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This text reflects my personal opinions and observations regarding the use of AI in Mathematics, how AI could be useful in the continuation of our science, and other thoughts among others.
It has come to my attention that many highly regarded mathematicians in my field feel a threat of the use of AI in mathematics. To be honest I feel the same. Here I will talk about what I feel is getting lost, what is the threat, and in what aspect AI would be useful.
When I was a high school student, especially when having tests or exams in mathematics, it was prohibited to use calculators. Later, as a university student, the same idea continued. Any use of such tools was prohibited. I always found this a little bit too much, to be honest. As long as I can save time and escape the tiring nature of calculations, why should I put myself in the position of losing time and also risk the correctness of the calculations?
When I started focusing on Pure Mathematics at university, the use of any calculator or assistance was minimized, but at the same time replaced by Math Stack Exchange for some help or hints with difficult exercises. And of course I was also giving back by answering questions too. Sometimes it felt like cheating in a way, due to me not having this glimpse, this spark that brought up the light of understanding in my brain.
Right now I am a PhD student and I have the luck of having a decent amount of publications. Some of them in decent or good journals, some of them not. For me it was always about having this spark, the initial light that helped prove something that in my eyes was nice or pleasant. I know my limitations as a human mathematician. And the recent advancements of AI worry me. I believe that we are experiencing change. Change in the didactics of mathematics, change in research, change in everything. In this text I try to put some order in my ideas, worries, thoughts and hopes about mathematics and AI.
My supervisor, my other colleagues and I have spent a significant amount of time discussing the progress of AI in solving math. Mostly we see it as a threat, especially for those of us who are very new to research and try to find a job on the field. I try to think positively, but I always see its use exactly like using a calculator on an exam. It is cheating in my eyes, but not in a very strict way. Let me elaborate. A perfect exam solved with calculator or other assistance and an exam done by hand by the same student are essentially the same. No difference. The time saved by assistance is important. But nothing could replace the human work hours behind the exam done perfectly only by hand. We admire the Pyramids of Egypt, the Parthenon and the Colosseum because they were made by human work and hands, even if we could rebuild them and construct them in much less time with machines.
I don't believe mathematics will die. The way we approached mathematics will change, will be different. I am of the opinion that even if we manage to have an effectively formalized LLM which can produce proofs by itself, we will still be needed to explain this to other humans. Somehow, it looks like we will become the tool, and AI the great mathematician, in such a scenario, solving problems and answering every question. Conjectures and problems are starting to be resolved recently, like the Jacobian conjecture or the Crouzeix conjecture, among others. These advancements might be important for the science of mathematics, but they take away the romantic feeling of sitting down with your colleague, spending chalk, paper and ideas. We have started to lose the romantic side of mathematics, the side which made mathematics an art. This is necessary for the future to exist, as this trait of mathematics is what brought people like me and my colleagues to this field. Of course we want to solve a problem and write a paper with our names on it. But what is the value if an LLM did the heavy work or also had the key idea? The value for the science is the same. The proof is correct, the problem is solved. The human aspect, though, is now missing.
Of course we should not be against this type of progress. One simply cannot still try to travel abroad by foot when airplanes exist. The progress of AI is the collective result of all of our work too, so one could say that we enjoy the fruits of years of hard labor. What it seems that I am against is the handling of the contribution of AI in such work. I have found myself observing that on arXiv the amount of papers uploaded every day has exploded. Many authors, among them top mathematicians in our field, acknowledge its use and, more importantly, specify when and where this is done. I believe this is the correct thing to do. Moreover, having used LLMs myself for different works, I have come to the point where I can recognize a paper that has ideas stemming from an LLM, or is even written by it. I get infuriated when I see such papers and no acknowledgement shared by the authors. It is unacceptable. We, as scientists, should have the decency to acknowledge its use. In my opinion, we should actually consider seriously putting the specific LLM model as an author and the people who used this model to resolve the problem as prompters or co-authors of second order. This sounds very radical, but it separates the human creation. Of course, if AI is used just for some calculations or to prove a technical lemma, then authorship should be claimed by the real authors.
Another topic I want to discuss with regard to the involvement of AI in mathematics is the referee process of submitted articles. I fully believe that we can actually take advantage of AI in this process, save time for everyone in the community and also have better-quality articles from now on in terms of correctness. I am fairly sure that there exist mistakes (non-terminal ones but still) in my own work. I am also sure that this is the case that similar type of mistakes exist in other people's work, which is fine in the sense that we are in the position where we push science forward, and in the process of pushing, things might get hard. At this point, AI can be used in the referee process; some people already do this as a last step possibly. Of course, not with just a simple prompt such as ``review this article for me,'' but rather by using it as a colleague with whom to discuss your opinion and find some more errors, typos or gaps that the human eye may not find. An article can be assessed by a referee along with an LLM model, and reviewing times can be significantly improved too. This will resolve many issues for young researchers who are stressed about finishing their PhD, or for senior mathematicians who apply for grants or higher positions.
A third point which I believe is worth discussing is teaching. Teaching mathematics always requires some type of human interaction. I have significant experience in teaching high school students (10 years). I always found that the relationship you develop with students is equally important to your way of explaining mathematics. If the students like you as a person, they will probably follow your example more easily. I am not saying that you should be a friend with them, of course. What I mean is to discuss human things with them. With boys, football. With girls, maybe let them tell you some of their issues when they feel like it. A good teacher, at least at high school level and with teenagers, should also be a good consultant for them. For university students, this is more difficult as the number of students in a class differs. Nevertheless, I still believe that a nice, understanding behavior towards students, an open approach, let's say, makes you likeable and students can be positive towards your classes too. This comes in direct opposition to AI. We may reach the point where an AI robot could teach math. This AI robot could also be able to discuss fun stuff with students to be more likeable. The thing is that this robot could never be the same as a human because we are flawed. A robot would behave the same as another of the same model. It would find similar jokes or ways to teach. But it would never be like me and you, or the math teacher you liked or disliked in high school, or that university professor who was very strict and you hated, or the other professor whose classes you loved to follow just because he had a calm voice and was very explanatory. We humans are different. We are flawed. Our mathematics have flaws and this is what keeps us different. The romanticism.
This is what we should keep alive. We should keep the artistic aspect of mathematics alive, and not only mathematics but many other aspects and occupations in life. It happens that I navigate myself to Reddit, and many posts from the subreddits "r/accelerate" or "r/singularity" appear to my feed. These subreddits belong mainly to the people who love AI and its progress. Don't get me wrong, I also admire and use this great new tool for my own purposes, sometimes for work, sometimes for fun. What I seem to not understand is the narrow sight the supporters of this new culture have. The threat is not just "losing a job". The threat we mathematicians feel and describe is the loss of the humanity. The pain and the effort one goes through to create something. The cooperation. The happiness of the achievement. The celebration. The thirst of continuing. The pain, frustrations, sleepless nights we go through when our problems are not solved are a part of being human. AI will move things fast, very fast. The question is "we move fast but towards where?" Why are we rushing to go there, when we don't even know what the destination is? It looks to me as though everything moves so fast that the human mind cannot comprehend and digest the consequences of this. We still put up papers on Arxiv, 30-40 everyday when in the past it was zero to 5-6 on a good day, we continue to give homework to students when they just solve it with the use of LLM's etc. We pretend that everything is the same but it is not. For many things there will be no need to exist, maybe even for mathematics at this point. But why is this good? Why making everything easy, fast, accessible to everyone, is good?
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