Exploring ideas in mathematics involves proposing conjectures and proving them with theorems. Historically, mathematicians have painstakingly constructed proofs line by line, a process that most researchers still use today. However, artificial intelligence is on the verge of revolutionizing this method. AI assistants, known as “co-pilots,” are now assisting mathematicians in developing proofs, offering the potential to solve problems that currently surpass human comprehension.
One particularly promising AI co-pilot is under development at the California Institute of Technology. This co-pilot can suggest next steps in a proof and assist in accomplishing intermediate mathematical objectives, thereby establishing the logical connections between major steps. Animashree Anandkumar, a professor of computing and mathematical sciences at Caltech, emphasized the utility of this co-pilot in providing multiple suggestions for advancing a proof. Notably, all suggestions made by the co-pilot are accurate due to being rigorously screened by Lean software, ensuring the validity of each statement.
The Caltech co-pilot is a large language model (LLM) similar to those powering other AI systems such as OpenAI’s ChatGPT and Google’s AlphaProof. Despite the potential for generating erroneous suggestions, the co-pilot’s recommendations undergo rigorous verification through Lean’s mathematical logic software. This reliability sets AI-assisted proofs apart from traditional informal mathematics, where human oversight can lead to errors being overlooked.
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Proof by Code
Lean, an open-source software increasingly popular among mathematicians, allows for the formalization of mathematics through coding. This approach eliminates the risk of error by automatically verifying mathematical statements. When working with the Caltech co-pilot, mathematicians can request Lean code suggestions to represent their work, enabling a streamlined and error-free proofing process.
Although entering mathematics as code in Lean may be challenging initially, AI co-pilots are expected to simplify this process. By automating the generation of code for obvious mathematical statements, co-pilots like the one at Caltech offer a faster and more efficient proofing experience. As these tools evolve, they may attract a new generation of mathematicians to adopt formal AI-assisted mathematics.
Experts believe that the integration of AI tools in mathematics will usher in a new era of collaboration and problem-solving. AI co-pilots can help break down complex problems into manageable subproblems, allowing larger groups of mathematicians to work together effectively. This collaborative approach, empowered by AI, holds promise for tackling challenging mathematical questions that were previously beyond human reach.
Mathematical Transformations
While AI systems like Google’s AlphaProof and AlphaGeometry 2 have demonstrated impressive capabilities at the International Mathematical Olympiad, they have not yet reached the level required for assisting research mathematicians with complex proofs. However, advancements in AI technology are expected to elevate human mathematicians to a higher level of problem-solving, analogous to the impact of the invention of the electronic calculator.
The future integration of AI co-pilots in mathematics is poised to reshape how mathematicians collaborate. By enabling larger teams to tackle major problems collaboratively, AI tools hold the potential to address long-standing mathematical challenges, such as the Millennium Prize Problems. This transformative shift in mathematical practice, driven by AI, may herald a new era of innovation and discovery in the field.
