Is There Actually a Good Way to Use AI for Math?

Yushang Cheng, Undergraduate student, China

SERIES · As application deadlines approach, Proofs and Prompts is publishing a series of posts from 19 to 23 October for people considering a PhD, doing a PhD or looking for a postdoc: guidelines, discussions and speculation about what to do and what careers in mathematics will look like. This post is one of them. Each is written independently and reflects only its authors’ views. All posts in the series

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The author first drafted these reflections in Chinese. AI helped organize them, suggested the mathematical example, and assisted with English translation and rewriting. The author’s full AI use disclosure appears below.

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I'm an undergraduate student in China, and much of what I know about research I have had to figure out for myself. I began using AI to explore theoretical questions last year; some of those attempts, including the ones that went wrong, are recorded in my GitHub archive and an SSRN preprint. AI has helped me move forward astonishingly quickly, though I sometimes worry that my mathematical foundations have not kept pace. I still need to put it aside, study on my own, check proofs, and learn to write without it. But I am genuinely curious about certain theoretical questions. Once an idea catches my attention, I want to know where it might lead.

Over time I developed a way of working with AI that feels useful to me. I suspect it could be better, though, so I started asking around. I talked to classmates, posted on forums, and even tried contacting professors and mathematicians. It felt a little like looking for a game walkthrough: surely people who have played much longer than I have must know a few tricks. Most of the AI advice I encountered was about coding, cleaning data, and other routine tasks. Mathematicians seemed a more promising place to look, but their advice was often just as brief: tell AI what you want to prove, ask it to find counterexamples, or have it argue against you. All sensible advice. Still, I had gone looking for a walkthrough and seemed to have come back empty-handed. Perhaps the steps in between are so familiar to experienced researchers that they hardly think to mention them. I thought I might as well share the habits I have developed and ask what you make of them.

Here is a very simple example of the part I find most interesting. Suppose we are studying a continuous function $f:[0,1]\to\mathbb{R}$ with

$f(0)=-1,\qquad f(1)=1.$

The intermediate value theorem tells us that there is at least one zero. If we also assume that $f$ is strictly increasing, the zero is unique. My instinct, however, is to ask whether removing strict monotonicity might make the problem more interesting. Consider

$f_1(x)=2x-1, \qquad f_2(x)=\frac{32}{3} \left(x-\frac14\right) \left(x-\frac12\right) \left(x-\frac34\right).$

Both functions are continuous and have exactly the same endpoint values, yet the first has one zero and the second has three. Of course, I could put strict monotonicity back, exclude the second function, and return to proving uniqueness. But if I wanted to understand this class of continuous functions in the first place, why should I throw the second one away? I would rather keep it. Now there are two interesting possibilities to study: when is the zero unique, and when can there be several? Can I find conditions that distinguish the cases, and are those conditions necessary? Sometimes I will try to weaken them again. When removing an assumption gives me an interesting counterexample, I want to see whether the counterexample can become part of the theory.

There is another problem I run into if I keep doing this. After enough rounds of removing assumptions, finding counterexamples, and obtaining new results, I may be left with a heap of conditions and conclusions without knowing what each condition is really doing. So I have become rather fond of building the dependencies between results, one step at a time. In the example, we can separate two claims:

$\begin{aligned} \text{continuity + opposite endpoint signs} &\Longrightarrow \text{at least one zero},\\ \text{strict increase} &\Longrightarrow \text{at most one zero}. \end{aligned}$

Together, they give a unique zero. I like this arrangement because I can see what each assumption contributes. If I later find another way to establish “at most one zero,” I do not have to change the existence argument at all.

For a more complicated theory, I might start with a primitive condition $A$, derive a meaningful property $B$, and then prove something new from $B$. I would rather have $A\Rightarrow B\Rightarrow C$ than keep proving everything directly from $A$. Sometimes the argument branches: $B$ leads to $C$ and $D$, and those results together lead to $E$. I just find this kind of structure beautiful. At first that was almost the whole reason I liked it. Later I realized that it can do real work. If $B$ is weaker than $A$, the statement $B\Rightarrow C$ may apply in settings where $A$ does not hold; and once $B$ has been proved somewhere else, I can reuse the same result. The intermediate properties have to mean something, of course. There is no point in making a proof longer just for the sake of a prettier diagram.

This also makes me wonder how far the assumptions in a theorem are from the original problem. If the result depends on a long list of strong technical conditions introduced along the way, I want to know whether they can be derived from more basic objects. In an economics model, for instance, can the conditions be expressed in terms of observed choices, compensated comparisons, or the support of the data? The two functions above offer a small illustration of the information problem. Even if we know the function is continuous and observe both endpoint values exactly, those observations do not tell us whether the zero is unique. The functions look identical to someone who can see only those two values. I often ask AI to construct pairs of models that agree on everything observable but disagree on the property I care about. When such a pair exists, my next question is what extra information would distinguish them, and what, precisely, that information would allow us to conclude.

I came to these habits by taking quite a few wrong turns. My first project concerned a rational-addiction model and what I called an exit mechanism. I was excited by the thought that a rational agent might have a distinct reason to invest in reducing or escaping addictive behavior. But in the model I had written down, the proposed exit or recovery value could be absorbed into the existing state-dependent utility. Behavior therefore could not identify it as a separate mechanism. An idea I had been eager to defend simply did not survive the test. That failure led me somewhere I had not expected: toward questions about what behavior can identify, what sparse compensated comparisons can reveal, and which additional observations really matter. The project eventually led to my SSRN preprint, What Do Finite Compensation Comparisons Reveal? I kept the failed earlier versions on GitHub; they are part of the story, too.

AI has been useful throughout this process. I ask it to remove assumptions, hunt for counterexamples, try proofs, construct observationally equivalent models, and help disentangle which lemmas actually depend on which conditions. I have also seen plenty of arguments that sounded complete until I checked the details. One other habit has grown out of these conversations, and I wonder whether anyone else has noticed the same thing. AI often gives me prose that seems determined to explain everything it has done: “we cannot guarantee uniqueness,” “this assumption is unnecessary,” “the important point is not this but that.” Those sentences can be useful while we are thinking. I just do not much like them in the finished mathematics.

What I like in textbooks is the way a good statement simply tells you the conditions and what follows. For the example above, I would rather write: continuity and opposite endpoint signs give existence; strict increase gives at most one zero; and continuous functions with identical endpoint values can have different numbers of zeros. I can then ask whether the sufficient conditions are also necessary, or how the remaining cases should be characterized. Sometimes replacing a conversational diagnosis with a proper mathematical statement even exposes a gap in my reasoning. The final paper should not have to reproduce every turn of the conversation that brought me there.

One thing about my own method worries me, though. As AI gets better at turning a guess into a mathematical theory, it becomes easier to treat research as a set of buttons to press. I can imagine a seminar where everyone asks the professor for their prompts, but nobody asks how the proof works. I would find that rather sad. The irony is that I’ve already benefited from this method myself, and sometimes I find it hard to stop using it. I know its downsides, but I also want to do this kind of research for a living. I don’t think I’m especially gifted at mathematics, academic jobs are hard to get, publication counts matter, and AI tools keep getting better. It’s very easy for me to tell myself that I have no choice. But that sounds a little like a thief looking at something in a shop window and convincing himself that he has no choice but to steal it. Of course, using AI isn’t stealing. What bothers me is how easily I can make excuses for myself. I can only try to stay aware of this, keep learning the mathematics, and make up my own mind about what AI produces. I shared my prompts because I hope they will help people explore questions they genuinely care about, not just produce more papers. Otherwise, we might end up with more theorems, but less interest in understanding them.

That is roughly how I work at the moment. Perhaps experienced mathematicians have always done something like this and simply never thought of it as a “prompt.” I would love to hear how you use AI to look for counterexamples, how you decide whether a failure is worth studying, and whether you consciously build theories out of reusable intermediate results. I would be just as interested in criticism: what am I missing, and what am I making needlessly complicated? I began this search hoping to find a good walkthrough for myself. It would be nice if sharing our actual experiences helped a few other beginners as well.

Research links. GitHub research archive; SSRN preprint.

AI use disclosure. The research experience, ideas, and initial reflections in this post are my own. I first drafted them in Chinese, in a rather fragmented form. AI helped me organize the original line of thought into a clearer structure, suggested the elementary mathematical example used here, and assisted with the English translation and rewriting. I reviewed the resulting text to ensure that it expresses what I meant and reflects my own experience and uncertainty.

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Received 7 October 2026, revised 22 October 2026.