Shulou(Shulou.com)11/24 Report--

Translator: Jiang Xun

Translator's note: this paper attempts to show that ChatGPT still has some defects in the field of mathematical proof. So except for the most basic questions below, the proofs given by ChatGPT are incorrect. Readers are requested not to take the proof here as a standard answer.

Chat-GPT is the impressive new AI chat robot released by Open AI. Its impressive applications can be found all over the Internet. But can it do math?

Through mathematics, we don't mean to simply do some calculations. Its own design ensures that there are computational problems that it cannot solve. I am more interested in whether it can solve problems that require some mathematical reasoning.

To select our question, we used the MMLU dataset. Galactica, a large language model recently released by Meta AI, has achieved some good results on this dataset. Their findings are presented in Section 5.3 of their paper.

Next, we'll pick a few questions and see how Chat-GPT handles them. If it fails, we will try to rewrite the problem in order to get some results.

We noticed that the MMLU dataset had multiple choice questions, but we just presented the problem to Chat-GPT. We just choose the problem from the abstract algebra part of MMLU. After that, we will stimulate Chat-GPT for intuition and proof. In the end, we will recommend some applications of Chat-GPT to mathematicians.

Question 1 I: the inverse in the multiplication group is

Chat-GPT: the inverse in the multiplication group is. The inverse element of an element in a group is a satisfied element, where it is the unit element of the group. In this group, it is the unit element, so the inverse element of the element is the satisfied element.

For example, the inverse is because. Again, the inverse is, because.

Note: Chat-GPT answered this question perfectly and gave an excellent explanation. To be fair, this is a relatively simple question.

Question 2 I: how many homomorphisms are there? Please write in latex format.

Chat-GPT: there are two homomorphisms of arrival. They are defined as follows:

Satisfy for all, set up

Satisfy for all, set up

Note: again, Chat-GPT is correct. However, I asked this question several times, once said it, and once said it for the wrong reason:

There are only two homomorphisms from (integer) to (integer module 2).

One homomorphism maps each integer to, and another homomorphism maps each integer to.

Question 3 I: P = (1, 2, 5, 4) (2, 3). Find it in

The index of. Please write down your answer in latex.

Chat-GPT:

The index in the appears in dictionary order in

The previous number of elements.

To find it.

The first step is to determine the loop type of p. The loop type of the arrangement is a list of the lengths of its disjoint cycles, in descending order.

The cyclic type of p is [2p2] because p consists of a 2-cycle and a 2-cycle.

Now we can use the cycle index to determine the number of permutations in Shop 5 with loop type [2jue 2].

The cyclical index is given by the following formula:

Where is the number of permutations of the loop, which is a variable that represents the loop of length.

Put the cycle type [2J2] into the cycle index to give:

It is simplified to:

The coefficient in this expression is, so the number of permutations in the cyclic type [2p2] is.

Due to

It is the only arrangement with cyclic type [2jue 2] in, so it is the rank in.

therefore,

The index in is.

Note: wow! This is very wrong! The first sentence is completely wrong. Also, its final conclusion is that it is not even an integer, which must be wrong. In fact, the indices of subgroups are meaningful, and most of them are positive integers.

But we can save something. Suppose we use algebraic techniques to calculate

The order of. Then we can use the following prompt Chat-GPT.

Me: now suppose it is a subgroup of order 5. What is the index in the middle?

Chat-GPT: the index of a subgroup in a group is the number of left cosets in the group. This is also equal to the number of right cosets in.

Since the order of a subgroup can be divided by the order of a group, the order must be divided by 120. The factors of 120 less than 120 are only 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, and 120. Therefore, the only possible order is 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40 and 120.

Because of the order of 5, the index in the middle is 24. 5.

Note: here, Chat-GPT answers correctly and gives a perfect explanation. Perhaps it is best to mention that the main fact used is the Lagrange theorem, although this can easily be done by asking it, "what theorem did you use above?" To get.

But what is intuition? What makes mathematicians great is some kind of "intuition" about the problem. Let's see if Chat-GPT can explain any of them.

Me: can you explain the intuition behind the Cauchy-Schwartz inequality? Please write the mathematical formula in latex.

Chat-GPT: of course! Cauchy-Schwartz inequality is a useful inequality, which appears in various fields of mathematics, such as vector algebra, inner product space and probability theory. It points out that for the sum of any two vectors, the following inequalities hold:

Intuitively, Cauchy-Schwartz inequality points out that the inner product of the sum of two vectors is "controlled" by the length of the sum of vectors. To understand this, let's consider the special case that sum is a unit vector (that is, a vector of length 1). In this case, the inequality becomes:

This means that the inner product of the sum is always less than or equal to 1. This is consistent with our intuition that if two unit vectors are "orthogonal" (that is, vertical), their inner product should be "small" and "large" if they are "parallel" (that is, pointing in the same direction).

I hope this helps! If you have any questions, please let me know.

Note: this is not bad, but it can definitely be improved. For example, explaining when the Cauchy-Schwartz inequality holds (when one vector is a multiple of another) helps to give a more precise meaning to the last paragraph. People can read the first few paragraphs of Tao Zhe Xuan's blog post to understand the explanation from one of the world's leading mathematicians. In addition, this is a mathematical discussion on the topic on Stackexchange.

Give proof. Me: can you prove this?

Chat-GPT: of course! This is the proof of the Cauchy-Schwartz inequality:

First, we can rewrite the inequality as follows:

Next, we can expand the left side as follows:

This formula is simplified as follows:

Now, we can apply the arithmetic-geometric mean inequality to each term on the right:

This is simplified to:

Finally, we can apply the arithmetic geometric mean inequality to the item on the right again:

This is simplified to:

We're done! This completes the proof of Cauchy-Schwartz inequality.

I hope this will help! If you have any questions, please let me know.

Note: unfortunately, this proof is not entirely correct. There are serious defects in both the proof method and the details (that is, its application to AM-GM).

To be honest, Chat-GPT is quite impressive for a model that is not tailor-made for mathematics in any way. I think, in the current format, it can already help mathematicians prepare handouts or even the simpler parts of the paper. Perhaps its greatest use is to automatically process some parts of funds and job applications.

This article comes from the official account of Wechat: meeting Mathematics (ID:meetmath), author: George Shakan

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