Out-numbered: AI’s contentious maths milestone

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0:54The Economist

1:00Hello and welcome to The Intelligence from The Economist. I'm Jason Palmer.

1:10Today on the show, a hard look at the counterterrorism machinery passed after 9-11 and a tribute to a champion of Chinese traditional opera.

1:24First up, though.

1:34We talked yesterday about how artificial intelligence was, on balance, creating jobs, a pattern you see whenever transformative tech breaks onto the scene. Today, we want to look at a transformation in the heady world of higher mathematics. OpenAI says it's cracked something called the Navier-Stokes existence and smoothness problem. It's one of the Millennium Prize problems, seven devilishly difficult mathematical mysteries first listed in 2000. The prize for cracking one is a million dollars. Look, AI is better than humans at lots of stuff, and it should come as no surprise that it can power through equations and theorems that have stumped humans for decades.

2:17But here's the thing about mathematics at this level. It's not just about the solution. It's very much about how you, or it, got there. One thing we can say for sure is that something significant has happened. But the way in which this becomes significant, I think, is very much dependent on who you ask. Sam Weigley is a science correspondent for The Economist. This really should be an amazing moment for mathematics. This is a major, major problem that they have been trying to work out for a long time. But the way in which it has happened has meant that there are a lot of complex feelings at play here within the mathematics community.

3:03And there are some serious questions that these mathematicians will be asking and answering over the years to come. So let's start in really basic terms here. What is the problem that's been solved? Right. So it has this somewhat complex name. It's called the Navier-Stokes smoothness and existence problem. And it relates to this set of equations called the Navier-Stokes equations. So what these are, are a set of equations that describe the motions of fluids. So here we're talking liquids and gases. So that means they're very useful in a broad variety of practical cases, like modeling how air is going to flow over the wings of aircraft or over Formula One cars, for tracking how blood will circulate around the body, for modeling how oil will squeeze through pipes, and so on and so on.

4:01And so that's all good and well. You know, physicists, engineers, they have a great grasp of how to use these equations in these practical scenarios. But what mathematicians are interested in is understanding the structure of the equations themselves. So they would like to understand these general properties of what is and isn't allowed within these equations. And this is where this problem that OpenAI say they have solved comes in, this existence and smoothness problem. It asks, in a broad sense, whether there are situations where these Navier-Stokes equations break down when describing seemingly well-behaved fluids.

4:42Is there a point at which they diverge from reality? The equations just malfunction. And this is one of the seven Millennium Prize problems. And what do we know about how OpenAI went about solving it, allegedly? Right. So they say that on August 28th, they started training a new internal model. And this model was showing unprecedented abilities. They say it far surpassed their latest commercial version. This is GPT-6 Astra, which itself was only released on September 3rd. So all of this is happening in a very recent time span. And using this new internal model, they spawned teams of AI agents.

5:25So these are AI models that can read the internet, they can run code, and they can even send messages amongst themselves. And on September 1st, they set this swarm of AI agents to work on the open millennium problems. Only 50 hours later, so very soon after they started working on it, a team of around 100 agents resolved this question that is very closely related to the Navier Stokes. problem. And noticing this, the scientists at OpenAI thought, okay, maybe this is a stepping stone to a full solution. Let's direct our full swarm of AI autonomous mathematicians to work on the Navier-Stokes problem.

6:08So on September 5th, and this is only about 88 hours of work later, but also 2.7 million messages being sent between the AI agents and at least$6.5 million worth of computing resources, a team of around 10 ,000 agents discovered a solution to the Navier Stokes problem. Now, normally when these kinds of centuries-old problems are solved, there is great elation in the mathematics community, but you hinted earlier that is not the case. Yeah, it's here that the story starts getting a bit messy. So OpenAI says that it only began working on the problem after hearing a rumour that a pair of mathematicians, Tristan Bookmaster, a mathematician at New York University, and Levant Alpoge, who happens to be an anthropic employee, but as I understand was not working in his capacity at anthropic, that they were nearing a solution and directed all of their resources to try and solve the problem first.

7:10It seems that there were talks held before the solution came out between Dr. Buckmaster and OpenAI. They don't seem to have gone that well. And they led to the mathematician publishing his work with Levant Elpoge early and in unfinished form online before OpenAI's announcement. Now, mathematicians, while certainly excited about having a solution to this, are saying that the OpenAI paper that outlines their solution is not going into the type or level of detail that one would usually expect in a mathematics paper into how they arrived at the solution. And certainly the kind of fanfare around this is about a solution being found, but not the insights that were necessary to get to that solution.

8:03This is slightly different to what mathematicians want from work. They're not doing it to just tick a box of a final solution. They want new mathematical insights to be generated along the way. But I think the real question that mathematicians want open AI to answer is not about the solution itself, but it is about how the solution was landed upon. What was needed to get to this solution? Because usually that is where the interesting mathematics lies. Well, this is what I wanted to ask you about, because I can tell you feel strongly about this story. Shouldn't we expect this story to play out over and over again, the rest of the millennium problems to fall, the rest of the conundra of the mathematics world to just simply be solved, skipped to, I guess?

8:53What does that mean for maths and where it goes? Yeah, it raises some very deep questions about what the purpose of mathematics is in the first place. So, and it is true that mathematicians have been working on this Navier-Stokes problem for a long time and have generated many new insights along the way. So if OpenAI comes in at the final moment and lands on a solution, there is a question of who deserves the credit, but there's also a question of what this is really good for. What have we learned from this new solution that the AI agents have come up with? And will this result lead to spinoffs of new fields of research, new interesting questions to investigate.

9:34There is also quite a difficult question behind all of this, which is whether the mathematician's work may have unintentionally, inadvertently snuck into the training data that OpenAI used to build their latest model. They were making use of OpenAI's products and OpenAI have been unable to rule out this as a possibility. What we can say is that OpenAI have said they will not claim the$1 million prize. And certainly mathematicians will now be pondering what all of this means for the future of their subject. Sam, thanks very much for your time. Thanks for having me.

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12:04It's 25 years today since two hijacked planes hit the World Trade Center in New York and another hit the Pentagon. It's hard to overstate how terrorism at that scale on home soil changed America in a day. With a speed and a congressional consensus we could only dream of today, President George W. Bush hastily created a vast counterterrorism machinery. New legal authority to wage war, far-reaching powers of surveillance at home and abroad, loosening the leash on agencies like the CIA and creating a whole new one, the Department of Homeland Security. That was then. Now the jihadist threat has mostly disappeared in much of the world.

12:48And as much of that legislation is being pressed into service to an imperial presidency, many think it's time to pare those powers back. 25 years ago, America's counterterrorism regime initially arose with the overwhelming support of Democrats and Republicans alike. These laws were not passed out of malice, but from an urgent fear and anxiety about potential follow-up attacks against America. Sam Beale is a Washington correspondent for The Economist. Fast forward a quarter of a century later, fewer than 1 % of Americans say terrorism is the top threat facing the country. And yet at the same time, that counterterrorism apparatus is largely still in place.

13:29And over the last year and a half, we've seen Donald Trump adapt that regime rather cynically for his own ends. And that's seeding a bit of a debate at the moment over whether America, 25 years after 9-11, should start reforming or paring back some of those legal authorities that were passed in the aftermath of 9-11.

13:49So talk me through those, the revisions that were set up after 9-11. I mean, there are two ways to think about this. There are laws that are passed to fight terrorism abroad, and there are laws that are passed to fight terrorism at home. So abroad, the main one is the authorization for the use of military force, which basically hands immense powers to the president to fight al-Qaeda and its associated groups abroad. And then at home, you have a dramatic expansion of the surveillance and police state. So think the creation of the Department of Homeland Security, the passage of the Patriot Act, warrantless surveillance powers, all those things.

14:20And you also have a range of financial statutes that are passed that allow the Treasury to go after terrorist financing groups. You have a broadening of the term foreign terrorist organization, which also lowers the threshold for the Treasury to go after finance networks that provide support to terrorists. And as regards what President Trump has been doing with those provisions more recently, what's been going on with the foreign end of things? I mean, what's the most striking thing about this administration has been the expansion of the term who is really a terrorist. And I think the best example of this has been in Latin America, where the president on his first day on office branded a group of Latin American narco gangs as foreign terrorist organizations.

14:59And pretty swiftly thereafter, we saw a pretty relentless boat bombing campaign against suspected drug running vessels, which carries on to this day. And the tactics and the way they've gone about that is entirely reminiscent of the war on terror, targeted killings. And at home, we've also seen sort of a broadening of the definition of terrorism. So the president has asked the FBI and law enforcement authorities to go after loose left-wing groups that the administration has branded as domestic terrorists. And they've used some pretty potent tools to do that. So federal agents have used the Patriot Act to seize financial information without warrants on progressive groups and charities.

15:36In Texas, anti-ISE protesters would sentence the president on rather spurious charges of providing material support to terrorists. And that's a statute that was really designed to go after Al-Qaeda and ISIS financiers. And then you have the repurposing of entire agencies. So DHS has gone from going from an agency focused on terrorism to one that is now at the center of Trump's deportation engine. So what about the nominal purpose of any of these laws, the terrorist threat itself? How has that changed in the quarter century since 9-11? I mean, it's changed quite a lot. So the threat from jihadist terrorism has really receded.

16:11Two decades of forever wars in the Middle East really degraded the ability of groups like al-Qaeda or ISIS to launch organized attacks on America. We have seen a rise in more lone wolf attacks by self-radicalized terrorists. But, you know, over the last five years or so, different administrations have really focused on the threat from domestic terrorist groups, whether that's far-right groups or also a recent surge in left-wing violence. And that has really changed the way the national security state sort of thinks about terrorism. But many of those authorities are still in the books. And what about the push that you mentioned coming from the Democrats' side about overreach and what might be done to pare back the powers of some of these laws?

16:52Obviously, Democrats have looked at the way the administration has used these tools aghast. And attempts so far have proven rather tentative and embryonic. So the big one was earlier in the year, Section 702 of the Foreign Intelligence Surveillance Act, which basically allows American spies to snoop on suspects abroad. But in the process, they will collect information on Americans. That authority lapsed in June, partly over fears that the administration was politicizing the intelligence apparatus. But many Democrats want to see that law reauthorized with stronger safeguards. Elsewhere, other members of Congress have said it's time to repeal that 2001 authorization for the use of military force, given how the presidency now can just go to war whenever it wants.

17:33But a lot of these are tentative, and plenty of people say Democrats might rail against the overreach now, but when they're in power, it's very hard to pare back those powers. So where do you see what you call embryonic movements heading in the future? One of the striking things about this 25th anniversary of the attacks is that 100 million Americans today were not alive at the time. And you've had a whole generation that has grown up under the specter of some of these very powerful tools which have curtailed their civil liberties and have also come at a great cost in blood and treasure. And there is a generation of lawmakers who are now entering Congress who are trying to think hard about how can you pare back that system and some of those abuses that we've seen under this administration.

18:14But it might take a while for that to really filter through. Sam, thanks very much for your time. Thank you so much for having me, Jason.

18:55The End Outside of a few social issues, their attitudes line up with those of their compatriots. So what is it that's setting so many Europeans against Islam? This week on Insider, our deputy editor Edward Carr and a panel of our expert journalists tackle that question head on. If you're a digital subscriber, Insider is already included in your subscription. And if you're listening for free, you can watch extended clips via the link in our show notes.

19:33Ann Wroe:In 2019, China's government bestowed the title of People's Artist on Guo Lanying. Anne Rowe is The Economist's obituaries editor. Guo Lanying's favorite song and signature song was My Motherland.

19:53Ann Wroe:She was an opera singer, but also she was a national treasure.

20:04Ann Wroe:At 15, she fell in love with a particular opera called The White-Haired Girl, which was the story of a peasant woman who was abused by a rapacious landlord who wanted to make her his concubine. and rather than submit she ran away and hid in a cave and there her skin got whiter until she looked like a ghost and then she was rescued and restored to life by the troops of the Chinese Communist Party. When she sang the songs from the white-haired girl she felt she was really taking part in the revolution. China, even under this new society, was still a place where women were on the very bottom rung.

20:49Ann Wroe:And she loved the role of the heroine in this opera because she could sing about the general repression of women. She sang most of the songs in this opera with a condemnation board strapped to her back and kneeling down, looking incredibly uncomfortable, but singing wonderfully. And her final aria in the opera was called Pathetic and Disgraceful, which was her attack on all her accusers and abusers and singing it for all women. But her way into this world of opera had been enormously hard. Her whole life had been a struggle.

21:35Ann Wroe:When she was born, her mother didn't have enough milk to feed her. In fact, in the house, her parents just had a handful of millet and an iron bowl. Her mother took her and left her in the woods to die. Her aunt, however, rescued her. Then, at the age of four, the whole family was starving again, and her parents, again, didn't know what to do with her. And at that stage, an uncle suggested opera, that if she became apprenticed to an opera master, she could then, as her mother said, be fed by a hundred households. So she was sold for five copper coins, or about$12, at the age of four, and went under contract to an opera master.

22:27Ann Wroe:And this regime was absolutely brutal. The art of opera was completely beaten into her. If she was insubordinate, she'd be bludgeoned with a rolling pin. If she failed to do a perfect somersault, she'd be beaten with a sheep whip. and from this regime she tried to run away but she'd been taken to another part of the north and she had no idea where she was so she had to stay also the feeling was that perhaps she had been spared to sing she hadn't been meant to die in the woods and she hadn't been meant to starve to death she was meant to be a singer so she persevered with this terrible training which even involved, in the wintertime, having to lie down on frozen lakes and sing and vocalise until she had melted a hole in the ice.

23:30Ann Wroe:The song My Motherland became famous from a film about the Korean War where she featured as a soldier hiding in a cave singing this song which became a monstrous hit in China.

23:49Ann Wroe:Her struggle had left her, she said, with a real layer of anger in herself. But she could also dissolve her anger and totally forget it when our whole audience roared her songs back to her. Anne Roux on Guo Lanying, who's died aged 97.