Mathematician Vicky Neale, senior teaching associate in the Department of Pure Mathematics and Mathematical Statistics in the University of Cambridge and director of studies at Murray Edwards College, is excited.
She’s been watching some recent breakthroughs that mathematicians around the world have been making in a huge and open collaboration on an ancient mathematical problem. Neale tells Adam Smith how she is now building the news into her work that aims to improve the ways maths is taught.
This podcast is produced and presented by Adam Smith
Adam Smith: I’m listening to a story about a lightbulb moment. That second when a school pupil’s eyebrows soar and her head lifts up: she’s got it. In this case, she’s found a solution to an algebra problem.
Vicky Neale …She suddenly realised that this algebra, which she’d sort of been introduced to at school, that she sort of half understood, that she could see why these manipulations worked. We drew a little picture, we talked about it, but she also suddenly understood how that had helped her to answer a problem. She’d been trying some numerical patterns, which is really important, that’s a lot of what mathematicians do, but we were chatting about how the next stage is to try and come up with a convincing argument.
I might use the word ‘proof’, that’s sort of technical jargon, and she said, “But I’m not going to be able to do that ‘cause I can’t check all of these examples.” And I said, “That’s right, you’re going to have to come up with some other kind of justification.”
And via this algebra, by a little calculation, she was able to do that and see that it was always going to be true and for her, I could see, “Oh, this is something a bit different from what I’m used to, I’m quite excited by that, I can see how this algebra gives me the capacity to do something much more than I ever thought I was going to be able to do…”
AS: I’m Adam Smith. Welcome to Pod Academy.
Vicky spends most of her time on a project with researchers and teachers trying to improve the ways mathematics is taught. Running beneath all of this work, like an underground river, is the enterprise of mathematics itself—the questions and the problems are flowing and bouncing off rocks and pushing forwards constantly through university mathematics departments across the world. I met Vicky at the college and started by saying that it seems to me that maths is a bit like Marmite—people either love it or hate it…
VN: That’s right and I find that very sad for two reasons. One is that I think very often the people who are saying that who don’t understand why I’m so excited about maths, that’s because they don’t know what it is that I’m excited about. They have this perception that’s very different from my perception. The other is that I think sometimes people have this perception that maths is an ability either you have or you don’t have. It just sort of depends how you were born. And I start from the perspective that everybody is capable of thinking as a mathematician, is everyone going to go and get a Fields Medal in mathematics—the equivalent of a Nobel Prize? No of course not, because not everybody is going to want to spend their time, immerse themselves in it, but I strongly believe that everybody has the capacity to make progress, to understand all sorts of things. And we see in schools, extraordinary examples where successful teachers, successful departments are able to have this impact, of course it’s trying to help everybody to have a positive experience so that whatever they go on to do they don’t feel that maths is not relevant to them, they don’t feel that they’re unable to engage with it.
AS: One of the other funny perceptions that a lot of people have about mathematics, probably myself included, is that it’s static, that there is a set of rules, your teachers try and teach you and that there are just these rules and you’ve just got to learn it, and yet I know there are mathematicians in universities up and down the country, so there must be new things to fnd out…
VN: That’s right, when people talk to me about mathematical research, some of the people I’ve spoken to I think have the idea that I must do because I’m doing this hard maths, that must mean doing long division with bigger numbers, something like that. I cannot tell you how far that is from the truth, OK. I am as bored doing long division with large numbers as everybody else is, I have a computer to do that if I need to divide large numbers so yet mathematicians up and down the country are doing all sorts of new mathematics now, that really isn’t at all like that doing bigger and bigger calculations.
AS: Can you give me some examples?
VN: Yeah, one of the things that I’m really excited about over the last 12 months is some progress on a really old problem in a subject called number theory, which is all about properties of whole numbers. That’s one of those subjects where it must feel like we know everything there is to know about the whole numbers because they’re whole numbers, how hard can it be? The answer is that it can be quite hard. Within that there’s this interesting sequence of numbers called prime numbers.
AS: I think I remember these from school.
VN: Aha, very good. Right, so a prime number is a number that’s only divisible by one and itself. Exactly. So 17 is a prime number because I can’t divide it by anything except one and 17. Eighteen is not a prime number. I can divide it by two and six and all sorts of things. So prime numbers turn out to be really fundamental in understanding whole numbers because they’re kind of like the building blocks of all the other numbers, so mathematicians are really interested in understanding the properties of the prime numbers.
AS: Why are they the building blocks?
VN: Because if you give me any positive whole number bigger than one, I can write it as a product of prime numbers, so for example, number 12 is two times two times three, so it’s a product of prime numbers. And it turns out I can do that for any prime number. So if you like, I can take my big number,whatever it is and split it up as a product of these smaller prime numbers, or maybe it’s a very large number that’s a prime and can’t be split up further and turns out that there’s only one way of doing that, so if you and I had sat down separately and tried to work out 12 as a product of prime numbers, we’d have come up with the same primes. You might have written it as two times two times three, or three times two times two, we might have written it in a different order but it’d be the same list of primes. And that uniqueness, the fact that there’s only one way of doing it turns out to be something quite important. That’s something special about prime numbers.
AS: So even for a really big number that’s not prime, there’s just one way to get there using the other primes?
VN: That’s right. For me, what’s fantastic is that I can say that to you with complete and utter certainty even without knowing what your big number is. Isn’t that fantastic? I’m absolutely, 100 percent certain that whatever big number you give me, I will,
- be able to write it as a product of prime numbers, maybe one prime number, maybe many. And
- there’s only one way of doing it.
That’s an example of a theorem, it’s a mathematical fact that we can prove. So it’s not just, that we’ve tried it with lots of examples, it’s not just that we’ve got a computer to do it with lots of examples, because you do that and there still might be another number where it might not work, who knows? So this is a great example of a theorem, that we can prove using mathematical arguments. That’s one of the reasons that prime numbers are so important because they somehow form this mathematical structure from which other numbers are built.
AS: That’s something that you can show. Are there things that you can’t yet show but you think might be true?
VN: Absolutely. What mathematicians do is they’re looking for patterns, they’re playing around, they’re trying things out and then they’re making predictions. And in mathematics we tend to call predictions ‘conjectures’. So they say, I think this is true, but I can’t prove it yet. I’ve done my examples and it seems kind of plausible, I can’t find a case where it’s not true, I don’t have a proof yet.
A fantastic example of that is called ‘the twin primes conjecture’. The twin primes conjecture is all about pairs of primes that differ by two. Eleven and 13 are both prime numbers, they differ by two. 107 and 109, you have to trust me, are both prime numbers, they both differ by two.
It turns out there are infinitely many prime numbers, that was known to the Ancient Greeks, 2,000 years ago. Euclid could prove there are infinitely many prime numbers, they keep going on and on. The question with the twin primes is: are there infinitely many pairs of primes that differ by two, infinitely many of these twin primes?
AS: Do all primes have a twin?
VN: Great question. No they don’t. There are examples of primes that just sort of live in a little gap by themselves and they sort of seem to become more spread out a bit. Intuitively it makes sense that primes on the whole become more spread out, because if you’re a big number there are lots of other things you might be divisible by, so roughly speaking it feels like it might be harder for you to be a prime as you go further along. We’ve got some arguments that can show that the primes on average become more spread out but every now and then you get these clusters of primes that are closer together, like these twin primes. And this is a really old problem, I’m not even sure how far back it goes, it’s one of those questions that’s so natural, it’s hard to imagine a time when someone wouldn’t have asked the question. As time has moved on it’s been easier to look for examples, we can get computers to look for examples, there are some really big examples known of twin primes, we’re never going to prove it by finding big examples, because maybe that was the last one. So mathematicians believe there are infinitely many of these pairs of primes that differ by just two, but nobody has a proof yet. But a proof looks a lot closer now that it did a year ago.
AS: Wow. All of a sudden.
VN: It’s really exciting progress. All of a sudden in the last year it’s amazing how quickly it’s all happened somehow. Yeah.
AS: What’s happened?
VN: May last year, a mathematician called Yitang Zhang, University of New Hampshire in the States, published a paper, and he showed there are infinitely many pairs of primes that differ by 70 million. And 70 million, you might think is a large number, and you’d be right, and of course that feels a bit rubbish because we wanted two [are there infinitely many pairs of primes that differ by 2], not 70 million. The reason that mathematicians were so fantastically excited by this was that it was the first time that anyone had proved any result like that. So it was the first finite number that anybody had. So this was amazing, it came completely out of nowhere, nobody had any idea this was coming, Zhang wasn’t terribly well known, he’s working on these things, completely out of the blue this paper appeared on the internet, and of course everyone gets very excited so they’re thinking about, is this paper right, they’re checking the arguments, it all seems to stack up. But then the question is, could we do better than 70 million, I mean what’s special about 70 million?
AS And you would eventually want to get to two?
VN: That’s right, the aim is always to get from 70 million down to two. And it wasn’t clear that Zhang had been ultra careful about the details, because 70 million is so big, that it doesn’t really matter whether it’s 70 million or 65 million, it’s still a long way off two, but what happened was a bunch of mathematicians broadly interpreted got together on the internet and very publicly decided to work on this problem.
AS: “Broadly interpreted”? Does it include armchair mathematicians?
VN: It includes anybody who wants to engage. So you don’t have to be a professor at an established university to qualify for this thing, it all took place on blogs on wikis and anybody who wanted to could participate. The level of conversation is pretty high, it’s all happening pretty fast, there’s a lot to learn if it’s not your thing, but in principle anybody could do it. Certainly anybody can read the conversations, it’s all there in public, there’s a kind of historical record, which is incredibly new actually because normally what mathematicians do is beaver away on bits of paper privately in an office or on a blackboard or something…
AS: Like Zhang had done.
VN: Like Zhang had done. And then you throw away all the rubbish ideas and the stuff that didn’t quite work and the stuff you’ve tidied up and made a bit better and you write up a shortest paper you can just to say, this is what I’ve done. And you normally don’t see any of that process that mathematicians go through all the time, of trying and failing to do things.
There’s a fantastic line I love from a mathematician called Julia Robinson, who was a mathematician in the 20th century in the States who had all sorts of challenges with getting employment because her husband was a mathematician and there were rules about not being employed in the same department but she did some fantastic mathematics. And there’s this brilliant line, when at some point an administrator asked her for some kind of dreadful form to describe what she did in a typical week, and her response was
Monday, try to prove theorem; Tuesday, try to prove theorem; Wednesday, try to prove theorem; Thursday, try to prove theorem; Friday, theorem false.
That’s somehow so typical of what mathematicians do a lot of the time. A lot of the time you’re trying to prove something not to be right, or you’re trying to prove something technically… and normally we don’t see any of that, normally it all goes on very privately, just with the individual or just between two or three or four people who are collaborating.
AS: Is it not published so that other mathematicians don’t need to go through the same pain?
VN: Well, this is one of the extraordinary things is that you would think, that say, I’ve tried this technique and I’ve tried really hard and this is what I did and it didn’t work out, you’d think that’d be really useful information but nobody wants to publish it. That’s not something that gets published. But space is not limited on the internet. I guess if you’re publishing papers and people are going to have to pay for it, I guess there’s an incentive to cut that out, but via these blogs and wikis, people have been collaborating completely publicly.
So going back to the twin prime conjecture, Zhang got 70 million, the goal was two. So there was this league table—literally a league table—on the wiki of ‘this is what I think I can do’, so one person will come along and say, “Well I’ve looked at this part of the argument and I’ve realised that if you do this thing slightly differently instead or you try a little bit harder, I got my computer to help with the calculation and that meant I could get a better answer, I think I can get this number instead of 70 million.” And then some other people come along and say, “Yeah OK, we agree with that, or hang on, there seems to be a problem here.” So from May last year for a couple of months, there was this kind of incredibly excitement or watching the numbers on the league table fall.
And I gave a couple of talks to the public at this point and I didn’t dare write anything on the slide about what the latest best known result was, ‘cause it was going to be out of date by the time I gave the talk, it was happening that fast. Mathematicians know this stuff happens because it’s what we do in our offices, but the fact that other people can share in that is really exciting. So by the time it got to July this number had gone from 70 million down to 4,680. I mean, that’s massively smaller. Still not two, but massively smaller. And the way this work is published, it’s published under a pseudonym that acknowledges that it’s done under this Polymath project.
AS: That’s the name of the wiki: Polymath?
VN: There are several Polymath projects. The idea is that ‘polymath’ seems like a clever name for it, right? July comes, nobody’s managing to get below 4,680 so this seems like quite a good moment to write up where they’ve got to. So the project writes up this kind of paper.
AS: Who led on writing that up?
VN: It’s all done collaboratively. So a lot of this was hosted on the blog of a mathematician called Terry Tao, who’s one of the most famous mathematicians in the world at the moment. He has a Fields Medal. He’s a world leader in various areas, he has a very widely read blog. So he was kind of hosting it, but it’s all done by file sharing, the people who were involved could go back and check and say, I’m not happy with this… and it sort of went quiet for a little bit, and everybody’s thinking, can we do better than 4,680, we’d like two… nobody really believe it was really going to get down to two.
AS: Realy? Why?
VN: Well… not without some new ideas. So we still believe the result, but it feels like this proof is never going to quite get down there, we need some new thoughts, and then in November… James Maynard. Very young postdoc, just finished his PhD, did his PhD in Oxford, now working in the University of Montreal for a year as postdoc, published a paper where he did even better. He got it down to 600.
AS: Wow. Another big drop.
VN: Another big drop, exactly. It was this kind of sudden kind of gear change again.
AS: So let’s just recap then. That 600…
VN: What this 600 is meaning is, there are infinitely many pairs of primes that differ by, at most, 600. And he puts it up on the internet and everybody jumps on it and says, is this correct, and they’re busy checking and it looks good, so they say, can we do even better? So the Polymath project is kind of revived in a new guise. So it’s now about, we’ve got this stuff that Zhang did, we’ve got our improvement on the stuff that Zhang did, we’ve got this new stuff from Maynard, what can we do?
AS: So had Maynard done something different to Zhang and the others in that first stage of the Polymath project in getting that number from 70m to 4,680?
VN: Yes, there were some kind of similarities because there were some kind of well established techniques that people are using in this, but he got some ingredients as well. So he’d taken some ideas from some new people, he’d come up with some clever twists on them, some new ideas himself, and he’d thrown all of those into the mix, That’s why there was this big drop, because there’s some new ideas going on, so then everybody jumps on this and they say, can we do even better? And the answer’s yes. And this number’s dropping and dropping. So the last time I looked I think it was down to about 270.
AS: This is all on Polymath. It’s there on Terry Tao’s blog, it’s on the wiki, there’s a new league table. There’s another game going on simultaneously.
AS: Go on…
VN: Because what mathematicians do sometimes is they say, we can’t do this yet, but if we knew this other exciting fact, then we’d be able to do it. So it’s like if I make this assumption of this thing that I sort of think is true but I can’t prove yet then I’d be able to do it.
AS: So it’s a bit like when you’re doing a crossword and you are really think you might know the answer to two-down but if you can only get four-across you can get an extra letter and that might be the thing that unlocks two-down.
VN: That’s right, you say, I can’t prove this yet, I’m not completely sure of this but I’m just going to assume it for now, I’ll be completely explicit about the fact that I’m assuming it, so this is what mathematicians do all the time. Kind of building on other conjectures, there’s particularly quite technical conjecture called the Elliott-Halberstam conjecture or the generalised Elliott-Halberstam conjecture, which has been around for a long time. No proof yet but there are lots of plausible reasons why it kind of probably is true.
AS: And what is the conjecture?
VN: It’s a technical conjecture about the distribution of the primes. So if you look at the primes, apart from two, they’re all odd. Odd numbers are all one or three more than a multiple of four, when you divide an odd number by four you get remainder one or remainder three. So you can ask yourself are there infinitely many primes that are one more than a multiple of four, are there infinitely many primes that are three more? And we know the answer to both of those. The answer’s yes in both cases. But then it’s like, how do they compare? The one more than a multiple of four primes and the three more than a multiple of four primes and we think there are roughly the same number of each.
AS: So that’s the conjecture?
VN: The conjecture is a precise estimate of how good… so we know some things in that direction, we know there are roughly the same number of each. But how precisely there are the same number of each, is where this conjecture comes in, and it’s not just about when you divide by four, it’s about when you divide by anything, but that kind of flavour. So James Maynard was able to get a slightly better bound if he assumed that conjecture and combined with his new Polymath work since November, that’s now down to six. So if you assume the generalised Elliott-Halberstam conjecture, there are infinitely many pairs of primes that differ by six. Isn’t that fantastic?
AS: That is fantastic. But I still want to say, but you are making an assumption!
VN: That’s right, of course and it might turn out that it’s not true, and it might turn out that somebody can prove it next week and it might turn out that nobody proves it for 50 years. And that’s the way maths works.
AS: I’m just thinking……. I think I’ve got four-across, which gives me an ‘a’ in the third space for two-down, but if I’ve got four-across wrong, then I’m going to get two-down wrong.
VN: That’s right, and it can be even worse, if you do two-down and then build something on that, and you build something on that, and you fill in three-quarters of the crossword all based on this assumption that you knew the answer to four-across and it turns out you got completely wrong then you have a problem.
AS: So Maynard’s work might be discredited eventually.
VN: There’s two parts to it. There’s the one that’s unconditional in the jargon, that doesn’t depend on the hypothesis, that’s fine. And there’s the work that does depend on this hypothesis, and it might turn out in the future – it doesn’t mean that there aren’t infinitely many pairs of primes that don’t differ by six, it means that proof wouldn’t work. You’d need something else instead.
But that’s happened throughout mathematics, there are all these kinds of houses of cards that build on things.
AS: So Maynard’s got it down to six.
VN: Maynard plus these other mathematicians working collaboratively. And actually James Maynard has been contributing to that conversation.
AS: But that’s only if…
VN: That’s six, conditional on this thing called the generalised Elliott-Halberstam conjecture.
AS: So what’s going to happen now?
VN: So what’s happening now is that that six is unlikely to shrink down to two without some new ideas, the 270 is unlikely to shrink without some ideas, so this is a good moment to have a pause, I think they’re writing up the paper at the moment, and then it’s a case of who comes up with the next idea, one of the exciting on the seat of the pants don’t know what’s happening is, nobody knows when the next idea comes.
The next idea might be next week, the next idea might be in three months times, the next idea might be thirty years down the line. Nobody has any idea. One of the important things to understand is that addressing these questions doesn’t close off, it’s not like that’s the end, because very often answering one question or making progress on a question leads to many more questions. So there’s all this excitement about the twin primes and it would be fantastic if people could prove the twin primes conjecture, but if you could do that then there are all sorts of other questions you could ask.
AS: So you might put that one to bed but there are lots of other problems.
VN: There’s plenty more to do. I think sometimes you think, well, these mathematicians have been doing it for 2,000 years, why haven’t they finished yet? And that’s because the questions that we’re able to ask now, Newton would never have been able to ask, Euclid would never have been able to ask, because it’s only by developing these new ideas, by answering questions by introducing new techniques, that leads to more and more questions.
AS: You’ve been very excited about all these…
VN: I am very excited.
AS: About all these achievements in recent years and months in mathematics research, so do you talk to students about that, can you build in that cutting-edge research into some of the education resources that you’re working on?
VN: Yeah, definitely, I’ve given talks to schools students about exactly this, I was talking to a bunch of 14- and 15-year-olds recently who were on a day thinking about might they want to go on and do maths at A Level, further maths at A Level, and I talked about exactly this.
AS: And what’s their reaction?
VN: It’s a bit of a mix, but I think it comes as a surprise to them that maths is happening, I think it comes as a surprise to them maths is being done by people, not dead blokes you sometimes see pictures of in the books sometimes, I think maybe it’s a surprise to them that I stand there and am as excited as I seem to be about it ‘cause they didn’t’ know people cared. They often have no idea that that’s what’s happening, which is sad.
It’s great, I love talking about this kind of stuff, but sitting listening to somebody talking about it, or maybe reading a book about it, or maybe reading an article, is not quite the same as doing mathematics, it doesn’t give you a taste of what is it really like to be a mathematician. So a large part of my time and energy goes into trying to give people an opportunity to have this experience of being a mathematician. I spent last weekend in act teaching a course with a colleague, our focus was, this was for adults, just interested adults, they were a complete mix of people, it was about trying to give them an opportunity to engage with ideas related to some of these things. So there are all sorts of questions that you can pose about prime numbers that are more accessible than the twin primes conjecture, so we didn’t say to these people, here’s the twin primes conjecture, you’ve got the weekend to solve it.
But instead we said, well twin primes are great, here’s another interesting pattern in the primes. If you think about the numbers three, five and seven, they’re all prime. So that’s three consecutive odd numbers, so it’s like two pairs of twin primes glued together almost. So you can say, are there any more examples of triples like that, of consecutive odd numbers that are all prime. And until you’ve thought about it, it’s not clear whether that question’s going to be easier than the twin prime conjecture or not. It turns out that it’s much more manageable and we addressed that on Friday evening with these students. ‘We addressed it’— they addressed it. They sort of sorted it out amongst themselves.
AS: But you knew the answer.
VN: We knew the answer. And we knew that they were going to have the mathematical tools to grapple with it. But that in a way gives you a taste of what it’s like, so I left the 15-year-olds with this at the end of my talk. I said, here’s something for you to think about on your bus on the way back to school because it’s exactly the kind of question that if they think about it a bit, if they play around with some examples, get a feel for what’s going on, make a conjecture, come up with a justification, they can do all of those things themselves and have that little bit of excitement about, well I’m a mathematician too, which I think is really important.
AS: Vicky, it’s been really great to speak to you about all these different things, I can definitely say that I’ve had a few light bulb moments myself, so thanks very much.
VN: Great, I hope you’re going to go and think about whether there are more triples like three-five-seven of prime numbers.
AS: What else am I gonna do on the train back to London?
AS: You can read the transcript or follow links to the Polymath projects and more at podacademy.org.
To follow progress in the twin prime conjecture: