Four Ways of Thinking: Statistical, Interactive, Chaotic and Complex - David Sumpter

thank you, thank you very much for the lovely introduction and for allowing me to come here for the third time. It is a real privilege to come and speak with you here. I previously worked at the University of Oxford. I think I left here around 2005, so it's really nice. to walk and all these old memories come back, so it's wonderful to be here, so what am I going to talk about today? Well, for me, this was a quote that Daryl took and used actually to publicize this talk and I hadn't given it much thought but I wanted to get up again because for me as an applied mathematician calculations are secondary now I'm pretty good at math I'm pretty good at calculating things and doing the required manipulations and I suppose I teach so I have to be reasonably good at it, but that has never been what motivates me first.

I'm not one of those people who likes to sit and do long calculations or well, maybe I like it a little bit, but not a lot. not as much as many of my pure mathematics colleagues the reason I became interested in mathematics and this really came from an early age the reason I became interested was that I wanted to understand the world around me and I felt that mathematics was the toolbox that I could use to gain that understanding and that's what we're going to look at today, that's the story I'm going to tell. I'm going to try to give you an idea of ​​how my own

thinking

works in

four

different stages, by analyzing the statistics.

chaotic

and

complex

interactive

thinking

and I am going to illustrate it with stories, some of them will come from football because Daryl has told me that I am very popular in my football talks.

I thought I'd add a little. For you, some of football will come from other parts of science, so I'm an active scientist working in a wide range of different areas and some of it also comes from my personal life, so how do I use mathematics to think about the types of social problems that I encounter every day how I interact with people when things go wrong how can I find a solution for them so I will take those three different branches Science Football and our personal lives and use them to Illustrate these

four

different

ways

of thinking, like this that the first way of thinking we have is statistics and when I did this I also went back in time and really Applied Mathematics is just over 100 years old, actually the things we use.

Today I went back in time and started with various historical figures who built these things and this is Ronald Fisher, this is him in 1912 when he was a student at Cambridge University and Fisher was an incredibly arrogant young man who believed he was smarter. that everyone around him and at school he was smarter than everyone around him and he went to Cambridge University, which is going to sound controversial, but at that time, if you wanted to study mathematics, Cambridge University was the best place to study mathematics in the world. and he went there and found out that he was much smarter than all the other students and he also thought that he was smarter than almost all the teachers, so you can imagine Fisher, he was sitting in his room a few weeks before the conference.

Tripos exams in which he, of course, got excellent grades. He was sitting in the room not studying for exams, but he was trying to figure out how the math he was using fit into reality. He felt that when the people who taught mathematics, the teachers who taught. He, when they used it, they didn't see the combination between what they were testing and the results they had and how that could actually be used and that's what he wanted to find. He was sitting there looking for that solution and it didn't work out. For him, he wrote an article, no one read it, no one was interested in his ideas and he ended up practically in the desert in which he wanted to fight in the First World War in 1914.

He couldn't because he was too short-sighted and he finished. he bought a farm and tried to run it, which was absolutely terrible in the terrible way that he just couldn't achieve, he wasn't very good at working hard, he was good at having theoretical ideas but he couldn't get anything done, but he was rescued and he was rescued by a guy called Sir John Russell and Sir John Russell ran the Rothemsted Experiment Station and he actually said he was looking for an eccentric mathematician to look at all his experimental results and that's why he recruited Fisher and here's Fisher in the photo. on the left side, that's how you would normally see him in Rothenstead, he would be fuming and explaining ideas to people and in this photo he is at a tea party, now Sir John Russell, um, he instigated the tea party because in Rothamstead in 1919, they started to have women working there and he felt that if they had female employees he didn't really know what to do with them, but he knew one thing: they needed to drink tea, so he made tea for them. uh, so they started having a tea party for these um uh, essentially for the woman's ostentation, but he continued, oh, well, in the end they all had the tea and they became a very central part of what was done. now in Rothenstead at one of these tea parties.

Well, although there was a Dr. Muriel Bristol who was one of the experimentalists and one of the people who did the culture studies and Fisher was about to serve her tea and she said enough, first I need to drink my milk and Fisher as usual in his arrogant way he just said nonsense this can't be true it doesn't matter if you drink your milk first in your tea you can drink your milk later you know it all mixes it makes no difference if you drink your milk first or if you drink your milk later and that and him no he wouldn't stop there he wasn't happy until he did an experiment and tested if Dr.

Muriel Bristol could tell the difference between uh if she drank her milk first. or her tea first in her tea and then she set up an experiment to do the test or she got some of her colleagues to suggest various methods on how they could do it and now I'm going to ask them, so we want to test If Dr. Muriel Bristol can distinguish between If the milk is put first in the tea or if the tea is put before the milk, I will allow you to consider three different

ways

or two different ways plus another alternative to do this. test, then the first one is to offer her a pair challenge, we offer her a cup with milk and a cup without milk and then maybe we randomize them in different ways and we have four tests for her, the other is that we present her a tray with tea with milk and no milk in the tray and we ask you to identify those that will be milked first and those that will not be milked first, okay, so raise your hand who thinks pair testing is the best way to do this. some people in the back with their hands up thinking that the first tray of the first milk tea is the best they can do better than this, okay, now we have a few more hands up, so I'll take the rest of you, uh, I'm going for the option.

C and you are also thinking that there is no difference between these two methods, they are both the same, so I am not sure if it is okay, raise your hand if you think there is no difference between the two methods. I think we've got a small majority for option b, okay, so let's take a look at this and this is the work that Fisher did if you have a pairwise challenge option A when you're setting up the experiment and This is the key to how Fischer was thinking when you're setting up the experiment. You have 16 different ways to arrange the glasses, so the black one, the black circle there is, is the T first, the white circle is the milk first, and you have four different pairs. and there are 16 different ways to arrange those pairs, so two to the power of 4 is 16 and the probability and this is the key here, the probability of Muriel Bristol getting this right is 1 in 16. she has to do it in order to get them.

Okay, if she can't tell the difference, the chance of her getting them right is 1 in 16. Now, if we press option b, the one she liked, this will turn out to be the best option there is. eight places to place the first cup seven places for the second cup these are the milk glasses um six places for the third cup and five places for the fourth cup and you have placed them randomly and then you fill them with these They would be the milk ones, then you fill them with the ones that do not contain milk and then you can also think about the order of the cups.

There are four times three times two times one ways to do it and using combinatorics you can find out well, there are 70 ways to arrange the glasses and if you don't believe in mathematics you can sit down and write them all down and I have, I have a little, yeah, I told you that I don't like to calculate, so I got a little bored before writing everything down. of the different ways you can arrange the glasses, but you can arrange them in these different ways, so if Muriel Bristol can't tell the difference, the probability of getting all four right is now only one in seventy, so this test is the best way to do it and I think this is a perfect example of using a good piece of mathematics in the form of combinatorics and that's what Fisher did.

He used different parts of combinatorics to solve a problem in experimental design and continued writing. a book that became a kind of manual that is used today on how to design different experiments. This is a bit. This is your Latin Square layout, which is a different layout than the random layout we just talked about, but then you could start from there. to spread his

statistical

ideas and I think there is incredible power in being able to think about the right way to do an experiment or the right way to analyze data. Now I like it so like Daryl said I've worked in football and I've worked in maths and football and this has taken me on some amazing journeys and it's very nice for example I had a thing with Gary Neville so now I like it brag about my type of football and my contacts, I think.

Gary Neville is the most famous footballer I have met on my journey so I just wanted to mention that and what always happens when I talk to footballers, ex-footballers or coaches about maths and football. It's just that they have this thing where they say, well, you know, numbers can tell us something, numbers can tell us some things, but you can't measure a player's attitude, you can't measure it, and that's why Gary said that. when we did this together. He said, oh, you can't measure if a team misses a goal, you can't measure the player who really gets everyone going and who really rallies the team and gets them going again and I was sitting there thinking that yes, actually that's exactly what you can do. measure with statistics and a few days after Gary Gary said this, I sent him an analysis where we analyzed exactly what we look at to see what happens when a team, so when a team can see the goal, this is Trent Alexander Arnold in a game in the 21-22 season and the line the center line here the center dotted line that is highlighted is when they conceded a second goal against United, they lost the game 2-1 at the end and just after half time they conceded a second goal. and the wavy line that goes up and down is Trent Alexander Arnold's performance with the ball and as time goes on you can see that he is getting better and better, he is actually producing a lot of good passes for his teammates and then the goal goes on the dotted line. and then you see his performance drop again and in this particular case, if we do the Gary statistic, we end up calling this the Gary Neville statistic, so if we do the Gary Neville statistic on this, his performance drops after of conceding a goal. compared to their performance in the 15 minutes before they concede a goal, so it's a measurable statistic and this way you can, for example, look at the best strikers in 2122 and it was quite interesting because you could say that Jamie Vardy he's a player who has a lot of attitude or character or something and then it turns out that's what we got when we measured him when when Jamie Vardy's team went down, they went down a little bit more often than some of the other teams, but he improved. 29 of the time it was worse, 13 of the time and his performance was the same about five of the time and you can see that yes, there is a ranking of the five best players of that season in these different situations and it is very good, I have put in parentheses. here because we used precisely one of Ronald Fisher's tests, Fisher's exact test, to test whether these players were

statistical

ly better when their team went down or not, so there are all kinds of ways we can use statistics, other example. and now we are moving towards the limits of the limits of statistics and I think this is a very important point because although I think Gary is wrong that you can't measure attitude at all, he is right in another sense.

Because you can't measure everything, you can measure certain aspects of how a player improves, it's information that you have, but you can't measure everything and this comes from a study, from a TED talk and during the writing of the book. I watched the 25 most popular TED Talks because I was very interested in how they use statistics to evaluate the validity of claims in TED Talks and this was a talk by Angela Duckworth and she said that Courage and Courage is the idea of ​​determination how determined. you must succeed determination is the strongest predictorof success and it comes from a study that she did with some colleagues and what they did was they looked at Ivy League college students.

I think at Yale they looked at US military cadets and they looked at people who were competing in a spelling bee and before they started doing these activities they asked them questions about how if you start something you always get it done, that type of questions, a series of 12 questions about whether they were determined and courageous people and they found that the answers that people gave to those questions were some of the biggest predictors of success and that's what he said in the talk and that sounds very impressive , a bit like I'm trying to persuade Gary Gary Neville that we can measure the attitude of football players, but if you look a little closer This is the real study, as opposed to yes, we shouldn't measure , we shouldn't measure success based on how many times people have watched the YouTube video because this YouTube video has been viewed 25 million times and it's not Angela Duckworth who wrote the headline. in it, but her article reveals it quite clearly and since she does not try to hide this in any way, the value only explains four percent of the variation between people, now four percent of the variation, how much is ?

It is not like this? It doesn't mean that only four percent of people explain this. I'm going to try to show you what four percent of the variants look like. Four percent of the variance looks like this, so if you measure the value on the scale of one to five, down here and look at the grade averages, for example, this is not real data, it's data that I just made up to illustrate what four percent of the variation looks like, four percent of the variants would have some kind of relationship a little bit like that and you can see if you squint carefully.

I can't see it from this angle, but I think you might be able to squint and see this. There's kind of a growing trend between value and GPA, but you also see. that some people do and there are many people who are very brave and are successful and there are people who are not brave and are not successful, there are also many people who are very brave and do not get a high grade and there are many people who are not very brave and they get a high rating, so when you interpret this you should not confuse, so I often say that you should not confuse the forest with the tree, you are a tree, right, every person in this.

In this room there is a tree, so if we tested all your bravery and your success in life we ​​would find some kind of relationship like this, but that wouldn't necessarily mean that you as an individual would have this relationship between bravery, so if you are not a person Brave if you never finish any project you start, you don't need to worry at all, you will be absolutely fine, there are so many ways your life can be successful and this is what I have tried to do here and I'm not sure I have the right art but The arc I want to describe here is the one I want to start by saying that statistics are very powerful.

I can show it to Gary Neville, but then, at some point, statistics don't give you all the right answers, and that's very well illustrated if we go back to Ronald Fisher, this rather arrogant young college student, because Ronald Fisher also has another scientist story. which was from a very early age. and this photo was taken in 1912 very interested in eugenics and he had the idea that we needed to raise people to be more like him, smarter and good at math etc. and he campaigned until the end. I think after the war he kept quieter about this, but all the way up to the war he campaigned, for example, to sterilize people who were considered weak-minded and this is, of course, horrible and I mean, it's something horrible to think about. but not only is it morally repugnant, it is also scientifically incorrect, they couldn't find any kind of gene for feeble-mindedness, there is no correlation between or there is a very weak correlation or no correlation between feeble-mindedness in one generation in mothers and in his daughters, so this was a scientifically dubious relationship that he continued to push forward and the reason he was successful is because or one reason he was successful in pushing us forward is that he would use statistics to attack his opponents, he called them all stupid, they didn't understand statistics, so he actually used them to undermine other people's arguments in a really counterproductive way.

He took a false theory and then used statistics to defend it. Not only did he do this once after the war, when he abandoned eugenics or at least stopped talking about it, but he later did the same thing with tobacco, so as we saw in the first picture, he was a very enthusiastic smoker and For him there was just no possibility that smoking could cause cancer, so he spent a lot of time researching very narrow areas of science, making statistics on that and trying to convince people that smoking doesn't cause cancer and I don't know what effect that would have. had this research, but we certainly have one of the world's leading statisticians advocating this position for a long time and that illustrates a lot of why statistics is a narrow focus, so I've written some There are sort of bullet points here, so that statistical thinking does not provide all the answers.

One problem is and I didn't really get into this, but I mean, what an idiot is right. I mean why do you need to test her if you know she can? First, tell the difference between milk, everyone was very happy, everyone was just enjoying his tea party and suddenly he's making a statistical day. I mean, you know she's a jerk, so that's one thing and I really think you know we joke about that, but. We see it at work all the time, you know, we're always told that we should do it, that statistical tests and metrics should be done on us and things like that, and that we don't need as much quantification as we have, so there's the effect size. and sometimes we talk about statistical significance, you can have statistical significance but still have a very small effect size and that is how confusing the tree in the forest is, many non-brave individuals are successful in the context of life, it is always important , so just because a player is more active when his team loses a goal does not mean that the team plays better just because Ronaldo demands that everyone give him the ball when they lose a goal or Jamie Vardy demands that he receive the ball, that does not mean that the team really does better.

As a result of that, it's very important to think about the context of these kinds of things and then, as we all know, correlation and causation are not the same thing, but that brings us to the next step that we need to think about. ways to discover causality we want to be able to discover our understanding of the world, one thing causes another and it leads us very well to

interactive

thinking and that is the next step of statistical thinking and I have another hero, I can reassure you. that this hero is not going to become a raving racist who bullies all his coworkers, so there are some mathematicians who have not done that in their lives, just a few of them, but they are out there and this is Alfred J Locker um and he was originally a chemist and he started his uh he was originally from Poland but he did his undergrad in Birmingham and I really like his story because he started working in this chemistry lab and he was a little bit disappointed.

With what you saw when you were doing your experiments, I mean, it's been a long time since I did chemistry in school, but you can be a little disappointed, you know you get the acid and the alkali, you mix them together and there's a little bit of salt and water. It's not always the most exciting thing you've ever seen, so he watched these stable reactions just come to equilibrium, but at night he was reading all these books like Charles Darwin's book and he was thinking about biology and all that. the exciting patterns that we see all the movement and movement of animals or the functioning of our brains, everything that happens in society and he was thinking why can't chemistry produce something like that.

I mean, we know that chemistry has to be the building block of this. but it's not something we see, we can't establish that relationship together and the way he solved the problem was he basically cheated and did the following, so if we've all done this in school, we've balanced equations along a reactions balanced and if you have two two h two, well, you have four hydrogen atoms, two oxygen atoms and they react to form two water molecules, so that's a standard chemical reaction and the important point here is that it's balanced. so there are four hydrogens and two oxygens on the left and there are four hydrogens and two oxygens on the right, but what Loter said is good, I'll forget about that balance thing even if I can't find a chemical reaction that is unbalanced.

I'll just think in my head. I'll do a thought experiment and this is where math is wonderful. I'll do a thought experiment where I ignore the fact that I can't balance my reactions, so he took and wrote down these equations. said imagine one r becoming two r's and imagine one R plus one F becoming two F's and you can see they are not balanced, there is one r on the left side of the first, two r's on the right, yes there is You can see that they're just not balanced and I've written down here below the way that you can think about these things: rabbits and foxes, so they're not, it's not a realistic model of rabbits and foxes that you have.

Thinking about the idea of ​​a bunny jumping and suddenly there are two bunnies jumping, we know it's a little more complicated than that, but we'll start with that idea and then a fox comes and eats a rabbit and then it makes Another Fox, that's what What the model says, it gives a rough idea of ​​how ecological interactions work and he took it and wrote differential equations. I wanted to put some of these equations here to give you an idea of ​​how they work. equations on the left here one of them. I'm not going to make you understand every detail of the equation, but what I want to give you an idea is that on the left is the exchange rate for the rabbits and the foxes.

Dr times DT is a rate of change of the rabbits DF times DT is the rate of change of the foxes and on the right are the things that cause that change, so I mentioned here that we want to include causality in our equations, so on the right of the things that cause that change and rabbits increase when there are a lot of other rabbits, they have a lot of bunnies and then the foxes eat them, so the more F they are, if we look at this term brf, that's the rate at which um , the foxes eat the rabbits and then when we go down, we have the opposite relationship, because for the foxes the foxes grow when there are more rabbits and then they finally die um of old, there's nothing hunting the foxes in this.

In this scenario, again, I'm not going to solve all of these equations, but I did want to mention a little bit about how you can think about them and understand them, and on the right, I learned this from Philip Maney when I was here. at Oxford about these kinds of methods, but he had a very lovely method, a professor of mathematical biology here, of solving these equations without solving them, so that you can divide the plane of the foxes and the rabbits and you can identify a point on that plane and look. to see if the rabbits increase or the foxes increase, then in the lower left corner the foxes go down because there are not enough rabbits to eat, but the rabbits go up because there are not enough foxes to eat them and that continues until there is enough number of rabbits and then the foxes start to increase so in the bottom right corner this arrow points up and if that arrow points right then the foxes increase and when the foxes increase they start to eat the rabbits and the rabbits go down and you start to get this going round and round of foxes and rabbits and you could basically show this without explicitly solving the equations that there will be cycles going round and round of these foxes and rabbits, we will get this interesting interaction and if we look over time, we have these periodic oscillations of the foxes and the rabbits, now this whole way of thinking that Latke introduced turned out to be useful in all kinds of situations, now the one thing we should try not to mention, but we can't.

It is not mentioned in any mathematical models, we try not to mention this, but it is also used in pandemic modeling and everyone has heard of these epidemic curves and our values ​​etc., but that is the same thing that becomes a susceptible plus an infectious one. too infectious and that is an example of one of Lotka's unbalanced equations that allows us to describe how an epidemic reaction will spread through a group of people. Now I'm not going to go into details like I said, I don't want to talk too much. about epidemics, but I really love this example, so I'm going to talk about this example.

This is a study that we did together with some colleagues and this is a very cruel experience. It's not that cruel, but we've had it. a group of university students we got a third, a university studentthird year, I gave a seminar to first year college students and we told the first year college students, you know, remember to give a round of applause after the seminar just to show your appreciation and So what really interested us was how people clap, what are the signals that make people clap and we could see that people start clapping when other people around them start clapping and you, basically, we have an epidemic of clapping and that's what the first green curve shows , it is the number of people who clap, it is the spread of the applauding virus that goes through the group, but this does not happen in real diseases, there is also a social recovery, so when people stop applauding, they look around and You hear the other people have stopped applauding and it was actually a bit like when Daryl left the room a moment ago, there was some sort of starting signal there that something might be happening that we might be about to start.

We started and everyone started. to turn down the volume and suddenly everyone went silent and that's the kind of social effect that we are very aware of all kinds of little social details and these ripple through us in a group and the conclusion that what we I love the thing about recovered is because we have that social recovery then and I try to always remember this that if at the end of a talk you have given or a presentation if the applause lasts for a long time that is not because you have given a good talk it is just because your audience is not particularly coordinated, so they couldn't, they couldn't, they couldn't stop together and I think that's what I encourage you to think and I said I also wonder about the personal aspects of this it's nice to sit down sometimes and think about the social reactions you have in your life and how they work and I have written down some of them.

I haven't told you what they are yet, so I'll tell you what the ones above are the one above that I was imagining this is a person p and then it's more of an O this is a couch that the person has outside the house so we have P plus or it goes to P plus or if you are just one person and you have a couch outside the house you will still be one person and you can't bring the couch into the house so what you have to do and this is the final equation you are You have to get a friend, so this one below is for 2p, 2 people plus a sofa that is outside the house, it's still two people.

You still have your people afterwards, but you've moved your couch into the living room and you can do those social media posts. interactions for each type of activity the one here on the right the one here on the right I was thinking about smiling, so if you are a smiling person, why? and you know a person who doesn't smile, whose ex then if you smile, hopefully they will become a smiling person too, but that's not always the case, sometimes you know you don't always start smiling because someone else is smiling, maybe just be an idiot who is just smiling for no reason what happens most often in human social interactions is like you think the next equator the equation at the bottom this is the most common equation I think it describes the behavior human and that is that one person who doesn't smile anymore two smiling people will become three smiling people because then they are convinced that it must actually be something to smile about and I use that a lot in my thinking if I'm thinking about how to do it in the book I take an example of if I'm trying to get a group if a group of friends are trying to get some kind of healthy activity maybe they spend a lot of time sitting together in the pub, they don't go out or do any exercise together, it's not enough for one of them becomes a reason to try to get them going.

You have to have two of them and they have to have really sustained effort and over time you get this tipping point effect where everyone starts moving and starts engaging in healthy activities, so those are the types of things. what do you have. thinking about what kind of chemical reaction, what kind of social reaction I'm involved in and that's been a lot, to be honest, this has taken up a lot of my adult life, studying this kind of thing and giving you a little bit of a taste of the type of things we do, this is just to give a kind of general representation, but when we model fish, for example, we create models that describe their social interactions and describe how a fish turns left if another fish turns left if another fish turn left. it turned right and so on, then we were building the top, there is a mathematical model, we have built a movement of fish, etc., then we would show that these simple rules of interaction would produce their collective behavior, then we would also study the movement of individual fish, that's the colored idea at the bottom, so we would actually scare all the fish and look at how they made an escape wave, measure that escape wave and then use models to understand that escape wave and it's a very powerful way. to think about all science.

You can build these interaction models, compare them with reality, and build an increasingly better understanding of fish behavior. We do a lot of similar things in football. is an example of an attacking run by Marcus Rashford and the model that we build for these types of situations, this red area here shows the territory that he controls and this is a physics based model where we say how far, how fast he can run, where can it reach and can we describe what area it occupies and also the value of that area? How likely is it that he will receive a pass at that particular point that leads to a goal and that allows us to explore players based on their runs and even allows us to explore runs where they don't receive the ball, so in this example where we are interested ​​on Luke Shaw here and he's running here on the left and he doesn't receive the pass.

I would love to have this pass, but you don't understand it, but we can still measure the value that that pass created so you can look at these counterfactual situations for football players and this is a very powerful method, the interactive way of thinking. It allows us to develop our understanding of systems. It doesn't have the same kind of I guess statistics have a more grounded kind of feeling. We use our imagination much more. We try to use our imagination to increase our understanding. and then build mathematical models to test that understanding. Now I wanted to go back to Lotka because there are also limits to this way of thinking and of course I wouldn't have four if we had figured it all out now.

There are limits to this and there are limits, they were limits that Lotter himself reached, he wrote a book called Elements of Physical Biology and he is one of these mathematicians who, and this happens to us a lot, we just go with the flow and believe that we can explain everything. with mathematics, there is nothing that we cannot explain and that is why he built models, he built models of Consciousness, he built models of our entire society and he believed that all of them could be understood using his Reaction Dynamics and Really yes, I didn't mean and I guess it was a very brave effort.

In 1922 he finished his Magnus Opus, so he didn't even have a computer or anything to simulate these kinds of models, but he never really managed to pin down an essential way in which all kinds of problems should be approached. It ended up being torn between a lot of different little things and I personally can relate to that very well because that's usually how I work with a lot of problems, there are a lot of different methods and you do a lot of different little things to get your solution in the day to day life of an applied mathematician, Isn't it like these theoretical physicists you know they have? like this beautiful Theory of Everything and you can come here and just say oh, it's all this and wow, but no, it's not like that, it's more like you're tinkering with different little problems in so many different ways, so Lotska never found her Grand Theory of Everything using interactive thinking and one of the reasons he never found the Grand Theory was because he didn't know about chaos, which is the third way of thinking.

Now, to introduce the chaos, I'm going to go. for another math hero, this is Margaret Hamilton and she was also, like the other two we met, prodigious in school, very talented college student, she wanted to go on and do a PhD in pure mathematics, but her husband also wanted to do a PhD and that. we're now in the 1960s and he ended up moving to Boston and he also had to get a job, he had a daughter to support and a husband to support, so he had to get a job to support them, but the job he got was programming this. machine the lgp 30 and he immediately fell in love with this computing machine because he hated making mistakes, he hated mistakes every time he calculated something, he calculated it perfectly and now he discovered that he could program this first computer to do the same calculations and he got access to this computer because I was working in the lab of a person named Edward Lorenz who was a professor of meteorology but also had a background in mathematics.

There are a lot of mathematicians in this talk, so she wanted to predict the weather she wanted. to predict the future weather based on temperature pressure and so on in different areas, could he predict the weather in the future and she started writing computer code to do this and this involved writing and making punch cards at that time and she had run her computer code and they did this, one thing is that they simulated, they simulated the weather one day and the next day they decided to verify their results by simulating by doing the exact same simulation on the computer to verify that everything worked, but they found the second day. they got a different result than the first day and Margaret was distraught because she didn't like making mistakes, she didn't want to think that there was a bug in her code, but she started reviewing the code and there were no errors in the code and what they found was that the The simulation's output was at six decimal places and the input they put into it was at three decimal places, so there was an error in the input at the fourth decimal place and this meant that the climate simulation made completely different predictions in the future, within about 10 days in the future in the simulated world, she made completely different predictions in the future and I didn't mention that this was a system of 14 differential equations that she solved, we have moved on. from Lockter and Voltaren too, so she solved these 14 differential equations and they make just this small error in the value that you enter, it makes a big difference and that is the first hint of the chaos butterfly that many of you will be familiar with and Lorenz.

He continued, he worked with um, I say Lorenz continued with Margaret Hamilton, we're going to find that he also did some very impressive things, but Lorenz continued with the help of Ellen Fetter, who replaced Margaret Hamilton as his programmer to produce what now. We know that we often think of this image, think of it or think of it as the chaos butterfly and what it illustrates is that if you start with two points very close to each other, we have now gone down from 14 dimensions to three. dimensions again, if you start with two points close together and they start to diverge, they will move on the same attractor in this way we have here, but they will never get closer or might get closer to each other for a short time. amount of time, but then they will live their own life and then when we go from two Dimensions to three, we have this

chaotic

movement where things never go back to the same place again.

I think I think we're gonna do. my experiment is fine, so I think let's do the experiment and then I'll start something else because you've listened to me patiently for 50 minutes, so you have to do the experiment, okay? Here we are going to let's do this I want you to work in pairs I want one of you should think about it so look at the person next to you and it could be a new friend. I have today, um, or it could be someone you came with and then I think I want one of you to think of a number between 1 and 99, then you tell that number to the other person and the other person does the next thing. rules, so if a number is less than 50, double it and this is the new number.

I chose 42 because you can never have a math talk without 42, so 42 times 2 is 84 and that's all you do, just double the number now. If the number is greater than 50, take it away from 100 and then double it to get the new number. So if I have 84, then I have 100 minus 84, it's 16 times 2, it's 32. Now tell your partner the new number and repeat. Step one and two, so we'll do this for um, do this with the person you came with or someone who's close to you, we'll do this for about two minutes and then we'll see where we get to, I think.

I've done very well, I can see, I can see, I hear the hum of numbers everywhere, very, very charming, um, I'm not going to make all of you come here and present your results, I just wanted to give. you get a sense of this kind of process, um, you're not generating purely chaotic numbers when you do this, if you had started with 20 for example you would have found yourself spinning around pretty quickly, but ifIf you had started with a number that is not divisible by five, you would probably have followed a fairly long path through different numbers and what I want to highlight about this process is this: numbers close together diverge very quickly, so if a group from there I would have started with 13 and another group from here had started with 14 at the end of this short period in which they had to tell each other the numbers; they would have been in very different numbers, so they had 13 26 52 96 8 16 32 64. 14 28 28 is not that far from 26 56 52 they are still together 88.96 they are starting to move away from each other but the big leap now is that one of them exceed the threshold and one does not, so you have 8 and 24. 1648 and then you have 32 and 96 and 64 and 8.

So in a few steps these numbers have diverged quite a bit from each other. I don't know if any of you took decimal numbers. um, they didn't think about that, but if you take decimal numbers, then you're going to get real chaos with this for almost any real number. choose, you will get if you take plus 0.1 just in this case, so this is 14.1 compared to 14.2. You start to stay together for a few steps, but after about seven, eight, nine ten, they separate and come back together a little bit for a while, but then they diverge and you have very different paths for those two numbers and we often illustrate this using something called a spider web diagram, so the idea here is that you take the One Step number and the number above could be around 20 for example, it will jump to around 40, then go to 80, then go back down to around 20 and then it will start moving everywhere and one of the reasons I wanted you to do this experiment is what it was. being what I could hear from his perspective was a murmur of uniformly distributed random numbers, he was essentially going through a lot of integers and everywhere in the room there was a different point in this distribution, he basically had this uniform distribution of numbers that were sort of It's coming up to me and I think it's really lovely to think about that, that everyone is doing the same process, everyone is doing exactly the same thing and yet they have this buzz, this distribution, this background of very different numbers and that's the chaos butterfly and to me it illustrates that there is an important point here.

I think chaos is wonderful. Margaret Hamilton, she hated chaos and left Lorenzo's lab and had learned a valuable lesson from working on these climate simulations. was that she doubled down and made even fewer mistakes and she wanted to work in the most extreme conditions possible where no mistakes could be made and that's why she got a job for NASA and became the head of software engineering who created the software she sent that was on the Apollo lunar mission and that's how she created the software that the astronauts used to tell them how to make navigation decisions to control the thrusters to update and know where the position of the ship and her was in the control room when they made the landing real on the moon, so I see this as a situation where you have to choose correctly whether you are going to control something due to chaos, if anything. you really care or there is something that is really important so you have to treat it like Margaret Hamilton does you had to treat it like the moon landing there is no mistake there is no room for any kind of error but you can't have control over everything so I often think about this in football because there will always be butterflies in other situations, so here this is not the uniform distribution as you generated, but the Poisson distribution.

There are many other situations where football is one of them where we just can. To avoid randomness Randomness will always appear. We can control a past as we talked to Pastor Mike Marcus Rashford, but we cannot control what happens in 90 minutes of a football match. Soccer is very big. random degree and I like to think of it that way. I like to think of it as the yin and yang theory of randomness, so Yang is order and on short time scales, the things we really care about we can control and we can build models of them. and we can understand them, but Yin is disorder and on longer time scales we simply have to admit that there are things that we cannot control and that are out of our control and in fact, I personally found this very useful because my wife is a little more organized and I'm a little more chaotic in a lot of situations and she wants to plan what we're going to have for dinner every night, she wants to plan a lot of details of what the week should be like and I'm like, oh, wow.

I have to go and give this talk at Oxford or not, I can't remember the things that I planned and said I'm going to do, and you kind of already know, so I have kind of a more random approach and, first of all, when I looked at Chaos Theory I thought chaos theory was perfect, it was a perfect justification for my chaotic behavior. I could just wander around because I didn't control anything, so you know why bother and Obviously I didn't explain this to my wife, but I thought about it a little bit, but actually, when I heard about Margaret Hamilton, I understood it a little better.

If there are things that really matter to you, then you make an effort to control them. and my wife and I find more of that balance and now I've discussed it a little bit more with her is that we make sure that there are things that are important, like we have dinner together every night and we make sure that we do those things. and some of the other things, like if there's a big pile of dirty clothes that need washing or mowing on the lawn, we let them turn into chaos when we feel like doing it, so I really think there's a good way to find Yin and Yang in your life to know that if you really care about something, control it well, if you don't care that much, just let the chaos take over, there is nothing you can do about it.

Time has passed and you wanted to know about it. Of course, what's the fourth? Well, we have found a yes, we have found the three ways of thinking. I will not review them because I am not going to extend the time. The fourth way of thinking I'm going to spend. a little less time on the fourth way of thinking for a reason and one of the reasons is that maybe you can go out and buy the book and discover the fourth way of thinking, but I would say a little bit about it because I also think that I think that it's something that should inspire younger researchers to think about how they're going to take the next step in this and this is a simulation produced by one of my master's students, Michael Hansen, and maybe some of you have heard of cellular automata . models before this existed The Game of Life is the most famous where there are rules of interaction and I gave my students the task of creating their own cellular automata that would make the most

complex

pattern possible and Michael Amical came up with the following rules.

Pit the white cells so they only interact with nearby cells, so this is a hundred percent cell grid, but the cells only interact nearby and the rule for white is that they become slimy, which is dark blue if there is less than four of its neighbors are bone, the white ones, otherwise they remain bones and then the sticky ones, the dark blue ones, these become fluid, which is light blue, if less than three of its neighbors are made of bone, otherwise they remain sticky and then, for the fluid, which is light blue. these become bones if two or more of their neighbors are bones.

Now the exact details of the rules are important. What I think is amazing is that you just have these simple rules that describe how cells interact with each other and all of a sudden you're creating these very splendid structures and the reason I called the white ones bone is because they stretch out and form this skeleton. and in between that skeleton you have alternating patterns of dark blue and light blue that pump back and forth, so with very simple local interactions you have extremely complex behavior and another example of this and this is something you can find on Twitter.

This has become a competition. These are all computer graphics and each of those computer graphics is produced with code that is a snippet of Twitter lens code. That means it has to be 240 characters long and there's a whole kind of culture on Twitter about who can create a graphic with 240 characters and each of these you can create, use fractals of course, and use shading, and you can create this guy. from the tree movies, they look like trees, they look like flowers, I seem to love them, these things look like a sticky substance and then you have some kind of bloom there too just with a very short code and that brings us to who the theory is, who is the hero of the last part of the book The Scientist, who is the hero of the last part of the book and that is Cole Magorov because he defined complexity or he did not define it as such, but he said that a pattern is as complex as the length of the shortest description that can be used to produce it and I think it's a very useful way to think about science and think about how we understand the world if we can find a way to describe something that doesn't lose the detail. the nuance but it captures it, then we can begin to capture the complexity and that's the thought I want to leave you with and thank you for being patient with me, thank you, thank you.