks such as defining concepts, naming theorems or definitions, and aiding in proof-checking, while struggling with more challenging problems such as olympiad-problem-solving and upper-undergraduate level mathematics exercises.
• The transformer architecture is the core piece of architecture that powers modern LLMs, allowing them to produce answers to mathematical questions using an autoregressive process.

Introduction

LLMs, such as ChatGPT and GPT-4, have received significant interest for their potential to assist mathematicians in various tasks. This paper explores the extent to which LLMs can aid professional mathematicians and outlines best practices, potential issues, and the mathematical abilities of LLMs.

Overview of Modern Language Models

• Language models have evolved over the years, from word embeddings to the introduction of the transformer architecture, which marked a significant advancement in neural network architectures.
• The transformer architecture enabled the development of models such as BERT and GPT, leading to the democratization of language models with increasing model sizes and training data.

Technical Background

• The transformer architecture operates in an autoregressive manner, where it predicts the next word token based on a given sequence of tokens. It involves tokenization, embedding, positional encoding, self-attention, and prediction layers.
• Training LLMs is a computationally intensive process and involves high energy consumption and CO2 emissions, but specific details on training costs and emissions are often not disclosed by LLM vendors.

LLMs for Mathematics

• LLMs have shown proficiency in tasks like defining concepts, proof-checking, and idea generation, but face challenges with tasks such as theorem proving and complex computations.
• More collaborative approaches, incorporating human expertise, are advisable when using LLMs for mathematical tasks, with potential strategies including using LLMs as a search engine, for idea generation, proof-checking, and collaborative writing.

Measuring LLM Performance on Mathematics

• Empirical studies evaluating LLMs’ mathematical reasoning abilities have demonstrated their strengths and limitations, with advancements in LLM versions leading to improved performance in certain tasks.
• LLMs’ performance varies across different types of tasks, with higher proficiency in simpler tasks and struggles with more challenging problems.

Conclusion

LLMs have shown promise in aiding mathematicians with various tasks, but their limitations, especially in more challenging mathematical problems, highlight the need for a collaborative approach combining human expertise with AI capabilities. The emergence of LLMs presents opportunities and challenges for mathematics education and research.

Appendix

Model	gpt-3.5-turbo-1106
Date Generated	2024-02-26
Abstract	http://arxiv.org/abs/2312.04556v1
HTML	https://browse.arxiv.org/html/2312.04556v1
Truncated	False
Word Count	13153