“mathematics is not about proofs, but about understanding“
Discussions about AI Math in the sub have been quite heated lately, and I’ve been following them every day. I am not a mathematician, in fact I come from the linguistics field. But I do want to chime in and share some thoughts.
Basically, there is this one question I cant shake out of my head: why are people so focused on the idea that mathematics is not about proofs, but more about understanding? Of course, I agree that mathematics is more than getting the right answer. But I can't help feeling that some of the arguments in this debate are based on assumptions that haven't been clearly discussed. That is, what exactly do we mean by understanding? How do you define it?
Suppose a mathematician spends ten years proving a conjecture. They go through countless failed attempts, discuss ideas with colleagues, and eventually publish a paper. Throughout this process, they undoubtedly develop a deep understanding of the problem. But who actually benefits from that understanding? First and foremost, the mathematician, and perhaps the colleagues who worked with them. Later researchers don't need to go through those same ten years of exploration. They can read the proof, follow the reasoning, recognize the underlying mathematical structures, and develop their own understanding. Isn't that how academic knowledge has always been passed on?
So if AI produces the same proof in a few hours, complete with detailed reasoning, why should we call it "an answer without understanding"?
I agree that discovering a proof yourself is certainly different from reading someone else's proof. The former doesn't just tell you that a particular method works. It also develops your ability to find methods in the first place. The failed attempts, unexpected observations, and changes of direction may help you solve other problems later. That's a fair point. But couldn't an AI-generated paper provide similar benefits to future research? It can show the mathematical community which methods work, reveal unexpected connections, suggest new directions, and pass those ideas onto future generations of mathematicians. Researchers may find inspiration in an AI-generated proof and develop theories that the AI itself never explicitly proposed. Shouldn't the value of a paper depend on what its readers can learn from it, rather than how many failed attempts its author went through?
I know that human research can lead to unexpected discoveries. But why should we assume that AI-generated proofs cannot contain equally unexpected ideas? Some people seem to assume that AI can only rearrange information from its training data. But even if that were true in a particular case, we would still need to explain why combining existing knowledge in new ways cannot produce genuinely new mathematical insights. After all, how much of human mathematical creativity also involves reorganizing and extending existing ideas? Of course, this doesn't mean every AI-generated proof is original or insightful. It still needs to be evaluated case by case.
I wonder what you think.