How to Think Like a Mathematician - with Eugenia Cheng

[Music] thank you thank you very much so I love maths and unfortunately not everyone loves maths as much as I do and there are many reasons for this and in my previous work I have spent quite a lot of time showing that maths can be fun that maths can be delicious that maths can be creative that math doesn't necessarily have right answers that math uses your imagination but the thing that I'm addressing today is slightly different because people often say well that's all very well but what's it useful for and I've often shied away from that question because I think we spend too much time going on about how useful maths is and especially to young people since when did any young person get interested in something because we said it was useful it's like introducing a friend and saying I'd like you to meet my friend they're really useful or it's like trying to get people to eat broccoli because it's good for you broccoli is delicious it is also good for you but then I thought about it and I thought well maths is useful and the way in which it's useful is because it's illuminating and the way in which it's illuminating is because it helps us to think more clearly and wouldn't it be useful if more people in the world could think clearly especially these days and so I realised that I felt a driving urge to try and help with this current situation we're in where we find ourselves often surrounded by fake news or claims of fake news maybe it's fake fake news and divisive arguments people yelling at each other people being victims exploiting bigotry prejudice blame shouting and tiny attention spans and I've always believed it important to think about all the things that I'm good at and do the best possible thing I can with them to help the world and I realized that in the current socio-political climate it was really important for me to do something about that and you might say what can a pure mathematician do about that and I thought well what I can do is I can help us all gain better clarity about the kind of arguments that are happening I'm not going to tell you what to think but I'm going to show you some ways that I find helpful in navigating the rather confusing world that we're in at the moment because how can you make sense in a world that doesn't well I think that we can and so this is a message of optimism I am an optimist that's why I'm in education you can't really be an education unless you're an optimist and we believe that that the key to everything is the young people and how they are going to to think better in the future when the rest of us have gone maybe that wasn't so optimistic but there so the point of my talk is to show how pure maths really does apply to our daily lives usually we think of pure maths as something abstract and remote but I am going to argue that pure mathematics is in fact a framework for agreeing on things and mathematicians have this curious ability to agree as a research field so we really make progress things become theories in other subjects people keep arguing and arguing and arguing say in philosophy there are many theories and you argue and there's never anything that's necessarily right if there's a philosopher in the room they may argue with me but mathematicians have a very clear framework for agreeing when something has been properly justified and this involves setting up what you're talking about carefully and then using logic carefully about it and this is sorely missing from many arguments in life at the moment so traditionally pure maths is thought of as remote and distant from the world and it's it's nothing to do with real life but it applies usually we think of it as being useful to applied maths where applied maths they're useful too maybe science and science is useful to have perhaps engineering and medicine and engineering and medicine are definitely useful and they are useful to you largely a numerical part of the world and many people think that maths is all about numbers and math is not all about numbers I do not do any numbers in my research as a category theorist and maths is about something more than that so while this view of the chain of implications of applications of mathematics this view is important there's more to it than that so I'm going to argue that pure maths is about how to think and how to think affects the entire human world and that this is a direct application it doesn't have to go through all those intermediate steps and in fact I was at a very interesting event recently about maths in the real world and everyone else was an applied mathematician except me I am NOT an applied medicine I'm a pure mathematician and one of the very interesting audience questions to the whole panel at the end of the day was what maths do you use in your daily life how do you use your research in your daily life and I felt very very validated or possibly vindicated or maybe both because the applied mathematicians all kind of said well we don't really use my actual research in my daily life but what I do use is the way of mathematical thinking and thinking through things I thought see that's pure maths I use it in my daily life all the time and I'm going to show you how including I'm going to talk about analogies the interconnectedness of things relationships of various kinds pivots between different contexts to help us understand other people and finally intelligence and what I think intelligence is or what it should be so first of all analogies I often like to think of maths there are many ways of thinking maths I often like to think of it as a theory of analogies because what we do is we look for things in common between different situations and then we say oh if we forget some of the small details about these different situations they will become the same so for example we might think about apples and two bananas and we go oh well these two things have something in common if we forget that these are apples and that these are bananas then they just become two things and now there are both examples of the same concept and that's what abstraction is it's moving from the kind of real things to something that they have in common and the thing that Maps does is it makes very precise what that thing is that they have in common we don't just say oh two apples are analogous to two bananas we say exactly what it is that we're thinking about that makes them analogous and there are different choices you can make for this so for example if instead we had said two fruits that would still be true right they're both still a type of two fruits but in making that analogy instead we have made something much more restrictive and so we will leave out say two chairs which are now not part of this analogy whereas if we say two things then we get to include all of that in the analogy so different levels therein one correct level for making the analogy but there are different levels that are more and less illuminating for different situations and the key in maths is that we say what level it is were using the problem in life is that we usually don't and so we end up in arguments about whether something is or isn't analogous and the answer usually is there is a sense in which it is analogous but is that sense illuminating and that's more what maths is about it's not this is right this is wrong no you're wrong you're right no I'm right it's about what is the sense in which something can be thought of in this way and how will that help us so here's an example and more profound example from maths where 1 + 2 + 2 + 3 are both examples of a + b and this is an abstraction which sometimes makes people shudder and sometimes people say to me I was fine with maths until the numbers became letters but the point of the letters is that we now get to include more things and we can study something plus something generally all at once and then we can understand a lot of different situations at the same time while only having understood it once and this is really good if you're lazy like me because I don't want to do the same thing over and over again and pure maths comes out of not wanting to do the same thing over and over again once you've done it a couple of times you go okay I'm gonna make a theory that does that for me so that I don't have to do it anymore just like if I had been born before dishwashers were invented I feel like I might have invented a dishwasher because I really hate doing the washing up so we can take this further because if we now look at 1 times 2 and 2 times 3 we can go oh yeah well that's a times B that's what those things have in common but we can now say well wait a minute on the one hand we have a plus B and on the other hand we have a times B both of those are just things we can do to a and B so that's a further level of abstraction which I might write a blob B where that blob is now an abstraction from plus and times and could represent any other kind of operation it's called a binary operation because it takes two things and sometimes that top level is really helpful because we can do things like thought talk about commutativity in general commutativity is where a plus B is the same as B plus a and also a times B is the same as B times a but that's two things if we just think about commutativity we only have to think about one thing and this is a very important way that pure maths helps us understand more things our brains of finite and a bit feeble so we have to be able to understand lots of things at once in order to save our brainpower and one way to save our brainpower is just to ignore stuff and simplify things by forgetting important details and many people do that but a better way to do it is to become more intelligent and as you become more intelligent you find ways of packaging things together as one concept and then you can understand it as one concept instead of having to carry around ten just like when you first learn to read maybe you spell out every letter one by one and it's very arduous but then eventually you see start seeing whole words and then it takes much less processing power for your brain than having to read every letter and then if you learn to speed read then you learn to see whole phrases at once as units and then you can read really first and this is this is how to think in that kind of way packaging things together so here is another example of some analogies where different levels of analogy are possible straight marriage and same-sex marriage are sometimes thought of as being just the same because their marriage and then other people get very angry and say those are not the same so let's think about what level of analogy we could use here if we think that marriage is just between an unrelated man and woman then indeed straight marriage is not analogous to gay marriage but if you think that actually it's between any two unrelated adults then they are analogous and instead of saying which one is right and which one is wrong by definition we should think about the different levels that people are using to make those connections now what happens next often is that the people who oppose gay marriage get very upset and say well the next thing we know we'll be allowing marriage between any two adults and because there are some young people in the room I've redacted what exactly is going to go on here and maybe you can work it out but then they say well with will out between any two adults well maybe then the next thing we know we'll be allowing between any two humans and then maybe the next thing we know will be allowing it between any two living creatures and then maybe the next thing we know is we'll be aligned between any two creatures now logically speaking any of those levels have sound logic that doesn't mean that they're all equally morally valid but what is certainly true is that just because we are ok with this level does not mean that we have automatically included the whole level all the way up to the top so if somebody thinks that thinking this implies thinking all the way to the top that isn't logical we can then have arguments about where is the correct place to stop but if we make precise what we're actually using as the analogy we can have a more sensible argument rather than just yelling at each other so now I'm going to talk about the intercom cydnus of things because everything is very interconnected we get told sometimes that growing up is about becoming independent but none of us is independent unless you're living in a in a cave as a hermit which you're not because you're all here we all interact with other people and nothing happens in isolation everything happens together with some other things that caused it to happen in huge chains and my field of research category theory is all about how thinking about how things are connected can really help you understand those things just like when you're studying a person and thinking about their writing their biography perhaps it would be very odd to just write about their character in isolation whereas if you write about you usually write about their family relationships and their friends and the people they work with and if you're getting to know someone if you just put them in a room by themselves it'd be quite difficult but you can tell a lot about someone by how they interact with different people how they interact with important people how they interact with serving staff in a restaurant how they interact with cashiers you know in the supermarket or something you can tell a lot from from how people interact and it's the same with any objects there's always some web of interconnectedness that adds insight to those thoughts so here is one of my favorite diagrams of interconnectedness I'm sure that we all know this since we are in London this is a fantastic example of something that is a logical diagram because this is not geographically accurate right the the stations are not in the correct places which means that sometimes hapless tourists end up taking taking the tube when it would have been way quicker just walk four minutes up the street it's not clear from the map how close things are and once you're on the train it doesn't really matter exactly where it's going you just want to know where you can get to and so this basically just shows connections between stations and that's what category theory does a lot it says it when we're not worried about physical location what we're interested in is the conceptual connection between things how things are linked according to what we're interested in right now now there may be situations when having a geographical representation of the tube is useful is a geographical Tube map I don't know if you've seen one before but it's pretty confusing it's very I find it very interesting thinking about how different it is from the original one and how fantastic it was whoever designed I've gotten who designed the the main one but once you start peering at this closely you can of what I usually does I find the Circle Line first and then I kind of orient myself using the Circle Line but this isn't to say that one of them is better than the other it's just that they're better for different purposes and this is the point of thinking mathematically you can think about certain aspects of a situation for some purposes and other aspects for other purposes and what that can help us do is get a certain flexibility to our thinking because one of the great problems with the world at the moment is rigid thinking and I sometimes think that strength physical strength isn't just about strength but flexibility right materials that have some flexibility in them can be very much stronger and when cars crash it's the give of the COG that crumples up that is actually making it safe for us because it's absorbing it's absorbing the impact and I think something's the same is true of the way we think as well that flexibility of thinking and the ability to absorb new ideas and process them is really important so relationship breakdown is something that I think about in terms of interconnectedness because often people want to blame one person and say who is at fault and there are many situations where where it's really not that simple okay so there are some in situations of domestic abuse where perhaps we can agree that it is really one person's fault but aside from that it's often some kind of vicious circle so maybe Alex feels disrespected and as a result alex is unable to show love and as a result Alex's partner Sam feels unloved and as a result sam is unable to show respect which is why Alex is feeling disrespected and what often happens is that one person in a relationship cares more about feeling loved and the other person cares more about feeling respected and then it goes around in these cycles and this vicious cycle something that we can we can then think more about by thinking about what the arrows are representing so these are is here are about actions and these arrows are more about feelings and once we've broken it down like that we can think about which things are more or easier to change is it easier to change your actions or your feelings now personally I think it's easier to change my actions because the feelings of things that just happen and so then for some people maybe they think it's easier to change their feelings and that's where it's there's no right answer to this but when you have a vicious cycle if you understand what's causing the cycle then you can figure out which arrow is the easiest one to break and maybe there's a theory that whoever has more power in the situation should be the one to take responsibility to break the cycle now this is a very important example when we think about police violence because this vicious circle comes back if we think about the fact that maybe police feel threatened by black people and that's why they show more violence towards black people but as a result black people feel threatened by the police and as a result maybe black people defend themselves aggressively against the police which means that the police feel threatened by black people and this cycle goes round and round and round and argh every time there is some police violence against somebody who isn't white and there is a lot of that in America at the moment and there always has been every time there will be people yelling saying well the police are just trying to do their job and if the black people hadn't been aggressive nothing would have happened and then the other people yell that the police shouldn't be aggressive and then but we what we should really do is think about how these things are all connected and then I think we can think about who has more power in this situation and personally I think that it's the people who have more power who should take the responsibility to change something and find an error to break but at least even if we disagree about that at least if we've understood this understood this interaction we can have a more sensible conversation about which arrows should be broken and who should break them another way of thinking about interconnectedness is where I draw myself diagrams of causation so when something happens like the egregious incidents a couple of years ago when someone was was ejected from a united flight because they needed a seat for cabin crew and he was ejected rather forcefully and sustained some quite serious injuries on the way out and many people started yelling on the internet as they always do oh well it was his fault if he had just obeyed them then he wouldn't be an injured and other people were saying well you shouldn't just just beat someone up because they didn't want to get off a plane and then they just argue at each other well kind of both of those things are true and there's really lots of things going on and I even read I think we're being filmed so I won't name this person but I read a rather curious editorial by somebody in a British newspaper who declared that it's actually our fault all of us because we're late for flights and the fact that we're late for flights means that they overbooked flights and because they overbooked flights they sometimes have to kick people off right and that's why that person got injured so it's your fault all of you I mean there was a tiny element of truth to that but it's a little bit tenuous and so I decided to draw a whole diagram about all the things causing this because it's never just one thing right so the end result was that someone was injured and well it's because he refused to leave and also that security used force but the airline called security someone had to call him in the first place why did they do that well why and why did he refuse to leave he needed to get to work why did the airline choose that person there are questions remaining about whether they were racially profiling then why did the airline even decide to kick people off in the first place well they had to kick people off because nobody volunteered well maybe the airline offered them too little money they didn't offer them very much money and maybe for some reason everyone really wants to get home well that's kind of understandable well also why did the airline even decide to remove people in the first place because the plane was too full well I was the plates u4o and also why did they need to do that because they needed some crew in Louisville and why did they need some crew in louisville maybe there was some issue with their scheduling maybe it was their fault for saddling badly and so why was the flight even - for well because they over booked flights why do they overbooked flights also not enough people she didn't show up and the flight was overbooked because people often miss flight so finally we have that reason up at the top there but then we can think about which are the most important contributing factors and we can think about which arrow shouldn't have happened and who should have taken responsibility for making sure that arrow did not happen so here is another example which is why I get fat now I used to be a lot fatter and I lost 50 pounds and I would quite like to keep it off and so I think a lot about this and whenever there is anything about there have been recent measures about trying to cut obesity trying to stop bog off deals on sugary things or trying to make people put the calories on the restaurant menus and there's always somebody I always read the comments online people say never read the comments I say always read the comments because if you don't read the comments you'll just stand your own echo chamber and you should particularly read the comments until you find something that makes you ill because it's so awful and I do that every day I honestly do because I think it's so important to stay aware of what people are thinking so anyway whenever there's something about losing weight someone will go oh well it's not rocket science you just have to eat less and exercise more well even rocket science is just applied maths so why do I put on weight well I put on weight because I take in more energy than I burn which is because I eat too much and because I exercise too little but it's also because of my metabolism and there are some things that change your metabolism so for example eating too little and exercising too much change my metabolism which means I put on weight so it's actually because I eat too much and also because I eat too little this is already getting a bit complicated right so my metabolism is that way well it's partly because of genetics and it's partly because of sleep patterns which can change your metabolism and why do I eat so much well I like food but it's also because of emotions and it's also see what's happening next so I eat too much also maybe genetics and also upbringing food a lovely thing to have and we love eating food and that's why I like food so there's also social pressure to eat too much people think you're boring if you don't just just gobble everything up but community with everybody and then there's also social norms but those cause emotions so there's also time pressure so time when I'm stressed I get emotional I also sleep less and that doesn't help and then there's also life life life causes emotions but then there's also the entire food industry which is if you think about it there's a billion multi billion dollar industry that is trying to get us to eat too much it's no wonder that they're doing really well at it so they do advertising the advertising makes us eat more and they're growing obesity rates fuel the diet industry because they all want us to go on diets that fail and this is all caused by money they're trying to make money out of us and they figured out that they can so moreover when I do gain weight then I do all these things which doesn't help so now we have all sorts of cycles all over the place so now tell me that it's simple it is simple if we become intelligent enough to understand a diagram of interconnectedness like this as one thing and that's a much better way of simplifying it than just saying oh you just have to eat less so if you that's if you do just eat less then you have just deleted the entire rest of the diagram that is one way to make this diagram into one thing but a better way is to become intelligent enough to comprehend the whole diagram as one thing and I think that's a better way so I did them draw a diagram for the 2016 US election a lot of people were yelling about whose fault it was some people saying oh it's all the people who voted third party oh it's Bernie the Bernie or bust people Oh Hillary's for Oh someone's for its a lot of things together and sometimes I feel like it's it's really not that profound to say it's a lot of things together and yet there are so many people trying to say it's just one thing so there is my diagram for that election which is one one of the the motivating reasons as I wrote this book because I was teaching in Chicago at the time and it was impossible not to talk about it and to try and use mathematical thinking to help understand the situation as much as possible so relationships we've talked about some literal relationships between people and the idea of relationships overall I'm going to show how thinking about relationships can really illuminate some situations and I'm going to start by talking about factors of 30 now if you've read my first book or seen me talk about my first book this might be a little bit familiar to you but we're going to take it further so the factors of 30 maybe you can remember what they are they are 1 & 2 3 5 6 10 15 and 30 very good ok uuuugh passed though it's not very interesting it's a bunch of numbers in a straight line and we can make it more interesting I think by showing which ones are related to each other as factors because some of these numbers are factors of each other and we can draw something a bit like a family tree showing how they are related and just like a family tree we don't need to draw lines between two generations because we can deduce that right we don't draw a line from grandparents to grandchildren we just go via parents so 30 years at the top like a kind of great-grandparents and then 6 10 and 15 go into 30 5 goes into 10 and 15 2002 6 and 10 3 goes into 6 and 15 and 1 goes into 2 3 & 5 so now we see it's a cube which is a bit more interesting than a bunch of numbers in a straight line we can think about why it's a cube there are lots of things you can do with this but if we think about it one is up the cotton because obviously nothing goes into 1 and then 2 3 & 5 are there because the only thing that goes into them is 1 under themselves you might remember that means that they're prime numbers so the prime numbers are at the bottom level and then the next level up we've got things that are products of two prime numbers so 6 is 2 times 3 10 is 2 times 5 and 15 is 3 times 5 and then at the top we have 30 which is a product of 3 prime numbers 2 3 and 5 so we have a hierarchy now that consists of how many prime factors each back to has and I can draw that one like this so at the bottom we have the empty set that set the symbol for the empty set and then we have 1 element sets and then we have two element sets and at the top we have all three of the prime factors 2 3 & 5 and that's how you make all the factors of a number you find all the possible combinations of its prime factors now maybe it becomes clear from this diagram that it doesn't really matter what those numbers were 2 3 & 5 so they could have been a B and C so now we have the same picture but with a B and C instead and all the possible subsets of a B and C structured using the hierarchy of ones with two things ones with one thing and then no things now we could try this again with 42 to see if we understood it so here are the factors of 42 there 1 2 3 6 7 14 21 and 42 and if we draw the diagram again now we can start from the bottom because we've understood the prime factors should go at the bottom that's 2 3 and 7 and then the products of 2 prime factors and then the products of all 3 prime factors and what you can see if we write it out using the sets of prime factors that it's the same diagram once you've gone all the way up to the abstraction of a B and C it's the same thing it's just that we swapped out a 5 for a 7 in the middle diagram and then to get back to the one on the far left we multiplied the numbers 2 in each position now there's something very crucial I want to draw your attention to in this new diagram which is that 6 is less than 7 now this doesn't sound very profound but if you look at the diagram 6 is higher up than 7 in this hierarchy but 6 is smaller than 7 on a normal number line where we're just doing things by size and there is often a situation where there are two different ways of placing things in a hierarchy and the clash between those two different ways is what causes tension and anted antagonism just like for example if at work somebody older is more junior than somebody younger who's been promoted over them that can cause antagonism now this has become very abstract we've turned the numbers into letters and I'm going to show you that there is a point to turning the numbers into letters which is it's now very much more widely applicable because those letters could be anything for example they could be three types of privileged rich white and male and then at the next level down we'll have the everything with two types of privilege so we have rich and white rich male and white male and then at the next level people with only one type of privilege rich white or male and then at the bottom people with none of those two types of privilege three types of privilege so if I put back in the missing types of privilege for emphasis we have rich white non male people so that we remember that we're including non-binary people we have rich non-white male people we have poor white male oh yes I'm using poor instead of non rich because it doesn't fit sideways we have rich non-white non-male people we have poor white non-male people and we have poor non-white male people and then at the bottom we have poor non-whites non male people so there are many things that we can understand from this diagram we've gone from a cube of factors of 30 to a diagram of interaction of different types of privilege now the first important thing to notice is that every arrow represents a loss of one type of privilege and it's important to notice that this just means that if everything about you stays the same but you just lose that one type of privilege then your position in society will get worse that's what the theory of privilege says it doesn't mean that all white people are better off than all non-white people sometimes people who are angry about this theory get really upset and point at some super rich black sports star and say see they're really rich white privilege doesn't exist but that's not what white privilege is about it's about saying that if that super rich sports star were all of the same things but also white they would be in a better position in society for example they would not have to fear for their lives in America or if they get stopped by the police now there is something more that we can understand from this diagram if we look along a row right there are no arrows going between things on a single row because they've all got different types of privilege but we can think about how much privilege they really have in society so maybe we can think that rich white women are actually doing much better than say rich non-white men who are probably actually doing better than poor white men because it turns out that money can mitigate for many many problems and so actually the rich white non men are kind of there and the poor white men are sort of there and the same is true on the bottom level but actually we can go even further than this because if we think about the difference between the two levels if we think about poor white men and we compared them with say rich black women we can maybe agree that it's definitely possible for rich black women to be way better off in society than poor white men and we can think about some extreme examples like Michelle Obama or Oprah Winfrey they're definitely doing better than poor unemployed white homeless men and so actually the diagram is like this it's a cuboid it's not really a cube and this has helped me understand why some poor white men are so angry in society at the moment because they are considered to have many types of privilege but they don't feel any manifestation of it in their actual lives and I think it's much more productive to understand the root of that anger rather than simply get angry with them in return and there's a bit too much of just getting angry with them in return so this kind of diagram seeing these abstract structures can also help us switch context so we can pivot between different contexts that have the same diagram in them and we can find analogous situations so for example in these cubes 30 and 42 and rich white men all occupied the analogous position and we can look at just one type of privileged relationship and so that the relationship that male people have over female people because they are the dominant in society is like white people over black people because white people are dominant in society over black people and that's also like rich people over poor people and so those the male people the white people and the rich people occupy an analogous situation in those situations whereas if you think about the relationship that male people have to female people compared with female people to male people that is no longer analogous because the arrow is pointing in the opposite direction now of course it is analogous if you go one step further and you just say oh well it's people and other people right but is that a useful level of analogy to go well we can have a discussion about that but there is a sense in which male people being prejudiced towards female people is different from female people being prejudiced against male people and the sense in which it is different is visible here because there is a power difference between them now we can then have an argument about how that power difference manifests itself but there is a power difference and in all of those situations when there is a power difference it also means that racism of white people against black people has something different about it from racism of black people against white people so there's another thing here which is that something might not be at the top in one context but be at the top in another context so these things aren't fixed we're all in different situations all the time and in some situations we're more powerful than in other situations so when I'm standing in front of a class with students and I'm going to grade their finals I have a lot of power over them but when I'm walking up the street and they're all taller than me and can beat me up then it's less less so not that my students would ever actually beat me up but that's how I feel when I'm walking up the street that more than half of the population is stronger than me and could beat me up so in this diagram here rich white men occupy the top position but if we restrict our attention to non male people in the diagram now it's the rich white non male people who are at the top and in fact we could restrict our whole context to women and we could use rich white and cisgendered as three types of privilege remember that cisgendered means that your gender identity does match the gender identity you were assigned at birth and we now see that rich white sis women occupy the analogous position that rich white men do in broader society and this has helped me understand why white women why there is so much anger against white women in some parts of the feminist movement at the moment because white women may be prone to seeing themselves as underprivileged relative to men because we all feel everyday sexism all the time but on the other hand they don't remember how much privilege they have over non-white women and so people feel angry about that now we all were all less privileged than somebody and more privileged than somebody else and so we should be able to perform these pivots to understand what it's like to be in both of those situations and this goes both ways because not only can we understand what it's like to be in an underprivileged situation we should also understand what it's like to be in a privileged situation not only because of how other people view us but also by what it will feel if somebody calls us out and how it what a useful way would be to call someone out without making them upset and defensive so here is a pivot that I have realized that I am able to do as an Asian person growing up in England as I did I have less privileged than white people because of white privilege but also I know that I am probably more privileged than most non-white people because Asians occupy a slightly different position than than say black people and so I can do this pivot where I can understand racism from both points of view and I think this helps me to see things from other points of view and so it helps me to empathize with people who have different experiences from me so another pivot that I can do is about being rich now when I do the cube of privilege and talk about being rich almost everybody puts themselves in the non bridge place because they will think that their non rich but we're all richer than somebody and we're all less rich than somebody else so yeah I'm not so rich that I don't have to work ever again but also I'm doing fine and there are people who are not doing fine they're struggling they're unemployed maybe they're working at minimum wage and so I can see aspects about richness from both points of view as well and so here is the one about white women where white women often see themselves as underprivileged compared with white men and they forget that they are over privileged compared with non-white women and I think we're often prone to see ourselves in the underprivileged situation and it would be it would be make better conversations if we can always pivot to remember how we are also over privileged in other ways so finally I would like to talk about intelligence and what I think intelligence is because sometimes people have arguments where they go oh you're just not being rational but what does rational mean you're not being logical if you study logic in math in maths or philosophy you just discover that logic starts with some assumptions and then you deduce things from those assumptions and if you just start with the you have to start with some assumptions otherwise you can't get anywhere so sometimes people say if you're using assumptions then that means you're not being logical that's not true but also if you only use those assumptions then that's not exactly illogical you just haven't got very far with your logic so some people say that they don't believe in gay marriage because they think that marriage is between a man and a woman which sounds logical but all they've actually done is said the same thing twice so there may be other ways of defending now that position but that isn't a very good one but it's not illogical so if I say I I like this table because I like this table that is logical it's just not very helpful it's not very productive and so that being logical is not the be-all and end-all of intelligence I think so I thought about it and because I'm doing will these diagrams of interconnectedness I try to distill what I think intelligence is about and I think it's about being reasonable being not just logical but powerfully logical by which I mean that you are able to make very long chains of deductions both forwards and backwards so you don't just say I like this table because I like this table that's not a long chain of deductions but if you can unpack a very very long way why something is happening which is what mathematicians do all the time because we're trying to trace things back to their first principles that's I think powerful logic where you actually get somewhere so the last thing I think is that to be intelligent you should really be helpful because if you just sit around making logical deductions that don't help anybody I don't think that's very intelligent we can argue about that but that's what I think and so then what does reasonable mean and what do all these things mean I thought well read what I think reasonable means is that not only do you use logic but you have some kind of framework for testing yourself and this is a really important aspect of sight the scientific method and mathematical theories that it's not just that we have a theory and we think it's true there is a very clear framework for when it is not true and if somebody wants to say it's not true there is a clear framework for how they do that and there's a clear framework for when we should agree so if they find a mistake in our proof then it doesn't actually mean the end result isn't true unless they can find a cancer example it might mean that our argument isn't very solid with scientific theory sometimes people say Oh scientific theories are just theories it's just some kind of statistical certainty about things but the point is that there is also a framework for telling when you're wrong so there is a framework for deciding when you're fairly sure about something and there is a framework for telling you're wrong and that makes it different from someone who says I just don't think the moon landings happened that's my theory it's as good as a scientific theory because they have no framework for admitting when they could be wrong there is no amount of evidence that would convince them they would think the entire thing was fabricated so if there is no possible framework for someone admitting that wrong then it's not really a framework and I would say they're not being reasonable in the sense that one cannot reason with them so being powerfully logical I think is about not just using logic but having techniques that you've developed for using logic that help you get further such as seeing things in an interconnected way pivoting between situations I also spend a lot of time in the book talking about grey areas and how we should deal with those because they're very difficult being able to see many different causes of something being able to have ways of unpacking things back and back and back and finally how are you helpful well I think that to be helpful you need to have techniques for being helpful but very importantly you need to be able to engage emotions because being helpful I'm talking about helping other people and if we don't engage our emotions then we can't be helpful to other people in as much as we could if we are to engage emotions and this works at and many levels it works at the level of education where if students don't feel things when they're learning something then it is much less likely to go in and this is one of the problems with a lot of science and maths education that it's just a bunch of cold facts and you're just supposed to follow these rules it's not the teachers fault if we think about a huge chain of causation there are all these these regulations imposed on the teachers they have to meet these curriculum standards they have to prepare them for these tests and these other tests and these other tests and these other tests and then it just becomes all about a bunch of tests instead of having feelings about what you're doing and and seeing the amazing stuff and the horrible stuff and the surprise things tough so emotions are really important part of this I think and too often we pit emotions against logic in a futile battle because if you pit emotions against logic I don't know about you but I think emotions will always win because emotions just are true and you feel them so strongly if you're afraid of something there no amount of somebody logically arguing with you is probably going to stop you being afraid of it well not me anyway maybe someone more logical than me or less emotional so I subscribe to Carlos to Pallas a theory of stupidity his theory stupidity is very interesting it comes in many in several steps one of the things he says is that every group of people has the same proportion of stupid people whether that is children adults politician maybe more with politicians Nobel Prize winners professors toilet cleaners criminals but then he says what does stupid mean and he also says oh and the proportion is always the same and it's always more than you're expecting even after you've taken that into account so this is what he this is how he defines stupid it's on a graph and this axis is how much you benefit yourself and this axis is how much you benefit other people so if you if you benefit yourself but hurt other people you're a bandit if you benefit other people but you hurt yourself he says that you're unfortunate so we could think of martyrs as well but some people think martyrs are benefitting them well anyway so this is where you you help loads of other people but you hurt yourself and I spent some time doing this because I believed that that was the noble thing to do and I think that quite a lot of women and men of a certain type also do that because we believe that being self sacrificial was noble when there are needs come last and that we put everybody first but that's not very productive in the long run now if you hurt other people and yourself at the same time you are stupid and so the final quadrant is where you benefit other people and you benefit yourself and that is what he calls intelligent and I agree with that and I like it because it has nothing to do with grades qualifications standards prizes money status it has nothing to do with any of those things it's about what's really important in society in my opinion and so to finish I would like to talk about this that what I've said is that logic can actually help us empathize with other people we take the emotions out but that can actually help us to understand how other people feel more instead of just getting upset with their conclusion that we disagree with and empathy can also understand other people's logic because once we understand how they're thinking we can understand the logic from their point of view rather than just trying to understand it from our point of view where it makes no sense because usually when somebody is yelling at somebody else that they're not being logical they usually are they're just using different logic and if we can find out where the logic is differing then we can have a more sensible conversation rather than just saying oh you're not being logical and so instead of a vicious circle we could set up this virtuous circle where our feelings and our actions work together to help everyone around us as much as we can and so I think that abstract mathematics really can help us with this virtuous circle and that is what I wish for the world my wish for the world is that we would all try harder to understand each other rather than competing all the time and trying to prove other people wrong and I really think that abstract mathematical thinking can help us achieve that and I hope that you'll join me in helping us all achieve that thank you very much you

Channel: The Royal Institution

Views: 148,896

Keywords: Ri, Royal Institution, maths, mathematics, math, eugenia cheng, lecture, fake news, art of logic, logic, crticial thinking, pure mathematics

Id: 8emPcpfqPRU

Length: 49min 59sec (2999 seconds)

Published: Wed Jan 30 2019

Reddit Comments

[u/CapKosmaty]

I really didn't know what I was getting into when I started watching this. I genuinely thought that this video would explain how to apply mathematics better in your everyday life, but Ms. Cheng 'exceeded' my expectations by far. Using factors to display privilege between different ethnicities and social groups? No problem! I don't even.

[u/xternal7]

Skips to 27:00, watches for few minutes: "that's not really that bad"

And then 31:something happens and shit goes from 0 to what the fuck.

[u/munsking]

goddammit... Ri used to be pretty cool...

[u/Neutral_User_Name]

The scary part is that on YouTube, there are 634 likes for that video, and 154 dislikes.

If we take the ratio of likes over dislikes (and again, has anyone ever considered the feelings of a number being on top of another one - come to think of it, we should ban the word "over") we get 4.11, which is the number one has to dial to reach the information line operator.

That means it is an informative video folks.

If you don't believe me, you are an intellectual terrorist.

(for anyone who does not want to watch the video, this sums it up pretty well - and I would like to point out that an addition is not divisive, it makes everything better).

fargin idiots.

[u/fuckeveryone________]

What the fuck is wrong with her?

[u/UUUU__UUUU]

I bought this lady's book on category theory and it's filled with her dating stories and baking cakes. I want a refund. I wouldn't even touch her book if someone paid me to.