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Direction: GPT-4 Advanced Tuning, Source: Qubit
Sister Goose, Tao Zhexuan, a genius mathematician , can no longer do mathematics research without the "mathematics chicken" GPT in the hands of ordinary people !
Just below one of his latest math problems, Terence Tao clearly states that he "used GPT-4," which suggested a possible solution to him.
With the help of GPT-4, he not only successfully broke through this problem, but also shared the answer on MathOverflow:
It provides me with the final idea of solving the problem, and then I just need to continue to calculate.
In order to share the convenience of working with GPT-4 to more mathematicians, Tao Zhexuan also posted his chat records, which fully recorded his conversations with GPT-4.
It can be seen that in this chat record, he called GPT-4 a "professional mathematics collaborator", not just an ordinary mathematics assistant.
This identity is really unusual. I don’t know if Tao Zhexuan will list GPT-4 as a co-author (manual dog head) when he writes the paper.
Not only that, but by poking the "continue this conversation" button at the bottom of the conversation record, you can also import the conversation record into your own ChatGPT with one click, and suddenly break into the conversation between the human master and the AI.
What new problems did Terence Tao solve?
The question Terence Tao studied was a question that was updated on the mathematics website MathOverflow two days ago.
This problem is called "A301897 of Elegant Recursion", and it looks like this:
Among them, A301897 is a sequence included by the integer sequence website OEIS:
In this sequence, there are a series of numbers from the arrangement b of length n. They have one thing in common, that is, they satisfy the Diaconis-Graham inequality and the equality sign holds.
The Diaconis-Graham inequality was jointly proposed by Persi Diaconis and Ge Liheng (yes, Ronald Graham, the father of the "maximum number") in 1977, which belongs to the research of probability and statistics.
Obviously, this long question not only contains a lot of professional mathematical formulas , but also involves web page information that needs to be redirected .
As a "mathematics chicken" that is not very good in mathematics, can GPT-4 really read formulas as quickly as a professional mathematician and give professional advice on solving problems?
In fact, compared to directly copying and pasting the question to GPT-4, Terence Tao did three things :
First of all, do some prompt engineering to clarify the "identity" and "target" of GPT-4. Among them, the identity is "a mathematical cooperation expert who is good at giving technical advice", and the goal is to "provide some advice" rather than solve problems.
Then, Tao Terence chose to convert the problem description into Markdown format instead of directly copying large sections of formulas into the GPT-4 question column (it can indeed be done) .
Finally, when it comes to content that requires GPT-4 to find information by itself, Terence Tao turns it into a reference note and gives the URL at the end.
The complete questioning process looks like this:
In other words, rather than letting GPT-4 directly become a mathematician to "solve problems", Tao Terence just regards it as an "expert" who provides a source of inspiration and suggests some research directions for himself.
GPT-4 quickly gave 8 specific suggestions:
1. Understand the problem: Before diving into the proof, make sure you understand each item in the problem. If you're not familiar with the Diaconis-Graham inequality, study it to see what each term in the expressions a(n) and R(n, q) represents.
2. Simplify expressions: Sometimes, complex expressions can be simplified. For example, you might want to see if the binomial coefficient of a(n) in the expression can be simplified, or if the sum can be computed in a simpler way.
3. Induction: Since R(n, q) is defined recursively, a natural way to prove the conjecture is to use mathematical induction. You might want to prove that if this conjecture holds for n, then it also holds for n+1.
4. Generating functions: Another way is to use generating functions. Try to find the generating functions of a(n) and R(n, 0) and prove that they are equal. This can sometimes turn a complex combinatorial problem into a simpler algebraic one.
5. Asymptotic analysis: As you mentioned, analyzing the asymptotic behavior of a(n) and R(n, 0) may give some insight into their relationship. This can sometimes be done using Stirling's factorial approximation, or by analyzing the dominant term in the expression.
6. Numerical evidence: Although not a proof, collecting numerical evidence can sometimes give us an idea of how to prove a conjecture. Since you have the PARI/GP codes and can check conjectures numerically, you might want to read the first few terms of each sequence to see if you can spot any patterns.
7. Look for similar problems: Sometimes a problem or conjecture is a generalization or variation of a known problem. Look for similar problems or theorems in the combinatorial literature and see if their proofs can be applied to this problem.
8. Seek feedback and collaboration: Don't hesitate to ask for feedback from colleagues, or post some of your results on forums like MathOverflow. Collaboration and peer review are often key to solving complex mathematical problems.
In the end, Terence Tao successfully solved the problem using suggestion 4, which is the method of generating functions .
Of course, the process of solving the problem does not need to use GPT-4, it can be done manually by using MAPLE worksheet + pen and paper. Tao Terence also tried the method of asymptotic analysis (suggestion 5), but it turns out that it is more effective to use generating functions.
To put it simply, Tao Zhexuan just uses GPT-4 to provide himself with different problem-solving ideas, that is, "problem-solving inspiration", but how to solve the specific problem, that is, a more mathematical method, he will not give it to GPT-4 to complete .
As for the application of GPT-4 such as "checking information", Tao Zhexuan has also brought it to the extreme.
For example, when thinking about a problem, he will also take out his "snapshot" and discuss it with GPT-4. For example, he thinks this problem is similar to Catalan numbers and hopes that GPT-4 can help him Find the corresponding information.
GPT-4 quickly gave a corresponding answer, which also prompted Tao Zhexuan to have new inspiration for another question.
To put it simply, Tao Zhexuan demonstrated the correct posture of mathematicians using GPT-4 in just two short conversations with GPT-4 - looking for inspiration and looking up information.
In this way, even "mathematical chickens" such as GPT-4 can become AI assistants for mathematicians.
How does the boss play GPT
In addition to sharing the chat records of human masters and AI, Terence Tao’s mastodon blog post also comes with a thoughtful guide, which is his experience in using ChatGPT and GPT-4.
According to his past practical experience, the most important first point:
Don't try to get the AI to answer questions directly, as that will almost certainly result in some professional-looking crap .
In order to prevent GPT from becoming the king of nonsense literature, the effective solution is as follows:
Let the AI play the role of a collaborator, and then let it provide strategic advice .
Like this：
In addition, what is the use of "mathematical chicken" GPT in the hands of great mathematicians?
Tao Zhexuan probably meant Aunt Jiang's:
Although ChatGPT's mathematical ability is not good, it is a good tool for people who do academic research to diverge their thinking .
(A bit unprofessional for the average person, but just fine for academics doing math)
How to explain the phrase "divergent thinking"?
The point expressed by Terence Tao is that since the answers given by ChatGPT on specific mathematical questions are not completely correct , it is better to simply use the partially correct characteristics of the answers it generates.
In short, let it help you find inspiration balabalabla:
When dealing with mathematical problems, large language models such as ChatGPT can be used to do some semi-finished semantic search work .
That is, ChatGPT does not provide exact answers, but only generates some possible hints.
In this way, based on the tips generated by GPT + traditional search engine search, the answer can be easily obtained.
And he also revealed that before the release of GPT-4 , he himself obtained access qualifications from Microsoft .
That is to say, it is the same paragraph as in Microsoft's 154-page "Spark of AGI", a full-blooded version that has not undergone security training but is more capable .
From Tao Zhexuan’s feedback, we can see that GPT-4 is very good at cosplay when talking to humans, such as acting as a sympathetic listener, enthusiastic feedback, creative source of inspiration, translator or teacher, Or the Devil's Advocate.
At the same time, Terence Tao made a bold but rigorous prediction about the performance of AI in mathematics research:
When integrated with tools such as formal proof verifiers, Internet searches, and mathematical notation packages, the AI of 2026, if used properly, will be a trusted co-author of research in mathematics, but also in many other fields .
In addition to mathematical research, GPT-4 is already an all-round assistant in Tao Terence's life.
He often uses GPT-4 to answer random, vaguely worded questions that previously required fine-tuning keywords in search engines.
There is also a colleague who is depressed because a relative has received a severe diagnosis. For this reason, Tao Zhexuan waved his big hand and asked GPT-4 to write a condolence letter eloquently.
The results of it? The colleague had tears in his eyes and was moved to cry.
Finally, let’s talk about Terence Tao’s use of GPT-4 to solve mathematical problems.
Under MathOverflow, some netizens think that he should not use GPT to answer math questions, and it feels like a very sensitive topic.
But some people still expressed Zici, saying that they think it is really Taiku spicy~
Tao Zhexuan did not hesitate to stand up and express his position. He didn't think there was anything wrong with it:
The current worry is no different from the focus of everyone's discussions in the early days of Wikipedia...
Now I get the initial thread on Wikipedia, and attach a link when citing it as an argument, showing that it is part of my argument, and it is everyone Things that children are used to.
And Tao Zhexuan's opinion is quite firm, that is, "I believe everyone will feel that there is nothing inappropriate to use GPT to support research in the future" ~
Join Terrance Tao in conversation with GPT-4:
https://chat.openai.com/share/53aab67e-6974-413c-9e60-6366e41d8414
参考链接：
[1]https://mathoverflow.net/questions/449361/elegant-recursion-for-a301897
[2]https://mathstodon.xyz/@tao/110601051375142142
[3]https://finmath.stanford.edu/~cgates/PERSI/papers/77_04_spearmans.pdf
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