We’re gonna need a lot more mathematicians

[This is a guest post by Amit Sahai. This blog post was initially written in a different file format and converted using AI. — T.]

When I was an undergraduate student, I remember talking with several students who felt that the pace at which the top students could understand new math concepts was far too fast for them. They, too, could understand the ideas, but it would take them much longer. Eventually, almost all of these students gave up their dream of pursuing research mathematics and found something else to do. I have been thinking about those students a lot in the last few days.

The research mathematics community consists largely of those of us who either rarely felt that way, or who felt it and managed to overcome it through hard work. We have had the good fortune to find a place in mathematics where we could make progress. But we are now entering a time for humility: a time when all of us are going to know what it feels like to be unable to keep up.

The AI systems I have worked with are already producing beautiful new ideas. They are doing far more than impressive calculations or quickly carrying out arguments that a strong human researcher would already understand. And we probably can’t even imagine the wonderful ideas that future systems will be capable of producing.

When we feel that we cannot keep up, will we take that as a reason to leave research mathematics, like the students I am remembering? As more of us experience this, there will undoubtedly be a temptation to draw the same conclusion as they did: If the machines can move so much faster than us, perhaps we should find something else to do.

For our community to give up the work of understanding would be a profound abdication of our responsibility to humanity. Each of us is entitled to choose a different life. The responsibility I am talking about belongs to us collectively: to build a future in which humans can understand and contribute to the discoveries that will change our world. A future with meaningful human agency.

Struggle is essential to understanding difficult concepts. Fortunately, this struggle can be shared. I have been blessed to experience this time and time again with my students and collaborators. Imagine a multitude of research groups, each with sustained support, each spending a term or a year trying to understand an extraordinary set of ideas produced by an AI system, with the help of AI systems. [1]

This may very well be among the most important mathematical work in the years to come, and we should support and prioritize it accordingly. This enterprise will require a significant expansion in the number of mathematically sophisticated human researchers available world-wide, as major breakthrough ideas accumulate.

Why should society want this? So far, this might sound like a utopian fantasy for us – a civilization focused on depth of human understanding, awash with mathematicians and physicists and the like. I would certainly love to live in such a world. And indeed there are deep philosophical reasons for society to move in this direction. But I think society has a much more immediate stake in making this possible, too.

Imagine that a future AI system proposes a radically new design for a one terawatt nuclear fusion power plant. It has found a way to sustain and control fusion that no human had conceived of. The design promises abundant, inexpensive, clean electricity. Robots stand ready to manufacture the components and build the plant.

A terawatt is an insane amount of electrical power. We would be deciding whether to construct a machine that handles extraordinary flows of energy using principles we have never conceived of, let alone put into practice. We would need to understand how failures can be contained, what happens to energy already stored in the system when it shuts down, how we can be sure that the materials that make up the power plant behave as expected, and what other questions we should ask before proceeding. The very novelty that makes the proposal exciting would mean that we cannot inherit confidence from decades of operating similar plants or experiments.

Before approving construction, I would want communities of humans to understand why the design works and what justifies confidence in its safety. I would hope that we all would.

Human involvement does not automatically improve a technical decision , and I see no reason to insist that humans manually repeat work an AI system might be able to perform more reliably, even including proving mathematical guarantees. But a theorem can only exist within a model. Understanding the guarantee means understanding the model, the experimental evidence for it, and our uncertainties about the accuracy of the model. This is demanding work, and mathematically sophisticated people must be available to engage with it.

One might respond that AI systems should handle those questions too, and ultimately decide whether the plant should be built. That is a serious position. But it asks us to accept a future in which decisions of enormous consequence rest on reasons that no human community understands.

I do not want us to arrive at that future simply because we failed to invest in our own capacity to understand. Human agency is a value of fundamental importance. We must retain the ability to meaningfully consider alternatives and decide what kind of world we want to be a part of building. I think it is worth the effort. [2]

To take on this responsibility, we may need to broaden our view of what a mathematician can contribute. I have in mind something like a “deployable intellectual reserve”: communities of mathematically sophisticated people that humanity can call upon to help understand consequential AI-enabled breakthroughs.

Our ability to understand difficult and unfamiliar ideas may become one of the most important contributions we can offer to society. We should be willing to bring that skill to problems far beyond our usual research interests. [3] Doing so asks us to expand our sense of our vocation.

A counter-argument might be that AI systems will make each of us so much more effective that fewer people could do this work, even as the pace of discovery accelerates. But each of us is merely human. We have fundamental limitations based on our biology. Depth of understanding needs time and a pace of life that humans can sustain. Each individual human can only be asked to do so much, but through earnest cooperation we can accomplish much more.

If AI fulfills its promise, we will encounter more beautiful and consequential ideas than we have ever seen. We must respond by building thriving human communities that can understand them together.

We’re gonna need a lot more mathematicians.

The ideas and opinions presented here are entirely my own, but GPT 6 Astra was instrumental in helping me draft this note. I also thank my former student Dakshita Khurana, my current student Isaac Hair, my colleague Terence Tao, and my family members Anant Sahai and Gireeja Ranade for valuable feedback. Note that there is much more to be said here, but I tried to keep this relatively short to focus succinctly on my primary thoughts.

Notes

[1] By this, I do not mean to imply that only AI-created results will be of interest in the future. But for major results generated by humans, we already have a tradition of spending extended periods of time studying them.

[2] And the relevant understanding cannot belong only to the organization proposing the technology. Imagine a public hearing at which the company’s experts are the only people capable of following the technical argument. Independent expertise is critical.

[3] Indeed, AI systems are likely to be very helpful in allowing researchers with diverse backgrounds to talk effectively with one another, and more generally understand unfamiliar concepts.