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Although many people are reluctant to admit it, it is likely that AI will catch up with human mathematicians within a decade.

A few days ago, a paper by California Institute of Technology and MIT researchers using ChatGPT to prove mathematical theorems exploded, which attracted a lot of attention in the mathematical circle.

Jim Fan, the chief scientist of Nvidia, excitedly retweeted that the AI mathematical Copilot has arrived, and the next one to discover a new theorem is a fully automatic AI mathematician!

The New York Times recently published an article saying that mathematicians are ready that AI will catch up with or even surpass the best human mathematicians within a decade.

Tao Zhe Xuan himself also forwarded this article.

Siobhan Roberts attended the IPAM seminar held by Machine Assisted Proofs this year, and then based on her own experience and interview, she wrote this article about AI and mathematics. AI has also subverted the mathematics community! Today, mathematicians have to face up to the latest revolutionary force-AI.

In 2019, Christian Szegedy, a computer scientist who is a former Google employee and now an employee of Bay area startups, predicted that computer systems would catch up with or surpass the problem-solving abilities of the best human mathematicians within a decade. Last year, he changed the target date to 2026.

Jeremy Avigad, a logician at Carnegie Mellon University, and Akshay Venkatesh, a mathematician at the 2018 Fields Prize at the formal Summer School of Mathematics and a mathematician at the Princeton Institute for Advanced Studies, are not interested in using AI, but he is keen to discuss topics related to AI.

In an interview last year, Venkatesh said, "I want my students to realize that this field is going to change a lot." "

Recently, his attitude is: "I am not opposed to the deliberate or even deliberate use of AI to assist human understanding." But I firmly believe that we need to be mindful and cautious about the way we use it. "

In February this year, the Institute of theoretical and Applied Mathematics at the University of California, Los Angeles, held a seminar on "machine-aided proof."

The main organizer of the seminar is Tao Zhixuan, a mathematician who won the Fields Prize in 2006 and works for UCLA.

He pointed out that using AI to assist mathematical proof is actually a phenomenon worthy of attention.

It was not until recent years that mathematicians began to worry about the potential threat of AI, whether it is the destruction of mathematical aesthetics by AI or the threat to mathematicians themselves.

Outstanding community members are putting these issues on the table and beginning to explore how to "break taboos."

Organizers of summer school, from left to right: Avigad,Patrick Massot and Heather Macbeth from Euclidean geometry to computer code for thousands of years, mathematicians have adapted to the latest advances in logic and reasoning. But are they ready for artificial intelligence?

A portrait of Euclid, a 17th-century Greek mathematician in the Getty Museum in Los Angeles: he is ragged and holds his geometric paper "elements". For more than 2000 years, Euclid's text has been the paradigm of mathematical argument and reasoning.

Jeremy Avigad, a logician at Carnegie Mellon University, said that Euclid began with an almost poetic "definition", based on which he established the mathematics of the time-using basic concepts, definitions and previous theorems, each successive step "clearly followed" the previous steps to prove things in such a way.

Some people complain that some of Euclid's "obvious" steps are not obvious, but Dr. Avigad said the system worked.

But after the 20th century, mathematicians are no longer willing to base mathematics on this intuitive geometry.

Instead, they developed a formal system with precise symbolic representations and mechanical rules.

Https://kilthub.cmu.edu/articles/journal_contribution/A_Formal_System_for_Euclid_s_Elements/6490703 finally, under such a system, mathematics can be translated into computer code.

In 1976, the four-color theorem became the first major theorem to be proved with the help of violent computing.

Four-color theorem: four colors are enough to fill the map so that there is no AI with the same color of two adjacent regions: sorry, I don't understand your theorem there is such a mathematical gadget called proof assistant, or interactive theorem prover.

Mathematicians will convert the proof into code step by step, and then use software programs to check whether the reasoning is correct.

The verification process is accumulated in a dynamic specification reference library that others can refer to.

Dr. Avigad, director of the Hoskinson Center for formal Mathematics at https://www.andrew.cmu.edu/user/avigad/Papers/formal_turn.pdf, said that this type of formalization laid the foundation for today's mathematics, just as Euclid tried to transcode the mathematics of that era to provide the foundation for it.

Recently, the open source proof assistant system Lean has once again attracted a lot of attention.

Lean was developed by Leonardo de Moura, a current Amazon computer scientist, when he was at Microsoft.

Lean uses automatic reasoning, supported by the old-fashioned AI GOFAI, a symbolic AI inspired by logic.

So far, Lean has verified an interesting theorem for rotating a sphere from the inside out, as well as a key theorem for a unified mathematical solution.

However, the proof assistant also has its drawbacks: it often complains that it does not understand the definitions, axioms, or reasoning steps entered by mathematicians, so it is also named "proof complainer".

These complaints can make research cumbersome, but Heather Macbeth, a mathematician at Fordham University, says the ability to provide line-by-line feedback can also make the system useful for teaching.

Https://leanprover-community.github.io/courses.html this spring, Dr. Macbeth designed a "bilingual" course in which she translated every problem on the blackboard into Lean code in handouts, and students were required to submit solutions in Lean and natural language.

Https://hrmacbeth.github.io/math2001/ "gives them confidence," Dr. Macbeth said, because they receive immediate feedback on when it will be done and whether every step along the way is right or wrong.

After attending the seminar, Emily Riehl, a mathematician at Johns Hopkins University, gave it a try.

Emily Riehl, a mathematician at Johns Hopkins University, has been using an experimental proof assistant program. She used Mini Program, a proof assistant, to prove the theorems in her previously published article.

After using it, she was shocked. "I now have a deep understanding of the process of proof, which is much deeper than my previous understanding. My mind is so clear that I can explain it to the stupidest computer. "

Violent reasoning, a group project that students participate in during math formal summer school-- this is not "math." another tool that computer scientists often use to solve some mathematical problems is called "violent reasoning". But the mathematical community often scoffs at this method.

However, AI scientists do not seem to care much about what mathematicians think, constantly using their own familiar methods to capture the mathematical "highland".

Heule, a computer scientist at Carnegie Mellon University, used a 200T "SAT Solver" file to solve the Boolean Pythagorean Triple problem in 2016.

Https://cacm.acm.org/magazines/2017/8/219606-the-science-of-brute-force/fulltext Nature magazine said in the article: 200T proof is the biggest proof process in history, is it really math to use these tools to solve problems?

But in the opinion of the problem-solving author, computer scientist Heule, "this approach is necessary to solve problems that are beyond the reach of human beings." "

Similarly, after defeating AlphaZero in the chess game, DeepMind designed a machine learning algorithm to solve protein folding (AlphaFold).

DeepMind published a paper arguing that the way they achieved these results was to guide human intuition through AI, thus promoting the development of mathematics.

Https://www.nature.com/articles/s41586-021-04086-x and Yuhuai Wu, a former Google computer scientist who is now starting a business in the Bay area, also said that the direction of his business is to use machine learning to solve math problems.

His current project, Minerva, is a fine-tuning language model used to solve mathematical models.

In the future, he hopes that the project will grow into an "automated mathematician" who can act as a general research assistant to "solve mathematical problems independently".

Mathematics is a touchstone on the other hand, many mathematicians who have been deeply exposed to AI technology also worry that AI is not taken seriously in mathematical research.

They believe that artificial intelligence technology can often "directly" help mathematicians "find" the answers they want.

Although mathematicians or AI experts don't know how AI found this answer.

Geordie Williamson, a mathematician who worked with DeepMind, once shared an experience of working with DeepMind.

While he was working with DeepMind, DeepMind discovered a neural network that could predict data values that he thought were important, and were extraordinarily accurate.

He tried very hard to understand how AI did it, because it could be the basis of a theorem.

But in the end, he couldn't understand the logic of AI, and neither could the people of DeepMind.

Like Euclid, neural networks find the truth in some way, but the logical reasons are difficult to understand.

On the other hand, from the mathematician's point of view, reasoning is the essence of mathematics, but it is a jigsaw puzzle that has been missing in machine learning.

In the tech world, if there is a black box that provides solutions in most cases, the tech community will be very satisfied.

AI is such a black box.

But mathematicians will not be satisfied with this situation.

According to the mathematician, trying to understand the principles of neural networks will lead to fascinating mathematical problems.

Solving these problems will make mathematicians "make meaningful contributions to the world".

If AI can prove mathematical theorems, what should we do if the hypothetical theorems generated by AI are all over the world?

Netizens send soul questions about this, and I am skeptical about the new hypothesis / formula of the AI system, because DeepMind has already done it in the knot theory.

I want to know how the community will respond to a large number of new assumptions about AI output. The logical argument created by check artificial intelligence is one thing; it is another to be overwhelmed by millions of "Oh, it may be true" advice. I don't think our existing review and publishing system is ready for this.

What impact will this have on people's trust in mathematics?

Some people think that machines can't do mathematical research quickly, but you can see how it has changed the way it studies, just as machine learning models and computing power have changed the field of biology.

Another netizen said that I have been thinking about this problem since AlphaDev, but the same program can build sorting algorithms or use automatic proof checkers to prove mathematical theorems. The real question is whether it can be used to prove something important, not just a trivial discovery.

However, some netizens are still skeptical about whether GPT-like tools can really find valuable truths.

Some netizens also pointed out that there may be a difference between human beings and AI in understanding and paying attention to mathematics. AI proves what is true, while human beings always pay attention to why it is true.

Reference:

Https://www.nytimes.com/2023/07/02/science/ai-mathematics-machine-learning.html

This article comes from the official account of Wechat: Xin Zhiyuan (ID:AI_era)

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