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"Coming up with a conjecture – a proposition that is suspected to be true, but requires unequivocal proof – is like a moment of divine inspiration for mathematicians. Mathematical conjecture is more than just educated guesswork. Formulating them requires a combination of genius, intuition, and experience. Even mathematicians have a hard time explaining their own discovery. Counterintuitively, however, I think this is the most transformative area of machine intelligence in the beginning." Thomas Fink, director of the Institute of Mathematical Sciences in London, UK, said.
In 2017, researchers at the Institute of Mathematical Sciences in London began applying machine learning to mathematical data as a hobby. During the COVID-19 pandemic, they found that simple artificial intelligence (AI) classifiers could predict the ranking of elliptic curves – a measure of their complexity.
Link to paper: https://arxiv.org/abs/2204.10140
Elliptic curves are the foundation of number theory, and understanding its underlying statistics is a critical step in solving one of the seven millennial puzzles, selected by the Clay Institute for Mathematics in Providence, Rhode Island, with a prize of $1 million each. Few expect AI to play a role in this high-risk area.
Artificial intelligence has made progress in other areas. A few years ago, a computer program called the Ramanujan Machine produced new formulas for fundamental constants, such as π and e. It does this by exhaustively searching for a family of consecutive fractions – the denominator is a fraction of a number plus a fraction, its denominator is also a fraction of a number plus a fraction, and so on. Some of these conjectures have already been proven, while others remain open.
Link to paper: https://www.nature.com/articles/s41586-021-03229-4
Another example is related to knot theory, which is a branch of topology in which a hypothetical string is tangled together before the ends stick together. Researchers at Google DeepMind trained a neural network using data from many different knots and discovered an unexpected relationship between their algebra and geometry.
Link to paper: https://www.nature.com/articles/s41586-021-04086-x
How can AI make an impact in the field of mathematics where human creativity is considered crucial?
First of all, there are no coincidences in mathematics. In real-world experiments, false negatives and false positives abound. But in mathematics, a counterexample can completely disprove a conjecture. For example, the Polya conjecture states that most integers below any given integer have odd prime factors. But in 1960, it was discovered that this conjecture did not hold true for the numbers 906,180,359. Polya's conjecture was immediately falsified.
Second, the mathematical data on which AI can be trained is cheap. Prime numbers, knots, and many other types of mathematical objects are abundant. The Online Encyclopedia of Integer Sequences (OEIS) contains nearly 375,000 sequences — from the familiar Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...) to the powerful Busy Beaver sequence (0, 1, 4, 6, 13, ...) that grows faster than any computable function. Scientists are already using machine learning tools to search OEIS databases to uncover unexpected relationships.
OEIS：https://oeis.org/
AI can help us spot patterns and form conjectures. But not all conjectures are consistent. They also need to improve our understanding of mathematics. In his 1940 article, A Mathematician's Apology, G. H. Hardy explained that a good theorem "should be an integral part of many mathematical constructs and used to prove many different types of theorems."
In other words, the best theorems increase the likelihood of discovering new theorems. Conjectures that help us reach new frontiers in mathematics are better than those that produce fewer insights. But distinguishing them requires an intuition of how the field itself will develop. This grasp of the broader context will be beyond the capabilities of AI for a long time to come – so it will be difficult for the technology to uncover important conjectures.
Despite these potential problems, there are many benefits to the wider adoption of AI tools in the mathematical community. AI can provide decisive advantages and open up new avenues for research.
Mainstream mathematics journals should also publish more conjectures. Some of the most important problems in mathematics – such as Fermat's theorem, the Riemann hypothesis, Hilbert's 23 problems, and Ramanujan's numerous identities – as well as countless lesser-known conjectures have shaped the direction of the field. The conjecture points us in the right direction, which speeds up the research. Journal articles on conjectures supported by data or heuristic arguments will accelerate discovery.
In 2023, researchers at Google DeepMind predict that 2.2 million new crystal structures will emerge. But it remains to be seen how many of these potential new materials are stable, synthesizable, and have practical applications. At present, this is primarily the task of human researchers, who have a broad background in materials science.
Link to paper: https://www.nature.com/articles/s41586-023-06735-9
Similarly, understanding the output of AI tools requires the imagination and intuition of mathematicians. As a result, AI will only act as a catalyst for human creativity, not a substitute.
Related content: https://www.nature.com/articles/d41586-024-01413-w