GoKawiilMathematicians are broadly not anti-AI. They’re more culturally open to using AI as a tool than, say, artists or writers. However, now that more and more genuinely prestigious problems have fallen to AI, that might be changing. Almost five thousand mathematicians (including twenty-five Fields medalists) have signed a declaration called A Severe Misalignment of AI in Mathematics. The core argument goes something like this:
In recent months, the success of AI in solving major mathematical problems has made headlines even outside mathematical circles. But solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight. Forgetting this in the world of AI may turn the tool against the primary goal. Indeed, the mass production at faster and faster pace of “true/false” statements could destroy fertile ground instead of breathing life into new ideas.
A lot of people online have interpreted this as the expected complaint from any field that gets automated: translators did it, artists and programmers have been doing it, and now it’s the turn of the mathematicians. I think this is too dismissive. Understanding the concrete problem mathematicians are upset about can help us better understand the impact of AI on our own fields, and what we’ll have to do about it.
Puzzle-solving and idea-generating
There are two types of mathematics. Most people are familiar with the first, which we might call “puzzle-solving”: you take a problem and try to find a solution to it. When you’re a student, these problems are typically easy, like simplifying some algebraic expression. When you’re a researcher, these problems can be nearly impossible, like proving Fermat’s Last Theorem. Puzzle-solving is easy to understand but hard to do, which makes it impressive to non-mathematicians, which makes it highly prestigious. In other words, puzzle-solving is legible.
The second type of mathematics is “idea-generating”: coming up with new ways of thinking about mathematics, and thus new terms or concepts. For examples of these, just glance down the list of arXiv mathematics papers. “Hardy spaces”, “Schatten exponent”, “Banach lattices” and so on are all concepts someone thought was interesting. This work is largely unimpressive to non-mathematicians, because nobody really knows if the concepts you come up with are particularly difficult or insightful. For instance, I have just generated the concept of a “Goedecke set”, which is the set of all natural numbers whose digits add up to a prime number. Who cares? The categories we want are the “natural kinds” of mathematics — the concepts that “carve nature at its joints” — and it’s almost impossible to tell what those are without years or decades of hard work.
How are the two types of mathematics related? We might say that generating ideas is the real intellectual work of mathematics. Puzzle-solving is important instrumentally: to identify which ideas can be used to answer longstanding questions, and thus which ideas are worthwhile. Over time, those worthwhile ideas become better understood and easier to use, until they reach the point where they can be used to advance science in general. Eventually the ideas become so well-understood that they can be taught to children: “zero”, “negative numbers”, “imaginary numbers” and “calculus” were all once rarefied mathematical ideas, but are now concepts we’d expect any precocious twelve-year-old to grasp.
There’s another, more prosaic purpose of puzzle-solving: to make mathematical skill and progress legible to outsiders. I can’t appreciate Terence Tao’s mathematical work, but I know what a Fields Medal is. I don’t have a good intuitive sense of what a Galois representation is, but I know about the proof of Fermat’s Last Theorem. We might say that puzzles like this have served as a way to indirectly reward skilled mathematicians for their more important idea-generating work (or for conclusively demonstrating that the ideas used in the proof are useful).
Do AI proofs undercut idea generation?
AI proofs undercut both of these purposes. I can now lay out precisely why I think mathematicians are so unhappy:
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