Large Language Models for Mathematicians

Abstract: LLMs such as ChatGPT have received immense interest for their general-purpose language understanding and, in particular, their ability to generate high-quality text or computer code. For many professions, LLMs represent an invaluable tool that can speed up and improve the quality of work. In this note, we discuss to what extent they can aid professional mathematicians. We first provide a mathematical description of the transformer model used in all modern LLMs. Based on recent studies, we then outline best practices and potential issues and report on the mathematical abilities of LLMs. Finally, we shed light on the potential of LLMs to change how mathematicians work.

• The paper demonstrates that LLMs can assist in mathematical tasks by leveraging transformer architecture for definitions and computations while facing challenges with advanced problems.
• The research details the inner workings of LLMs, emphasizing the role of embeddings, positional encoding, and self-attention in processing mathematical language.
• The paper advocates applying LLMs under human oversight, suggesting that their integration can enhance proof-checking and collaborative writing in mathematical research.

Introduction

The field of NLP has experienced a significant transformation thanks to the advent of LLMs. These models, which include widely known variants like ChatGPT and their subsequent iterations such as GPT-4, are reshaping how we approach and process language-driven tasks. This paper explores LLMs' mathematical applications, examining their utility and effectiveness in enhancing the work of professional mathematicians. The discussion takes us through an in-depth analysis of these models, their potential implications for the practice of mathematics, and the particular challenges faced in applying them to the field.

Transformer Architecture

The foundation of an LLM such as ChatGPT is the transformer architecture, a model comprised of several layers designed to process sequences of data. In essence, this structure is trained to predict text by considering the input provided, known as the prompt. Through layers of computation involving embeddings, positional encoding, and self-attention, the transformer can handle sequences of tokens (word pieces) and provide contextually enriched outputs. Despite this complexity, there remains a fundamental difference between how an LLM and a mathematician arrive at a solution to a mathematical problem.

Assessing Mathematical Capabilities

When applied to mathematics, LLMs exhibit varied levels of performance based on the complexity of the tasks they are given. Their ability to function as a search engine for definitions and mathematical concepts proves to be one of their strongest suits. However, their competence significantly drops when faced with more demanding questions, such as those from mathematical Olympiads or high-level functional analysis problems. The models also demonstrate a decent ability to handle computations, though with limitations due to their lack of an in-built numerical solver—a gap slowly being bridged by integrating external tools.

Best Practices and Perspectives

LLMs can be utilized in several ways to supplement the work of mathematicians, from proof-checking and collaborative writing to serving as a brainstorming tool. Yet, these approaches are not without their pitfalls. LLMs can produce erroneous proofs, fail to correct them, solve different problems than prompted, and struggle with arithmetic. These limitations suggest that while LLMs can be valuable tools, they should be used in tandem with human oversight and expertise. Future developments may see purpose-built models for theorem proving that could significantly impact mathematical processes and education, although replacing mathematicians remains far from reality.

In conclusion, the exploration of LLMs reveals a technology with promising capabilities and significant scope for further innovation in the mathematical domain. The models' increasing sophistication hints at a landscape where the fusion of artificial intelligence and human insight will likely reshape the future of mathematical problem-solving and research.