Alex Davies, Petar Veličković, Lars Buesing
Sam Blackwell, Daniel Zheng, Nenad Tomašev, Richard Tanburn, Peter Battaglia, Charles Blundell, András Juhász, Marc Lackenby, Geordie Williamson, Demis Hassabis, Pushmeet Kohli
2021 · Nature
Abstract The practice of mathematics involves discovering patterns and using these to formulate and prove conjectures, resulting in theorems. Since the 1960s, mathematicians have used computers to assist in the discovery of patterns and formulation of conjectures 1 , most famously in the Birch and Swinnerton-Dyer conjecture 2 , a Millennium Prize Problem 3 . Here we provide examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning—demonstrating a method by which machine learning can aid mathematicians in discovering new conjectures and theorems. We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures. We outline this machine-learning-guided framework and demonstrate its successful application to current research questions in distinct areas of pure mathematics, in each case showing how it led to meaningful mathematical contributions on important open problems: a new connection between the algebraic and geometric structure of knots, and a candidate algorithm predicted by the combinatorial invariance conjecture for symmetric groups 4 . Our work may serve as a model for collaboration between the fields of mathematics and artificial intelligence (AI) that can achieve surprising results by leveraging the respective strengths of mathematicians and machine learning.
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5 selected excerpts · 5 total mentions
I'm thinking this might have broader use than artistic appeal. From what I've heard, knot generation is a young but increasingly important topic in knot theory, since it can be used to generate data to train ML models on, and subsequently (hopefully) discover new algorithms for knot classification. See https://www.nature.com/articles/s41586-021-04086-x for example.
DeepMind published a system that does sort this with a backend theorem prover a year ago. My point is, I don’t think transformer based text prediction systems are the right model here. I could be wrong, but it think about how formal systems work, they seem a far cry from what decoder architectures are doing. https://www.nature.com/articles/s41586-021-04086-x
Personally, I haven't moved the goalposts a millimetre in 30 years, and I won't in future. When a computer does maths - not as a tool wielded by a human mathematician, but in its own right discovers/invents and proves significant new theorems, advancing some area of research mathematics - I'll take seriously the idea that we've reached AGI. Maths in and of itself doesn't require any physical resources. It's possible that doing maths in practice requires knowledge of the world to extract some kind of product from (I'm skeptical, but it's possible), but in principle a rack mounted server could demonstrate its mathematical ability to the world with nothing more than the…
ability to send and receive messages. This hasn't been done so far, not because there are obvious missing prerequsites, or because nobody's tried it, or because it has no value, or because there's a prohibitively high barrier to entry for people to have a go. It hasn't been done because nobody knows how to make a machine be a mathematician, and I've seen little evidence of any progress towards it. That's my goalpost, always has been. Reach it and I'll be overjoyed. And FWIW, I strongly believe it can be reached. I don't see the latest round of ML (or any ML, really) as a step towards it, but I'd love to be proven wrong. When I mention this someone always points at some bit of recent research, such as [1], but it's invariably just a new way for a human mathematician to make use of a computer. If anybody knows of any progress, or serious attempts, towards a true AI mathematician I'm very curious to know. [1] https://www.nature.com/articles/s41586-021-04086-x
Link to the actual paper: https://www.nature.com/articles/s41586-021-04086-x This is a twist on an old theme (just add machine learning!), where, previously, the machines would make the conjectures, and humans would prove them. In this case, it's more along the lines of the machine noticing that certain things are related, which inspires humans to prove that relationship, which eventually leads to the solution of a big, outstanding conjecture. Here is a paper about automated conjecturing in graph theory, if you want to see the other side of the coin. No cool ML stuff, but fun nonetheless: https://www.sciencedirect.com/science/article/pii/S002437950...
Direct links to the papers: https://www.nature.com/articles/s41586-021-04086-x "Advancing mathematics by guiding human intuition with AI" https://arxiv.org/abs/2111.15323 "The signature and cusp geometry of hyperbolic knots" https://arxiv.org/abs/2111.15161 "Towards combinatorial invariance for Kazhdan-Lusztig polynomials"
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