AI can partner up with human intuition to make new discoveries in mathematics
The first step towards finding a new law, Nobel Laureate Richard Feynman once famously said, is to guess it. It may sound funny, but the statement still rings very true. Feynman was mostly referring to physics, but his statement can be applied to other fields as well — including mathematics. To find new laws or new theorems, this “guessing” (or rather, intuition) plays a key important role. But this intuition isn’t only reserved for humans anymore.
It may come as a shock to those studying school-level mathematics, but research in mathematics is a deeply imaginative and intuitive process, says Geordie Williamson, a Professor of Mathematics at the University of Sydney, one of the authors of a new study that combined human and artificial intuition.
If you want to work on things like the shape of the universe, quantum computing, or what would life be like in 20 dimensions, there’s no straightforward way of doing that — you have to get creative. To do that, mathematicians often look for patterns in existing laws and try to crystallize their ideas and guesses into conjectures (conclusions formed on the basis of incomplete information). They then try to prove these conjectures and turn them into theories and laws.
For decades, mathematicians have used computers to aid them in this process, but it’s mostly been for computation. The most classic example is the Birch and Swinnerton-Dyer conjecture, one of the seven famous Millenium Prize Problems. The conjecture describes the set of rational solutions to equations defining an elliptic curve and was addressed in the 1960s with the help of machine computation. The Four Color Theorem, which states that you don’t need more than 4 colors to color a map so that no two adjacent regions have the same color, was also proven with the aid of computers. Katie Steckles also has an interesting post breaking down the Four Color Theorem in more detail. But we may be at the dawn of a new age, where computers help mathematicians with more than just raw computational power.
Coffee and knots
Machine learning, a type of artificial intelligence (AI) that mostly focuses on predicting outcomes based on patterns, is increasingly considered a potential tool to aid mathematicians. A paper published recently in Nature describes how DeepMind (the company behind the AIs that mastered chess, Go, and shogi) had one of its algorithms look for patterns in the fields of knot theory and representation theory. The algorithm was able to suggest new connections, which mathematicians then examined and proved. Essentially, the algorithm provided the “intuition” part of the research.
The results suggest that computers can now complement mathematical research not just by calculating and computing outcomes, but also by guiding humans’ intuition. Professor Andras Juhasz, of the Mathematical Institute at the University of Oxford and co-author on the paper, commented in a statement:
“Pure mathematicians work by formulating conjectures and proving these, resulting in theorems. But where do the conjectures come from? We have demonstrated that, when guided by mathematical intuition, machine learning provides a powerful framework that can uncover interesting and provable conjectures in areas where a large amount of data is available, or where the objects are too large to study with classical methods.”
“While mathematicians have used machine learning to assist in the analysis of complex data sets, this is the first time we have used computers to help us formulate conjectures or suggest possible lines of attack for unproven ideas in mathematics,” Williamson also said.
Knot theory (as the name implies), is the mathematical study of knots. Mathematically speaking, a knot is an embedding of a circle in 3-dimensional Euclidean space. Initially, the founders of knot theory in the 19th century were eyeing the practical applications of knots. But since then, the field has taken a different direction. Not only have more than 6 billion knots been tabulated, but the field has practical applications in fields like physics or biology. Understanding how to unknot and reknot various types of DNA could shed light on enzymes to do their jobs, and knots can also be used to calculate the orbits of quantum particles.
Thanks to computers themselves, we now have access to more data than any mathematician can reasonably analyze alone, so this line of research could be fruitful for years to come. But it all started with a coffee.
Williamson met Demis Hassabis, chief executive of DeepMind, and the two discussed possible applications of machine learning to mathematics over coffee. “Could machine learning lead to discoveries in mathematics, like it had in Go?”, Williamson recalls pondering. This conversation kickstarted the collaboration.
The algorithm focused on something called Kazhdan-Lusztig polynomials, the building blocks of representations — general relationships that express similarities (or equivalences) between mathematical objects or structures. As the mathematician explains it, representations can be thought of as molecules in chemistry, and in this analogy, Kazhdan-Lusztig polynomials are the atoms that make up the molecules.
Surprisingly, the algorithm trained to predict these polynomials was able to deliver with “incredible accuracy”, Williamson says. The researchers worked with the algorithm’s suggestions, which led them to a new conjecture: Kazhdan-Lusztig polynomials can also be obtained from much simpler objects — mathematical graphs.
In parallel, mathematicians Andras Juhasz and Marc Lackenby at the University of Oxford worked with DeepMind and used a similar approach to break new ground in knot theory; specifically, to uncover a relationship between traits (or “invariants”) of knots.
In the short and medium-term, this approach of blending AI and human work can lead to significant discoveries. But in the long run, using machines for their “intuition” could teach us a bit about intuition, or intelligence itself, Williamson concludes.
“Our paper reminds us that intelligence is not a single variable, like the result of an IQ test. Intelligence is best thought of as having many dimensions.”
“My hope is that AI can provide another dimension, deepening our understanding of the mathematical world, as well as the world in which we live.”