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

[Music] thank you thank you very much for the lovely introduction and and being allowed to come here for the third time it's a real privilege to come and and talk to you here I worked earlier in Oxford University I think I left here about 2005 so it's very nice for to walk around and all these old memories come back so it's lovely to be here so what am I going to talk about today well for me and this this was a quote that Daryl took and used actually in advertising this talk and I hadn't thought so much about it but I wanted to to lift up again because for me as an applied mathematician the calculations are secondary now I'm quite good at maths I'm quite good at calculating things and doing the manipulations required and I suppose I teach it so I have to be reasonably good at it but that's never been the thing that motivates me first I'm not one of these people who likes to sit and do long calculations or well maybe I like it a little bit but not a great deal not as much as many of my pure mathematics colleagues the reason I got interested in mathematics and this really came from an early age the reason I got interested was I wanted to understand the world around me and I felt that mathematics was the toolkit which I could use to get that understanding and there that's what we're going to look at today that's the story I'm going to tell I'm going to try and give you an insight into how my own thinking Works in four different stages looking at statistical interactive chaotic and complex thinking and I'm going to illustrate it with stories some of them are going to come from football because Daryl has told me that uh I'm very popular in my football talks I thought I'd throw in a little bit of football for you some of it is going to come from other parts of science so I'm an active scientist working on a large range of different areas and some of it also comes from my personal life so how I use mathematics to think about the types of social problems that I encounter every day how how I interact with people when things go wrong how I can find a solution for them so I'm going to take all of those three different branches Science Football and our personal lives and use them to illustrate these four different ways of thinking so the first way of thinking we have is statistical and when I did this I also went back in time and really really Applied Mathematics is just a little bit more than 100 years old really the the things that we use today so I went back in time and I 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 he believed that he was smarter than everybody around him and at school he was smarter than everybody around him and he went to Cambridge University which was this is going to sound controversial but at the time if you wanted to study mathematics Cambridge University was the best place to study mathematics in the world and he went there and he found that he was pretty much smarter than all the other students and he also thought that he was smarter than pretty much all of the professors and so you can imagine Fisher he was sitting in his room a few weeks before the tripos exams which of course he aced he was sitting in the room not studying for the exams but he was trying to work out how the mathematics he was using was coupled to reality he felt that when the the people who taught in mathematics the professors who taught him when they used it they didn't see the coupling between what they were proving and the results they had and how that actually could be used and that's what he wanted to find he was sitting looking for that solution and it didn't go well for him right he wrote an article nobody read it nobody was interested in his ideas and he ended up pretty much in the wilderness he wanted to fight in World War One in 1914. he couldn't because he was too short-sighted and he ended up buying a farm and trying to run this Farm which he was absolutely terrible at the terrible at he just couldn't manage to he wasn't very good at working hard he was good at having theoretical ideas but not getting anything anything done but he was rescued and he was rescued by a guy called Sir John Russell and Sir John Russell he ran rothemsted experimental station and he actually said he was looking for an oddball mathematician to look at all their experimental results and so he recruited Fisher and here is Fisher pictured on the left-hand side this is how he could be typically be seen at rothenstead he would be sort of puffing smoke and explaining ideas to people and in this picture he's at a tea party now Sir John Russell um he instigated the tea party because at rothamstead in 1919 um they started to have women working there and he felt that if they had lady employees he didn't really know what to do with them but he knew one thing they needed to drink tea and so he made he made tea for these uh so they started having a tea party for these um uh for the woman ostentation essentially but he went on oh well everybody had the tea in the end and they became a very central part of what was done at rothenstead now at one of these tea parties um though there was a a Dr Muriel Bristol who was one of the experimentalists and one of the people who did the studies on crops and Fisher was about to serve her tea and she said stop I need to have my milk first and Fisher as usual in his arrogant way he just he said nonsense this can't be true it doesn't matter if you have your milk first in your tea you can have your milk afterwards you know it all mixes up together there's no difference if you have your milk first or if you have your milk afterwards and so and he wasn't he wouldn't just stop there he wasn't happy until he'd done an experiment and tested if Dr Muriel Bristol could tell the difference between uh if she had her milk first or her tea first in her tea and so he set up an experiment to do the test or he got his some of his colleagues they suggested various methods how they might do that and I'm now going to ask you actually so we want to test if Dr Muriel Bristol can tell the difference between if the milk goes into her tea first or if the tea is put in before the milk and I'm going to allow you to consider three different ways or two different ways plus another alternative for doing this test so the first is to offer her a pairwise challenge we offer her a milk cup and a non-milk cup and then we maybe randomize these in different ways and we have four tests for her the other one is we present her with a tray with milk and non-milk tea on the tray and we ask her to identify the ones that will milk first and the ones that were not milk first okay so hands up who thinks that the pairwise test is the best way to do this we've got a few people at the back hands up who thinks that the milk first tea first tray is the best you can do better than this okay now we've got a few more hands up so I'm taking the rest of you uh going for option C and you're also thinking that there's no difference between these two methods that they're both the same okay so I'm not sure if that's okay hands up if you do think there's no difference between the two methods I think we've got a little bit of a majority there for for option b okay so let's have a look at this and this is the working that um that fisher did so if you've got a pairwise challenge option A when you're setting up the experiment and this is key to how Fischer was thinking when you're setting up the experiment you have 16 different ways you can organize the cups so the black one the black circle there is is T first the white circle is milk first and you have four different pairs and there's 16 different ways of arranging those pairs so that's two to the power of 4 is 16 and the probability and this is the the key here the probability that Muriel Bristol gets this right is 1 in 16. she has to in order to get them all right well if she if she can't tell the difference the probability she gets them all right is 1 in 16. now if we do hit option b the one that you liked and this is going to prove to be the better choice there are eight places to put out the first cup seven places for the second cup these are the milk cups um six places for the third cup and five places for the fourth cup and you've put them out at random and then you fill up with the these would be the the milk ones then you fill up with the non-milt ones and then you can also think of the ordering of the cups there's four times three times two times one ways to do it and using combinatorics you can find out well there's 70 ways of placing the cups and if you don't believe the math you can sit and write them all out and I got I got a little yeah I told you I don't like calculating so I got a little bit bored before I wrote all of the different ways you can arrange the cups out but you can arrange them in these different ways and so if Muriel Bristol can't tell the difference the probability of getting all four right is now only one in in seventy so the the this test is the better way to do it and this I think is a perfect example of using a nice 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 he went on to write a book which became is still a sort of handbook used today of how you design different experiments this is a slightly this is his Latin Square design which is a different design than the randomized design we've just talked about but he could then from there start to spread his statistical ideas and there's just an incredible power I think in being able to think about the correct way to do an experiment or the correct way to analyze data now I like to so I've as Daryl said I I have worked on football and I've worked in mathematics and football and this has taken me on some amazing Journeys and it's very nice to for example I was I had a um thing with Gary Neville so I like to show off now about my sort of football and contacts I think Gary Neville is the most famous uh footballer I've I've met on my journey so I just wanted to sort of throw that in there um and what's what's invariably happens when I talk to footballers or former footballers or coaches about about mathematics and football is that they have this thing where they say oh well you know numbers can tell us something numbers can tell us a few things but you can't measure a player's attitude you can't measure and so Gary said that when we did this thing together he said oh you can't measure if a team goes a goal down you can't measure the player who really gets everyone going and really rallies the team and gets them going again and I was sitting there thinking yeah actually that's exactly what you can measure with Statistics and so a few days after um Gary Gary said this I sent him an analysis where we analyzed exactly that we look to see what happens when a team um so when a team can see the goal so this is Trent Alexander Arnold in a game in the 21-22 season and the line the central line here the central dotted line which is highlighted is when they conceded a second goal against United they lost the game 2-1 in the end and just after half time they conceded a second goal and the squiggly line that's going up and down that's Trent Alexander Arnold's performance on the ball and as time goes on you can see that he's getting better and better he's he's actually producing lots of good passes for his teammates and then the goal goes in dotted line and then you see his performance drops back down again and so in this particular case if we do the Gary so we ended up calling this the Gary Neville statistic so if we do the Gary Neville statistic on this his performance goes down after they concede a goal compared to his performance in the 15 minutes before they concede a goal so it's a measurable statistic and in this way you can for example we looked at the top Strikers in 2122 and it was quite interesting because you might call um Jamie vardy a player who's got a lot of attitude or character or something like that and then it turns out that's what we got when we measured it when when um when Jamie vardy's team went down they went down a little bit more often than some of the other teams but he got better 29 of the times worse 13 of the times and his performance was the same about five of the times and you can see yeah there's a ranking of the top five players for That season in these different situations and it's very nice I've put in parenthesis here because we used precisely one of Ronald Fisher's tests Fisher's exact test in order to test if these players were statistically better when they're when their team went down or not and so there's all types of ways in which we can use statistics another um example and now we're kind of moving over to the edges of the limits of statistics and I think that this is a very important point because while while I think Gary is wrong in that you can't measure attitude at all he's right in another way because you can't measure everything you can measure certain aspects of how a player gets better it's one piece of information you have but you can't measure everything and this is from a study uh from a TED Talk and during the writing of the book I watched the 25 most popular TED Talks because I was very interested how they use statistics in order to assess the validity of claims in TED Talks and this was a talk by Angela Duckworth and she said that Grit and grit is the idea of determination how determined you are to succeed the grit is the strongest predictor of success and it comes from a study that she conducted um together with some colleagues and what they did is they looked at Ivy League undergraduates I think at Yale they looked at U.S 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 do you always see it through it those types of questions a series of 12 questions about if they were determined gritty types of people and they found that grit this the answers that people gave in those questions were some of the biggest predictors of success and that's what she said in the in the talk and that sounds very impressive a bit like me trying to persuade Gary Gary Neville that we can we can measure um attitude in football players but if you look a little bit more closely at this the actual study as opposed to the yes we shouldn't measure we shouldn't measure success on how many times people have watched the YouTube video because this YouTube video has been watched 25 million times and it's it's not Angela Duckworth who wrote the headline on it but her paper reveals quite clearly and as she doesn't try to hide this in any way at all the grit just explains four percent of the variance between people now four percent of the variance how much is that it isn't it doesn't mean that only four percent of people are explained by this I'm going to try and show you what four percent of the variants look like four percent of the variance looks like this so if you measure grit on the scale one to five on the bottom here and you look at grade point averages for example now this isn't real data this is data I've just made up in order to illustrate what four percent of the variance looks like four percent of the variants would have some type of relationship a bit like that and you can see if you squint carefully I can't see it from this angle but I think maybe you can squint and see this there is a kind of increasing Trend there between grit and grade point average but you also see that some people who yeah and there's there's lots of people who are very gritty and are successful and there's people who aren't gritty and unsuccessful there's also lots of people who are very gritty and don't get a high grade and there's lots of people who aren't very gritty and do get a high grade and so when you're interpreting this you shouldn't confuse so I often say that you shouldn't confuse the forest for the tree you're a tree right every person in this in this room is a tree so if we tested all of your grittiness and your success in life we'd find some kind of relationship like this but it wouldn't mean that you necessarily as an individual um had this relationship between grit so if you're not a gritty person if you never see through any projects you start you don't need to worry at all you're going to be absolutely fine there's lots and lots of ways in which your life can succeed and this what I've tried to do here and I'm not sure if I've got the art quite right here but the the arc I want to describe here is I I want to start by saying statistics is very powerful I can show off to Gary Neville about it but then at some point statistics doesn't actually give you all of the answers right and that's very well Illustrated if we actually go back to Ronald Fisher this rather arrogant young undergraduate student because Ronald Fisher also has another scientist story he was from a very early age and this picture was taken in 1912 very interested in Eugenics and he had this idea that we needed to breed people to be more like him like to be more clever and smart and good at maths and so on and he campaigned all the way up to the war I think after the war he kept more quiet about this but all the way up to the war he campaigned for for example sterilizing people who were considered feeble-minded and this is this is of course horrible and I mean it's a horrible thing to think about but not only is it like morally repugnant it's also scientifically wrong they couldn't find any kind of Gene for single uh for feeble-mindedness there is no correlation between or there's a very weak correlation or no correlation at all between feeble-mindedness in one generation in mothers and in their daughters so this was a relationship that was scientifically dubious that he continued to press forward and the reason he was successful is because or a reason he was successful in pushing us forward is is thing forward was that he would use statistics in order to sort of attack his opponents he would call them all stupid they didn't understand statistics so he would actually use statistics to undermine other people's arguments in a really counterproductive way he took a fake Theory and then used statistics to defend it and he didn't just do this once after the war when he'd given up on on Eugenics or at least stopped talking about it he then did the same thing on smoking so as we saw in the first picture he was a very keen smoker and for him there was just no possibility that smoking could cause cancer and so he spent a lot of time investigating very narrow areas of science doing his statistics on that and trying to convince people that smoking didn't cause cancer and I don't know the effect that this research had but certainly you've got one of the leading statisticians in the world who's defending this position for a long long time and that illustrates a lot of why statistics is a limited approach so I've written down a few a few sort of bullet points here so statistical thinking doesn't provide all the answers um one problem is and I I didn't really get into this but I mean what a dick he is right I mean why do you need to test if she you know can tell the difference between the milk first everyone was very happy they were all just enjoying their tea party and suddenly he's doing a statistical day I mean you know he's an idiot so so that's one thing and and I really think you know we joke about that but we see it at work all the time you know that's always being told that we should be there should be statistical tests and metrics made about us and things like that and we don't need as much quantification as we have then there's the effect size thing and we sometimes talk about statistical significance you can have statistical significance but still have a very small effect size and that's the confusing the tree in the forest many non-gritty individuals are successful in life context is always important so just because a player is more active when their team goes a goal down does not imply the team plays better just because Ronaldo demands that everyone gives him the ball when they go a goal down or Jamie vardy demands that he gets the ball that doesn't mean that the team actually does better as a result of that so it's very important to think about the context of these types of things and then as we all know correlation and causation they're not the same thing but that leads us to the to the next step we need to think about ways to tease out causation we want to be able to tease out our understanding of the world one thing causes another and it brings us very nicely on to interactive thinking and that's the next step on from statistical thinking and I have another hero I can reassure you that this hero is not going to turn out to be a raving racist who bullies all of his co-workers so um there are a few mathematicians who haven't done that in their lives just a few of them but they're around and this is Alfred J Locker um and he was originally a chemist and he started his uh he's he was originally from Poland but he did his undergraduate degree in Birmingham and I I really like his story because he started working in this chemistry lab and he was kind of disappointed with what he saw when he was doing his experiments I mean it's a long time since I did chemistry at school but it can be a little bit disappointed you know you get the acid and The Alkali you mix them together and there's some salt and water it's not always the most exciting thing you've ever seen and so he'd see these stable stable reactions just come to equilibrium but in the evening he was reading all of these books like Charles Darwin's book and he was thinking about biology and just all of the exciting patterns we see all the motion and movement of animals or the firing of our brains everything that happens in society and he was thinking why can't chemistry produce anything like that I mean we know that chemistry must be the underlying building block of it but it's not something we see we can't make that relationship together and the way he solved the problem was he basically cheated and he he did the following thing so if we've all done this in school we've balanced equations right through a balanced reactions and if you've got uh two two h two well you've got four hydrogen atoms two oxygen atoms and they react to get to make two water molecules so that's a a standard chemical reaction and the important Point here is that this is balanced so there's four hydrogens and two oxygens on the left and there's four hydrogens and two oxygens on the right but what loter said is well I'll just forget about that balance thing even though 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 it's lovely with mathematics I'll do a thought experiment where I ignore the fact that I can't balance my reactions and so he take he wrote down these equations he said that imagine an r that becomes two r's and imagine an R plus an F which becomes two F's and you can see that these aren't balanced there's one r on the left hand side of the first one two r's on the right yeah there's you can see that they're just not balanced and I've written down here below the way you can think about these things are rabbits and foxes so and and they're not it's not a realistic model of rabbits and foxes you've got to think of the idea of one little rabbit hopping around and suddenly there's two little rabbits hopping around we know it's a little bit 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 that's what the model says it gives a a rough idea of how ecological interactions work and he took that and he wrote down differential equations I wanted to put in a few of these equations here to give you a feeling for how they work the the equations on the left here one of them I I'm not going to get you to understand every detail of the equation but what I want to give you a feeling for is on the left is the rate of change of the rabbits and the foxes so Dr by DT is a rate of change of the rabbits DF by DT is the rate of change of the foxes and on the right are the things which cause that change so I mentioned here that we want to get causation into our equations so on the right of the things which cause that change and rabbits increase when there's lots of other rabbits they have lots of little bunny rabbits and then they are eaten by the foxes so the more F they are if we look at this brf term that's the rate at which the um the rabbits are eaten by the foxes and then when we come down that we have the opposite relationship for for The Foxes the foxes grow when there's more rabbits and then they eventually die off um of old age there's nothing which hunts the foxes in in this in this scenario now 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 here actually learned this from Philip Maney when I was here in in Oxford about these types of method but he had a very lovely method a professor of mathematical biology here for solving these equations without solving them so you can split up the plane of foxes and rabbits and you can identify a point on that plane and look to see do the rabbits increase or do the fox's increase so in the bottom left-hand Corner the foxes go down because there's not enough rabbits to eat but the rabbits go up because there's not enough foxes eating them and that goes on until there's a sufficient number of rabbits and then the foxes start increasing so in the bottom right hand corner this Arrow points up and if that Arrow points up well 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 circling round and round of foxes and rabbits and you could do basically show this without explicitly solving the equations that there's going to be Cycles round and round of these foxes and rabbits we're going to get this interesting interaction and if we look at it over time um we have these periodic oscillations of the foxes and the rabbits now this whole way of thinking which latke introduced turned out to be useful in all sorts of situations now the one thing we should we try not to mention but can't be unmentioned in any mathematical modeling thing we try not to mention this but but it's also used in pandemic modeling and you've all heard about these epidemic curves and our values and so on um but that's the same thing that a susceptible plus an infective becomes too infectives and that's an example of one of lotka's unbalanced equations which allows us to describe how an epidemic reaction epidemic will spread through a group of people now I'm not going to go into as I said I don't want to talk too much about epidemics but this example I really love so I'm going to talk about this example this is a study we did I did together with some colleagues and this is a very cruel experience it's not it's not it's not that cruel but we have we had a group of undergraduate students we got a third a third year undergraduate student to give a seminar to first-year undergraduate students and we told the first year undergraduate students you know remember to give a round of applause after the seminar just to show your appreciation and then what we were really interested in was how people applaud it what are the cues that make people applaud and we could see that people start applauding when other people around them start applauding and you basically have an epidemic of Applause and that's what the first green curve shows that's the number of people clapping that's the spread of the Applause virus going through the group but then and this doesn't happen in real diseases you also have a social recovery so when people stop clapping they look around and they hear the other people have stopped clapping and it was actually a little bit like when Daryl left the room just now there was a sort of start signal there that there might be something going on that we might be about to start and you all started to go down in volume and suddenly everyone was quiet and that's the type of social effect we're very aware of all kinds of small social details and these spread through us in a group and the the conclusion that what I love about the recovered thing is because we have that social recovery so and I try to always remember this that if at the end of a talk you've given or a presentation if the Clapping goes on for a long time that's not because you gave a good talk it's just because your audience aren't particularly coordinated so they couldn't they could they couldn't manage to to stop together and I think that's what that's what I would encourage you to think about and I said that I also wondering about the personal aspects of this it's nice just to sit sometimes and think about the social reactions that you have in your life and how they work and I've written down a few of them I haven't told you what they are yet so I'll tell you what what they are the top the top one that I was imagining this is a person p and then it's plus an O this is a sofa that the person has outside the house and so we have P plus o goes to P plus o if you're just one person and you've got a sofa outside the house you're going to still be one person and you can't get the sofa into the house so what you have to do and this is the bottom equation is you have to get a friend and so this bottom one is 2p 2 people plus a sofa that's outside the house is still two people you've still got your people afterwards but you've moved your sofa into the living room and you can write those social interactions for every type of activity the one on the right here the ones on the right here I was thinking about smiling so if you're a smiley person why and you meet a non-smiley person whose ex then if you smile then hopefully they become a smiley person too but that's not always the case sometimes you know you don't always start smiling because somebody else is smiling they might just be an idiot who's just smiling for no reason at all what happens most often actually in human social interactions is they as you believe the following Equator the one the equation at the bottom this is the most common equation I think that describes human behavior and that's that a non-smiling person plus two smiley people will become three Smiley people because then they're convinced that actually must be something to smile at and I use that a lot in my thinking if if I'm thinking about how to 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 going with some kind of healthy activity maybe they spend a lot of time sitting in the pub together don't really go out and do any exercise together it's not enough for one of them to become a why to to try and get them going that you have to have two of them and they have to have a really sustained effort and over time then you get this Tipping Point effect where everybody starts to move over and starts it starts to engage in the healthy activity so those are the types of things you have to think about what type of chemical reaction what type of social reaction am I involved in and that's been a lot of to be honest this has taken up a lot of my adult life is studying these types of things and to give you a little bit of a flavor of the sorts of things we do this is just to give a sort of overall representation but um when we modeled fish for example we would create models which described their social interactions described how one fish turned left if another fish turned left if another fish turned right and so on then we were building the top there it's a mathematical model we've built a fish movement and so on so we'd show that these simple rules of interaction would produce their Collective Behavior then we'd study also the the movement of individual fish that's the colored idea at the the bottom then we'd actually frighten all of the fish and we'd look at how they made an escape wave we we'd measure that escape wave and then we'd use models to understand that escape wave and it's a very powerful way of thinking throughout science that you can build up these models of interaction you compare them to reality and build a better and better understanding of fish Behavior and we do a lot of similar things in football so this 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 can he run where can he get to and we can actually describe what area he he occupies and also the value of that area so How likely is he is getting a pass at that particular Point going to lead to a goal and that allows us to actually Scout players based on their runs and it even allows us to scout runs where they don't get the ball so in this example where we're interested in Luke Shaw here and he's doing a run here on the left and he doesn't get the pass he'd love to have this pass but he doesn't get it but we can still measure the value that that pass created so you can look at these counterfactual situations for um for for football players and this is a very powerful method the interactive way of thinking allows us to build up our understanding of systems it doesn't have the same kind of I I suppose the statistics has a sort of more of a grounding feeling to it this 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 um there are also limits to this way of thinking and of course I wouldn't have four if we'd if we'd solved it all now so there's there's limits to this and there's limits were limits that lotker himself hit he wrote a book called elements of physical biology and he's one of these these mathematicians who and this happens a lot to us is we sort of just get carried away and we believe that we can just explain everything with mathematics that there's nothing that we can't explain and so he built models he built models of Consciousness he built models of of our whole society and he believed that all of them could be understood using his reaction Dynamics and it really yeah he didn't I mean and this was I suppose it was a very Valiant effort this is in 1922 he finished his his Magnus Opus so he didn't even have a computer or anything to simulate these types of models on but he never really succeeded in pinning down one essential way in which you should approach all sorts of problems he he ended up kind of split between lots and lots of different small things and that I can personally I can relate to that very well because that tends to be how I work with lots of problems there's lots of different methods and you're doing lots and lots of small different things in order to get your solution the day of day of an applied mathematician isn't it's not like these theoretical physicists you know they have like this beautiful Theory of Everything and they can come here and just say oh it's all this and wow but no it's not like that it's more that you're sort of tinkering around with small different problems in lots and lots of different ways so lotska never found his Grand Theory of Everything using interactive thinking and one of the reason one of the reasons he never found is Grand theory was because he didn't know about chaos which is the third way of thinking now to introduce chaos I'm going to go to another mathematical hero this is Margaret Hamilton and she was also like the other two we've met prodigious at school very talented undergraduate student she wanted to go on and do a PhD in pure mathematics but her husband also wanted to do a PhD and this is now in the 1960s and she ended up moving to Boston and she also had to get a job she had a daughter to support a husband to support and so she had to get a job to support them but the job that she got was programming this machine the lgp 30 and she fell immediately in love with this Computing machine because she hated making mistakes she hated errors whenever she calculated anything she calculated it perfectly and now she found that she could actually program this first computer to do the same calculations and she got access to this computer because she was working in the lab of a person called Edward Lorenz who was a professor of meteorology but also with a mathematical background there are a lot of mathematicians in this talk so um and he he wanted to predict the weather he wanted to predict the future weather based on temperature pressure and so on in different areas could he predict the weather into the future and she started writing a computer code to do this and this involved writing and doing Punch Cards at the time and she'd 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 check their results by simulating making the exact same simulation on the computer to check that everything worked but they found on the second day they got a different result than on the first day and Margaret was distraught because she didn't like making mistakes she didn't want to think there was a mistake in her code but she started going through the code and there was no errors in the code and what they found was that the output of the simulation was in six decimal places well the input they put into it was in three decimal places so there was an error in the input in the fourth decimal place and this meant that the weather simulation made completely different predictions in the future going like a few 10 days into the future in the simulated World it 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've moved on from lockter and Voltaren too so she solved these 14 differential equations and they make just this small error in the value you put in makes a massive difference and that is the first indication of the butterfly of chaos which many of you will be familiar with and Lorenz went on he worked with um I say Lorenz went on Margaret Hamilton we're going to find also went on to do some very impressive things but Lorenz went on with the help of Ellen Fetter who replaced Margaret Hamilton as his programmer to produce what we now know as the we often think of this picture I think or I think of it as being the butterfly of chaos and what it illustrates is if you do start with two points very close to each other we've moved now down from 14 Dimensions to three dimensions again if you start with two points very close together and they start to diverge they'll move around on the same attractor on this shape that we have here but they'll never come close or they might come close to each other for a short amount of time but they'll then live their own life and so when we move from two Dimensions up to three we have this chaotic movement where things never come back to the same place again I think I think I think we're going to do my experiment okay so I think I think we're going to do the experiment and then I'll I'll boot out something else because you've listened to me patiently for 50 minutes so you have to get to do the experiment okay so here we're going to we're going to do this I want you to work in pairs I want you one of you should think of it so it's a look at the person next to you and you might be a new friend that you've got today um or it might be somebody that 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 follows the follow the the following 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 in it so 42 times 2 is 84 and so that's all you do you 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 is 16 times 2 is 32. now say the new number to your partner and they repeat step one and two so we'll do this for um do this with either with the person you came with or somebody who's nearby to you we'll do this for about two minutes and then we'll see where we get to I think you've done it very nicely done I can see the I can see I hear the murmur of numbers everywhere very very lovely um I'm I'm not going to get you all to come up here and present your results I just wanted to give you get get you to get a feeling of this type of process um you're not generating purely chaotic numbers when you do this if you'd started with 20 for example you would have found yourself cycling around quite quickly but if you started with a number that's not divisible by five you would have probably been on quite a long trajectory through different numbers and the point I want to make about this process is the following is that close together numbers very quickly diverge so if one group over there had started with 13 and another group over here had started with 14 by the end of just this short period where you got to say the numbers to each other you would have been on very different numbers so you have 13 26 52 96 8 16 32 64. 14 28 28 is not so far from 26 56 52 they're still together 88.96 are starting to get away from each other but the big jump is now one of them sort of goes over the threshold and one of them doesn't so you've got 8 and 24. 1648 and then you've got 32 and 96 and 64 and 8. So within a few steps these numbers have diverged quite far from each other I don't know if any any of you took decimal numbers um you didn't think of that but if you do take decimal numbers then you get true chaos from this thing for almost any real number you choose you will get if you take plus 0.1 in this case only so this is 14.1 compared to 14.2 you start to they're together for a few steps but after about seven eight nine ten they go apart they come a little bit together again for a while but then they diverge and you've got very different paths for those two numbers and we often illustrate this um using something called a cobweb diagram so the idea here is you take the number from One Step and the previous number might be around 20 for example it will jump up to be around 40 then it will go to 80 then it will crash down to around 20 again and then it will start to move around everywhere on this and one of the reasons I wanted you to do this experiment is what was being what I could hear from your perspective was a Mumble of uniform distributed random numbers you were essentially going through a lot of integers and everywhere in the room there was a different point in this distribution you basically had this uniform distribution of numbers that were sort of kind of coming up to me and I think it's really lovely to think of that that you're all doing the same process you're all doing exactly the same thing yet you kind of have this hum this distribution this background of very different numbers and that is the butterfly of chaos and and for me it illustrates there's an important Point here I I think chaos is wonderful Margaret Hamilton she hated chaos right and she she left Lorenzo's lab and she'd learned a valuable lesson from working on these weather simulations and it was that she doubled down and made even fewer errors and she wanted to work in the most extreme conditions possible where you couldn't make errors and so she got a job for NASA and she became the head of the software engineering which created the software that sent that was on the Apollo moon mission and so she was she created the software that the astronauts used to tell them how to to do navigational decisions to control the thrusters to um uh to update to know where the position of the ship was and she was in the control room when they when they made the actual landing on the moon and so I see this as a situation where you sort of have to choose right in if you're if you're going to control something because of chaos if there's something you really care about or there's something that's really important then you have to treat it like Margaret Hamilton does you had to treat it like the moon landing there's no error there's no room for any type of error but you can't have control over everything so I often think about this in football because there's always going to be butterflies in other situations so here this isn't the uniform distribution as you generated but it's the poisson distribution there is lots of other situations football being one of them where we just can't avoid Randomness Randomness is always going to pop up we can we can control a past like we talked about with the Pastor Mike Marcus rashford but we can't control what happens in 90 minutes of a football match football is to very to a very large degree random and I like to think about it like this I like to think about it as the yin and yang theory of Randomness so Yang is order and on short time scales things we really care about we can control them and we can build models of them and we can understand them but Yin is disorder and the on longer time scales we just have to admit there's things that we can't control that are beyond our control and I actually found this personally very useful because my wife is a bit more order and I'm a little bit more chaos in many situations and she wants to plan like what we're going to have for dinner every evening she wants to plan quite a lot of details of how the week should look and I'm like oh whoops I've got to go off and do this talk in Oxford or uh I don't I can't quite remember the things that I've planned and said that I'll do and I'm kind of you know and so I have a kind of more random approach and first of all when I looked at Chaos Theory I thought chaos theory was the perfect it was a perfect justification of my chaotic Behavior I could just go around because it was no controlling anything so you know why bother and I didn't explain this to my wife obviously but um I I thought a little bit about that but then actually when I heard about Margaret Hamilton I understood it a little bit better if there are things you really care about then you make effort to control them and my wife and I we found more of that balance and now I have discussed it a bit more with her is that we make sure there are things that are important like that we eat dinner together every evening and we make sure that we do those things and some of the other things like if there's a big pile of dirty laundry that needs to be done or cut in the grass we let them go over to chaos when we feel like doing them so it's really I think there's nice 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 as much just let the chaos take over there's nothing you can do about it now I have gone over time and and you wanted to know of course what is the fourth well we've we've found a yeah we've found the three ways of thinking I won't review them because I'm not going to go over time the fourth way of thinking I am going to spend a little less time on the fourth way of thinking for a reason and one of the reasons is that you can maybe go out and buy the book and find out about the fourth way of thinking but I would say a little bit about it because I also think I I do think it's something which should Inspire younger researchers thinking about how they're going to make the next step in this and this is a simulation produced by one of my master students Michael Hansen and maybe some of you will have heard about cellular automata models before there's this The Game of Life is the most famous one where there's interaction rules and I gave my students a task to make up their own cellular automata that made the most complex pattern possible and Michael amical came up with the following rules he had the bone the white cells so they only interact with the cells nearby so this is a hundred by hundred grid of cells but the cells only interact nearby um and the rule for the white ones is that they become goo which is dark blue if less than four of their neighbors are bone the white ones otherwise they remain bone and then the goo ones the dark blue these become fluid which is the light blue if less than three of their neighbors are bone otherwise they remain goo and then for fluid which is the light blue one these become bone if two or more of their neighbors are bone now the exact details of the the rules are important what I think is amazing is that you just have these Simple Rules which describe how cells interact with each other and suddenly they're creating these very Splendid structures there's a and the reason I called the white ones bone is that they stretch out and they kind of form this skeleton and between that skeleton you have alternating dark blue light blue patterns pumping backwards and forward so with very simple local interactions you have an extremely complex Behavior and another example of this and this is something you can find on Twitter this has become a competition so these are these are all computer graphics and every one of those computer Graphics is produced with code that is a Twitter lens piece of code so that means it has to be 240 characters long and so there's a whole kind of culture on Twitter of who can create a graphic with 240 characters and each one of these you can create use fractals of course and use shading and you can create these kind of movies of tree they look like trees look like flowers it seems I love these things look like goo and then you have some sort of flowering thing there as well just by a very short code and that brings us to who is the theory who is the hero of the last part of the book The Scientist who's the hero of the last part of the book and that's um Cole magorov because he defined complexity or he didn't Define complexity as such but he said a pattern is as complex as the length of the shortest description that can be used to produce it and I find that a very useful way for thinking about science and thinking about how we understand the world if we can find a way of describing something that doesn't lose the detail doesn't lose the Nuance but captures it then we can start to actually capture complexity and so that's the thought I I want to leave you on and thank you for being patient with me thanks thank you [Applause]

Channel: Oxford Mathematics

Views: 337,628

Keywords: maths, math, statistics, data analysis, David Sumpter

Id: PPCfDe8TfJQ

Length: 56min 7sec (3367 seconds)

Published: Wed Oct 04 2023