In the last few months, OpenAI has stunned the mathematical community with the publication of hundreds of math findings. Many of these proofs, which span number theory, theoretical computer science, and logic, had stumped the world’s best mathematicians for decades, or centuries.
The reaction from the math community has been a mix of indignation, astonishment, and sorrow. “Tuesday night, I cried and it woke up my daughter,” said Hugo Duminil-Copin, the recipient of the 2022 Fields Medal, after OpenAI’s October release. “They have annihilated my field of research.”
“I literally cried when I spoke to Wired a few weeks ago,” said Steven Strogatz, a mathematician and author at Cornell University. “People are using language like ‘they’re strip-mining our subject’ or ‘they’re nuking the landscape.’ The feeling is disorientation. I’m discombobulated some days. I’m exhilarated some days. And there is enormous grief.”
I wanted to talk to one of the mourners to understand their technological grief. I wanted to know if theirs was a familiar case of bruised ego or disappointment—akin to, say, a boy who spends an hour building a tower out of blocks and watches his big brother smash the thing to smithereens—or something stranger and more profound.
To be honest, I also wanted to know what it might feel like for me. Today, the mathematicians are weeping, because the machine smashed their tower first. But tomorrow, it may be the physicists, the economists, the illustrators, or the writers who weep over their fallen blocks. I am not one of those AI people who believes that hundreds of millions of jobs will be wiped clean from existence in the next five years. But I am a realist who recognizes that a rapidly improving technology whose output is intelligence is destined to make violent contact with a modern sensibility that prizes human intellect and often equates it with purpose, dignity, and self-worth.
I wanted to know our future. So I called Steven Strogatz.
Craft + Conversation
To prepare you for where my conversation with Steven went, I want to step back from math to ask a question that feels a bit embarrassing in its bigness: What makes humans special?
Some might say intelligence and sociality. We are the smartest creature and Aristotle’s “social animal.” But we share each quality with other species. Crows solve multistep puzzles. Ants congregate and specialize. Elephants appear to mourn their dead.
What sets humans apart is the way these two capacities multiply each other. Human intelligence is unusually suited to imitation and iteration. We imitate our parents and peers with uncanny fidelity, and then we tinker. We copy, and we create. Our sociality spreads this copying-and-tinkering process across groups and generations, so that one discovery becomes the floor for the next. Chimpanzees in West Africa have cracked nuts with stones in the same way for 4,000 years. In roughly the same period, humans went from Babylonian clay tablets to space rockets to machine intelligence. What makes us special, it would seem, is this capacity to compound. Other animals craft and converse, but only humans blend craft and conversation into a culture that builds on itself.
Science itself is both a craft and a conversation. As a craft, science is a mode of thinking; a special way of paying attention to the world; a lens that regards phenomena as something more than mere appearance. To the scientist’s eye, the universe is akin to a cosmic attic of treasure chests, each filled with hidden gemstones that can be pulled into the light if we ask the right questions, find the right keys, turn the right locks.
But science is not a solitary treasure hunt. It is a social experience, akin to a game. Like many games, it is both collaborative and adversarial. Newton famously compared his achievements to “standing on the shoulders of giants,” and a part of science is extending that human pyramid infinitely upward. Feynman preferred to describe science as “the belief in the ignorance of experts”—which is to say, a game of giant-slaying. In both cases, science is a relationship to other people. Whether we’re asking “why, professor?” or insisting “no, that’s not quite it, Mr. Big Shot,” science is not just an act of discovery but a relation to fellow discoverers. Science is a conversation.
End of Conversation
When I called Steven to ask why he cried, he said he wanted to tell me about Archimedes.
Two thousand years ago, the Greek mathematician Archimedes calculated the area of curved objects and pinned down the value of pi by measuring the areas of dozens of tiny polygons that filled out a circle. Writing to the mathematician Eratosthenes about his work on curved shapes and infinity, he wrote that “there will be no lack of discoverers to add to the number of theorems which have fallen to our share.” To Strogatz, this sentence contains the essence of mathematics. I could hear his voice break even as he told this part of the story. “I feel like Archimedes is writing to me, or to people like me,” Strogatz said. “Archimedes is saying, ‘I can’t solve these things, but maybe you people in the future can stand on my shoulders.’ That is a very emotional, existential thing. There’s this feeling that we’re in this together, questing for something beautiful and transcendent and sublime.”
A hard-boiled tech optimist might tell Strogatz to get a grip. Math doesn’t care about your feelings, scientific reality is indifferent to your tears, and so forth. He knows it, too. “We’re being derided as guild protectors: Your guild is getting wiped out. Boo-hoo. This has happened to other people before. Suck it up.”
I know why the mathematicians weep. Yes, some are crying about bruised egos, the feeling of wasted years, a creeping sense of status obliteration, the frustration of an unfair competitor, the indignation of feeling cheated and mistreated, and uncertainty about the future. But they are also mourning, because they think something alien and disembodied is setting fire to Archimedes’ letters and dancing in the ashes. Humanity is being stripped from human science. When OpenAI publishes several hundred proofs that end several hundred treasure hunts, it is succeeding brilliantly along the craft axis of science, but it is interrupting the thousand-year conversation. The machines and their architects are saying: From here on out, it’s our turn to speak.
Knowledge Without Understanding
And what if OpenAI is right? What if we ought to treat math mostly as a quest, rather than as a conversation where feelings matter as much as discovery?
Strogatz compared the relationship between the math community and the AI labs to the two biggest tennis tournaments of the summer. “I was thinking about Wimbledon versus the US Open,” he told me. At Wimbledon, the players dress in white, the fans are respectful, and while winning matters, sportsmanship is at a premium. The U.S. Open, however, is brash, individualistic, and disruptive. “People wear their own colors and fans heckle,” he said. “It’s more democratic, and it’s less elitist.”
Like Wimbledon, the field of mathematics is old, and a certain decorum is still cherished among its modern practitioners. Like the US Open, the labs are young, brash, and even a little rude. But AI’s ability to democratize access to mathematical genius may be a virtue of its own. Earlier this year, Shanmu Jin, a Chinese medical resident, used ChatGPT to solve a problem called Crouzeix’s conjecture, which professional mathematicians had failed to solve for decades. (Several mathematicians examined the result and found that it works, but the paper is awaiting formal peer review.)
Think about a world where non-mathematicians leapfrog the field after a few AI prompts. It would be a world with more knowledge. But it might be a world with less understanding. AI models have a mind of their own, and it is not even what I would call a “mind.” Their mathematical proofs often take on alien shapes, and their reasoning can be inscrutable to experts. “There is a Twilight Zone aspect to all of this,” Strogatz said. “We are interacting with electrons moving in circuits, and we don’t understand the intelligence that they create.” He raised the example of Google Translate, which has essentially solved the problem of live translation between languages. “Google Translate is amazing, but it has taught us essentially nothing about linguistics,” Strogatz said. “We don’t understand why it works. It just works. That’s the Twilight Zone. It is knowledge without understanding.”
That might be a huge loss. Understanding isn’t just a source of pleasure, the dopaminergic rush of the aha! moment. Understanding is power. When Maxwell understood the laws of electricity and magnetism that other scientists had built up through the 1800s, he showed through calculus that electricity and magnetism were not forces between two objects but something more like a field. Understanding Maxwell, Einstein could look at gravity—commonly assumed to be a force between two objects—and prove that gravity, too, is a field that extends across the universe. One might argue that it required a human understanding of fields to come up with the most important theory in physics.
The ability to understand the components of each new scientific breakthrough has allowed people to build mighty branches off the tree of knowledge. But Strogatz is prepared for a future where the tree grows beyond our reckoning. He shared the story of an MIT computer scientist, Regina Barzilay. After surviving breast cancer in her early 40s, she was stunned to discover how little her doctors knew about breast cancer diagnosis. She led a team that trained AI on hundreds of thousands of mammograms to predict breast cancer cases many years in advance.
Strogatz asked Barzilay, “Do you understand what the AI is seeing?”
“No,” she said.
“Does it worry you that you don’t understand your own invention?” Strogatz asked.
Her response: “Why would we limit a life-saving model to what humans can understand?”
Lost + Found
Barzilay’s question is hard to answer. I don’t want to die of the cancers that killed my parents. I also don’t want to lose my job to a tangle of GPUs. What do we want from technology? Solve every problem we have without creating any problem we don’t have. Cure diseases, but safely. Jailbreak the secrets of the cosmos, but do it with some respect. This is a lot to ask for, but it is not too much to want. Give me the benefits of superhuman intelligence; keep for me the things that make human intelligence meaningful.
When I got off the phone with Strogatz, I thought some more about Archimedes. He left problems for other people to solve, and the unknown served as an invitation. The danger is that by automating the search for knowledge, AI might obliterate the human connections that sustained the search. But there is another possibility. If intelligence becomes less uniquely human, we may come to value more highly what remains uniquely human: our capacity to relate and to care for one another.
Archimedes solved problems, and Archimedes wrote letters. He discovered, and he shared. AI may take over much of the first job. The second is ours, if we’re wise enough to keep it. Mathematicians may still teach proofs, argue over what they mean, translate alien logic into human insight, and pass that understanding to students. The challenge will be to protect that half of science: the pleasure of encountering another mind and being moved by the encounter. When the machines have discovered every truth there is to find, there may be nothing left to find except each other.
Innovation creates chaos in the short term, and progress in the long term.
In the future there will still be mathematicians, and there will be no 'end' of math to be discovered. Ask Gödel.
There is a frontier in science/knowledge. Every new breakthrough and tool just pushes that frontier back a bit.
And scientists move out to that frontier, and continue pushing. Sometimes, when a big move happens, a whole older generation is left behind, and a new generation is required to replace them.
And so it goes.
What’s happening now is very different to the waves of innovation that came before it. Past innovation required mathematicians to actually do the work and build understanding through that process. With this recent development, we have essentially ceded that work to the machines, which limits our understanding. It’s as if the frontier has been advanced but our ability to actually process that advancement and make sense of it has more or less stood still. And to make matters worse, the frontier will only get further pushed out at a faster rate by these rapidly improving models, while our ability to keep up lags.
I do agree there will still be a place for mathematicians but it’s hard to see this as a continuation of existing patterns of evolution. It’s more like mathematics has come in contact with aliens and we now need to figure out where/how humans fit in the new paradigm.