The $1 million prize from the Clay Institute will go to AI. The rules of the mathematics world have changed drastically. How will mathematicians deal with 'massive conjectures' in the future?

In the world of mathematics, a complete proof of "an unproven conjecture" often requires a combination of talent, intuition and experience. Even mathematicians have a hard time explaining their discovery process. .

However, with the rise of large models in recent years, we have witnessed a new force of change. AI not only surpasses humans in predicting the complexity of elliptic curves, but also A breakthrough was made in exploring new formulas for fundamental constants.

Recently, Thomas Fink, Director of the Institute of Mathematical Sciences in London, published an article in the world view column of Nature, exploring how AI plays a role in the field of mathematics. Its unique role and how it helps mathematicians move from conjecture to proof. In this article, Fink mentioned the potential of AI in mathematical reasoning and proof, and its impact on progress in the field of mathematics. Fink pointed out that AI can discover patterns and laws hidden in it through analysis and reasoning of a large number of mathematical problems. For example, through machine learning algorithms, AI can learn from millions of mathematical problems

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Article link: https: //www.nature.com/articles/d41586-024-01413-w

The richness and uniqueness of mathematical data provide fertile soil for AI training: from prime numbers to Knot theory, AI is helping us discover new connections between mathematical objects.

For example, through the Online Encyclopedia of Integer Sequences (OEIS), AI tools can be used to search nearly 375,000 sequences to find those unexpected relationships. The article reveals how AI can innovate in mathematical data. Sail the oceans and discover treasures that humans have yet to touch.

However, although AI has broad application prospects in the field of mathematics, it is not omnipotent.

As G. H. Hardy said in his 1940 paper "A Mathematician's Apology", a good theorem should be an integral part of many mathematical structures. .

AI can help us discover patterns and form conjectures, but distinguishing the importance of these conjectures requires mathematicians’ intuition and a deep understanding of the development of the field.

The author explores how AI can serve as a catalyst for mathematicians' creativity, rather than a substitute, and the two can work together to push and expand the boundaries of mathematics.

Thomas Fink is a researcher at the Institute of Mathematical Sciences in London, a not-for-profit institution engaged in research in physics and mathematics. He is working with BHI on topics such as repairability and recombinant innovation, and his research interests include discrete dynamics, complex networks and fundamental laws of biology.

Mathematics+AI

In 2017, researchers at the Institute of Mathematical Sciences in London, among others Including me, as the director, I began to apply machine learning technology to mathematical data analysis as an exploratory attempt, which also marked the beginning of the initial exploration of the application of artificial intelligence (AI) in the field of mathematics.

During the COVID-19 pandemic, we made an unexpected discovery: a simple AI classifier is able to predict the rank of an elliptic curve (a measure of elliptic curve complexity) .

Elliptic curves are the basis of number theory. The Clay Mathematics Institute once selected seven major mathematical problems in the millennium and provided a prize of US$1 million for each problem. Predicting elliptic curves is a key step in solving these problems, but at the time, few people were optimistic that AI could play a role in the field of mathematics.

In 2021, the Ramanujan machine designed by the researchers generated new formulas for fundamental constants, such as π and e, by exhaustively searching families of continued fractions. fractions) to implement this algorithm, where continued fractions are a special fraction representation, consisting of an infinite number of stacked fractions. The denominator of each fraction is itself a fraction, forming a denominator chain.

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

Some formulas generated by Ramanujan machine have been proven correct by mathematicians, adding new knowledge points to the field of mathematics, but not all formulas have been proven, Some formulas are still unsolved problems facing the mathematical community, waiting to be explored and solved by future mathematicians and AI technology.

Knot theory is a field of topology that studies how lines or ropes are twisted and knotted in space. In this field, we usually consider an idealized rope that is glued at both ends to form a closed loop.

Recently, researchers at Google DeepMind used neural network technology to conduct data analysis on various knots and trained the neural network to identify and understand knot patterns.

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

Most surprisingly, the model discovered a previously unknown connection between the algebraic properties of knots and geometric shapes, meaning that, through the mathematical algebra and geometry With this method, we can gain a deeper understanding of the structure of knots and their properties, which is of great significance to research in fields such as mathematics and physics.

The Impact of AI on the Field of Mathematics

Mathematics is an exact science that does not accept any contingency. Unlike experiments in the real world, mathematics A counterexample is enough to overturn a conjecture.

For example, the Pólya conjecture once held that most integers below any given integer have an odd number of prime factors, but this conjecture was proven wrong in 1960 because of the number 906,180,359 If this condition is not met, it will be falsified with just one counterexample.

In addition to this, the cost of data acquisition in the field of mathematics is relatively low because mathematical objects such as prime numbers and knots are ubiquitous, for example, the Online Encyclopedia of Integer Sequences (OEIS) It contains nearly 375,000 sequences, from the well-known Fibonacci sequence to the rapidly growing Busy Beaver sequence. Scientists have begun using machine learning tools to search the OEIS database to find new mathematical relationships.

Artificial intelligence can also help us discover patterns in mathematics and come up with new conjectures.

But not all conjectures are equally important. A good conjecture should be able to advance our understanding of mathematics, help us build more mathematical structures, and prove different types of theorems. play a role in.

However, to distinguish which conjectures are more valuable requires a deep intuition and understanding of the development of the field of mathematics itself, and a grasp of the overall development of mathematics. For artificial intelligence, it may be It will be difficult to achieve for a long time.

So while AI can help us spot patterns and conjectures, it may still have a long way to go in identifying which conjectures actually matter.

Despite concerns about the application of artificial intelligence in the field of mathematics, the introduction of AI has undoubtedly brought a positive impact to the mathematics community, not only providing key advantages for mathematical research, but also It can open up new research avenues and stimulate innovative thinking.

Mathematical journals should increase the number of publications on mathematical conjectures. Historically, many major mathematical problems, such as Fermat's last theorem, Riemann's hypothesis, etc., as well as many lesser-known conjectures, have greatly promoted the development of the field of mathematics. These conjectures provide researchers with correct answers. Research direction accelerates the process of mathematical research.

Therefore, publishing journal articles about conjectures, especially those with data support or inspiring arguments, is of great significance in promoting scientific discovery.

Take Google DeepMind’s research as an example. Last year they predicted 2.2 million possible new crystal structures. However, the stability, synthesis possibilities and practical application value of these new materials still need to be explored. For further validation and research, the work currently relies primarily on the expertise and understanding of a broad background in materials science by human researchers.

Additionally, the imagination and intuition of mathematicians are critical to understanding and interpreting the results produced by AI tools.

AI plays a role in promoting and stimulating human creativity in this process, rather than replacing humans. It is more like a tool to help mathematicians explore unknown areas faster and discover new mathematical truth.

References:

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

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