Why AI won’t replace mathematicians

The New York Times has jumped on the “AI will take our jobs” bandwagon, in an article titled “A.I. Is Coming for Mathematics, Too.” As the author Siobhan Roberts states, “For thousands of years, mathematicians have adapted to the latest advances in logic and reasoning. Are they ready for artificial intelligence?”

The analysis in the article is based on a fundamental misunderstanding about what research mathematicians do.

Automating proofs

If you remember math from high school or college, you probably remember a focus on solving problems. Find the value of x. Compute the definite integral. Solve this differential equation. The work was focused on learning existing techniques and applying them in practice.

I was trained as a mathematician, so I can take this a little further. As you move on to more advanced mathematics, the focus shifts from problem solving to proofs. Can you identify a mathematical object (typically, a formal abstraction of objects in the real world like numbers, shapes, patterns, and so on) and draw conclusions about properties of such objects? Can you show, for example, that there are more real numbers than rational numbers, even though both quantities are infinite? Can you prove that any reasonable map of countries on a plane can be colored with just four colors so that no two countries of the same color share a border?

As Roberts’ article explains, with a sufficiently formal set of language, you can now translate some mathematical concepts into symbolic language that an AI can understand, and it can determine if a proof is valid, or even attempt to come up with a proof on its own.

If you think that mathematicians focus mostly on manipulating symbols in formal ways as if they were machines, then you might think that such work could now be automated. But that fundamentally misunderstands what mathematicians do.

The real work of mathematicians

To explain how creativity pervades mathematics, I’ll share a story from a class I took while studying for a Ph.D. at MIT.

The professor of this class was certainly focused on showing us the main concepts of the branch of mathematics we were studying. And much of the class would include him walking through proofs of the main concepts at the blackboard.

But often, part way through the proof, he would stop and say, “Let’s go to the intuition board.” He’d swing around to a different blackboard at the side of the classroom and start drawing illustrations of the concepts he was explaining. This was not formal reasoning. This was insight. It was our chance to see the ideas behind the proofs and get a deeper understanding of how the mathematical concepts worked in practice.

This dive into mathematical intuition was essential, because now we, as students, could not just see that a given proof was valid, but how and why it was valid. And that gave us a fighting chance to build our own ideas on the ideas in the class, not just to parrot back the proofs that the professor was sharing with us.

It is true that mathematicians do proofs. It also true that computers can now help with those proofs. But that is nothing new. Mathematicians (including my brother) regularly work with computers to help them explore mathematical concepts. The mathematician that proved the four-color theorem that I mentioned earlier used a computer to help complete the proof.

But the truly powerful creative work of mathematicians involves questions like these:

• What new concepts can we define that will lead to fruitful and productive explorations in mathematics?
• What new questions can we seek the answers to?
• As we explore those questions, what new and promising areas of work do they suggest?
• Is there a different way to prove the things we are trying to prove? If so, what does that new proof suggest that might open new areas of inquiry?
• How do the new proofs we execute relate to problems people are now trying to solve in the real world — like quantum physics, computation costs, optimum ways to get work done, and so on?

As you can see, these are questions that require judgment and creativity. Mathematicians are not just working on proving that proposition x is true or false. They are wondering about what sorts of beautiful, elegant, novel, or productive directions their concepts and proofs can take them.

Such work is as much art as science, even if it uses the tools of mathematics and logic.

And because it so heavily involves art, judgment, creativity, and insight, I doubt that an AI will ever do it at a useful level.

Mathematicians’ jobs are safe, because computers, even with AI, just can’t do that kind of work.