Q&A: Knuth, curry and kettles

Chris Smith put the question to the University of Nottingham's Tony Padilla...

Tony - I know the story. You have to buy 27 tickets and if you buy 27 tickets, you will win. You won't necessarily win the jackpot, but you'll win something, maybe just the lowest prize that you can get which is if you get two numbers. What was really impressive in what they did was that they actually were able to show that if you buy 26 tickets, you're not guaranteed. Proving that 27 is enough was actually the easy part.

Chris - If I bought one hundred percent of the tickets, I have to win, don't I?

Tony - So I think there's something like, I can't remember the exact number, 45 million, 47 million combinations or something. So if you bought a ticket with every single combination in, you're obviously going to win, right? That's guaranteed.

Chris - But the jackpot isn't a hundred percent return, so therefore you still lose money?

Tony - Yes. But you can bring that number down to 27, and actually they showed it using some really funky finite geometry, some really cool maths. And they were able to show that there were 27 combinations of numbers which will always have at least two of the numbers that appear on Saturday's draw.

Chris - Are they specific? Have you got to choose a certain sequence of numbers?

托尼——不仅仅是一组,你必须choose. You don't have to put the same 27 as me. We just have to arrange them in a certain way. The way they actually were able to show this, they were able to package your choice of numbers into a bunch of clever diagrams. And there were five of these diagrams and actually there's six numbers on the lottery ticket. So it was always guaranteed that you couldn't have all the numbers appearing in all the different diagrams, there always had to be one diagram which had two of the numbers. And that was the key to winning. But the hard part was proving that 26 was not enough. Actually the calculation for proving that 26 was not enough in principle should have 10 to the 165 steps, which is one with 165 zeros after it, which is clearly just completely impractical. There's just not enough time in the universe for that calculation. But they used some really clever software called prologue, which was able to help them with their calculation.

Chris - And when it comes down to making a profit or not, have they done any modelling to see what would happen if you did this every week at the end of a year of playing the game? But you won't necessarily win your money back each time because it's going to cost you x amount of pounds to enter.

Tony - It costs you 54 quid to buy the 27 tickets.

Chris - It's two pounds a time. Have they modelled that? How long would you have to play the lottery so that you've got a chance you're actually going to be at least cost neutral?

Tony - That's not discussed in the paper.

Chris - That's the key question, isn't it? I mean, what's the point of doing the research?

Tony - I was telling my wife about this yesterday and she was like, "well, prove it then." I was like, "what do you mean prove it? They proved it! We're going to lose. We might win, but we'll ultimately lose because it's going to cost us 54 pounds. We're very unlikely to win more than 54 pounds.

Emma - Homo naledi is really, really interesting. They're a species of human ancestor that dates to about 235 to 335 thousand years ago. So they come from one site only, called Rising Star Cave in South Africa. And they're really remarkable because actually by this point in evolution, so 300,000 years ago, we're seeing the origins of our own species Homo sapiens, most of our evolutionary relatives that are close to us already have quite large brains. They're quite tall, and yet naledi has a small brain, not much bigger than a chimpanzee, quite small in terms of their stature, averaging under five foot. And yet there's this remarkable site at Rising Star Cave where up to 15 individuals or more have all been found deep in this really hard to access cave, and it's the only site we know them from. So this is really, really fascinating. What it does is really challenges some of our assumptions about the evolution of things like complex behaviour, so making tools, burying the dead, things like this. And the papers that have come out a couple of months ago now, there's three that have been published together and they're presenting some really quite controversial new evidence from the site. The authors are claiming that naledi was intentionally burying their dead. That they're making tools and even one of these burials, the individual has a stone tool clasped in their hand and that they're also making art. So making engravings on the wall of the cave deep down in this really inaccessible place. And it's generated a lot of discussion. I mean, those suggestions that a species with such a small brain would be capable of things that we've assumed that you need a really big brain for, is quite controversial. But the study's also quite controversial because the level of evidence that's being presented to support these big claims is not as robust in the minds of many in the scientific community as perhaps it ought to be. And so this is also kind of frustrating, I think, in terms of the presentation of science, particularly in popular formats. And there's even a Netflix documentary that's just come out about this site and there's very big claims being made, but unfortunately not big evidence at this stage.

Chris - Well, we'll watch this space. Thanks for that Emma.

Andrew - Yeah. My reaction to that is that there are real risks in some areas of education, particularly if you focus on plagiarism and things like essay writing and marking and assessment. On the other hand, a lot of teachers are now getting quite interested in the possibilities of AI for improving the teaching materials that they use. There's all sorts of interesting graphical methods that represent geometry in maths, for instance, so there's a great opportunity for enhancing the materials teachers use. And there are risks, but my feeling is that it really goes to the heart of what we mean by education, the benefits of education, which in particular learning for many people. Learning is an accumulation of factual knowledge. And this can take people a long way. It can take them through exams. I even heard a story today about somebody who got three A's, without really necessarily understanding what they were doing, but just accumulating the necessary minimal knowledge based on things like past examination papers and so on to get through exams. Whereas of course, what we really want to achieve through learning is understanding. We want people to understand so that they can apply in novel, unpredictable situations, material that they've learned, not just memorising facts. So of course, the kind of AI software we're talking about now is based entirely on previously written or previously drafted or music that already exists. It doesn't have creativity, it doesn't have originality, it doesn't have emotion, it doesn't, et cetera.

Chris - What Catherine's saying though is are we effectively robbing the opportunity to have those skills from school children? It sounds to me like you are saying, well, as the machine doesn't have that, it can't replace that in people. But might it make people lazy learners as in, because they don't have to go through the process of learning to get that information into their mind so they understand it and can, and can see it from multiple viewpoints, which is what really understanding a topic is because you don't have to go through that to regurgitate a something that looks like a cogent answer, therefore it could undermine your understanding, your ability to really engage with the subject.

Andrew - I agree there is that risk, but my hope would be that it reorients us more towards people learning to understand. So for instance, when I used to teach, it wasn't just a matter of marking and getting percentages and preparing them to succeed in exam questions that can be almost predicted. But it was a matter of sitting around in group discussions and probing people's understanding of whether they could apply a material that they'd learned to a new situation. In other words, understanding rather than just memory and knowledge.

Toby - It is certainly true. They're buzzing around very fast in the air. They're buzzing around at about a thousand miles an hour or so, which is not that strange because of course when we think about the speed of sound through air, sound waves travel through air by the little molecules collectively getting together and propagating pressure waves through the air. So the speed of sound is quite closely related to the speed of the molecules.

Chris - And why are they going so fast?

Toby - They're going so fast because they're hot and they're very little. And the hotter they are, the faster they go.

Chris - Because it's not much to accelerate. They don't need much energy?

Toby - They don't need much energy. But I have to say, they don't go very fast in the same direction for very long. I think it's less than a billionth of a second before they hit someone else.

Tony - That statement is true. The size of an atom is about a hundred thousand times bigger than the nucleus. That's the kind of ratios that we're talking about here. I went to see Oppenheimer last night. It is amazing. There's a really nice scene in it where Oppenheimer is talking to his future wife Kitty, and he's talking about how most of what's in their hand is empty space. There's nothing there between the nucleus, the electrons that they're orbiting around, except of course the fields. We say it's nothing, but there's actually the electromagnetic field that's there. And that's why, when we perceive a solid hand or a solid table or whatever, we're really perceiving the electromagnetic field. So it's not really empty. There's a field there.

Chris - So when you press down on a surface, the reason your hand doesn't go straight through a surface, despite the fact that 99.999999% of the surface is empty space, is because my electrons around my atoms in my hand are coming close to the electrons around the atoms in the desk, and like charges repel. So the minus electrons repel the minus electrons and push back on me as hard as I'm pushing on them.

Tony - Exactly. And that's the field that you are feeling. But actually what's really interesting is we can ask about actual empty space and stuff that's got nothing in it, and ask, is there really anything there? If I go out into space and I create a vacuum, is that really empty? And actually perhaps not, because we think it has something which is called vacuum energy, which is actually the thing that's accelerating the universe. So it all comes together in one wonderful place.

Chris Smith asked University of Cambridge palaeoanthropologist Emma Pomeroy...

艾玛——是的,我也喜欢咖喱。没有想be a bit of a boring scientist and say, 'well, what do we mean by curry?' I think it's actually important to start out by asking that.

Chris - Chicken tikka jalfrezi made just a little bit hotter with the pilau rice and a peshwari naan, please.

Emma - It is actually a term that we use to refer to cuisine from many different parts of the world, including the Caribbean, India, Southeast Asia, China. And it probably comes from the 17th century, and actually the British travelling out there, to those parts of the world, and trading and seeing food that was basically a sauce with spices containing meat and vegetables and just calling it all the same thing: a curry.

Chris - They call it our national dish these days, don't they? I think it is more popular than anything else, isn't it?

Emma - Indeed. And the fact that it's so widespread across many parts of the world, it makes this a really interesting question. So where did this kind of cuisine come from? So there's evidence from this site in Vietnam that you mentioned, it's from about 2000 years ago. It comes from samples from a sandstone slab that's probably been used for grinding spices down. And what they do is they take these samples of the starch grain, tiny particles within the plant cells, and they actually differ in their shape between different plants. So a palaeobotanist can look at those under the microscope and say to you whether this is evidence of a particular plant. And they found a whole range of different spices, including things like turmeric, ginger, galangal, sand ginger, things that we associate with being in a curry. And there was evidence that they'd actually been ground because the starch particles were damaged. So they're saying that perhaps this is some of the earliest evidence of curry.

Chris - Can it also, apart from obviously being interesting in telling us about what people must have been eating, can it tell us a bit about trade and that kind of thing? Because if you know that some of the things they can see are, say, not native, they must have come from somewhere else, that must be a connection to another bit of the Earth at the same time arguing there was trade going on. So it unlocks the door to an understanding of what humans were doing, trading, et cetera, in that era.

Emma - Absolutely. And this is one of the really interesting things that came out of the same study, that a lot of these spices that they were using don't come locally. They're actually coming hundreds of kilometres away over the sea. So it's not just telling us about the history of a food that we're particularly keen on, but it is telling us about those ancient trade networks. And I think it's also interesting to think about how people flavoured their food and how they enjoyed their food in the past, because perhaps we think of it as being quite utilitarian in past populations: you just ate to keep you going. And actually evidence for flavouring foods and adding things like mustard to foods goes back far further than you might think. So we've got evidence perhaps even as early as 40,000 years ago of people adding mustard seeds into their food, presumably to give it some flavour, because they're tiny little things so probably wouldn't have much other function.

Chris - But the standard of cuisine was also quite advanced from fairly early on, wasn't it? I remember about, almost 20 years ago, covering a story where researchers working in China uncovered a pot with noodles in it. And it was obviously the remains of someone's last meal. And I remember one commentator I interviewed about it said, these noodles are 2000 years older than Jesus!

Emma - Absolutely. It goes back even further. So at the site of Shanidar Cave where I work in Northern Iraqi Kurdistan, we have some evidence that Neanderthals were actually pounding particular pulses and grass seeds, soaking them perhaps to make them more digestible as well, and then mushing them together into some kind of paste which they were then cooking. So probably this kind of complex food processing goes back far further than perhaps we imagine, back to Shanidar 70,000 years ago, and we might see something that is not too different from some of the foods that perhaps we might eat today.

Chris Smith asked Tony Padilla...

Tony - Graham's number, for a time, was the largest number ever to have appeared in a mathematical proof. It's named after Ron Graham, who is an American mathematician. He was solving some mathematical problems, these kinds of problems that mathematicians like to solve. It was very abstract. It's in some branch of mathematics called Ramsey theory. It involved hyper cubes in extra dimensions and all sorts of wonderful things. And he was trying to find bound for the results. Is this thing I'm trying to find, is it less than some number? And he was able to show it was less than this number, Graham's number, but this Graham's number is just ridiculously big. You take a number like a googol for example, which is a one with a hundred zeros. Well, Graham's number dwarfs that. Or take a googolplex, which is a one with a googol zeros, well, Graham's number dwarfs that. It's just off the scale, enormous. You can't write it down, you would run out of space in the universe to even try and write it down. So, to express it, we use a special type of notation that you have to invent to describe this thing. It's called Knuth's arrows, and it's like an extension of powers that you learned at school. Three squared, three cubed, that kind of thing. It's an extension of that that allows you to very rapidly get to very, very, very big numbers.

Tony - One of the things that I discussed about it in the videos and in the book was, I was trying to think about how big it is. And how can I express that? Well, what if you tried to imagine this number in your head, if you actually tried to picture its decimal expansion written out. Well, if you try to do that, Chris, then the inevitable outcome is that your head would collapse into a black hole. There's just too much information that, long before that happened, your head, it'd probably blow up, but if you managed to not blow up, it would collapse into a black hole.

Chris - So how did that guy's head not collapse into a black hole then?

Tony - Well, he used this clever notation, which allowed him to compact it. He wasn't thinking of it as a decimal expansion, right? Let's take the number 27. I could write down 27 or I could write down three cubed as a different way to express that number. So it's like that but in a much more enhanced way. So he had a way of expressing it that wasn't the big massive decimal expansion and so his head didn't collapse into a black hole.

Chris - But how did it change the world, the fact that he proved this?

托尼在吉尼斯世界纪录——它了!

Chris - That really matters, okay.

Tony - I think what was important about it was that it showed what was provable. Proof theory for mathematicians is a really important thing. Understanding what's provable, what isn't provable, can you show that this can actually be answered, these are important questions in logic.