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# Artificial intelligence aids intuition in mathematical discovery

Mathematicians have been developing theories by studying examples throughout history. For instance, by looking at a cube and a pyramid, one might realize that the number of vertices, edges and faces are related. A mathematician recognizes such a pattern, extends it to more-general shapes, and then starts to think about why this relationship might hold. Parts of this process involve computations, for which mathematical software has been useful since it first became available in the 1960s. However, human creativity enables mathematicians to instinctively understand where to look for emerging patterns. Writing in *Nature*, Davies *et al.* now describe a way of using artificial intelligence (AI) techniques to help with the creative core of the mathematical-research process^{1}.

The relationship between the properties of convex polyhedra (3D shapes with flat faces, straight edges and vertices that all point outwards) was found centuries ago, and the formula describing this relationship is named after the Swiss mathematician Leonhard Euler. Regardless of the shape, the number of vertices (*V*) minus the number of edges (*E*) plus the number of faces (*F*) is equal to two: *V* − *E* + *F* = 2 (Fig. 1). Can you arrive at this formula by studying a few examples of different shapes with a pen and paper? In this case, it’s possible, but mathematical ideas that are more complicated require more-extensive computations — for which a computer can be extremely useful.

Mathematical research based on finding and studying examples typically follows a cycle (Fig. 1). First, the researcher identifies a few relevant examples (a cube, a pyramid and perhaps a dodecahedron), then computes some of their properties and analyses the possible relationships between these properties. These relationships are then refined until a pattern emerges. The researcher continues by testing these relationships on more complicated examples (from icosahedra to huge, randomly shaped polyhedra) and discards any properties that aren’t relevant. If the relationships do not hold, or the reasons why they hold remain unclear, the researcher redefines the criteria used to determine which examples are relevant. And the cycle continues.

All except one of the phases in this process require both human creativity and computation. For instance, analysing the properties of the examples chosen involves creativity in identifying which properties might be relevant and then computation to calculate them. The only phase without computational tasks is the refinement step, which could be considered the core of the creative process. This phase requires the researcher to extract general phenomena from concrete examples — based mainly on intuition. In the case of the polyhedra, this step might involve extending the pen-and-paper exercise above to different dimensions: does the pattern also hold for 2D shapes? And what about higher dimensions?

Although AI methods are not yet widespread in mathematical research, in the past few years, several groups have shown that machine-learning tools can, in principle, be used to find relevant examples in large data sets^{2}^{,}^{3}. Others have used such tools to estimate the properties of mathematical objects with high accuracy in efforts to better understand these data sets^{4}. Davies and co-workers have now shown that machine learning can be used to assist researchers in the refinement step of the research cycle, previously regarded as a task mainly based on human intuition. Their approach could, in principle, be used in many different areas of mathematics.

The idea is to identify two structures — perhaps lists of numbers or networks — from the properties of mathematical objects of a certain type. It is then possible to hypothesize that these structures are related in the sense that one structure can give us information about the other. Machine learning is ideally suited to this task for large data sets, because it can use one structure to guess details of the other with greater accuracy than would be expected on the basis of chance.

Euler’s polyhedron formula offers a simple way of illustrating Davies and colleagues’ approach. The first structure in this case would be a list of four numbers representing the number of vertices of the polyhedron, the number of edges, its surface area and its volume. The second structure would be the number of faces. Euler’s formula can then be written as a simple linear relationship between these two structures. The process of arriving at the same formula also makes clear the fact that the volume and surface area are not relevant to this relationship. Applying this approach to Euler’s formula is straightforward, but things become more complicated when the relationships aren’t as simple to identify. In such cases, machine-learning techniques can help.

The real advance was demonstrated when the authors successfully applied their approach to two separate areas of mathematics. They used it to identify previously unknown relationships in knot theory and in combinatorial representation theory. Neither result is necessarily out of reach for researchers in these areas, but both provide genuine insights that had not previously been found by specialists. The advance is therefore more than the outline of an abstract framework. Whether or not such an approach is widely applicable is yet to be determined, but Davies *et al.* provide a promising demonstration of how machine-learning tools can be used to support the creative process of mathematical research.

*Nature* **600**, 44-45 (2021)

*doi: https://doi.org/10.1038/d41586-021-03512-4*

## References

- 1.
Davies, A.

*et al.**Nature***600**, 70–74 (2021). - 2.
Peifer, D., Stillman, M. & Halpern-Leistner, D.

*Proc. Mach. Learn. Res.***119**, 7575–7585 (2020). - 3.
Lample, G. & Charton, F. Preprint at https://arxiv.org/abs/1912.01412 (2019).

- 4.
He, Y.-H.

*The Calabi–Yau Landscape: From Geometry, to Physics, to Machine Learning*(Springer, 2021).

### Competing Interests

The author declares no competing interests.