Paper address: https://www.ams.org/journals/bull/2024-61-02/S0273-0979-2024-01828-X/S0273- 0979-2024-01828-X.pdf

This article explores the current performance of AI technology in solving word problems that combine basic mathematics and common sense reasoning. Capabilities and limitations.

The author reviews three methods developed using AI natural language technology: directly giving answers, generating computer programs that solve problems, and generating formalizations that can be used by automated theorem testers Expression.

The author believes that the importance of these limitations in the development of AI technology for pure mathematical research has not yet been clearly defined, but they are extremely critical in mathematical applications and in developing computers that can understand human-written software. Mathematics content is also important during the program.

What is proof in the machine age

Paper address: https://www. ams.org/journals/bull/2024-61-02/S0273-0979-2024-01826-6/S0273-0979-2024-01826-6.pdf

In this article, the author explores the nature of proof and its evolution in the machine age, analyzing it by comparing the values ??in traditional verification and computer verification.

The method ultimately proposed in the article may allow computers to prove successful strategies borrowed from human experience.

Automation allows mathematicians to reflect on their own value

Paper address: https://www .ams.org/journals/bull/2024-61-02/S0273-0979-2024-01825-4/S0273-0979-2024-01825-4.pdf

In the paper, the authors harshly criticize colleagues for their lack of thinking, especially when considering the mechanized future of mathematics, and for ignoring important debates about technology and artificial intelligence at a wider level in society.

Continued fractions in p-adic number field

Paper address: https://www .ams.org/journals/bull/2024-61-02/S0273-0979-2024-01819-9/S0273-0979-2024-01819-9.pdf

Continued fractions It has a long history in number theory, especially in the field of Diophantine approximation.

This article aims to outline the core results of p-adic continued fraction theory, which is a continued fraction defined on the p-adic number field Qp.

The content will start from basic concepts to introduce the latest progress and open issues currently faced.

Tao Zhexuan posted: Machine assisted proof

By the way, Tao Zhexuan also made use of the "Machine assisted proof" of the paper he wrote before.

Paper address: https://terrytao.files.wordpress.com/2024/03/machine-assisted-proof-notices.pdf

In this paper, Tao Zhexuan said that with the help of LLM's ability to process natural language input, they are likely to become a user-friendly platform, allowing mathematicians without specific software knowledge to use advanced tools.

Today, he and many scientists are accustomed to using these models to generate simple code in various languages, including symbolic algebra packages, or to create complex diagrams and images.

Currently, formal proof verification (formal proof verification) relies heavily on human effort, which makes it impractical to fully formalize a large number of current research papers in real time.

In the field of partial differential equations, it is often necessary to go through multiple pages of calculations to estimate integral expressions involving one or more unknown functions (such as the solution of a PDE).

This involves using the bounds of these functions in different function space norms (such as Sobolev space norm), combined with standard inequalities (such as H?lder inequality and Sobolev inequality), and such as Differential identities in integral or integral notation.

Although this type of calculation is a routine operation, it may contain various degrees of errors (such as sign errors). For reviewers, carefully checking these calculations is tedious and time-consuming. , and the calculations themselves hardly provide deeper mathematical understanding or insight other than the final estimate being correct.

It is conceivable that in the future tools may be developed to establish mathematical estimates in an automatic or semi-automatic manner, and replace the current lengthy and uninspiring estimation proofs with a pointing formal proof Link to the certificate.

Going a step further, we might be able to expect that, based on an initial set of assumptions and methods, a future AI tool will be able to come up with the best estimate it can, without having to do a paper first Pen calculations were made to predict what this estimate might be.

At present, it is estimated that the possible state space is too complex to be explored automatically; but with the development of technology, the possibility of realizing such automated exploration is not out of reach.

Once achieved, we will be able to conduct mathematical exploration at a scale that currently seems unfeasible.

Let’s take partial differential equations as an example. Current research usually only studies one or two equations at a time; but in the future, we may be able to study hundreds of equations at the same time.

For example, first develop a complete argument for an equation, and then let the AI ??tool adapt these arguments to a large number of related equation families, if necessary, when the expansion of the argument presents unconventional situations , the AI ??will ask the author questions.

Now, in other areas of mathematics, such as graph theory, the first signs of such large-scale mathematical exploration are beginning to appear.

However, these current preliminary attempts are difficult to promote on a large scale because they rely on AI models with extremely high computational costs or require a large amount of expert-level human participation and supervision.

However, Terence Tao believes that in the near future, we will witness the birth of more innovative machine-assisted mathematical methods.