Tao Zhexuan: ChatGPT has joined my math workflow

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ChatGPT has become the research assistant of the genius mathematician Tao Zhexuan!

More than ChatGPT, he also announced directly online:

A variety of AI tools are incorporated into their workflows.

Recently, Tao Zhexuan has a green eye for AI, and even talks about only one topic on the Internet: AI, especially the application of large language models in mathematical research.

During this period, the "hidden functions" of various ChatGPTs were dug up by Tao Zhexuan:

as big as finding formulas and auxiliary proof theorems; From rewriting essay sentences to querying the pronunciation of mathematical terms in small languages.

And why is there so much attention to AI-assisted work all of a sudden? Tao Zhexuan used his old line of mathematics to make an analogy to AI:

Traditional computer software is like a standard function in mathematics, which is relatively rigid;

AI tools are more like probabilistic functions in mathematics and will be more flexible.

For this analogy, Bao Yungang, a researcher at the Institute of Computing of the Chinese Academy of Sciences, directly praised it very graphically.

Some netizens said:

AI-generated content sometimes really has a "stroke of God" that helps people work better.

However, some netizens accepted Tao Zhexuan's inability to use ChatGPT to assist in mathematical research, after all, for a long time, the public's complaints about ChatGPT focused on mathematical ability.

So, when Tao claimed that ChatGPT could complete some semi-finished work in mathematics, someone asked directly in the comment area:

Are you serious? I'm a huge fan.

Then again, what use can a "math rookie" ChatGPT have in the hands of a big mathematician?

Let's take a look~

ChatGPT is "just right" for academics

In general, Tao Zhexuan roughly means:

Although ChatGPT math ability is not a drop, it is a good tool for divergent thinking for people doing academic research.

(A little unprofessional for ordinary people, but just good for academic staff engaged in mathematics)

So how does this just right scale ChatGPT work?

Tao Zhexuan directly gave several examples of how he solved mathematical problems with ChatGPT:

At first, he threw the questions asked by his colleagues word for word to ChatGPT.

ChatGPT also answered in a mock way, mentioning a highly related term: log-moment generation function, and even discussing a specific example in the answer given.

This term, this example... At first glance, he even deceived Tao Zhexuan's "magic eye", but after checking it, Tao found:

The answer is wrong!

Emmmmm, it stands to reason that the average person's logic ends here—to conclude that ChatGPT is not very good at math.

But Tao Zhexuan did not stop, he carefully analyzed the solution process given by ChatGPT, and found that it was not completely wrong, but still had merit.

For example, ChatGPT uses the lmgf formula in the solution process, while the Legendre transformation of the lmgf formula is used in the standard answer given by Kramer's theorem.

Although it is not the correct solution idea, it is also close to the correct answer.

Then he tried another math problem using the ChatGPT SMS version that his son helped do:

How do I prove that there are infinitely many prime numbers?

Although the proof given was not entirely correct, Tao found that the idea of argument given by ChatGPT could be fixed, and this idea he had never seen before.

This trial directly opened Tao Zhexuan's train of thought.

Since the answers given by ChatGPT on specific mathematical problems are not completely correct, it is better to simply play the characteristics of generating answers that are partially correct:

When dealing with mathematical problems, you can let a large language model such as ChatGPT do some semi-finished semantic search work.

That is, ChatGPT doesn't have to provide exact answers, just generate a few possible hints (similar to balabala for inspiration).

In this way, according to the prompts generated by ChatGPT + traditional search engine search, you can easily find the answer.

Subsequently, Tao Zhexuan also demonstrated it specifically.

First of all, to skillfully throw out a question, suppose Tao wants to find Kummer's theorem but can't remember the name of the theorem, he asks:

I'm looking for a formula about (balabala), it's a classic theory but I can't remember the name, can you give me an answer?

The final ChatGPT answer is the Legendre formula (a relevant result), and based on this answer, Kummer's theorem can be easily found using a traditional search engine.

Then again, since they are only used as a tool in mathematical research, why is AI more suitable in Tao Zhexuan's eyes, while traditional search engines are not very good?

AI "thinking logic" is more divergent

Tao directly analyzed the internal operation logic of traditional computer software and AI tools.

Let's start with traditional computer software, which runs logically like a function: →, this is a very standard mathematical concept.

Specifically, if the input is in a given domain, the software can reliably give a single output in the range (), and if the input is not in the given domain, it cannot give results or give some results.

AI tools, on the other hand, will not be as rigid as traditional computer software, and the logic it runs is not based on classical functions, but is similar to the probability kernel μ:→Pr().

input, AI samples from a probability distribution μₓ and then outputs randomly. And this probability distribution is concentrated near the perfect outcome ().

However, this can also lead to some random biases and inaccurate results.

But in comparison, AI tools still have certain advantages.

On the one hand, it is more flexible and can handle noisy or poorly formatted input more gracefully than traditional software tools.

On the other hand, to a certain extent, the "way of thinking" of AI will also be more divergent.

After announcing the inclusion of AI tools in his workflow, Tao also updated his posts on working with AI on mathstodon.

For example, write an email with AI:

Or discover the highlight of ChatGPT when dealing with mathematical problems: the ability to recognize transliterated versions of mathematical concepts in different languages.

AI semi-automatic proof theorem, reviewers difficult

Tao Zhexuan's observation of AI has also caused a discussion in academic circles.

He pointed out that the experience of reading papers written by AI is completely different from papers written by humans.

Reading a paper written by a person can usually capture some clues in context and style, through which the "meat" of the paper can be quickly separated and the reading speed can be accelerated.

AI-generated mathematical papers, the text looks very convincing, and you must look closely line by line to find the flaws.

New York University professor Marcus interpreted it as saying that peer review is harder from now on.

However, some netizens also believe that it is a good thing for reviewers to read carefully line by line, rather than relying on superficial format signals.

Others are wondering if AI can come up with some entirely new mathematical conjectures.

Whether AI can make a conjecture is unknown, but the AI automatic and semi-automatic proof theorem is already a reality and is moving towards practical use.

Or Tao Zhexuan, who in February organized a seminar on machine learning-assisted proofs at UCLA's IPAM (Institute of Pure and Applied Mathematics), which showcased many cutting-edge achievements in this area.

IBM researcher Jason Rute demonstrated an approach inspired by AlphaGo, comparing theorem proofs to Go, with the next move equivalent to the next hand and proof equivalent to winning.

Google scientist Yuhuai Wu shared the progress of large language models in automatic formalization.

And Tao Zhexuan's own interest in AI does not stop there, and the use of machine learning to solve practical problems is also within his scope.

Just at a lecture in Australia, he also shared how to use mathematics combined with machine learning to predict fire changes.

Reference Links:

[1]https://mathstodon.xyz/@tao/109945628011027107

[2]http://www.ipam.ucla.edu/programs/workshops/machine-assisted-proofs/

[3]https://twitter.com/GaryMarcus/status/1632191991021965313