Welcome back to the six events of the Oxford Mathematics Public Lectures Home Edition. My name is Alan Goretti and I'm in charge of external relations for the Mathematical Institute as usual. Well, thanks to a sponsor's execs, markets execs, market leading, quantitative driven electronic market maker with offices in London, Singapore and New York. Tonight's speaker is Professor David Sumpter from the University of Uppsala in Sweden.
David is actually closely connected to Oxford as the Old Orile Society Fellowship with us before heading to Sweden, used and applied mathematician with a wide range of interests and has made seminal contribution to the problem of collective behaviour, which cover everything from the inner working of fish schools and colonies, segregation in society, machine learning and artificial intelligence to the analysis of passing networks for football team.
A few years ago, you actually wrote an entire book on the mathematics of football called Soccer Metrics on which give us a brilliant lecture last year. Since then, David has been very prolific and has written yet another book, which is a general reflection about the role of mathematics in the world. All summarise two 10 equation that rule our lives and society. Typically, for such event, we would have a big box of book for the other to sign up to the lecture.
Obviously, we will not be able to do this tonight, however, do not despair. You can still obtain a signed copy of his book from Blackwood's Bookshop. All you need to do is email Oxford at Blackwell's the Echo Dot UK by 15 November, and they will provide you with all the information that you need. I'm very keen to hear which a questioner should know so that I would be able to forget all the other ones that currently quote my brain. So please, David, start now and tell us all about learning.
10 Equation can improve my life. Thank you. So it's not every day that you ask a mathematician for advice about how you should live your life. But that's what I'm going to try and do today. I'm going to talk about 10 equations, which I think can make you happier. Make you a better person might even make you richer and more successful.
Now, when a mathematician starts trying to offer this advice, it's worth starting by looking at some of the alternatives that are already on the market and there's lots of them. So this is just a small sample. There's Marie Kondo, she tells you how you should tidy up and make things better. I was leaning in this Jordan Peterson and his 12 rules of life. You can learn like Ice Einstein, you can declutter your mind.
A lot of them involve a few swear words where you just have to stop doing these extreme things you can solve to be more happier. There's lots and lots of different, I think, the seven, 12 and 15 laws for confidence, and this one's my favourite down in the corner here. This is gangster confidence. So the 16 laws for gangster confidence, apparently. So what can a mathematician add to this?
Well, one kind of clue and maths is really an abstract well, but one kind of clue can be found in the book The DA Vinci Code. Dan Brown. The da Vinci Code is a very different type of books than these self-help books, and it's a work of fiction, of course. But in the first 60 to 100 pages of this, it brings out a lovely idea that mathematics can underlie everything.
And it describes this using the number PHI the golden ratio one point six one eight and the idea of PHI comes up a lot in mathematics that comes up in the golden rectangle. It comes up in the Fibonacci sequence and in sunflowers. And the idea proposed in the book, and it comes up with a lot of fake examples as well as real examples. But the idea in the book is that there is a secret held in mathematics and it's held in a single number.
So the problem or the question I'm going to pose in this talk is could mathematics hold the answer? We are all looking for? Could we just throw away or leave all of these other self-help books and concentrate on finding a mathematical answer? And I'm not going to tell you every one of the 10 equations today, I'm going to give you a feeling for a few of them. The first one that I'm going to talk about is called the judgement equation.
Now. I'm going to introduce it by thinking about an example, and it's quite a scary example, but one we're all familiar with, or we were at least familiar with before the corona times. Imagine you are an experienced traveller, having flown 100 times before the flight you are now on is different. As you descend, the plane starts to rattle and shake in the way you've never experienced before. The woman next to you lets out a gasp. The man sitting across the aisle grips his knees.
Everyone around you is visibly scared. Could this be it? Could the worst possible scenario be about to unfurl? So here is the nightmare scenario. You feel that the plane you're on is possibly going to crash. This is maybe a overdramatic representation of what you're currently experiencing. We want to know the probability of a crash, given the plane is shaking, we can feel the plane shaking. We're worried about the crash. The shake is the data, the crashes, the model.
OK, so let's deal with that problem. You've been on a hundred flights before in this scenario and what's happening in this particular flight? Is your flight shaking? So this is a 10 by 10 grid of flights. There's only one of them that's ever shaken as badly as this particular one that you've been on. And so what you know from this information is the probability of this much shaken given that you're not going to crash if you assume you're not going to crash is one in a hundred.
This is the worst possible flight that you've ever been on. So it could be you're going to survive. And a good estimate for the probability of shaking, given you're not going to crash is one in a hundred. It's worth just thinking about what might happen if you went on two thousand flights like this. So if you have a probability of shaking this badly of one in one hundred, then if you go in 2000 flights, you'll have 20 of those very bad shaky rides. And then let's go one step further.
The background rate of plane crashes, plane crashes are very unlikely, so you don't know exactly how many plane crashes occur, but the chance of being in a plane crash is something like one in 10 million. You actually have time to look this up on your phone and in your current situation, I can tell you it's around one in 10 million. It might be a bit less than that. It might be a bit more than that. So now start thinking about all of those possible plane journeys you can be on.
What I've done here is I've made a thousand times, ten times 10000 thousand array of journey, so imagine a thousand times 10000 journeys and that is 10 million journeys. An inside all of those journeys, there's always going to be these shaky journeys and you happen to be on one of those shaky journeys right now. But the probability you're going to crash is only one in 10 million before you got onto the plane if you weren't shaking.
So here's the crashing plane somewhere hidden somewhere in between all of those 10 million journeys. So that's horrible for the people on that flight, that train, that that plane. So now you actually have all the information you need in order to work out your probability of being in a plane crash. In fact, there's going to be 100000 shaky journeys in this in these 10 million journeys, and only one of them is going to end in the crash journey.
That's essentially 10 million divided by 100 gives you a hundred thousand. And that means roughly. The probability your shaky journey is the crash journey is approximately one in 100000. You really aren't going to die. It's the worst thing you've experienced, but you don't actually need to be worried given the circumstances. Now what? What's the judgement equation? I've gone through this problem with which I've gone through it from a frequent his point of view.
I've looked at the frequency of these different things occurring, but we can actually use an equation which I call the judgement equation. But it originally was found by Thomas Bayes. It's called Bayes rule came from the end or mid end of 18th century. And what it does is it gathers up all the information we have, so you notice that while I was doing my calculation, I find the probability of crashing the background crash rate for planes is one in 10 million.
If your plane is going to crash, it will shake. So the probability of it shaking given a crash is one. And probability of Shay Given not crash, that's based on your experience. We estimate that to be around one in 100. It's the worst flight you've been on. But what we really want to know is the probability of crash given shake, and that is given by base equation, which looks like follows as follows the here.
We just put in all of the numbers and we work out, we come up with this thing being approximately equal to one in 100000 now. Bayes rule and the judgement equation have come up a lot recently because they also come up with regard to the coronavirus. One of the problems that's come up during the COVID crisis is there are false positives in tests for the disease.
So if you go and get yourself tested for the virus sometimes and the rate is about one in a thousand, you'll come up with a false positive example. Even though you haven't got the disease that you'll get a test result that comes back, that says you have got the disease. And we can also think about this using the judgement equation. So here we've got 100 people. I've just said that it's more like a one in a thousand, but imagine it was a one in 100 error rate for the disease.
So we've got one in 100 error rate. Well, one in 100 people are going to get back a positive test even though they're healthy and don't have the disease. So that's one in a hundred there. But imagine also that we have one in 100 people who have got the disease. So here's 100 people. And one of them. Has got the disease. That's this one here. Now, what we can see here quite clearly, actually, is that in this case, it's just 50 50 if you get a test result.
So if you get a test result and it tells you you could be that person who has the disease or you could just be this person who got a wrong test result back, in fact, there's a 50 50 chance that you're one of these two people. Now, the one in 100 the I've I've put up there. It really depends on a lot of different factors. Now one of them, for example, is you only go and get tested if you've got symptoms.
If you look at these, these little guys here, all of the ones with face masks on, they don't go and get tested. But there's a few individuals here who have got got symptoms. So if only one in seven individuals have, even if only seven out of 100 individuals have symptoms, then it's much more likely that a test result is correct. There's all sorts of parameters that you have to think about when you're thinking about the coronavirus and the testing of that.
And so I don't want to give I don't want to give a conclusive discussion of that just now that's raging over Twitter and in the news, these types of discussions. But it's the same mathematical problem based on the judgement equation, which underlies both of those things. There's a general way which I think mathematicians think about the world, which is very useful or applied.
Mathematicians think about the world, which is very useful. And I like to think about how we break the world down into model and data. I often see I personally see the future or I have a sort of model in my head, which is a little bit like a film. I see different things happening. So when I'm in an aeroplane and it starts to shake and I start to get worried, I start to imagine the bad things that could happen to me.
And that's a model of the future of the world. The data of the world is the shaking experience that something bad is happening, and all of the time, what we how we should be thinking is updating our model on the basis of the data we get in, and that applies to everything you can think about. If you're swimming in the sea where you know there might be a shark attack and you see something moving, you need to your model is the shark attack.
Your data is the movement in the water. It's the same when you get an interview for a new job. Your model is that you'll be successful and you'll get the new job. The data is the outcome of the interview and how the interview went and the same. When you get a medical test, you've got a model that you'll be infected with the disease and you get the test and that improves your understanding of if you will actually have the disease or not.
And all of these things can be solved by putting the collecting up the data that you have and putting them into Bayes theorem, putting them into the judgement equation. And. The judgement equation isn't just about. Objective things that happen in time in our lives, it's also about our feelings. You can also use the judgement equation in order to update and have a better understanding of the personal things that happen in your life.
So I'm going to give an example of this. So you probably can't help but notice the change of scene for this particular segment of the talk. And that's because when Darrell, who organises the talk talks at the institute, and Alan, who presented me and myself when we reviewed the original lecture that I did in a kind of cold, empty lecture theatre, we felt that the setting wasn't quite right for the message that I'm going to try and bring out now.
And I think it works very well when I'm presenting things to do with aeroplanes and coronavirus, when we're thinking about factual situations, when we're handling hard data and trying to understand that. But what I'm going to talk about now isn't so much objective. Data is much more subjective data. It's things about feelings and it's things about emotions. Now, I think all of us have a lot of feelings and emotions generated by our interactions with the people around us.
And I think that the judgement equation based rule actually suggests that we should be less judgemental and more forgiving of other people. And I'm going to illustrate that through an example. So if I think about my own life, I mean, I have lots of frustrations. For example, I'm teaching a course just now with students emailing me about different things to do with the lectures,
and I just feel I've answered these questions 100 times. And of course, I know it's different students, but I get frustrated. And then you get frustrated. I also work in a football club giving data analytics and doing analysis, and I get frustrated by the demands on me. And the idea is that there's a lot there's just a lot of passion inside football, for example, and it's all about winning. And things do get very heated and people do get angry with each other.
And also, if I think about in my personal life, I mean, you know, everybody I have a family wife and two kids, a teenage kids. And there's, you know, there's a lot of debate and discussions that go on and how can we actually use mathematics to be better at dealing with those types of situations? Now, I think when I started studying mathematics, I had a very kind of hard idea about it.
I saw it as like a very precise logic. And if you go back to the idea of the da Vinci Code, I talked about to the star. It's this idea that there's a code, a secret which gives you the correct right answer. But it's not really like this because this is a list of some of the bad things that can happen to you on an everyday basis. I made the list here of when friends or apparent friends let you down.
And this is something which all of us have experience, sometimes we overhear a friend say something nasty about us. Sometimes your friend steals an idea or a joke that you've made up and they Petraeus is your own. Sometimes your friends have parties and they don't invite you. And these things make us all sad and they make us all well. They nobody can have these types of things without feeling a little bit upset. And the question is, Well, what should you think?
And one thing that you can think in these types of situations is that the people who do these things to us, the apparent friends, well, actually, they're idiots. And you question you have to think about is, is this friend who's let you down in this way, an idiot? Well, I do want to say that I don't go around. I don't really go around thinking that people are idiots. You know, if you meet me for the first time, it's not that I have this percentage probability that you're an idiot,
but it's a useful way of thinking about the problem. And the numbers here aren't that important. So I've said here that I have a sort of baseline rate of people that I meet are likely to be idiots one in 20. So five percent of people are just unhelpful idiots. And then 95 percent of people are nice and easy to get along with. Now again, these numbers are subjective. I don't really think people are actually, maybe I do think some people are idiots.
Anyway, the point is that we're going to work through this problem and we're going to find out the probability that somebody who said something not very nice to you is actually a total idiot. And this is a a representation of this problem. We've got PM, which is the probability of our model, and we've got PMC, which has a probability of the complement of our model.
And there's a five percent chance this little grey box here is the five percent chance that the person is an idiot and the 90 percent, 95 percent chance here that they're not in India. That's before they do anything bad to you at all. So now comes the data now comes with we've set up the model. Now comes the data. The data is the bad thing that this person has done to you. They've let you down by not inviting you to a party or by saying something bad behind your back.
What the data tells us. Is that? We can what we can say is that, well. Idiots spend about 50 percent of their time doing nasty things, talking about people behind their backs, letting other people down. But what we have to remember is that even known idiots can have bad days, in fact, 10 percent of the time. I've estimated that perfectly nice people do something rather regrettable and not very nice.
So these idiot type people in the well, they spend 50 percent of their time making nasty comments and non idiot spend 10 per cent of the time. And now we can break this down. Also in this square, so this this has area 100 percent. And these are the ideas we've divided. Those five percent of people are idiots into 50 percent of the time saying nasty things and 50 percent of the time just being generally OK.
Now, for the non-media as well, we can also divide them and we've got 90 percent of the time. They're being nice and just doing nice things, being really great friends. But 10 percent of the time, they're also making mistakes. They're not idiots, but they've made a mistake. And when something nasty happens to us, when we've collected the data, we know we're in either this situation or this situation.
So the other situations disappear because something bad has happened to us and we just left with these two situations. Now, what's interesting? Is that if you look at the relative size of that box? So if I go all the way back here, this was the idea of being nasty. This was the nice person making a mistake. We've just got these two boxes left. Those are the two possibilities of where we lie after we've had this nasty comment or we've not been invited to the party.
We put them together, and I want you to look at the relative sizes of these. So the idiot being nasty is just this size here. The not idiot making a mistake is a much larger box, and this is the key to what Bayes rule tells us, Bayes rule says. What are the comparative areas of these two particular boxes? And we can see that this one is about actually one fourth of the size of this one here.
And when we put the numbers that I told you into Bayes equation, the nought point five times the Note .05 is calculating the area of this box, which is the idea of being nasty. And that's also put down here because we add up the two areas together and the nought point one times nought point nine five is adding up the area of this box. And to find out the probability that your friend is an idiot, we need to find out the area of this divided by the area of this, plus the area of this.
That's what Bayes base rule tells us. That's for the judgement equation tells us. And the conclusion is that whatever this person did to you, the probability that they are an idiot and you should stop talking to them immediately is that only 20 percent? And this tells you that you should be a much more forgiving person. Most of the bad things that happen to you are just because nice people are making mistakes. Now, the numbers I've given in this example are not really the important thing.
The important thing is thinking about the alternative hypotheses in all of these different situations, and I think we don't have it. We have a tendency not to do that immediately. We think we take problems personally. We immediately start to get this idea that people are idiots. But when we just calm down for a minute, we think through the alternative that actually even nice people make mistakes now and again,
we come to a very, very different conclusion. A key in this example is that it started with me allowing myself to have a wrong model of the world. I allowed myself to have some emotional reaction where when my friend let me down, I concluded that he was an idiot. But then after that initial response, this initial model that I had created, I worked through the judgement equation and was able to see that there was another explanation behind all of this.
So what's important to emphasise is this imagining wrong models or feeling frustrations and being disappointed with other people is actually a good part is actually a key part of making a good judgement. OK, I think I've got the point of that one across now. I think we'll go back to the lecture theatre and listen to what more I've got to say in this case. One thing I worry about personally is screen time with my kids. My kids use their mobile for their teenagers 15 17.
They use their mobile phones a lot and to be honest, their parents also use their mobile phones and their screens a lot. But what we are interested in knowing is, what's the effect of that? Is there a big negative effect of that? And this article I came across, which was from Berkeley, is published in a Berkeley website. Outlined and it gave this graph about how mental wellbeing of of teenagers decreases with the amount of time they spend looking at their screens.
Now this looks quite worrying because you've got a negative trend here, but what I think you need to remember? As well, what's the effect size? So what's quite unclear when you first look at the picture is that this is a scale of zero to 100. They've asked people zero to 100, how do you feel? And the scale is only 40 to 50 here. In fact, from the average answer to the most extreme answer. There's just a one percent effect size.
So you become one percent less happy because you've because you've you're using your screen and this or because of teenagers using the screen. And you can think of this in terms of the. Of the judgement equation. In fact, also in this same in this same study, they found a three percent effect size for if you got a good night's sleep, if teenagers got a good night's sleep of teenagers, eating breakfast had a three percent effect size.
So not eating breakfast reduces your happiness by three percent. Not having a good night's sleep reduces your happiness by three percent. And that in the context of the phone usage shows that it's quite a small effect. So when I talk to my teenagers about using their phone too much, I also talk to them or I spend three times more time talking to them about having a good night's sleep and eating breakfast than I do, nagging them about how much they use their mobile phones.
It's a very nice article by Candice Ortigas in Nature, where she balances all sides of the debate, and that's how, you know, when you're reading something that's proper science, when you actually hear all sides of the debate. And she comes to the conclusion that smartphones are bad for some teams, but not for all of them.
So that's the judgement equation, and I've gone through some examples of how it can be used, it can be used for thinking about disasters, it can be used for thinking about diseases. It can be used for thinking about your personal life. And it can also be thinking used for thinking about the things that you read online and evaluating evidence for them. One of the nine equations now what the other nine? Well.
These are the names of them. I've got the betting equation, the advertising equation, the confidence equation, learning equation, influencer equation, skill equation, market equation and the reward equation. Since we're at the mathematics lecture, I'll tell you the actual equations themselves.
These are what they look like. And my argument is that if you have these equations in your life, you will both be able to understand the situations around you better on a more personal, on an everyday level. And also, if you learn the mathematics behind these equations, they lie behind a lot of the financial industry, the social media industry, the betting industry and lots of other work inside public service as well.
So I'm going to give one more example, I think I'm yeah, I'm going to give one more example of of this. I'm going to take out the reward equation and tell you a little bit about that. The reward equation is widely used by social media companies, and it's used to decide what types of rewards and information it should give us if it finds that we like something a lot. It gives us more of that stuff so that we keep clicking on it.
My idea is actually, you can reverse engineer the reward equation both to monitor what's going on in your life and also and also to decide what you should do and what you shouldn't do. So here's my scenario. Imagine you lying on the couch, rewarding yourself by bingeing on TV and you start watching a Netflix series, the first episode is brilliant, as they always are. The second is OK and the third is slightly better again. The question is how long should you keep watching before you give up?
I think this is a problem that's familiar to all of us. We're stuck with so many TV series out there always feels you could be watching something else that might be better. But how long should you keep watching the series before you decide now? I'm just going to give up. So here here's how it works. First, you should rank the first episode you watch out of 10. So, yeah, give it a score zero to 10. This is your tracking variable.
Keep this in your head. And what I'm going to propose is for all the activities you do. Maybe it's going to the gym, meeting friends, everything you think you're doing, you can have this track invariable one number, which describes how much you enjoy this, this activity. But just now we're watching Netflix. And the rule is following, if you're tracking variable is seven or below, then stop watching. So this is Netflix, you. There's lots of series available.
I'm going to be quite strict here. If it's below a seven, then stop. If the first episode is below a seven, then you stop. Then watch the next episode and you give it a score out of 10 at the school for this episode to the quality score and divide by two. And then you round this number up to get your new tracking variable Kutty +1. Then you go back to step two. OK, you start. Here's how it works in practise and you start with the first episode.
You watch it. You give it nine out of 10. Kutty is equal to nine. This is one of my favourite favourite of series Big Little Lies and enjoy the first episode. Nine out of 10. The second episode isn't quite as good. So I give it a six out of 10. Now, what I do to update my track, invariably is I take the original nine, I add the six to it. So I've got 15 and I divide by two and I get seven point five. And remember, I said I round up, so I round up to eight.
I think it's good to round up because then you just need to keep one hole number in your in your head. So it's still an eight. So even if it got six, it survives at six and goes on to the next next round, you're going to watch the next one. And that's lucky because the next one you watch is a nine out of 10. So you're tracking variable is updated. You had eight from your previous time, plus nine divided by two is an eight point five.
Now round it up to nine. So the main advantage of this method is a good show will survive a six whenever you have a six out of 10. It pops back up to the eight and you keep watching the show. Of course, you can change the parameters in this model. You can make things decay more slowly. You can change this divided by two, or you can change your threshold, but it's a good model for keeping track of things. But it won't survive a five, if you have a nine, then you have nine plus six.
Divided by two is seven point five. But if you have an eight series plus a five, you're just not going to survive it. This was another series which I watched, which wasn't quite so good. Very quickly, even though it started very well, I stopped watching it quite quickly. Now, the general form of the reward equation is as follows. You have Kutty +1. This is your tracking variable in the next time stat is equal to one minus alpha, we set alpha equal to two in the original.
Sorry, we set out for equal to half in the original set up, plus a half times the reward that we got. The reward is that is the score we have for the show. The cute is our tracking variable and what's very nice in this general form of the reward equation, which has been around since the 1950s. We know that the tracking variable is going to converge to the average reward. So this is a relative series watching.
We can actually see that whatever we start off with our tracking variable at, it will eventually converge to the real world that we have. And this is something that we can use to monitor all parts of our life online. We can give each thing that we use on social media, not just Netflix series a score. And we can keep track of those scores in order to see, is this something that I want to continue doing or is this something that I should get rid of and get out of my life?
There is a problem here, and it's that these scores actually keep changing over time. How do we keep track of the changes in these scores when we don't know the underlying value of something? And here actually there's an answer we can look to the answer. So it's. Use firemen trail. And what The Ferryman trial does is whenever the ants find something good, they find some food, they leave FireMon Trail going to that food, and it recruits other aunts to that foods.
So the ants pheromone trail is like a tracking variable, which follows after where the food is. And you can do experiments with ants. Where you can offer them two alternatives. This is where they've been offered some red food and some blue food. And there's a y shape junction here. This is a close up of the junction where the ants have to choose if they will go to the red food or to the blue food.
Now, how do the ants manage to get the balance right because they're trying to work in this dynamic environment where things are changing all of the time? And a good way to think about this from a mathematical point of view is to think of the ants as balancing on a seesaw. So here is my first aunt. She's on this seesaw and this seesaw is like the left right choice where the food to look for the food. And she maybe makes the left choice, she goes in that direction.
Now what she does is when she makes a left choice, she leaves Ferryman going towards that choice. And so that bias is the decision for the next stamps which come along. The ants don't actually walk on seafood on a seesaw. They're on this Y-shaped maze. But the idea is to say that once they start going in one direction, they bias the choice towards that direction.
So the next time comes along and she follows the previous one down next one, she does the same thing, and they keep following one after the other down towards the food. Now this is good. They've managed to find this lollipop, which was the food that they were looking for. But the problem is, once they've tipped the seesaw in this direction, if the food moves up here, how do they get back to the to the food, to the new food source?
And they can't. They're sort of stuck in the nuance which will come along will follow the other end, so follow the trail down in the wrong direction. So. What you can do if you're an end is you can tip a bit less, so here we'll have the first and. And so this is equivalent to having less reaction to the ceremony. She's making a decision. She goes that way. And you'll see that the seesaw is balanced slightly less so that she's left less ferryman.
Or she's reacted less strongly to the ferryman than the next and maybe goes the other way and balances a seesaw up again. And then you start to get off in that direction. But when it starts to get it, it never really gets too unbalanced because they're making decisions to go left or right, basically in a random way. But the problem now is if you don't tip enough, then you never really get focussed on the best food, so the food is moving around in different directions.
And the answer following the food, but they never really they never really tip over strongly to One Direction or the other. And it turns out the answer for the ends is to kind of get their seesaw on a balance like this. So by getting that see saw it is sort of tipping point, they just enough, but not too much. What they can do is that they can track where the lollipop or the food is moving to. And this means getting exactly the right responsiveness to the food.
The mathematical model behind this is is consists of two different equations which are based on the reward equation. But what we've done is we have this feedback term here as well. This is a feedback and this is a feedback here. And this model's both the the seesaw idea that I've talked about. It models the Y. May's decision, and it can also model how we make our choices about social media.
I'm not going to go into the details of solving these equations, but when you solve them using a theory called bifurcation theory, you can actually show that if you have two weaker response, if you are not very sensitive to the Ferryman trail, you get the situation like this that they just sort of randomly distributed. If you get too strong a response, they can tip and go in the wrong direction.
But if you get this just right response, then the answer can be very flexible when the food sources change position. And this is something you find in all animals, including ourselves. So. This is something you find in all animals, including ourselves, this is just a few examples here. This is humans applauding after a talk. This is fish escaping when we frighten them.
And over here we've got bird flocks flying around. And what animals have is a collective FOMO, a collective fear of missing out that always balanced at this point where they can switch direction to to move towards food, to move towards shelter, to move towards a mate. And that's kind of what we are in the internet, on the internet as well.
If you want to collect all the rewards that are available to you, then you have to be there all of the time, clicking and searching and trying to find out what's going on. And that's just incredibly stressful. It kind of lies in our collective nature to do this, but it's a very stressful thing to be doing on a daily basis. So I have a recommendation that I've also learnt from the hands. Another thing the ants are very good at is just taking it easy.
The ants do this running around like crazy. They put themselves at tipping point when they're working for ants also like to rest. They like to sleep. And so I have this picture of my wife plays a lot of Pokemon Go and have this picture of Snorlax resting because as well as you can at times, go crazy and try and follow all the information. But other times you want to take it easy and have a sleep like Snorlax does.
OK. I've told you about two of the equations today, I've told you about the judgement equation and the reward equation. There are if there's 10 equations, of course, eight other equations in the book, if you've been looking carefully, you'll see that I've listed seven of them here.
These equations are used throughout our lives, the learning equation was used by YouTube when they wanted to learn about what videos we like to increased massively our use of YouTube by finding out what we liked and showing us more of it. The advertising equation is used by Facebook to find out what we're interested in and target adverts at us. It's the correlation equation as well. The learning equation is a derivative of maximisation.
The advertising equation is the correlation equation. The confidence equation is based on the confidence interval. It's used both in gambling to find out if you really have got a winning strategy, and it's also used widely in research in order to find out if our results are really true. The influencer equation dice was first invented by Google or used by Google.
It's actually based on mathematics, which is 100 years old, and it's used to measure who are the most important people in the social network or on on the internet. The market equation is used in finance. It's a diffusion equation which describes as I break it down into signal noise and feedback there the way ways the market work and how you can use that to make a lot of money.
The skill equation is used in sports to evaluate the quality of different football players, for example, and the betting equation is well, it's used in betting, but also about how how we can use probability in our own lives. So there's all these equations or these equations, which allow you to get a better understanding of the world. And they all combine models where you describe things as a mathematical equation and data that you take in the data can be things from social media,
from finance, from banks and so on. Combining models and data is the secret to successfully using these equations. There's a tense equation as well. And so what's that? Well, all of these equations, they involve combining different types of models with data. Now, the tenth equation is different.
I call it the universal equation, and I like to say it as if then, because eventually this is going to end up as a logical statement, but it goes back all the way to what I mentioned at the start of the talk. The DA Vinci Code. Now, the da Vinci Code is a fascinating idea that there could be a single equation that unifies everything and brings the world together. The single theory of everything which all revolves around, in this case, phi a single number.
But I believe that this can't be the case. And I'm just going to give you a brief idea about why we can't have a single number, a single mathematical idea that unifies the entire universe. If we take, for example, fine, where does it come from? Well, fi, I mentioned at the start, it comes from the golden rectangle. One side of the rectangle has lent a the other side of the rectangle as a a plus b. And this is a golden rectangle.
If the rectangle that's left over when you put in the square, the ratio of those sides B to A is the same as the ratio of A-plus beat A. And that's given by this equation here. A-plus B divided by a equals to a of a b and is equal to PHI, which is the golden ratio. So that's one place in which fire comes up and what Dan Brown emphasised in his book, he came up with quite a few examples which weren't true, but he came out with quite a few that are true.
Is that fire causing quite a lot of different settings? And here's one of them. Here is the Fibonacci sequence. This is this the Fibonacci sequence itself, and the rule is as follows you have one plus one is equal to two one plus two is equal to three to plus three is equal to five, three plus five is equal to eight and so on, up and up and up. And where does that golden ratio occur? Well, it occurs here. So 13 divided by eight is equal to one point six two five.
So we take consecutive numbers and we divide them by each other and we get closer and closer as we go off into infinitely large numbers. That ratio gets closer to one point six on eight. And this is one of a number of different mathematical results where the same number appears in lots of different settings, and it's this that gives the kind of semi mystical feeling about mathematics that there's some sort of hidden truth behind those numbers.
And it also it's always confused mathematicians right through history. So it led Henri Poincaré to reflect on this, and he said, and this is a rhetorical question really by him. He said that if all the assertions of mathematics put forward can be derived from one another by formal logic, just like we've we've looked at with or I'm going to look at now with fine. The mathematics can't be anything than some sort of immense tautology, so that just means everything in mathematics is true.
Logical inference can teach us essentially nothing new. But can it really be the case that all of these books that we look at in mathematics have no other purpose than in a roundabout way to say that A is equal to a? Now this is a rhetorical question on the part of CAR. This is. He can't really believe, I think when he states this, that all of mathematics just says that a is equal to a. So if we take the golden rectangle here and and the equality we've set up for the different sides of it.
What we can do is we can rewrite this by we have a divided by a plus b divided by A is equal to a as a basis. That's fine. But we've also said Overby, well, that's equal to the golden ratio PHI. So let's just put Phi in here and be over is one over a baby. So we have one over five there and we have one. Yeah. And then if we multiply through by five, we have five here, plus one is equal to five squared.
And we always like to have everything over on one side, so we have a nice quadratic equation and the quadratic equation for this, if we put up zero on this side and we've got phi squared minus five minus one. And if you remember the roots of the quadratic equation, your views, the quadratic formula, and that will be if we've got it right here because we have it here, but it's a one plus the square root of.
One plus four. Divided by two, which is one plus or plus a minus plus or minus the square root of five, divided by two. So that's how we come up with the value of PHI. Now. That's the problem from the point of view of the golden rectangle. If we look at the other problem. So then if we look at the Fibonacci sequence, what we know about the Fibonacci sequence is we have x t plus one is equal to x t plus x T minus one.
So what I mean by that? Well, if you remember the sequence one plus one was equal to two, one plus two is equal to three. And I'm just writing that same thing out in an algebraic form. Now what we can do is we can say, well, what happens when T gets very large? And first of all, if I just write out x t plus one divided by x t. Then that. Is equal to x t plus x T minus one.
Divide it by x t. And this is also equal to one plus x T minus one divided by x t. Now, if I take the limits of tea goes to infinity in this, and that means I just keep going in lots and lots of steps. Exactly as I've done here, I keep going forward and forward in time. So if I want to know what this limit is, I can replace it with five. And that's going to be equal to one plus. I'm here because I'm going off to the limits of time equals infinity. This is just one over five.
Once we come up in further and further in enough time steps and that equation, you should recognise from there because it's the same equation again. And it's going to have the same solution again. And so the occurrence of these types of relationships. Isn't so much, it isn't so much a deep secret. It's more exactly as Henry Henry Pryor, said Henry Ford. It's it isn't so much. It isn't so much a deep secret, but it's more exactly as Poincaré originally proposed.
What I've really shown here is that a quasi we get the same solution for these two different mathematical problems. Ultimately, for problems like this, for pure mathematics problems which aren't coupled to data, then they do ultimately just prove that a quasi and rhetorically was a rhetorical question that Poincaré posed. The answer is yes. And when you look at philosophy of mathematics, they come to the same answer.
What you're doing when you decouple a problem from reality is you're just proving that a equals eight. And in that way, you can never expect a secret number of any type to inform what your form something about the structure of your life. But. Then the question could mathematics hold the answers we're all looking for? I believe it can. But only when we let data into our lives as well is when we combine. Data in the form of all the things that happened to us.
With a model, a mathematical description of the world. That we can actually start to understand what's happening and better control our lives and maybe become more successful and happier people.