Why do we need to study math? Once upon a time, the answer was generally something like “well you won’t have a calculator in your pocket all the time!” – but smartphones solved that one. Nowadays, if you ask a professional mathematician to justify their existence, they’ll probably say something about needing human intuition to solve the greatest mysteries of the universe. “A calculator may be able to compute the cube root of 14.7 in under a second,” they’ll say, “but it takes something special to prove the four color theorem.”
Well, sucks to be them, because DeepMind researchers have now created an artificial intelligence capable of proving – and even suggesting – abstract mathematical theorems. DeepMind is the team of computer scientists that made history in 2016 with AlphaGo, the first computer program to successfully defeat a world champion in the game of Go.
“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,” said mathematician Geordie Williamson, co-author of a paper on the AI mathmo that was published today in the journal Nature.
Williamson is a world-renowned representation theorist, meaning he works with mind-bogglingly abstract objects and tries to come up with clever tricks to transform them into merely extremely abstract objects. Basically, if you were searching for an area of math that would be safe from AI, this would be reasonably high on the list.
“Working to prove or disprove longstanding conjectures in my field involves the consideration of, at times, infinite space and hugely complex sets of equations across multiple dimensions,” Williamson explained. “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.”
One of those conjectures, now looking quite a bit more provable than before, involves what’s known as Kazhdan-Lusztig polynomials. These are mathematical expressions that have some pretty deep and fundamental connections to a wide range of abstract math. The conjecture has been unsolved for 40 years – but thanks to DeepMind’s help, Williamson thinks a solution is just around the corner.
That's impressive enough on its own, but incredibly, DeepMind’s talents aren’t limited to simply tidying up human mathematicians’ leftovers. It turns out the artificial arithmetician is quite the prodigy in the field of knot theory – the math of, well, knots (we promise, it’s much more important than it sounds). It has been helping co-authors Marc Lackeby and András Juhász discover and prove an entirely new, never-before-seen, and best of all for a mathematician, surprising theorem that connects algebraic and geometric invariants of knots.
“It has been fascinating to use machine learning to discover new and unexpected connections between different areas of mathematics,” said Lackeby. “I believe that the work that we have done […] demonstrates that machine learning can be a genuinely useful tool in mathematical research.”
So, far from being worried about their future employment, the mathematicians hope that the future will see more collaboration between humans and AI. Intelligence, Williamson explained in exemplary mathematical terms, is “best thought of as a multi-dimensional space with multiple axes: academic intelligence, emotional intelligence, social intelligence.” When you look at it like that, artificial intelligence is just one more axis to explore a problem along, he pointed out – and more axes means more directions from which to approach a tricky problem.
“AI is an extraordinary tool. This work is one of the first times it has demonstrated its usefulness for pure mathematicians, like me,” Williamson said. “Intuition can take us a long way, but AI can help us find connections the human mind might not always easily spot.”