I’ve been thinking about the future of mathematics and what AI might change about it.
AI is getting better at solving problems. It can write code, help prove theorems, find patterns, and do many things that used to require a lot of human effort. I think it is reasonable to expect these systems to become much better in the coming years.
But there is one thing I think we should talk about more.
Math is hard. And making AI smarter does not necessarily make math easy.
Of course, AI might solve thousands or even millions of mathematical problems that humans have not been able to solve. It might solve problems that people have spent their whole lives working on. That would be amazing.
But does solving more problems mean that we will eventually run out of hard problems?
I don’t think so.
One reason is that mathematics is not just a collection of questions waiting for answers. Solving one problem can lead to new questions. Sometimes a result gives us a new way to think about something, and that leads to problems we could not even ask before.
So even if we solve a lot of the problems we know today, there is no good reason to believe that we will be close to the end of mathematics.
But I think there is another, more interesting point.
Some problems are hard not simply because we are not smart enough to solve them. They can be hard because solving them requires a huge amount of computation.
Computer science has a whole field dedicated to this question: computational complexity. It studies how the resources needed to solve a problem change when the problem gets bigger.
For some problems, making the input twice as big might not be a big deal. For others, it can make the problem much harder. In some cases, the amount of work grows so quickly that even a very powerful computer cannot solve problems of a realistic size.
This is one reason I find the P versus NP problem so interesting. We still don’t know whether every problem whose answer can be checked efficiently can also be solved efficiently.
I don’t want to make a big claim about what the answer will be. But the question itself shows how much we still don’t understand about computation.
And even if AI becomes much better at mathematics, it does not automatically mean that it can solve every problem we give it. Being very good at solving problems and being able to solve all problems are two different things.
Maybe future AI systems will be able to do things that we cannot even imagine today. Maybe they will be better than humans at almost every part of mathematical research. I’m not ruling that out.
Actually, I think we should take this possibility seriously instead of assuming that humans will always be better at something.
But then what happens to us?
If AI can solve problems better than we can, what is the point of doing mathematics ourselves?
I think the answer partly depends on how we decide to do mathematics in the future.
If we want to discover new mathematics, we should probably learn how to use the best tools available to us. If AI is good at searching for patterns, we should use it for that. If computers can check proofs much faster than humans, we should let them do it.
There is no reason to make our work harder just to prove that we can do it without machines.
At the same time, we should not assume that the way we do mathematics today is the only way to do it. New tools might change the kinds of questions we ask and the way we work on them. Maybe the role of mathematicians will change a lot. Maybe some parts of mathematical research will become almost completely automated.
I don’t know.
But I think there is a difference between saying that AI might do mathematics better than humans and saying that AI will eventually finish mathematics.
The first seems possible to me. The second is a much bigger claim.
We don’t even know how to describe all the mathematical questions that might be interesting in the future. And we certainly don’t know whether the resources available to us will ever be enough to solve all of them.
This is what makes me less worried about the future of mathematics.
I’m not saying that humans will always be necessary. Maybe one day we will build systems that can do almost everything a mathematician does, and do it better. Maybe we will not even understand some of the mathematics they discover.
But why should that mean there is nothing left to do?
I think we sometimes look at progress as if the goal is to solve every problem and eventually have nothing left to work on. But I don’t see why mathematics has to work that way.
Every new tool gives us new possibilities. Some problems become easier, some become possible to solve for the first time, and others become interesting because of what we have just discovered.
Maybe AI will change mathematics more than anything before it. Maybe it will completely change our idea of what doing mathematics means.
Still, I have a pretty strong feeling that we are very far from a future where there is no interesting mathematics left.
And if AI really does become better than us at mathematics, I think we will have much more important questions to think about than whether we are still the best at it.
I want to end with a quote I came across in a Telegram channel:
“If there is only one job left for humanity, it will be some form of doing mathematics.”