How mathematicians are incorporating AI assistants into their work

The evolution of artificial intelligence is transforming how mathematicians develop and prove complex mathematical theorems, with new AI co-pilots emerging as powerful collaborative tools that could potentially extend human mathematical capabilities.

Current state of AI mathematics: AI co-pilots are emerging as promising tools for mathematical proof development, with Caltech leading development of advanced systems that can suggest proof steps and validate mathematical reasoning.

• The Caltech team has developed an AI co-pilot that can automatically propose next steps in mathematical proofs and help complete intermediate goals
• This system operates within Lean, a software environment that rigorously validates mathematical statements
• Google’s AlphaProof and AlphaGeometry 2 have already demonstrated silver medal performance at the International Mathematical Olympiad

Technical infrastructure: The underlying technology combines large language models with specialized mathematical validation software to ensure accuracy and reliability.

• The co-pilots are built on similar technology to ChatGPT and Google’s Gemini
• Lean, the software platform housing the AI co-pilot, allows mathematicians to formalize proofs through coding
• Unlike traditional mathematics where human error is possible, Lean automatically checks statements for accuracy

Adoption challenges: Despite the potential benefits, widespread adoption of AI mathematical tools faces several hurdles.

• Many academics haven’t yet embraced Lean due to its coding requirements
• Converting mathematical concepts into code can be time-consuming and requires extensive detail
• Current AI systems, while impressive, haven’t yet reached the capability level needed for complex research mathematics

Future implications: The integration of AI co-pilots could fundamentally transform how mathematical research is conducted.

• AI tools may enable larger groups of mathematicians to collaborate on complex problems
• These systems could help tackle notorious challenges like the Millennium Prize Problems
• David Silver of Google DeepMind suggests that within three years, AI-human partnerships might approach problems as complex as P versus NP

Shifting paradigms: The evolution of mathematical practice through AI assistance points to a fundamental transformation in how mathematical work is conducted.

• Young mathematicians are increasingly embracing AI systems for competitive advantage
• The transition mirrors the revolutionary impact of electronic calculators on mathematical computation
• AI co-pilots could enable a more collaborative approach to mathematical research, challenging the traditional image of mathematics as a solitary pursuit

Strategic horizon: While current AI mathematical assistants show promise, their true potential lies in augmenting human capabilities rather than replacing human mathematicians, potentially leading to breakthroughs in previously intractable mathematical problems.