Transcribed by Molly Benson
Jini Palmer: Welcome to Town Hall Seattle’s Science Series. In a special doubleheader, bestselling author Dr. Eugenia Cheng and historian, author, and academic Amir Alexander came to our Forum Stage to talk about how math, logic, and geometry make our everyday life more quantifiable, and how these tools can help us comprehend the world around us. Eugenia Cheng took the stage first, addressing the contemporary problems of fake news and the futility of rationality. She presented her book, The Art of Logic in an Illogical World. Afterwards, Amir Alexander recounted the use of geometry throughout history, and how it has shaped our landscape, design, and society. He spoke about his book, Proof: How the World Became Geometrical.
Eugenia Cheng: Thank you very much. First of all, thank you so much to the Town Hall for inviting me back here to speak on my latest book. It’s always very affirming to be invited back somewhere again, and this is the third time I’ve been invited here, so that’s triply affirming. And of course, it’s wonderful to be in Seattle, and the sun does shine every time I speak at the Town Hall. So thank you.
Thank you for joining us this evening for a math evening where I’m going to talk about my latest book, The Art of Logic in an Illogical World. And the point about that illogical world is that it can sometimes seem like we’re drowning in the current world, where the world is awash with divisiveness and conflict and fake news, victimhood, exploitation, privilege, blame, bigotry, shouting, and minuscule attention spans. And it can seem that we’ll never agree with each other ever again, and that we’re doomed to be stuck in echo chambers and just yelling backwards and forwards into nowhere. And is all hope lost? That’s the question. And I say, no, all hope is not lost. It can seem like that, sometimes. And this book arose from my teaching art students at the School of the Art Institute, and in that Fall semester of 2016 some things happened, various things on both sides of the Atlantic. And the morning after the election, I did what many people did. I got depressed, I cried, I drank, and I thought, “But what can I do that’s productive?” Because I truly believe in doing something rather than sitting around complaining. And I also believe in looking at your own combination of abilities and trying to use them in the best way you can to do something to help the world in a way that you see fit. And I thought, “Well, what can I do as a pure mathematician in this political situation?” And I realized what I’d been doing with my students all semester was using the principles of mathematical thinking to find greater clarity in those divisive arguments. And I felt that I could share that more broadly with everyone who wants to find that clarity. Some people, of course, aren’t interested in finding any clarity in those arguments, but I believe that there are people who do want to understand what’s going on. And understanding what’s going on on all sides of the argument is the first step. I’m not saying it will solve all the problems, but certainly if we don’t understand it, then we can’t solve the problems. And that’s why I wrote this book, and it grew out of the discussions I had had with my art students, where I teach abstract mathematics. It’s not remedial mathematics; it’s not the things that they’ve forgotten from high school. It’s how to think, how to use mathematics as a way of thinking. And so I teach art students. And the art students at the School of the Art Institute are very interesting students. They are not mathematicians at all. And it’s my dream job because I really want to share what I feel about mathematics with more people. And there are so many myths about math. that it’s just about numbers and equations. That it’s not for everybody. That some people are math people, and some people aren’t math people. And that if you can’t do your times tables then you can’t possibly be a good mathematician. Things like that are myths that I am trying to dispel. And so teaching art students is a really wonderful place for me to find out more about what put people off math in their past, if they were put off, and what can, as it were, put them back on math. And I really believe, and I’m sure if there are any educators in the room you’ll all agree, that it’s really important — thank you — to tap into what motivates your students in order to motivate them in what you’re teaching, rather than trying to impose your motivation on top of them. And I realized that what really motivated my students is questions of social and political justice. And that’s why this book came out like this. My students in Sheffield, somehow the thing that most motivated them was food. And that’s why my first book was about math and food.
So, pure mathematics, I think, is not just about numbers and equations, and it’s not just about getting the right answer, and it’s not just about solving problems, even. I believe it’s a framework for agreeing on things, and that every academic discipline has a framework for assessing what should count as good information. And in the current world, I think that’s what we really need from education because there’s tons of information everywhere. Information is no longer at a premium. And so actually what’s more important is a way of deciding what counts as good information, and pure mathematics has one particular framework based on logic. So, I believe that there has been an old fashioned, traditional view of what pure mathematics is, but pure mathematics applies to applied mathematics. And that applied mathematics is useful for science, and that science is then useful for engineering and medicine, which is then useful for the numerical, quantitative parts of the human world. And that is true. And I used to believe that was the extent of how my research would ever be useful because my research is so very abstract. But the thing is that this narrow view enables people to declare that they don’t need math. People can be glad that it exists, but they can say, “Well, I’m not ever going to go into any of those fields, so I don’t need to do it myself.” Whereas, I believe that pure mathematics is actually about how to think, and that, therefore, it is about the entire human world; at least the part of the human world that thinks, and sometimes these days it seems like some of the human world doesn’t think very much. So anyway, I am going to talk about, first of all, analogies and what role they play in mathematics, and the interconnectedness of things. Then I will talk about how abstract math enables us to see relationships that we maybe didn’t see before, and that we can use those abstract relationships to pivot between different situations so that we can understand more things than we previously did. And finally, I will talk about how this fits in with what I believe is intelligence.
So first of all, analogies. Another thing that we can say is that pure mathematics is a theory of analogies. And here’s what I mean by that. That, supposing, we have two apples and two bananas, for example. Then we can say, “Oh, well, there’s something that they have in common.” And if we forget the details about them being apples and bananas, we go, “Well they’re both two things.” And this is fundamentally how we come up with the idea of numbers in the first place. Numbers are an abstraction from sets of objects that have something in common. And if you teach children how to count — I love helping children with math — and if you’re trying to teach them how to count, you just have to wait until they make that abstraction leap. And you can’t do it for them; they just have to sort of see what’s going on. And every time we make another abstraction leap in the process of math education, some people don’t quite make it. And there are various reasons for that. And I think one is that it can seem pointless, if it’s not well motivated. And another is that you just have to wait until you can do it. Nobody can do it for you. So, the other thing is that there are different ways to do it. This is not a completely automatic process. So, if, for example, we wanted to — Oh, this is very exciting. That if instead of saying two things, we said that what these had in common was two fruits, that would also be true and that is also an abstraction. However, in that case we would not be able to include, for example, two chairs in that situation. That is not an example of two fruits. So in that case we have to go up one level further to two things, in which case we can encompass those examples. And what I’m going to argue, one of the things I’m going to argue, is that although abstraction seems to take us further away from real life, it actually enables us to bring in far more examples than we could before.
So here’s a more mathematical example where if we look at 1 + 2 and 2 + 3, they are both examples of a + b. And people often say to me, “I was fine with math until the numbers became letters.” I’m going to show you what the point is of numbers becoming letters. So we could also say, we could look at 1 x 2 and 2 x 3, and they’re both examples of a x b. But now a + b and a x b — oh, oh, thank you very much. Thank you. So, a + b and a x b are both examples of a something b, and that is a yet further level of abstraction. And these levels are abstraction, the bottom level of what you might do in elementary school when you’re doing arithmetic. And this level is what might happen when you start to meet algebra. And then this level is kind of what happens if you go to be a math major in university, and you take abstract algebra, maybe group theory. It’s about binary operations. And none of these levels is right or wrong. That’s not the point. Sometimes people think math is all about right or wrong, and it’s not really. It’s about what light you’re shedding on any particular situation. And one of things that my PhD advisor, Martin Highlands, taught me was that the aim isn’t to find the most abstract possible approach. The aim is to find a good level of abstraction for what you are trying to do in this moment. And what happens in normal life typically is that we talk about things being analogous to each other, but we don’t focus on exactly what is making them analogous. And that ambiguity that we leave leaves open the possibility of disagreement, just based on using different levels of abstraction that we’re not making clear. Whereas in math, we’re very explicit about which level we’re using, so that we remove that particular ambiguity.
So here’s an example of how that ambiguity comes in. So, if we talk about straight marriage and same-sex marriage, some people say, “Well, there’s no difference really between those.” And some people go, “Oh no, it’s different, it’s terrible.” And so, what’s really going on is that people are using different levels of abstraction. So if you think that marriage is about an unrelated man and woman, then indeed same-sex marriage is not part of that picture. However, some of us believe that it’s actually really about two unrelated adults, in which case same-sex marriage is part of that. And people disagree because they’re using different levels of abstraction. And the next thing that happens, because we’re not being precise about which level we’re using, the people who disagree about the upper-level can hallucinate that we’ve gone further than we have, and they get upset and say, “Oh well the next thing we know we’ll be allowing any two adults, even if they aren’t unrelated.” I’ve redacted what’s going on just in case there were children in the audience. Because then we could go up further, and we could say, “Two humans,” or we could say, “Two living creatures,” or we could say, “Two creatures.” And the point is that just because some of us have decided that we want to go to here, does not automatically mean we have shot all the way up to the top. But when we’re not being precise about what level of abstraction we’re using, it can open us up to those arguments about saying this is the same and this isn’t the same. So I’m not saying that this approach solves that problem, but I’m saying it gives us an opportunity to have a slightly clearer and more sensible argument about it.
So, the next thing I want to talk about is how things can be seen as being interconnected. Here’s my favorite diagram of interconnectedness, and it is an abstraction of the London Underground system. We have forgotten many details about where things are, and, in fact, it’s not geographically accurate at all, but it’s very useful for seeing how stations are connected to each other by which lines. But because it’s not geographically accurate, you can end up with slightly hapless tourists trying to take the train from, say, Leicester Square to Covent Garden, even though they’re only about two minutes walk away, because you can’t really tell from this picture. So, here is the geographically accurate picture of the London Underground, which is a different abstraction. And it’s not better or worse. It’s probably less useful if you’re actually trying to take the train somewhere. But it’s quite interesting seeing where everything is. And so the point is that these are two different abstractions that illuminate different aspects of the situation. And this is what math does. It temporarily abstracts from a situation to see what we can learn from it.
So here’s an abstraction that I find quite interesting to do with relationship breakdown. And so I think this is what often happens when relationships break down. That maybe one person, I’ll call them Alex, feels disrespected, and when Alex feels disrespected, Alex is unable to show love. And then Alex’s partner, Sam, feels unloved, as a result of which Sam is unable to show respect, so Alex feels disrespected. And we have a vicious circle that can escalate. I can further abstract and label these arrows as kind of action arrows and these arrows as feelings. And this doesn’t solve the problem, but it makes a start because we can think about how we could break at least one of these arrows because you only have to break one to break the circle. And you can say, “Well, is it easier to break an action arrow or feelings arrow?” Maybe we can’t control our feelings, we just feel them. But maybe we can decide not to act on them. So, perhaps, even when Alex feels disrespected, they could concentrate on still showing love regardless of that, and then the situation won’t spiral out of control. So then we’ve reduced it to these two action arrows, and we could argue about who should take responsibility for breaking the arrow. And one possible theory is that whoever is more mature should do that. So we have this vicious circle.
And this vicious circle, at an abstract level, is very similar to even more tragic things. For example, the situation of police violence, which one could try and say happens like this. That police feel threatened by black people, and so they defend themselves against black people; which makes black people feel threatened by the police, and so they defend themselves against the police. And so the police feel threatened by them. Now, I would like to point out, I’m not saying that this is what happens; it is a view of what happens. But it has been shown that even when black people don’t do anything to defend themselves, and they do everything that they’re supposed to do, there is still police violence against them. So, we can again say, “Well, should we break the action arrows or the feelings arrows?” And we might argue that maybe the police are the police. Why do they feel threatened? They’re the police. Could they be trained to feel less threatened? It has been shown that they feel less threatened by white people than by black people, so maybe we can train them. But we could also maybe train them to take action differently and de-escalate things rather than immediately escalating things. And we could say, “Well, who should take responsibility?” Some people yell and say, “Well, black people should just obey the law and then they wouldn’t (inaudible).” But I would argue that it’s clearly the police who have the power in this situation, and so I think that they should be the ones taking responsibility for changing it. Of course, if they don’t, that doesn’t help the situation. But maybe we can find some more clarity about what is going on. And this is something that I think abstraction can help us with.
So, another way I use interconnectedness to help me is when there are very many factors contributing to the same thing in an interconnected way. So, there was the egregious United incident when they needed to kick someone off a flight because it was overbooked. And they asked someone to leave, and he didn’t want to leave. And so they called security and dragged him off, and he got injured on the way out. And there were wonderful arguments on the internet going, “Well, if you do what you’re told, then you won’t get injured. It really is that simple.” And whenever someone on the internet says, “It really is that simple,” it really usually isn’t that simple. It’s like if someone says, “Fact.” If someone says, “Fact,” it usually means they just don’t have an argument to back themselves up. And I actually read an editorial saying, “You know whose fault this is? It’s your fault. All of you because you sometimes miss flights, and that’s why they overbooked flights. So it’s your fault.” And I thought, “Well, let’s think about this, shall we?” The end result was that injury was caused. Injury was caused because the guy refused to leave, and also security used force, and also because the airline called security. Now, why did he refuse to leave? He needed to get to work. He was a doctor, and he needed to get to a shift at the hospital. One might say that’s a reasonable reason to want to get to work. There’s also why the airline chose that person. Questions remained about racial profiling. But also why did the airline decide to kick people off in the first place? Well, because nobody volunteered, really. And why didn’t nobody volunteer? Two things: The airline didn’t offer enough money, and also somehow people really wanted to get where they were going. And why did they even need to remove people? Well, because the flight was too full, and also they needed to get some crew somewhere. And why was that? Did they have an issue with their crews scheduling? And why was the flight too full? Well, two things, again: The flight was overbooked, and also not enough people failed to show up. And so here, finally, is the fact that people often miss flights. So, this interconnected system is really what happened. It’s not the fault of any one of these things.
Now I understand the world is a complicated place, and in order to understand it we need to simplify it. But simply forgetting most of the information, ignoring it, is not a very good way of simplifying it. I think a better way to simplify the world is to become more intelligent because then the world becomes simpler relative to your brain. And one way I think that abstract math can help us is because it gives us ways to understand interconnected systems as a single unit. So that if we can understand this as a single unit, then we don’t have to be afraid of it. Now, it’s still very complicated, but if we can understand the whole thing, as a unit, then math gives us a way to move things around in our brain that have been packaged up in this way. I like to think it’s like those vacuum cleaner bags where you put all your clothes in a bag, and you suck all the air out, and then it makes them easier to move around or put under the bed. Things like that.
So, here’s another one that I drew about why I gained weight. I used to be larger, and I don’t want to be larger again. I lost 50 pounds, and I’d like to leave it that way. And so people often say, “Oh, well, you know, it’s not rocket science. You just have to eat less and exercise more.” Now the thing is even rocket science, that’s just applied math. Gaining weight. Why do I gain weight? Well, yes, it’s because I take in more energy than I burn, which is because I eat too much, and I exercise too little. Yes, but it’s also because of my metabolism, and my metabolism slows down if I eat too little and exercise too much. So already we have that it’s because I eat too much, and I eat too little; and also I exercise too much, and I exercise too little. Also my metabolism is controlled by my genetics and also sleep. And I eat too much because, well, I like food. And also because I emotionally eat, and both of those are caused by my genetics and my upbringing. And there’s also a social pressure to eat too much. And social norms cause social pressure. And emotions and time pressure causes me to get emotional stress and sleep less. Life happens, too. And then there’s the entire food industry that’s spending tons of money trying to get us to eat more. And the supposed diet industry that doesn’t want any of us to succeed because then how will they make money? Plus, when I do gain weight, then I start eating too little and exercising too much and getting stressed. So, it’s this simple. It’s not that simple. It’s this simple. But understanding this helps me see where the vicious cycles are, and helps me understand which links I can try to break so that I stay the way I want to be. Which doesn’t mean that everyone should try and do that, but that’s what I want to do.
So I did draw a diagram for the election. Here it is. I got tired of people saying, “Oh, it’s just the fault of. It’s just the fault of the people who voted third party. It’s just the fault of the Bernie or bust people. Well, it’s just Hillary’s fault for running in the first place.” And so I think it was all of these things, including, for example, the voting system. The third party voting thing wouldn’t have been an issue if third party votes actually counted for anything. There’s some other stuff maybe in this gap over here that has come to light since then.
And so maybe I’ll move swiftly on and talk about how abstraction can help us understand relationships between things. So here’s something that is more obviously a piece of mathematics that might seem a bit irrelevant. Factors of 30. So, maybe we can remember what the factors of 30 are. The numbers that go into 30 are 1, 2, 3, 5, 6, 10, 15, 30. Very good. It’s not that interesting. It’s a bunch of numbers in a straight line, and I always say, “We live in a three-dimensional world.” So we write on two-dimensional pieces of paper and one-dimensional straight lines. And so we stuff our thoughts into a one-dimensional straight line, where maybe they have natural geometry in higher dimensions. And I also like to say, “This is why I do not tidy the papers on my desk because they have natural geometry and high dimensions they don’t want to (inaudible).” So, we can find some of the natural geometry of this situation, by looking at which numbers are also factors of each other and drawing a kind of family tree of those relationships. So, 30 is at the top, like a kind of great-grandparent. And if any of you came to hear me talk about how to bake pie, I did show this, but now I’ll take it a little bit further. So 6, 10, and 15 go into 30; 5 goes into 10 and 15. And I don’t need to draw an arrow directly from 30 to 5 because, like in a family tree, we don’t draw grandparents relationships because we can deduce them from two levels of children. So, 2 goes into 6 and 10, 3 goes into 6 and 15, and 1 goes into 2, 3, and 5. So, now we see that it’s really a cube, which is a little bit more interesting than a bunch of numbers in a straight line. And if we think like a mathematician, then we go, “Why did that happen? What other numbers make cubes? Does every number make a cube? What other shapes can we get?”
So there are various ways that we can take this, but maybe you can see that it’s because these three numbers are prime numbers. Those are the ones that don’t have any other factors except 1 and themselves. And that gives us three dimensions for the cube. So, at this level we have numbers that are a product of two prime factors and then 30, which is a product of those three. So if I draw it like this, I get the actual prime factors at each level, and at the bottom that’s the sign for the empty set where there are no prime factors. And then we can see, maybe, that it didn’t matter that this was 2, 3 and 5; that they could have been any other numbers. Like, we could say, “Well, it could have been some a, b and c.” Okay, I’ve turned the numbers into letters. And the point is now we can immediately try this with something else, say 2, 3 and 7. Now, 2 x 3 x 7 is 42. Here are all the factors of 42. And now if I build a diagram up from the bottom, I have 2, 3 and 7 at the bottom, then the products of two things, and then the product of three things at the top. The middle diagram is analogous to the previous one, but where every 5 has been replaced by a 7. And when we go up to the level of abstraction of a, b, and c, it is not actually the same diagram. And so abstraction shows us something that is the same about different situations deep at their route. And I’m going to show you how that is powerful. But first I want to stress something about this diagram, which is that 6 is less than 7. That might not sound very profound, but 6 is less than 7, and yet — whoops — 6 is higher than 7. Six is higher than 7 in this hierarchy. And whenever you have two different hierarchies that disagree on the same thing, that can be a source of antagonism. For example, if somebody is older at work but more junior than someone else, that can be a cause of antagonism.
So, now I want to show what the point is of going to this level of abstraction because a, b, and c can now be anything. They don’t even have to be numbers. They could, for example, be three types of privilege, such as rich, white, and male. And so then, what we have is here, the people with two of those types of privilege, and then at this level, the people with one type of privilege, and then at the bottom, the people with none of those types of privilege. And if I fill back in the missing words, I’ve got rich, white, non-men; rich, non-white, men; and non-rich, white, men. And then these, and at the bottom, the people with none of those types of privilege, the non-rich, non-white, non-men. And so the first thing I’d like to stress is that whenever I talk about this everyone in the room seems to have a tendency to identify themselves as not rich. While it is true that there are many people richer than all of us, let’s remember that there are also many people who are much less rich than all of us. And so it depends how we want to see ourselves. So, this diagram is a diagram showing direct losses of one type of privilege along those arrows. And I think this is important to remember because sometimes people get very upset about the theory of privilege, and they say things like, “Well, look at that super rich, black sports star. He’s much richer than me. And so that shows that white privilege doesn’t exist.” And the thing is that’s not what white privilege means. White privilege means that if everything about you stayed the same, but you moved along one of these arrows just by, hypothetically, not being white anymore then we would expect you to be worse off in society. It doesn’t mean that all white people are better off than all black people in society.
And there’s something else that I can learn from this because, just like in the previous thing where 6 was less than 7, we can compare the people at this level. Now there are no arrows at this level because there is no direct loss of one type of privilege along here. But we could consider how well we think those people are doing in terms of absolute privilege. And I think that the rich, white non-men, including rich, white women, are probably doing better than the rich, non-white men. Who are, in turn, probably doing better than, say, poor, white men and non-rich, white men because money, you know, money. And the same along this level. But even further than that, if we compare between the levels, I think that rich, non-white non-men are probably doing better than non-rich, white men. For example, rich people like, say, Oprah Winfrey or Michelle Obama are definitely doing better than poor, white men who are maybe unemployed or homeless or really struggling. And so, it’s actually like this, it’s a cuboid of privilege, not a cube. Skewed by what I think is the absolute level of privilege in society at the moment. And this has helped me to understand why, particularly, some white men are so angry about the theory of privilege. Because they are told that they have two types of privilege, but they don’t actually feel the manifestations of that privilege. And they see people who are considered to have less privileged doing better than them in society. And I think it’s much more productive to understand this structural source of their anger rather than simply get angry with them in return. And it almost actually surprised me that some abstract mathematical thinking helped me understand that.
So, I would like to talk about how we can use this kind of abstract thinking to pivot, to help us understand different situations. Because in these situations, 30, 42, and rich, white men, were analogous. They occupied the analogous position in those diagrams, and we can look at more analogous situations. For example, the structural power that male people have over female people is analogous to the structural power that white people have over black people, which is analogous to the structural power that rich people have over poor people. And again, I’m not saying that all male people have power over all female people, but the structures of society are skewed in that direction. Whereas, if we look at male people relative to female people, this is not analogous to female people over male people because the power structure goes the other way up. And this gives us a sense in which, for example, if men are sexist towards women, then that is different from women being sexist towards men. If we think that that counts as sexist, which depends what you take your definition of sexism to be. But we can say, there is a sense in which it is the same because both cases are people being horrible to other people. But there is also a sense in which it is different. And once we acknowledge that, then instead of just shouting about whether it’s the same or not, we can look at why we’re disagreeing, and what the manifestations of that difference are, and whether it is more useful to think about them as the same or different. And this is another thing that my PhD supervisor taught me. That math isn’t about writing wrong. It’s about the sense in which something is right in this situation, and the sense in which something else might be right in another. And when we’re having arguments with people it can be very productive to think about the sense in which they have a point even if we disagree with it.
Now, something might be not at the top in context, but in the top and at the top in another context. For example, in this diagram the rich, white men are at the top, but if I focus my attention on this portion of the diagram, now the rich, white non-men are at the top there. And so we could restrict our whole context, say, to just thinking about women. And then we could have an analogous diagram where we take three other types of privilege among women such as rich, white, and cis-gendered. And then we see we have an entirely analogous cube involving the losses of those types of privilege with poor, non-white, trans women at the bottom here. And this has helped me to understand why there’s so much anger at the moment, especially towards rich, white women in some parts of the feminist movement. Because if they are prone to considering themselves as underprivileged relative to men, and that it will especially happen if they spend most of their time surrounded by white people, then they will not understand how privileged they are relative to all the other people. And I heard some murmurings about cis, and I’ll just remind you that cisgendered means that your gender identity matches the one that you were assigned by doctors at birth. So, this has helped me understand that anger rather than simply getting angry in return. And we can all perform pivots in this way because we’re all more privileged than somebody and less privileged than somebody else. And so we can understand what it’s like to be in different parts of this diagram, and that can help us understand other people’s experiences who are in different parts of the diagram relative to us.
So here’s some pivots that I do myself. So, I think that as an Asian person, I have some lack of privilege compared with white people. But I also acknowledge that I think that Asian people are probably among the most privileged of non-white people. And so I can pivot between these two situations. One where I’m lower down, and one were I’m higher up. So that I can understand the experiences of different people, and also think about how I like to be treated when I’m feeling lower down so that I can then try and treat people well in situations when I’m higher up. And another one is about riches, right? So I’m not so rich that I never need to work again. Not that I would ever be because it’s not about riches why I work. I work because I want to help make the world a better place. But I am doing fine. I’m working hard and doing fine. Some people are really struggling. They may be working hard and still unable to make ends meet. Maybe they have many jobs that don’t pay them enough or all sorts of things: health problems, homelessness. And so we’re all less rich than someone and more rich than someone else, so we could perform that pivot. And here is the one about white women, where white women are less privileged relative to white men, but more privileged relative to non-white women. So, everyone can do those pivots. And I use these pivots to help me empathize with other people, which brings me to the, perhaps, surprising conclusion that abstract mathematics helps me with empathy. And often abstract mathematics might not be something that you put in the same sentence with empathy, at least not in a positive way. But I think this is an important part of mathematical thinking, actually.
And I would like to conclude by talking about what I think intelligence is, and how I think this can help us be intelligent. So, I thought about this and I thought about another diagram of interconnectedness. So I think that intelligence involves being reasonable, being powerfully logical, but also being helpful. And so what does reasonable mean? Well, reasonable means that you are able to be reasoned with. There are some people who hold views where nothing, no evidence, logic, reasoning, nothing at all would ever get them to change their mind. And that is, in fact, unreasonable. And so I think that reasonable means that you have a framework for deciding why you believe the things that you believe, and especially a framework for deciding when it’s time to stop believing them, so that you will change your mind. And that, I think, the reason part of it involves using logic. Now being powerfully logical means that you don’t just use logic, but you use logic with some kind of techniques to build your logic up. Because if you say, for example, that some people say, “I don’t believe in same-sex marriage because I think marriage should be between a man and a woman.” Now that is not illogical. It’s just that you haven’t actually got anywhere. You’ve said the same thing twice, basically. And so that’s not illogical, but you haven’t used any steps of logic to develop your argument. And that’s what I think being powerfully logical is about. And finally, I think that being helpful is really important because I don’t think there’s any point in sitting around using your brain a lot if it’s not going to help anybody. And this is my opinion, but I think that being helpful involves not just using techniques but also actually engaging emotions and understanding the emotions of other people. Because if we keep yelling logic at people who are feeling emotions, then it won’t help. And that we need to engage and empathize with people to understand why people disagree with us and to access some form of discussion that involves making human connections. And we know this even when we’re teaching mathematics, that if we don’t understand why a student thinks what they think, then we’ll never persuade them of anything. And if they don’t feel emotions while they’re learning, then everything will just kind of wash over, and they’ll move on and never want to do it again.
So, I believe in Carlos Cipolla’s theory of stupidity, which says this — it’s a two dimensional theory. It’s a graph like this. And this axis is how much you benefit yourself, and this axis is how much you benefit other people. So, there are various different quadrants here. And so if you are at the top left, you hurt yourself. Now, let’s see, I’m not quite sure which one I’ve done first. Maybe it’s the bottom right. Right. So, if you hurt other people while benefiting yourself, then he says you are a “Bandit”. Whereas, in the top left one you are benefiting other people while hurting yourself. And he calls that “Unfortunate”. We might think of it as being a martyr. And I used to believe that that was a good thing to be. And I think that many women have been taught by society that we should sacrifice ourselves for the good of other people. And that’s one of the reasons I kept working in a job that was making me miserable because I thought that I was doing something good for society. But then, you see, what about the bottom left hand corner? That’s where you hurt other people, and you hurt yourself at the same time. And that is stupid. And Cipolla’s theory goes on to say, he reckons that there is the same proportion of stupid people in any group of people, whether it is professors, students, children, convicted criminals, politi— maybe there’s more (inaudible) maybe stupid (inaudible). And he says it’s always more people than you’re expecting, even when you take that into account. And this is what he says is stupid, people who hurt themselves and other people at the same time. So this top right hand quadrant is where you benefit yourself and you benefit other people at the same time. And he says this is “Intelligent”. And I agree. I think that’s what intelligence is. I don’t think it has anything to do with the grades you get, or the number of degrees you have, or how much money you earn, or how many houses you own, or how many people you have power over in your company. I think it’s about how and to what extent you are able to benefit others and yourself at the same time.
And I think that abstract mathematics can help us with this. And I think that we can create a virtuous circle where logic can help us by feelings. It can help us to understand the feelings of other people by doing those pivots. And that empathy can also help us understand other people’s logic because we need to empathize with them in order to understand their thought processes. And so, I conclude that abstract mathematics can, I think, help us create this virtuous circle, and help us go out into the world and be intelligent. And I hope that we will all want to do that. Thank you very much.
Amir Alexander: Thank you. And thank you also, thank you, Eugenia, for a fascinating presentation. And one of the advantages, you worked out all the technical bugs now, so I’m told this works now. There it is. Yeah. Okay. Beautiful. Also, and as Eugenia argued, I think, and I was entirely convinced mathematics and mathematical thinking really is very useful and very helpful in our world, even in our chaotic world.
And so I will start with a person who completely disagreed with Eugenia, who was this man here. He’s a mathematician. Carl Gustav Jacobi was a very prominent mathematician in the 19th century. And in 1842, Carl Gustav Jacobi was invited to speak at the British Society for the Advancement of Science that was meeting in the industrial city of Manchester. And, as he proudly wrote his brother when he came back, he stood there before all those British men of science, and he told them, “It is the glory of science to be of no use.” And in particular mathematics. Mathematics, he said, mathematics, he declared, that the only purpose of mathematics is the honor of the human spirit. That it really has nothing actually useful in it. And he said, “That is a great thing.” Completely, completely, completely useless. Now, the fact is that Jacobi did not make a lot of converts in Manchester. This was what a city like Manchester looked like in 1842, and the people he was talking to were people who were making money from all of those new technological and scientific innovations, and they understood were all based on math. And here comes this German with his funny accents and telling them that mathematics is completely useless. He did not make many converts among his audience there.
However, the view that mathematics is, in fact, useless. And that as the great British mathematician G. H. Hardy said, “If it has to be justified as art, as beautiful art, if it is to be justified at all.” It is actually a view that was quite prevalent among mathematicians at that time, and, in fact, ever since. Hardy, for example, he thought that, yes, of course mathematics can be used for various things like to describe the laws of motion, MC squared, of course, to make airplanes fly, cell phones work, build skyscrapers, and send rovers to Mars. That is all well and good. And to this we can also add what Eugenia said, it also teaches us how to think properly and think correctly and think in productive ways. But all of that, Hardy said, “That was not interesting. That’s not real math. Real math is really useless; is really pointless.” And there is some truth. I mean, you can at least see what they see, what somebody like Hardy means. I mean, after all, whoever actually used Archimedes’s method for calculating the area of a parabola? How useful is that? How useful, really, are Cantor’s transfinites? This theory that you can count that different infinites have different values, and that we can rank them. It’s a beautiful, amazing, amazing theory. But how useful really is it, really? How useful? What use did anybody actually ever make of it? Or more recently the Langlands program. I’ll say, one day maybe it will be useful, maybe, maybe not. In the case of Archimedes, we’ve been waiting more than 2,000 years, still, still waiting. So, it’s not so clear that perhaps mathematical thinking has its value and teaches us to think correctly, but it’s far from clear that the actual mathematics is in fact useful in itself.
What I’d like to offer here is a different kind of perspective here. That mathematics is fundamentally this: Mathematics is the science of order. That is, if there is something that is the deepest order, the deepest order in the universe, something that deep down is absolutely true, cannot be wrong, and orders everything in the universe, something that is unshakeable, and somehow true and nessarily true, then that is, if we can prove that, then that is what mathematics is. And that has enormous implications because that means that what we say about mathematics, the kind of order that mathematics is, that our world is different, depending on what kind of mathematics we ascribe to, depending on what we think proper and true mathematics is. The natural world is different, but also the human world is different. If we think that mathematics is one way or another way then our whole world, whether natural, but also social, political, religious, philosophical, everything changes. If we think that the deepest order of things, the one that includes everything that goes down to the very roots of creation, is different. And that is mathematics. So today I’d like to talk about the kind of mathematics that has a particularly long and illustrious history. And because it told us about the particular kind of order in the world, it shaped not just our understanding of the natural world, which it certainly did, but also changed our understanding of our relationships to each other, our institutions, our political institutions, and our social relationships. That is the great and ancient science of geometry.
So, geometry, I say, matters. It matters a great deal, and to show how it matters, let me tell you a little story about some famous personages from the past. Perhaps some of you have watched the Netflix series on Versailles. No. Maybe. Okay, I just want to say, it’s a nice series. But, anyway, Versailles. Versailles, it has Louis XIV there. Let me tell you, if you ever do watch the show, that Louis XIV looked nothing like this Louis XIV. No, there’s no relationship, no connection there. That Louis XIV of the show is a very, kind of, very modern, democratic kind of guy. This guy was not at all.
The year was 1661, and Louis XIV, he was a king already for 18 years at 23 years old. He was a king since he was a child, five years old. But only a few months he was a real King because up until that point he was under the tutelage of the regent Cardinal Mazarin. Mazarin died, and immediately Louis declared he will rule by himself. He will not have a minister anymore, a chief minister; he will rule personally. He will rule by himself and usher in the great and glorious age for France. And on August 17th, 1661, Louis came to visit the estate of his own Superintendent of Finances by the name of Nicolas Fouquet. This is this gentleman over here. And Nicolas Fouquet had just finished building a beautiful, beautiful estate called Vaux-le-Vicomte in France. You can see it here with beautiful chateaux and beautiful gardens. And the King came towards evening. He descended at the entrance to the Chateaux over here. And he was led by his host through the rooms of the Chateaux decorated by the famous people, by the greatest artists of the day. And after that, he proceeded down to the gardens — this is seen from the Chateaux. They walked down this central alley past those beautiful symmetrical, geometrical shaped parterres to the circular pond here, and then down past those pools of Triton to this mirror pond here, and then to this grotto here where they were presented with a new comedy by Molière. It was a grand entertainment. After that, all of them were served with a lavish dinner, including the 5,000 soldiers of the Royal household who accompanied the King. And just when night fell, and they thought that everything was over, they had fireworks shoot up from the Chateau, from the roof of the Chateau, and then descend on the garden like midnight suns. So, it was a grand entertainment. And the host, Fouquet, was sort of glorying in this moment of Royal approval. However, the next day he saw the King away. The King was going back to his Chateau nearby in Fontainebleau. And he turns to his mother, Anne of Austria, and he tells her once Fouquet is (inaudible), “Madame, ah Madame, shouldn’t we disgorge these people of all of that?” That’s too much, basically. And sure enough, he was a man of his words. A few months later, he summoned Fouquet to an audience, and in the other room he asked the captain of Musketeers, a man by the name of the count d’Artagnan. And at a sign from the King, d’Artagnan sprung up, grabbed Fouquet, who was his former friend, put him under arrest. and Fouquet spent the rest of his life in a prison cell in Pignerol, in the Alps, and never saw his beautiful estate again. Died in 1680 a forgotten man.
So the question is, why did the King react so strongly? What raised the wrath of King Louis XIV? A man who had been loyal throughout his life, stood by him in uprisings, stood by him during the Fronde, never expressed anything but love and admiration and loyalty to the King. What was it that raised the King’s wrath? And a clue is what happened to this beautiful— Okay. All right. So the King, he says, “Okay, now we got Fouquet out of the way.” He summons Fouquet’s gardener, André Le Nôtre, and he tells him, “What you did there, now you do for me. Now you do for me. But you do it on a scale that is 10 times or more greater than anything you saw in Vaux-le-Vicomte. You will create a garden, a garden like that, that will make everyone forget that garden.” That garden that he saw at Vaux-le-Vicomte. And sure enough, anybody who’s been to the gardens of Versailles will admit that certainly the scale of that place is far above and beyond any garden that had ever been before, and I suspect since. It’s not exactly a pleasure garden, it’s called a pleasure garden, not exactly pleasurable. It’s huge, but it is certainly something that will make anyone forget, and completely put any competitor in the shadow. He went to Vaux. He pulled out the trees. He took the fountains. He took the statutes. Brought them all to Versailles, brought the people to Versailles, put them to work and told them, “You do that now for me.” Because in the end it wasn’t really Fouquet’s wealth. He was a very wealthy man. He had a fleet of ships on his own. It wasn’t that. It wasn’t his patronage of the art that doomed Fouquet. It was, ultimately, his geometrical garden. That’s what it was because Fouquet, in Vaux, built the greatest and most beautiful geometrical garden in France, and that was something that Louis would not tolerate. He would build his own geometrical garden, and that will be the one that will set the standard and forever erase the memory of the upstart king. Why is it the geometrical garden that doomed Fouquet? Why was that so outrageous to the King of France? And, to understand that, actually, we have to go backwards quite a bit of time. In fact, 2,000 years of time to understand why geometry was so important, and why geometry was even so dangerous at the time of Louis XIV.
We don’t know who created the first geometrical proofs. There is some unknown genius. We know he was Greek. We know he — presumably he, or they, who knows — lived on the shores of the Mediterranean, in one of the Greek cities that dotted the shores of the Mediterranean. And it was probably something simple about lines and angles, something that seems very trivial to that. But very soon others joined in and started producing proofs that were, we know by the year 400 BC, were quite sophisticated. And proofs are interesting. Why did the Greeks invent geometrical proofs? It’s not because the Greeks had the only mathematical tradition. We know they are amazing mathematical traditions, you know, from the Babylonians to the Egyptians, the Mayans, in India, in China. Remarkable mathematical traditions. Remarkable mathematical traditions. Not one of them thought of inventing proofs. Because proofs are not about measurement. Proofs are not about doing astronomical measurements or land measurements or a counting house. It’s not about that. Because proof is about finding truth. Once you’ve proven something, when that something is proven, it is absolutely, irrevocably true. Not because God said so or tradition or anything like that, but simply reason tells you that it is absolutely and irrevocably true. And, basically, no one can argue. That’s the end of the argument. It is proven. It is absolutely and necessarily true. And that’s kind of a stunning thing. You know, there’s something quite very radical about that. And that was a discovery, that you can actually prove something, that was a discovery that was made only once in human history. And never again.
And the first proofs were, in fact, kind of haphazard about different things. The person who then united them and systematized them was Euclid of Alexandria. Of course, we know very little about him. We know he lived around the year 300 in the Library of Alexandria. And how did he do it? What did Euclid do? What did Euclid do? Euclid, he starts out by a set of definitions and a set of postulates and common notions that are very simple and self-evident things. The whole is greater than its parts. If something is equal to another thing and that is equal to another thing, the two things are equal to each other, and so on. Things that nobody can deny. They’re obviously and self-evidently true. That’s where he starts. And from there he starts creating, based on those assumptions, he starts creating proofs, starts building proofs, and based on those proofs he creates more proofs, and so on. And every proof is connected. It’s not just that it is true in itself. It is also interconnected to all those other proofs. Proofs about lines and triangles and circles and angles and so on. It’s a whole world of those kinds of geometric objects. And it is a perfect world. It is a perfect world like no other. Because not only is everything true, everything there is true, everything there is always true, eternally true, but it is all interconnected. They’re all in very specific fixed relationships to each other, and they’re all in a particular hierarchy to each other, right? The postulates are the simplest at the top and then everything. And then there’s one layer of proofs and then based on them, another one and another one and another one. And they’re all interconnected in one fixed, eternal, unchanging network, a whole world of mathematical truth. That is the accomplishment of Euclid. That is what his world is like. A perfect world of truth and rigor that is eternally and unchangingly true. So that is quite an accomplishment in itself. In fact, it’s often said it’s the most influential book that was ever written. Perhaps. There are other competitors, I guess, the Bible, for example. But it has a claim, let’s put it that way, and that’s not bad. It’s remarkable. I mean, Euclid is not known for any particular innovation. He’s known for putting it all together and creating a world.
And the problem was, this is a beautiful world, an eternal world, but the little flaw is, that is amazing, but it’s not our world. Right? Plato thought that it’s the way to lead to the perfect world of the forms. Our world is a shadow world, the world of shadows and imperfections. Aristotle also thought that mathematics can’t really describe our world. And not only that, later on, when the Christian Church also agreed, “Geometry is all well and good, very nice. But our world is a fallen world, is a corrupt world, certainly not something that can be described by geometry.” So geometry gets a lot of praise. It is amazing. It is true. It is, some say, the only signs it pleased God to bestow on mankind. But it’s also kind of irrelevant because our world is nothing, in fact, like that.
And that lasts for about 1,700 years; 1,700 years from this breakthrough by Euclid until, pretty much, the year 1413. Not a famous year for most people, but it is, as it happens, the year in which a man in Florence, by the name of Filippo Brunelleschi, conducted experiments on perspective, on linear perspectives. And then he and his friends, famous artists, Masaccio and, later on, his friend Leon Battista Alberti, they developed, established, and popularizee the theory of linear perspective. I mean, initially, it is a theory of how to draw things, how to paint three dimensions on a flat surface. So, basically, there is a vanishing point. This is from Alberti’s book Della Pittura. You see there’s a system. There is a vanishing point. All parallel lines point to one particular point that is on the horizon. This is Masaccio in what is essentially a perspective exercise because you can see, I guess— Okay, so this is basically, this whole image, is basically an exercise in perspective in which you see all those parallel lines from the bottom. They all point to a single position.
But it is much more. I mean, it is not just a trick of painting because the implication is that space itself — Can I have next? That space itself is, in fact, geometrically structured. That those lines of perspective that go to the horizon, they are, in fact, they are real. They structure space itself. So you see the difference, for example, in the art between — this is a perspectival painting by Masaccio in which he just does a few touches, not like the first one, a few touches to create those parallel lines that already give depth to the picture versus just a few years earlier you have this other image which doesn’t have that. It doesn’t have internal space. It has all the images. I mean, it’s a very powerful image in itself, the Monaco picture, but it simply does not have that inner space. It is not that one is more realistic than the other. This is a realistic picture that is, in fact, very close to how we experience our life, surrounded by a lot of people. We don’t think of it. And you don’t think of it as a geometrical space. And yet, but this one already has this geometrical space built in — next one, please. And so, this is the difference.
So these are the two maps of Florence, one from 1352. This one from the 1480s. You can see, I guess, from the 1480s. And they both depict the same city, and the city had not changed much except for the great dome of the cathedral, which Brunelleschi is also famous for building. The city does not change much, but the world, the world itself had changed. This city, the medieval city, is in fact very much how you experience a medieval city, surrounded by buildings and towers and churches behind every corner. Perhaps even more than this; perhaps even more than this one. But the difference here is in using perspective, the space itself has become imbued with your geometric principle. Every point in the second image is predetermined by geometrical principles. And that is how you can tell this was a turning point. This was the time when geometry came down from the sky in a little city of 30,000 people. And just a few people that we actually can name, ahey made the connection and said, “No, the world itself can be structured by geometrical principles.”
Now, it’s one thing to say that the world is structured by geometrical principles, the natural world, whether in perspective or as Galileo said, “The world is written in the language of mathematics.” And science itself, of course, is always looking for the deep mathematical principles in the world. But what about the human world? What does it mean to say that our world is mathematically structured? And it was not long before some princes in Europe realized that this was the significance of this idea that the world is geometrical. Because if they present themselves, or if they believe themselves to be not just, “I am King of France because I will put you in jail if you say otherwise or I will cut off your head.” But because being King of France means that you are an expression of the deepest order in the universe. That the hierarchy of your kingdom is not just because you have military force, but because you are an extension of a deep order in the world that has enormous, enormous power and enormous implication.
And so, it was indeed the kings of France that were the first to adopt, first, not the last, but the first, to adopt this idea. That geometry is power. Geometry is power, is legitimacy on a scale and with implications far beyond anything that was offered previously. And they did so in many ways, presenting themselves as the apex of a geometrical order. Just to mention a few: They did so in their courts, the whole structure, the whole order of the court. It was not just the old medieval jumble of different people vying for power. There was order now. There was strict, hierarchical order in which everyone had their place from top to bottom, from the King at the top to the princes of the blood, to the mere dukes, to the counts. Everybody had their place, and everybody constantly negotiated their place very, very precisely. That was the essence of life in court, finding your way in a predetermined, orderly society that was ordered geometrically. In the arts, the French courts invented the geometrical dance that we know as ballet, but was in fact invented precisely as a stylization of those kinds of negotiations. Of court etiquette that determined who was on top, who was on bottom, who sits, who stands, how do you greet, and so on. And it was all based on geometrical gestures and geometrical movement. There was philosophy justifying the Royal that was structured as philosophical treatises, but nothing, nothing equaled the importance of geometrical gardens.
Geometrical gardens were the emblem of French royalty. It started very simply. King Charles VIII brought a couple of gardeners from his failed campaign in Italy, and they produced a very, very simple garden in his favorite Chateaux of d’Amboise. And over the next two centuries, this bond between the kings of France and geometry simply increased and grew. This is, for example, the Tuileries. They were in the 14th century already on a larger scale. In fact, you see this geometrical order presenting a perfectly ordered geometrical land under the gaze of the palace at the top. And there were others, there was the Luxembourg, there was the Saint Germain, and so on. It became the emblem of royalty, of French royalty, was geometry and nothing more so than the geometrical garden. It was the emblem of their sovereignty, of their right. Why? Because that perfect world that came from Euclid, that perfect ideal world that is orderly and hierarchical; that is what they were. That is what they were determined to create. That is how they presented themselves. That is how they saw themselves.
And so when Louis came over to visit his Minister of Finance, he saw this. He saw this. Now this is a geometrical garden, right? This is a geometrical garden in all respect because you see there is this perfect, perfect symmetries, the geometrical pattern, the circles, the straight lines, the squares, but even more so. In some ways it is traditional. In some ways it is completely new because it is, in fact, structured as a perspectival painting, right? Leading up to the point on the horizon marked by a statue of Hercules. So it’s not just the geometries of the patterns that holds it together. It is the geometry of the world. It is the deep geometry that structures everything in a perspectival painting and holds it all together, makes it one unified, interdependent unit. It was the best geometrical garden that has ever been, both artificial and natural, right? So you can see the similarities here in the structure of Vaux-le-Vicomte as a perspectival garden. This was a Royal garden. This was a Royal garden in all respect because it was a geometrical garden, and the greatest geometrical garden that ever was. Except for one thing. Except that it didn’t belong to the King. Except that it belonged to a commoner who was presenting himself as the apex, as the top of this necessary, immovable hierarchy. So this was an offence. This was not just untactful. This was not just somebody, you know, a little bit ambitious, an overly grand underling. This was an attack on the foundation. This was a geometrical attack on the foundations of the regime as he saw it.
And so at Versailles, Louis determined not just to crush it, but to create something that would be his own. To create his own geometry that would present the proper order of the world and him at its apex. Which is what he did, precisely, at Versailles. Versailles is very much a garden in the style of Vaux-le-Vicomte except about a hundred times larger, if you take all the areas surrounding. Its immediate area here is very much in the tradition of the Tuileries and so on, and the kind of geometrical patterns that you were familiar from there, presenting an orderly, fixed, hierarchical world. And from there, again, you have this main access leading to the horizon, creating it as a perspectival painting. But the real power of Versailles is not just — this is called the Petit Park here. The palace is here. This is the Petite Park, which was, actually, an older garden that preexisted, that came from the age of Louis XIV. The real magic of Versailles is what happens here, beyond. This is the Grand Park over here in which you have the Grand Canal and the surrounding. Because what you see there, if you look from the palace, what you see there is simply open forest, open woods. You do not see any of those very elaborately carved, beautiful parterres that you see here. But because it is, once again, this perspectival painting, you see going out to the horizon, then all of this, even these open woods are structured together through the deep geometrical order of the world. And underneath it all, beneath these wood, this looks like open woods from the palace, but underneath it all, there are these geometrical paths. All of these straight arrows intersecting, intersecting at right angles, and all of them, together, forming an arrow — an arrow here, arrow here, arrow here — that is aimed directly at the palace, at the center of the palace. And in the palace, ultimately at the center, is the King’s bedroom.
And what this tells you is that beneath all this variety, all the chaos that we see in the world, all this mystery that we see in the world, there is a fixed underlying geometrical order. There is. You don’t see it when you look at it from afar, but underneath it all it structures everything. The deep geometrical order of the world. And this deep geometrical order is not random. It’s hierarchical because it leads, layer upon layer upon layer, it leads all the way up to the King’s palace. In which it supports the King’s palace as the natural, necessary place of all authority. So if you think of yourself as someone, you know, today, Versailles is a museum, we go there, we see, “Oh, this is all very lovely. Yes.” But at the time, for somebody walking those paths in Versailles, these claims, royal supremacy as the natural geometrical order of the world was not just an abstract claim, it was self-evident truth. It was all around you. The entire world around you proclaimed this deep geometrical order of the world. Everything had its place. Everything had its place in the grand order. And who presides over it? Of course, the King himself in his palace.
Now, I’d like to end with another, a little more contemporary, gesture here because you look at here and this is this, perhaps, I hope, it was interesting. That’s for you to judge. But it was a long time ago, right? Louis XIV died in 1715. You know, his great-great-grandson, Louis XVI was beheaded in 1793, you know? So this order, this supposedly eternal geometrical order, we go to it and say, “Okay, that’s how things were.” This geometry and the power of geometry, does it still shape our outlook today? And my answer is yes, but I’ll give just one example here. I think people are familiar with this view of Capitol Hill and the Mall; there’s Pennsylvania Avenue. Washington, D.C. Washington, D.C. is not a museum of a dead, ancient monarchy. It is the capital of the greatest republic in the world. It is the greatest, I think, and the most spectacular geometrical city in the world. Because the story Versailles is Versailles was then imitated, first in gardens and also in cities. But no city matches the grandeur of Washington, D.C. Washington, D.C. was designed by a man called Pierre L’Enfant who was, himself, not only a Frenchman, but somebody who grew up in the court of Louis XV and Louis XVI and knew Versailles and Tuileries very intimately. And that’s what he used when he designed Washington, D.C.
And when you look at this picture, you say, “Okay, well there’s Versailles,” right? You stand in the Mall, you look up at Capitol Hill, like the palace on the Hill, all of the arrows, all streets lead to the Capitol Hill. Here we have this obvious hierarchy with houses of Congress at the top. But of course the US is not a monarchy. And what L’Enfant was trying to do was use the language to present a republic. So, you have this Versailles garden here. You also have, at a right angle to the Mall, you have what he called the “President’s Palace”. He was very disappointed by the scale of the White House which he called a “pathetic country house”. But that was supposed to be the other node of government. So they’re already two, and they intersect at right angles, each one with their own Mall leading up to them. And then they are connected by Pennsylvania — next please — and then they are connected. So you have, this is Capitol Hill. What we know is the Mall. This is the president’s house, the White House over here, and the South Lawn and the garden there. They intersected right angles. They are connected by Pennsylvania Avenue over there. Each one of them is in itself a great node of all lines converging on it. So already you have two lines. You have not one center, as you had at Versailles. You have two centers, two great centers that are, themselves, competing with each other, but also in this careful dynamic balance. You have Capitol Hill, and you have the White House. Not by accident but by design because L’Enfant designed it that way.
So you have those two great federal powers, but that’s not the end of it because on top of that L’Enfant also created all these 15 squares. Why 15? Because, at the time, there were 15 States to the Union, and he called each one of those by the name of a different state. And each one of these dominates its immediate area. So it’s a local power. And they’re all connected by this network, a rigid network that is unchangeable that overlays those two. That overlays the entire city and balances those two centers of power. So what we get here, this is 1791, two years after the ratification of the Constitution, and L’Enfant uses the language of Versailles, the language of geometry to create a capital that is, in fact, designed to be the Constitution in stone. So presenting the constitution not as just a compromise that was reached and so on, but as a necessary, inevitable, based on the deepest order of geometry, unshakable, eternal, stable, because it cannot be changed and it cannot be moved. And that’s how it was designed.
Now I know we go, and Eugenia talked quite a bit about those kinds of problems. I mean, we talk about our political situation today. I won’t presume about your political stances, but I think most of us will agree that this is a time when many of our assumptions are challenged. And many of our institutions that we thought were secure are being challenged and, in fact, seem much more vulnerable than we had expected. Whether any of those institutions, federal, state, Congress, White House are all seemed in crisis and challenged. But I have to say, this is just my opinion, you know, when I go to Washington DC, and you walk those geometrical streets, and you go to the mall, and you look up at the houses of Congress. You go look up at the White House, the grand boulevards, that order, the geometrical order that L’Enfant put there. And I kind of think, you know, I think the message is still there. The living message, not the dead message that is now Versailles, but the living message. That there is more there, you know, that this order, this deep order will survive. You know, there is something more here than the particular politics of a particular time or the particular resident of the White House. And the message of geometry is still alive. So, thank you very much.
Question 1: Well thank you both. Wonderful mathematics presentations. The question I have is, actually, there are pieces from both of these that come together. One of the things is that when the psychologists, sociologists, and so forth tell us that as human beings our rational part of the brain, our logical part of the brain is really not driving us nearly as much as the other parts of our brain, the things we respond to emotionally. When Louis XIV shows up and decides he’s, I interpreted as, so furious that his accountant person, his finance minister, or whatever it was, could not have a better garden. That didn’t sound particularly logical to Louis in one sense. But, as you kept going, maybe you’d have different views about that. But what do you do with the kind of work that’s coming out of psychology saying, logic and reason are, really, sometimes nice, but they’re not really that and important. And driving people by creating emotions of fear and hate are much more effective than reason in terms of gaining power, controlling, and so on.
Edward Wolcher: I think that’s a great question for both of you, actually. If you’d both like to answer it. Yeah.
EC: Thank you. Well, I agree. And I think that one of the things I say in my book is that if we keep pitting logic and emotions against each other in a battle, then emotions will keep winning. And that, I think, is the situation that we’re seeing across the world in political situations at the moment. But I don’t think that the two are mutually exclusive. And I think that if one side keeps telling the other people that they’re just being emotional, not logical. Actually the other side is saying that about this side, as well. And so we won’t get anywhere. And I think we can use logic to understand the emotions because I think emotions do have reasons inside them. It’s just that we have to understand them from the point of view of that person’s emotions, not from the point of view of our own logical system. And one of the ways in which mathematics constructs it’s logical systems is it always starts with some axioms, which are the basic assumptions that we don’t try to prove in the system. And then we build our proof from those axioms using logic. And if you start with different fundamental beliefs, you will get to different conclusions even though you’re still using logic. So, it’s not that some people aren’t being logical and some people are; it’s that they might have different starting points. And so I find that I can use abstraction to understand people’s emotions. I’ve done this on myself. So, for example, there are things I do that seem to be irrational. For example, I used to be extremely afraid of flying even though statistically it’s much safer than driving. Whatever. I was still completely afraid of flying. And instead of saying, “Well, that’s just emotional,” I thought about what it was that was going on, and it’s that my fear isn’t based on statistics. My fear is based on being reminded of death, and that’s the thing that I’m afraid of. And so it’s the fact that it was linking with that. And so using an actual process where you unpack somebody’s process and find what their axioms are. I, basically, always find that I can see some logic in emotions, and so that they’re not separate, and that we can actually use both at the same time.
AA: I very much agree with that. And I think, yeah, I think that you cannot really dissociate emotions and logic and reason. And just, you know, to give some of the examples that I talked about today. Louis XIV, in building this garden, he was a great geometer, he was an ideologist, but he was also a great psychologist. Because when people go there and experience that grandeur of the King, and that absolute necessary order that he is king, people react accordingly. People won’t necessarily enjoy it. The doge of Venice, sorry, the doge of Genoa, for example, visited the gardens in 1685. He did not enjoy his visit because the French fleet had just pulverized his city. But he got the message, and that is a very clear emotional, psychological message. You go there, you understand, you accept emotionally, both logically and emotionally, what the proper order is. So I think they are very much working, together, and the same, I think, in Washington, D.C. I have to say, like I said, when I go there the reaction is “Wow, grand city.” But the reaction is also very deep; It’s emotional. It was planned that way, and geometry is what creates it.
EC: And in fact, to take it one step further, I think it’s very difficult to understand someone else’s emotions using emotions. Unless they’re actually the same emotions as yours. Because we can get so caught up in our own emotional reaction to whatever it is that they’re saying. There are people who I disagree with vehemently, and if I let the emotional response take over, then it’s very difficult to see what’s going on. But if I abstract from it and use abstract mathematical thinking, kind of logical steps instead, then I can separate out my own emotional disagreement with them and understand it from their point of view. And understanding someone else’s point of view is, I think, really the starting point to a more unifying and less divisive world for us.
Question 2: Yes. Thank you for your talks. They were great. Eugenia, I have a question for you. I really enjoyed your application of logic to social issues. And if we were going to adopt that framework, how would you envisage that happening in the real world? Let’s say we decided to use a logical framework on social media or in our newspapers. Would that mean there’s no opinion pieces anymore? Would our articles have to be like scientific papers with, kind of, the framework? More of a lighthearted question. How would that work?
EC: That’s an interesting definition of lighthearted, but thank you. I mean, I think that not all opinions are equally valid, and that’s something that the world seems to be losing sight of a little bit. And all opinions should really be backed up by something. And that it’s not about right and wrong, but it’s about the extent to which a backup has been provided. And there are many different ways to back things up, and I’m not trying to claim that mathematical logic is the only good way to back things up because the scientific process, for example, is slightly different from the logical process. The scientific process is based on evidence and replicability and more statistics. And then there are other disciplines that use different things. So the way that truth is assessed in history is slightly different, and the way it’s assessed in archeology is slightly different. And all of these disciplines provide frameworks for assessing how valid we should consider a truth to be. And I think that if you just state an opinion with no backup whatsoever, in a way, that’s what an opinion piece is. “Well, my opinion is this,” and that contains no framework. I don’t see a place for that in my ideal world. But if we can understand where that opinion is coming from and provide some sort of justification for it, then I think that it’s very interesting. So, in order for people to understand thelogical frameworks or scientific frameworks, of course we have to improve the education around these things. And so then we have to change the entire education system. And in order to do that we have— And then we get into a sort of other vicious cycle where we can’t change the education system until we change government. We can’t change the government until we change the education system. And so then what we do? Well that’s why I have been writing books to try and bypass all that. Because I would like to change the entire world, but in the meantime, I’ll write a few books and try and help people understand some things outside of that system.
Question 3: Okay. Two questions. Eugenia if you’ve got a classroom of reluctant geometry students, which one of your books would you start them on? And then the second question is, when the women in France march on Versailles demanding bread, do the gardens change their tune at all? Or is it something else?
AA: Shall I try to start with that? Okay, I’ll start with that. Something very interesting happens to the gardens of Versailles in the later decades of the old regime, even before the French Revolution. This notion, this idea of the garden, of this perfect, perfect geometrical world with the king on the top is being challenged. And it’s being challenged, particularly, by the rise of the philosophy of Rousseau. And Maria Antoinette, the queen, who adopts that philosophy of Rousseau, and starts creating enclaves within the garden that are not at all what Le Nôtre had in mind, that were not at all about the supremacy of the King. But little enclaves of supposed nature, where she and her courtiers can play milkmaids and get away from that rigid geometrical order. So it’s a consciously anti-geometrical reaction. And you see it on the ground. I mean, that’s what’s amazing with geometry. You can see it on the ground in the design of cities, in the design of gardens, it is imprinted. That order is imprinted on the ground and shapes are our environment. And you know, it’s a matter of, kind of, seeing it. After what happens, ultimately, the revolution. After the revolution then Versailles, the palace and gardens, are made into museums. So, they are preserved, as are most of the old Royal geometrical gardens in Vienna and St. Petersburg. But they’re preserved as museums. Their power is curtailed. They are no longer the ways of world is, they are just something that we visit and quaintly say, “That’s how things used to be.” So their power is neutralized. But not in Washington because Washington is alive.
EC: So, in answer to the other part of the question, for a reluctant geometry class, it sounds like I would recommend Amir’s book. Out of my books, I think my first one, How to Bake a Pi, which is about what math is for, and the way in which it can be for everybody. And that it can be fun, and is all around us and is related. And it’s something that you can do for yourself, even if it hasn’t seemed like you can do it for yourself. So I think that’s what I’d say.
Question 4: Thank you. I have a question for Dr. Alexander. And my question is whether you can tell me the alignment of Versailles, the Chateau in relationship to the garden, and whether there’s an east-west alignment. And if so, whether this suggests a closer medieval tradition in that the east-west alignment depicts the medieval cathedral, which in turn, again, is a symbol for the medieval world. So it’s like a very medieval, over all, framework, and a tradition that, yeah, again, suggests that we are looking not into something new, something that was generated in the Baroque tradition, and the Baroque time, but something that is a proto-development of medieval thought.
AA: Well, fascinating, because, yes, first of all, yes. The gardens are pointing westwards, like a cathedral. Which is very interesting because Louis’s representation was as the sun king, and he had the statue of Apollo at the end of the Petit Park, and initially he had the home, the Grotto of Tethys near the palace. And supposedly, Apollo, the sun king, would transverse the sky, except that he was doing it backwards, which was always a little bit of a quandary. You know, the sun was moving eastwards rather than westwards. So, yeah. The pagan elements are definitely there in the presentation as the sun king. I’ve never heard the idea of the cathedral. That’s, yeah, that’s enormously, that’s very interesting. That the fact that it, in fact, did preserve the old alignment of the cathedral. That’s very interesting. Thank you. Thank you for that.
Question 5: There is a tragically increasing number of members in our society that have begun to reject the premise of science or math or logic whatsoever. People who deny evidence and say, “The globe isn’t warming. The planet is flat. That we have never been to the moon.” Things like this. What do you say to those people who reject that premise to begin with and how do you think our society can move forward from that?
EC: Thank you. It’s very easy to get very depressed about that kind of situation. And what I remind myself is that we cannot reach everyone at once, and that’s okay. And that there’s a whole range of people and that maybe those are the most far away people from where I am. Whereas, there are some people who are less far away. For example, there are people who really do want to believe those things but don’t really know how to deal with it, or who don’t do it quite as well as they want to. So they’re trying to be logical, but they make mistakes. The kinds of people who are trying to believe in science, but then they don’t quite fact check all the articles that they just immediately repost. And so I think that we can try and reach those people first, and that if you immediately try to reach the people who are the most difficult, then, yes, you’re doomed to get depressed and feel like everything is hopeless. And the thing is that if we decide that everything is hopeless, then it will be hopeless. That’s definitely for sure. So I try to ascertain whether there’s any chance that I can make any progress, and I try to understand where they’re coming from rather than try and change their mind. But then also it’s important to preserve your own mental health. And that if you’re really going to get depressed and be attacked by people, then I think it’s okay to decide you’re not going to engage with that right now, and try and improve everyone else. Because, for example, I don’t think, now I don’t actually have the stats to back this up, but I don’t think that it’s a majority of people who think the earth is flat. And so I think it’s all right. I think there are more people who we can reach, and that if we can just shift the kind of center of gravity of logical and scientific thinking a bit further back towards the logical and scientific thinking, I think that could make a huge difference. We don’t have to change everyone at once. We just need to shift things because I think things only shifted a bit. That’s what I suspect. And that maybe if we should just try and shift where we can then, in fact, we can make progress and maybe change things a bit.
Question 6: So when you look at things like the golden triangle. They’re very two-dimensional. And you look at a lot of art with a vanishing point, the simulation of three dimensions, but it’s really two dimensional. When we look at things that are actually three dimensional, when you’re talking about spherical geometry, for instance, we look at them as somewhat chaotic, really. Do you see any evidence that we’re moving towards art that actually encompasses the three-dimensional-ness of symmetry and sees more order in these things that we create as chaos? Whether it’s talking about art or whether it’s talking about architecture?
AA: Hmm. Oh, that’s, yeah, that’s an interesting question because of course you’re right. When you use this linear perspective, it’s not just a plain description of the world, it’s a certain view of the world. It’s also not just describing the world as geometrical, it’s just telling. This is what the real world is. It’s geometrical in that sense. Of course, there have been developments in geometry that have moved away from that single, necessary, fixed order. Talking about non-Euclidean geometry, and the implication that there can be, in fact there is not just one necessary, single truth and one true point of view, but in fact, there is an infinity of possible geometries, rather than a single truth. Which was a very challenging and, in fact, disturbing idea. I mean, I think in some ways you can say we’re not living in a Euclidean world. We’re living in a post-Euclidean world, in which we’re all living in our own bubbles with our own different perspective. And they’re all equally true because we hold such different assumptions. And art, in fact, modern art certainly did try and deal with that. I’m no expert of that, but clearly early 20th century art, like Cubism, they have this effort to portray things from different angles and different sides at the same time. It’s clearly also a response to this move away from this single, unified, Euclidean view. So that’s what I can say.
EW: Alright. We all exist in our own non-Euclidean bubble. Thank you so much.
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