Why artificial intelligence could revolutionize mathematics

Editor | Cabbage Leaf

"Proposing a conjecture - a proposition that is suspected to be true, but requires explicit proof - is like to a mathematician are moments of divine inspiration. Mathematical conjectures are more than educated guesses. Counterintuitively, even mathematicians have trouble explaining their own discovery process. The original most transformative field of machine intelligence is Thomas Fink, director of the Institute of Mathematical Sciences in London, UK.

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 a simple artificial intelligence (AI) classifier could predict the ranking of elliptic curves—a measure of their complexity.

Elliptic curve is the basis of number theory, understand it Basic statistics are a key step in solving one of the seven Millennium Puzzles, selected for $1 million each by the Clay Mathematics Institute in Providence, Rhode Island. Few expect artificial intelligence to play a role in this high-stakes field.

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 families of continued fractions—fractions whose denominators are a number plus a fraction, whose denominators are also fractions where a number plus a fraction is a fraction, and so on. Some of these conjectures have been proven, while others remain unresolved.

Paper link: https://www.nature.com/articles/s41586-021-03229-4

other One example relates to knot theory, a branch of topology in which a hypothetical rope becomes tangled together before the two ends stick together. Researchers at Google DeepMind trained a neural network using data from many different knots and discovered unexpected relationships between their algebraic and geometric structures.

Paper link: https://www.nature.com/articles/s41586-021-04086-x

Artificial Intelligence How to make an impact in a 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 will completely overturn the conjecture. For example, the Polya Conjecture states that most integers below any given integer have an odd number of prime factors. But by 1960, it was discovered that this conjecture did not hold true for the numbers 906,180,359. Polya's conjecture was immediately disproven.

Secondly, mathematical data that can train artificial intelligence 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, ...), which grows faster than any computable function. Scientists are already using machine learning tools to search the OEIS database to discover unexpected relationships.

OEIS: https://oeis.org/

Artificial intelligence can help us discover patterns and formulate conjectures. But not all conjectures are consistent. They are also needed to improve our understanding of mathematics. G. H. Hardy explained in his 1940 article "A Mathematician’s Apology" that a good theorem "should be an integral part of many mathematical constructs used to prove many different kinds of theorems."

In other words, the best theorems increase the likelihood of discovering new ones. Conjectures that help us reach new mathematical frontiers are better than those that yield fewer insights. But distinguishing them requires an intuition about how the field itself will develop. This kind of grasp of the broader context will be beyond the capabilities of artificial intelligence for a long time—so the technology will struggle to spot important guesses.

Despite these potential problems, there are many benefits to wider adoption of artificial intelligence tools in the mathematics community. Artificial intelligence can provide decisive advantages and open up new avenues of research.

Mainstream mathematics journals should also publish more conjectures. Some of the most important problems in mathematics—such as Fermat's Last Theorem, Riemann's hypothesis, Hilbert's 23 problems, and Ramanujan's many identities—as well as countless lesser-known conjectures have shaped the development of the field direction. Conjectures point us in the right direction, speeding up 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, synthesizeable and have practical applications. Currently, this is primarily a task for human researchers with broad backgrounds in materials science.

Paper link: https://www.nature.com/articles/s41586-023-06735-9

Same , understanding the output of artificial intelligence tools requires the imagination and intuition of a mathematician. Therefore, AI will only act as a catalyst for human creativity, not a replacement.

Related content: https://www.nature.com/articles/d41586-024-01413-w

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