It’s a thrilling time to be engaged in the mathematical community! The computer programming language Lean is quickly becoming the go-to proof assistant, enabling mathematicians to write and verify proofs with a level of detail never before possible. Collaborations such as Formalising Sphere Packing in Lean demonstrate the unparalleled synergy between artificial intelligence (AI) and human mathematicians. Combined, they make our work with mathematics more innovative and rigorous. Sidharth Hariharan is a first-year PhD student at Carnegie Mellon University. He’s served as the linchpin—driver, negotiator, visionary—to this transforming project that formally kicks into gear in March of 2024.
Lean’s capabilities offer mathematicians a unique opportunity to organize their proofs in a general way that computers can verify for absolute correctness. Indeed, this process is useful not just for affirming the correctness of mathematical claims, but for developing the understanding of deep and far-reaching concepts. Hariharan, who has trained himself through extensive practice using Lean to formalize proofs, mentioned the importance of this relationship. The ambitious project, which has achieved remarkable outcomes so far, shows the promise of AI to help with the human pursuit of mathematics.
The Innovations Behind Lean
In this way, Lean serves as a highly specialized language model. What is sometimes known as a reasoning agent. This new, hybrid framework combines classical natural language reasoning with rich, formalized reasoning. Even Jesse Han, the CEO and cofounder of Math, Inc., which developed Lean, undersells what Lean can do, saying
“It’s a particular kind of language model called a reasoning agent that’s meant to interleave both traditional natural language reasoning and fully formalized reasoning.”
Lean is extremely good at doing literature searches. It’s equally exceptional at ho-hum tasks such as writing code, notes or even running compilers. These functions make it easier for mathematicians to automate the verification of basic facts, so they can concentrate on more challenging aspects of their proofs.
That powerful collaboration between AI and mathematicians was seen in the successful autoformalization of Maryna Viazovska’s 24-dimensional sphere packing proof. This monumental task took the team more than 200,000 lines of code and was done in a matter of two weeks and possible by Lean. This collaborative process raises the bar in our profession. Yet looking beyond those gaps, it vividly demonstrates in the most severe ways how AI could greatly increase human productivity.
Collaboration Between Humans and AI
Hariharan’s experience with Lean has been nothing short of revolutionary for his approach to teaching mathematical concepts. He’s committed to hygienic formalizations of sphere packing proofs. This points to a larger theme in mathematics wherein working with AI delivers real-world dividends. He noted,
“When they reached out to us in late January saying that they finished it, to put it very mildly, we were very surprised.”
This surprising result serves as an example of the potential of AI to assist deep mathematical discovery. Hariharan and his collaborators are harbingers of an exciting new era in mathematical proof verification. In this new partnership, humans and machines operate together more fluidly.
In addition to this, Han went into detail about the challenges they encountered during TNC’s formalization process. Unfortunately, he noted, the 24-dimensions case requires a significantly greater level of work. This is due to the fact that it carries additional subtleties related to the structure of the Leech lattice. He stated,
“And it was actually significantly more involved than the 8-dimensional case, because there was a lot of missing background material that had to be brought online surrounding many of the properties of the Leech lattice, in particular its uniqueness.”
This unusual collaboration produced spot-on results. That made a difference in creating an environment in which mathematicians jumped into learning about and playing with the new AI technology.
The Future of Mathematical Proofs
The implications of these developments are deep and far-reaching for the future of mathematics. As AI evolves, I couldn’t be more optimistic about AI’s role in mathematical research and its future expansion. The integration of AI into mathematical proof verification promises to free mathematicians from some of the more tedious aspects of their work. Han expressed optimism about this shift:
“I think the end result of technology like this will be to free mathematicians to do what they do best, which is to dream of new mathematical worlds.”
The ground-breaking discoveries made via programs such as Formalising Sphere Packing in Lean promise swift development in AI-human partnership. The success stories of identifying and overcoming these errors during these collaborative efforts serve as testaments to the reliability that these systems can elicit. Hariharan noted,
“One of them helped us identify a typo in our project, which we then fixed.”
Gauss has already proven itself a significant research tool, and its creators—from Math, Inc. This momentum heralds an exciting new era for deep, creative mathematical inquiry. Gauss has already shown its usefulness, by autoformalizing Viazovska’s proof and finding typos in published papers using Lean.