Will AI complete mathematics?

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AI in mathematics

1:14Hello, welcome to Inside Science from the BBC World Service with me, Tom Whipple. On Wednesday morning, the world's mathematicians arrived into work. They poured themselves a cup of black coffee. They wiped the blackboard clean. They sat down in the common room and started another day thinking about theorems. Then they learnt that half of those theorems-ish had just been apparently solved. This week, there was a mathoclism. Quite possibly the biggest day in the 10,000 year history of maths.

1:48On Inside Science this week, we're looking at that, but we're also celebrating human achievement too. It's Nobel Week. From mice with blue lights in their brain to massive ice cubes with blue lights underground, what do you need to know? Someone who likes maths, but maybe, I presume, not mice, is Caroline Steele. Caroline has been looking through the other science news. What have you got for us? Well, did you know that this week is World Space Week? I did not. Well, it is. And I've got some space news for you. Excellent. Excellent. There's always space news and I'm always excited to hear it.

2:21But first of all, let's do some maths news. Now, in 2023, ChatGPT arrived and the mathematicians laughed. Give it a simple sum and it would get it wrong. It was less mathematically skilled than my spaniel Wellington. Then, this week, overnight, OpenAI dropped hundreds of mathematical papers on the internet. Decades of maths progress overnight. PhDs of work done in hours. My name is Yang Kui He and I am a mathematician here at the London Institute for Mathematical Sciences.

2:55I suppose I see this coming, but it is still surprising to me how fast this has come. So, within about nine years or so, I've now finally seen the point where LLMs are doing very, very serious mathematics. My name is Sophie McLean and I am a final year maths PhD student at King's College London. One of the things that I love most about maths is the humanity and the need to know the answers and the love of the puzzles and working together to find a solution. And all of these things are lost.

3:26This isn't to say that AI is an entirely bad thing for maths, but I think it will get rid of some of the things that matter to me most about maths. What does all this mean? Well, as we heard, some mathmos are feeling apocalyptic. Others, though, are dazzled by the possibilities. We spoke to Kevin Buzzard, Professor of Pure Mathematics at Imperial College London. I'm an expert in number theory. I don't know all of mathematics, but about 30 of these are papers for an algebraic number theory. And of those, you know, about six or seven of them look quite impressive.

3:59On the other hand, I would say those papers were really theorems that it wouldn't have surprised me if a human had come along and said they could do it. But the vast majority of what's happening here is stuff that you would expect a serious researcher to have produced. A lot of mathematicians are just now beginning to wake up to AI. But some people in the field have been watching it, you know, for the last five years. You know, when ChatGPT came out, it was roundly mocked by mathematicians at being terrible at mathematics.

4:29It couldn't multiply together four-digit numbers reliably. And we just happen to be at the point now where it's sort of zooming past us. The real question is, what's going to happen in a year's time? Because you have to understand, these machines are going to get better and better and better. At some point, things will start to tail off, but we don't really know when. So the future looks very interesting for mathematics, I think, right now. Interesting, or if you are one of those decent researchers, or indeed one of those PhD students who might be feeling rather crushed that you've described this work as rather minor.

5:03What does this mean? I think that these new tools are going to accelerate the field, just like the computer. The computer accelerated parts of mathematics, so these new tools will also accelerate parts of mathematics. But they're certainly not doing everything that humans do right now. Intelligence is not a linear thing, right? We can't say, I'm more intelligent than you, or you're more intelligent than me. It doesn't work like that, right? For example, I'm much better than my daughter at mathematics, but she's much more emotionally intelligent than I am.

5:33Intelligence is multidimensional. And these language models, they're just sort of a new dimension of intelligence. When the chaos dies down a bit, I think we'll just understand that we're now able to move faster. What do you think is the human domain that's going to be left, where you can say, I'm applying my AI tool, and it's merely solving the Riemann hypothesis, but actually, I've got this little bit of mathematical emotional intelligence that means that I'm still valid as a maths researcher.

6:06So AI has not proved the Riemann hypothesis, right? I'm sure you don't. We've made some kind of incremental progress towards the Riemann hypothesis. It's just a famous mathematical problem that currently has a million dollars attached to its head, that it says something about the zeros of a certain function, the Riemann-Zeter function. As of yesterday night, we now know a slightly bigger region where there are definitely no zeros of this function. So in some sense, it's incremental progress towards the grand hypothesis, but it's certainly not a proof of the hypothesis.

6:39I mean, Hume has proved thousands, tens of thousands of theorems, and AI has proved hundreds, right? So, you know, AI is still way behind. It will get better at proving theorems, almost certainly. But where did the Riemann hypothesis come from, right? That was a human came up with the idea that this thing might be true. This was a question, a mathematical conjecture. And right now, AI has not come up with any interesting conjectures. This might change, but it hasn't changed yet. Who knows that the tech companies are currently very interested in doing mathematics,

7:10because doing mathematics is currently giving them big wins because it's making their models better at all sorts of things. But when those big wins stop, they'll start trading on other things, right? And maybe we'll find that they won't keep getting better and better. And so maybe they won't get to the point where they start making profound conjecture. Conjectures are, you know, they're like lighthouses telling us where to go and what to avoid. And they're explaining where the future of mathematics is. And at the minute, humans are still driving that.

7:40Didn't you give me a sense of the atmosphere in the maths common room at Imperial at the moment or on the maths WhatsApp chats? So it's really wildly different. I mean, mathematics is going to change. And so some people's reaction to this change, they're going to lose the thing they love. It's going to be very different. And so some people grieve. There's this theory that there are five stages of grief. You can absolutely pinpoint on every stage. You'd see people who are in denial. They just want the companies to go away.

8:12They think that AI should not be doing mathematics at all. You see angry people. You see bargaining people. You see the people that are telling the tech companies what they should be doing. And you certainly see very sad people who are just sort of confused about where they fit into things. Because unlike me, I think that these new tools are coming along. They're going to accelerate mathematics. And this is going to be great. And in a couple of years' time, I can completely envisage the best teams being sort of human computer teams working together.

8:43But other people just think it's not going to be like that. The computers will just dominate us on all axes and there'll be nothing left for humans to do. And especially if you're a young mathematician, if you think that that's where we're going, then you will be depressed.

Nobel week prizes

8:57Thanks, Kevin.

Nobel week prizes

8:58Now, as promised, we are going to get onto humans doing spectacular things. I got a call and he told me, are you Peter Hegemann? And I said, no, I'm Georg Nagel. Oh, sorry. So it was that Monday morning of Nobel week began with a wrong number call to Professor Georg Nagel of Würzburg University, Germany. Luckily, it turned out that he had, after all, won the Nobel too, sharing the prize in physiology or medicine with Peter Hegemann and his colleague, Karl Dyseroff.

9:32Then we were on to Tuesday when physicist Francis Halsen also had his life changed by a phone call. He barely had time to get in the champagne before Wednesday came and it was the turn of the chemists. Henri Cargon and Kenzo Suai for discovering why life isn't symmetrical. So what's been going on? Here is your Nobel debrief. Kit Chapman, chemistry writer and a man who's also won a very distinguished prize. He was winner last week of the Royal Society Trevadi Book Prize.

10:03He's with me. And so given you're here, Kit, shall we start with the chemists? Did you see this one coming? I didn't. And this one's quite controversial, but not for the science. The science is fantastic. So we're looking at chirality in life. And that's really easy to understand. So if you get your hands out, you've got your left hand, you've got your right hand. You can't rotate your right hand over your left without flipping it. It's a mirror image. And life doesn't behave that way. All of your proteins are made left-handed.

10:34So how does life order out those molecules? When you're doing chemical reactions, usually you get 50-50. Well, this work has looked at how you can make sure every reaction every time is always producing the right type of handedness. And they've figured out how life does it. You teed this up by saying this is very controversial. What's the controversial bit? There's two elements to the controversy. The first is that two people have won the Nobel Prize. The Nobel Prize allows up to three.

11:06And a lot of chemists would have suspected, if we're looking at chirality, Donna Blackmond is going to get a nod. So Donna Blackmond is considered pretty much the doyen of this area. The other reason it's controversial is we've kind of done it before. So in 2001, we had chirality as a Nobel Prize. Barry Sharp, this was one of the people who won. He's actually won two Nobel Prizes now. He's a bit gone off and won one for CH activation. And we had one in 2021 as well.

11:37So we've done chirality now three times with the Nobel Prize. Look, let's move on to physics. Caroline, physics is your thing. And this is a great prize, isn't it? It's such a good one. Francis Halsen, so he's a researcher who basically turned a giant block of ice in the South Pole into a neutrino detector, which is just such a cool experiment. Do giant for us. How giant is giant? I think it's a square, a kilometre cubed. So pretty big, not like an ice cube you might find in your drink. And it's called ice cube, which is also very sort of on brand for physicists.

12:10Just say it how it is. It's an ice cube. So neutrinos are these tiny subatomic particles. They were first posited by Wolfgang Pauli back in 1930 when he famously said, I've done a terrible thing. I've postulated a particle that can't be detected. But ice cube has detected neutrinos. So they come from stars and other cosmic events. They hurtle through space. They arrive on Earth. And they're much more common than you might think. In fact, there are trillions passing through your hand every second.

12:42But they just pass through and do nothing. They just pass through. Yeah, they basically really don't interact with matter much, which is why they're so incredibly hard to detect. So Halsen came up with this sort of bizarre way to detect them. It was quite sort of controversial when he came up with the idea. A lot of people thought it would be a waste of time and money. But when ice cube was switched on, it almost instantly detected a neutrino back in 2013. And my favourite fact about these detectors is that they're all facing downwards into the Earth because they're not looking for neutrinos that have passed through the ice.

13:14They're looking for neutrinos that have gone through basically the North Pole, gone through the entire Earth and come out the other end at the South Pole. So this ice cube is basically using the Earth as a way of getting rid of noise from other particles that would sort of disappear as they pass through the entire planet Earth. What I like about this prize, and I think this leads into the next one, is it is really practical. It's a prize for a tool. I think the last one I can think of like this is CRISPR for the genetic engineering tool.

13:46But the next one is for a tool as well. Let's hear what optogenetics is from Georg Nagel. Optogenetics is the manipulation of a cell or a complete animal by illuminating it with the correct wavelength. So, for example, in our case, it was the blue light. In red light, it would not respond. But with blue light, it responds. And you can do this only via genetically modifying this cell or animal or plants.

14:19He's left off his bit very modestly there, which is he found these proteins that respond instantly, essentially, to light. They were found on alga. He helped characterize them. And then they will send an electronic electric signal. And it basically turns cells into light responding cells. I think it has really important practical applications. So, I think the researchers have said, you know, there might be potential to use this tool in the future to help people who have lost their sight in a specific way regain some of their vision, which would be a really exciting practical application for it.

14:59And there's another one of this. And I mean, this always happens with Nobels. But, Kit, you alluded in the first one to Donna Blackmond not getting the Nobel. For this, I think that the name that's missed off, there's always a name, but Ed Boyden is someone who was one of those sort of fateful second authors on papers who hasn't. But this is a broader problem with the Nobels, isn't it? It is. I mean, the way that the Nobels were structured, initially, Alfred Nobel wanted it to be the best research of the year.

15:29And they've become kind of a lifetime achievement award. The other problem you've got is it's very much dependent on the Swedish Academy of Sciences and who is actually on the judging panel. And, of course, they take in every suggestion, you know, people who have won the Nobel Prize in writing and say who they think should be nominated. And they try and be as fair-minded as possible. They're all fantastic and capable scientists. But you're going to have a natural leaning to areas that you're interested in. But this year, I think it's a pretty good mix this year, actually. Fabulous. Look, thank you very much, Kit.

16:00And if people want to know more about chemistry, do read the other prize winner from this month, the Royal Society Trivedi Book Prize, The Age of Alchemy, by Kit Chapman. And I'm not even just saying this. It is absolutely fabulous. You're listening to Inside Science from the BBC World Service. Tell us what science you think we should be investigating. Our email address is insidescience at bbc.co.uk.

16:26They say everything happens for a reason. But, I suspect, everything happens for a Reese's. Like this commercial break. Did you need 15 seconds away from music? Or 15 seconds to eat a Reese's? Perhaps it's true. Everything happens for a Reese's. 9-1-1, what's your emergency? What? Please help us. They're everywhere. We think they're murder hornets. 9-1-1 Nashville returns. There's a homicide detective down here. Three bodies on the ground. With emergencies of biblical proportions.

16:58I will send swarms upon you and upon your people. That's one of the plagues for the Bible. And somebody's unleashing them on Nashville. Brian Phillippe joins 9-1-1 Nashville. All new Thursdays on ABC.

Margaret Hamilton legacy

17:09Next day on Hulu. Now, Caroline, there has been a, well, there's been a death this week, but a wonderful lady to celebrate. In 2016, President Obama reflected on the moon landings, on the delicate, dangerous task of putting humans on the moon, of the split-second timing, and all the things that could have gone wrong. And then he gave former MIT programmer the Presidential Medal of Freedom. He said, our astronauts didn't have much time, but thankfully they had Margaret Hamilton.

17:46Hamilton led the programming team for the Apollo missions, and she was a pioneer in every sense of the word. Now, we've just, before we get to chatting about her legacy, we've got a clip of her talking about the sense of responsibility she felt. This is with Kevin Fong on 13 Minutes to the Moon. A person's life was at stake. The astronauts' lives were at stake. And that's when I began to worry about everything working together and what the astronaut might do by mistake. What would happen as a result of hooking up the wrong radar, the radar in the wrong position

18:18could affect the software. How do you let him know it when he's busy doing his stuff and he doesn't even notice that there's a problem? Um, so it became a mission. Caroline, I mean, she was, she had an extraordinary job. Unbelievable. Yeah. An incredible woman and, you know, carried a lot of responsibility, as you said, and you can really hear it in her voice in that clip there. And it did all sort of come to a head. There was a hairy moment. So in 1969, minutes before Apollo 11 was set to touch down, an onboard computer alarm sounded,

18:56indicating that there's an emergency, which I can't imagine what that must have been like for the astronauts on board. But basically, Hamilton's software was able to compensate for this hardware failure. And the mission was able to continue rather than having to abort. I tried to have a bit of a look into the engineering behind it. It seems that she kind of specialised in, in a way, pioneered priority-driven software, which is where you basically shut down anything that's unnecessary so that you can prioritise the absolutely essential parts, in this case, to land on the moon.

19:27And it all happened. I mean, largely thanks to her. There's a really famous photo of her stood next to her code. The piled-up code. Yes, the pile of code. And oh my gosh. I mean, you know, it's basically like imagine stacking maybe 15 really fat books on top of each other. And the code reaches as high as she is tall. And just briefly, because we have to say, I mean, I'd rather sort of just concentrate on the thing she did, which is incredible. But you can't leave aside the context, which is she was a woman leading a bunch of programmers

19:58in the 60s. Yes. And I think she was, you know, she was one of the first women employed by NASA entirely. I think she was the only woman working in this area at MIT at the time. You know, it was a big, it must have been a big deal. And if you think about it, there will have been a lot of additional hurdles she had to overcome to get there. So, yeah, I imagine she was an absolutely incredible scientist and incredible person as well. Well, look, staying with space.

Star death planets

20:25Yes.

Star death planets

20:26News keeps happening in space. News keeps happening in space. And it's World Space Week, which I can't believe you didn't know. No, I will go home and remedy it and put out the bunting. Well, actually, I think World Space Week's origins are quite sweet. So they sort of, it's always between the 4th and 10th of October. The 4th of October, 1957 was Sputnik's launch. 10th of October, 1967 was the Outer Space Treaty coming into effect. So that is, that is what we're sort of celebrating. Fittingly, there's been some exciting space news published in Nature Astronomy this week.

20:59So astronomers have found a planet, probably, hopefully found a planet, born from the ashes of a dead star. And that dead star is now slowly eating the planet. Oh, goodness. So it's like sort of birth, death, cannibalism. Exactly. Yeah, so as a star like our sun reaches the end of its life, usually it swells into a red giant, sheds some of its outer layers, and then you end up with a sort of dense hot core called a white dwarf. And usually we think of this process as being quite destructive for planets.

21:30You know, if and when that happens with our sun, it's not going to look good for Earth. But in this case, astronomers think that this discarded material actually came together to make a new world. So that makes it a second generation planet. First generation planets like Earth come from the birth of stars. Second generation planets, this is the first one we've found around a white dwarf, come from the death of planets. And they worked out using two sort of clever clues. Basically, there are some unusual chemical fingerprints coming from the white dwarf.

22:01There's a lot of niobium. Do you know when and where niobium comes from? I do not. I didn't until I was reading about this. Niobium is formed in the death of stars. And that basically suggests that this star is receiving material from a planet that's formed since the star has died, and it's sucking it out of the atmosphere. And then also the brightness pattern of the star. So yeah, pretty cool. Because, you know, we often think of the death of the star as kind of the end of a planetary system, but actually, it turns out a star's death could be the start of a new chapter.

22:34Rebirth. Thank you, Caroline.

DNA typewriter technology

22:36And now here's Roland Pease with news of a remarkable way of recording the individual fates of millions of cells in a developing embryo. Yes, think of it as a set of birth certificates issued as new cells are born, but written not on paper, rather on strips of DNA inside each cell. The reason to want this is to track how the single cell of a fertilized egg, or zygote, proliferates into hundreds of specialized cell types and millions of cells for our different tissues and organs, how they're related, and how different paths got chosen.

23:11And with the rapid developments in gene editing, says project leader Jay Shendure, the natural language for this record is DNA. DNA is digital, right? So there's four letters, A, G, C, and T, analogous to the characters on a keyboard. And, you know, we call this technology that we use in the paper DNA typewriter, and it works very much like a typewriter. We're punching keys, except, you know, the keystrokes that we're making are being written into DNA, and we're going left to right, right?

23:41Which in the parlance of molecular biology, we call five prime to three prime. Those being the labels for the beginning and the end of a piece of DNA. I mean, just give me a sense how this sort of works, and I'm sure it's going to be more complicated than I can really imagine. But in a sense, what you're doing is you're putting into the embryonic cell some kind of DNA that's additional to all the chromosomes that they have, and then some mechanism by which this writing is done? Correct.

24:11We're essentially using genome editing. So I imagine you and many of your listeners have heard of CRISPR by now. So we're using a kind of CRISPR technology called prime editing. And the enzyme that does prime editing, we're putting it in to the mouse zygote. And we're also putting in, you know, essentially the paper, we call it tape, that is going to get written to, as well as other bits that allow the editor to write the keystrokes to the tape.

24:41As it were, start on page one, then page two, page three, and so on. Exactly. But page one gets split into two pages, right? There's a copy machine in there. That's the cell division. And so maybe you punch a few keys when you're on page one, and then that cell divides. And now you've got, you know, whatever you've written up to that point is passed on to the daughter cells. And then now each of the cells punches a few more keys independently, and then those divide, and so on and so forth. So we're taking advantage of the cell's machinery to kind of copy our records forward as development

25:15proceeds. And so this means that each cell has a log that carries its history. Exactly. That's exactly right. I mean, the numbers must be phenomenal because even a small embryo presumably has millions of cells that you're trying to keep track of. Yeah. So we profiled this embryo, and we actually didn't know if the experiment worked until we started looking at the embryo. So we chose the time point when the embryo was big, but not too big. And we should say this is a mouse embryo, not a human one. It's a mouse embryo, and we stopped it around two weeks in, and that's when we took a look.

25:47And yeah, it's millions of cells. We were able to profile over a million of those to build a family tree from what we read out there. And the point is that by this stage, the elements of all their organs, their hearts, their kidneys, the eyes, the brain, all these different tissue types are already there. And the idea would be to work out how they got to be the way they are. Exactly, right? So at this point, there are hundreds of cell types, right? There's different kinds of heart cell types, different kinds of neurons, blood vessels,

26:17et cetera, hundreds. And we are trying to figure out from these, you know, the typewriter information, we can reconstruct their relationships to one another. And from the resulting tree, we can kind of, we can time, you know, when these cell types emerged and also what their relationships are to one another. I mean, for me, I'm endlessly obsessed by the magic of the way that simple cell at the very beginning of this whole process become a complex organism with millions of cells doing

26:48completely different but complementary things. And I guess at the moment, this is a sort of a proof of principle of what you might be able to discover. But you have ambitions to go really deep into those processes. Yeah, I think when the genome was sequenced, we thought it was very big, right? It was 3 billion of these letters, right? Now, you know, we look at what will fit on a thumb drive and it feels quite small. So it's miraculous that something that is a small file on your thumb drive can essentially encode the complexity of not just a human, but all of these other species that we see

27:23out there in the world. And, you know, I do find it miraculous. And yeah, I think even with the data we've generated, this is just kind of the first step along a path. But the nice thing is that, you know, a mouse is a mouse, right? It develops in a certain way. And, you know, we can look at it with this technology or that technology through the lens of lineage, through imaging. But there's one underlying reality that we're trying to capture, you know, analogous to blind men, you know, feeling an elephant from different angles. And I think we're approaching the point where really trying to put all this information together

27:57and come up with a real description of how it works from single cell to free living organism running around is on the horizon. And I'm very grateful to be around for that phase. Jay Shendure's DNA typewriter was just published in Science and Jay led the project at the University of Washington and the Seattle Hub for Synthetic Biology. Thank you very much, Roland. And that's all we've got time for. You've been listening to BBC Inside Science with me, Tom Whipple. See you the other side of the mathpocalypse.

28:30It's goodbye from me and goodbye from Caroline. Goodbye. How's a movement that started off as a meme taking on India's prime minister? I'm Tristan Redmond, co-host of the BBC's Global Story podcast. On today's show, we hear how the so-called cockroach movement has motivated young people across the world's biggest democracy to protest for their rights. How could they reshape the country? For more, listen to The Global Story on bbc.com or wherever you get your pods.

29:03I'm Tristan Redmond.