Math doubleheader with Eugenia Cheng and Amir Alexander. Many of you might be returning for the first time after a couple of years and for many more of you, this might be your first timeever townhall. This is the first, exactly the type of thing we exist to do. Im pleasedto present these two fantastic authors. We love our lecture programs, all the political thoughts, all of the art you hear but i am especially fascinated by a mathematical approach to thinking about the world. Im a reformed college math student having a chance to present Eugenia Cheng and Amir Alexander on a doubleheader is, was a great opportunity for this special homecoming festival im pleased to see everyone here tonight and thanks especially to our partners so the format is unusual tonight if youve been to other townhall because it is a doubleheader. Opening with a solo presentation from Eugenia Cheng, then we will take a short break while we switch the presentations over to Amir Alexander who will get his talk after which we will have a joint q a to keep questions in mind from both talks. There is this beautiful. [bleep] dee residence tonight where eugenius talk focuses on mathematical thinking and its urgent present political dynamic and then we take this broader historical view of the sort of Historical Development of mathematical thinking with Amir Alexanders talk so hopefully there will be residences, both of them will be welcomed so you can pick up a copy of either book at the table and we will have signings afterwards and the bar will be afterwards open if you want to play great zeno games. The firstthing ill say about townhall is we are a member supported organization. But now to introduce eugenia, the first of our two speakers and to say a few more words about zeno, i would like to welcome julie marl. [applause] thank you. I am the executive director of zeno zeno is a seattlebased nonprofit. Our vision of the world is one Everybody Knows they can do math. We achieve this through programming for families ages 3 to 5 with a focus on families of color and low income communities. Our work is all about making math fun and playful because we know that there no such thing as a life lived without math. And believe that a strong Math Foundation is key to a life of opportunity and success area were so excited to be sa Community Partner to townhall for tonights lecture. So doctor Eugenia Cheng is a scientist in residence at the school ofthe Art Institute of chicago. She one tenure in pure mathematics at the university of sheffield and is honorary visiting fellow at the university of london. She was previously taught at universities of cambridge, chicago denise holds a phd in pure mathematics from the university of cambridge. E. Alongside her research and category theory and her undergraduate teaching or aim is to rid the world of math phobia. Her first popular math book how to bake a pie was raised by the New York TimesNational Geographic and Scientific American and she was interviewed around the world including on the bbc, npr and the late show with stephen colbert. Her book eybeyond infinity was shortlisted for the Royal SocietyInvestment Science book prize 2017. Eugenia was an early pioneer of math on youtube and her videos have been viewed over 18 million times today. Eugenia is also math columnist for the wall street journal , a concert pianist and founder of the leader to, join me in welcoming doctor Eugenia Cheng. [applause] thank you very much. Thank you to the townhall for inviting me to speak on my latest book. Its affirming to be invited back somewhere again and this is the third time ive been invited here so that strictly affirming and its wonderful to be inseattle and the sun doesnt shine every time i speak at a townhall. Thank you for joining us this evening for a massive evening where im going to talk about my latest book, the art of logic in an illogical world and the point about that illogical world is this. It can sometimes seem like we are drowning in the current world. Where the world is awash with divisiveness and conflict and fake news, victimhood, explication, privilege, blame, bigotry, shouting and minuscule attention spans. And it can seem that we will never agree with each other ever again and we are doomed to be stuck in an chambers and just yelling backwards and forwards into nowhere and is all hope lost . Thats the question and i say no, all hope is not lost. It can seem like that sometimes and this arose from my teaching at the Art Institute and in that fall semester of 2016, some things happened, various things on both sides of the atlantic. The morning after the election i did what many people did. I got depressed, i cried, i drank and i thought thats what can i do thats productive because i truly believe in doing something rather than sitting around complaining and i also believe in looking at your own commendation 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 what can i do as a pure mathematician in this political situation and i realize what ive been doing with my students all semester was using the principles of mathematical thinking to find greater clarity in their divisive arguments and i felt that i could share that more broadly with everyone who wants to find that clarity. Some people of course are interested in finding any clarity in their arguments but i believe there are people who do want to understand whats going on an understanding whats going on on all sides of the argument is the first step. Im not saying it will solve all the problems but if we dont understand it we cant solve the problem and thats why i wrote this book and it grew out of the discussions i hahad had with my heart students i teach abstract mathematics, not remedial mathematics. Its not the things that theyve forgotten from high school, its how to use mathematics as a way of thinking and so theres someone here who has a question. Does this, should this work . Its on. Thank you. Perhaps the volume be turned down . At least when im interested. Whenim not interested i speak a little less loud. Well, ill just keep going i suppose until we can findwhat my next line is. I teach art students and the students at the Art Institute are interesting students. They are not mathematicians at all and its my dream job because i want to share what i feel about mathematics with eemore people and there are so many myths about math that its just about numbers and equations that some people are math people and some people arent math people and if you cant do a science table you cant be a good mathematician and things like that are myths that im trying to dispel so teaching art students is a wonderful place for me to find out more about what of people have math in their class and what can put them back on math and i really believe and im sure if there are any educated in the room you will all agree that its really important. It happened to what motivates your students in order to motivate them and what youre teaching rather than trying to impose your motivation on some of them and i realized alwhat motivates my students is the presence of social and political justice and thats why this came out like this. I students , the thing that motivated the most was about food and thats why my first book was about math and food. So it still doesnt as i get closer to it. Should i go and stand over there . I could give the rest of the talk without my slidesthat i like my slides. Okay, thats going to be complicated because i have a lot of transitions but i guess we will try it. Its not just about numbers and its not just about getting the right answer and some are just about solving problems even. I believe in a framework for agreeing on things and every academic discipline as a framework for assessing what should count as good information and in the current world i think thats what we need eafrom education because theres tons of information everywhere. Information is no longer at a premium so whats more important is a way of deciding what counts as good informationand pure mathematics as one particular framework based on logic. So i believe that there has been an oldfashioned traditional view of what pure mathematicsis but pure mathematics applies to apply mathematics. And applied mathematics is useful for science but science is then useful for engineering and medicine is then useful for the numerical quantitative parts of the human world and its true and i used believe that was the extent of how my research would ever be useful because my research is so very abstract but the thing is this narrow view enables people to declare that they dont need math. People, math can exist but they say i am not going to go into any of those fields so im not going to do it emmyself whereas i believe your mathematics is about how to think that therefore it is t about the entire human world. At least the parts of the human world that bank and sometimes these days, it seems like some of the human world doesnt think very much. So anyway, i am going to talk about personal analogies and what role they play in mathematics and the interconnectedness of things. And i will talk about how abstract math enables us to see relationships we maybe didnt see before and that we can use those abstract relationships to pit between different situations so that we can understand more things and we previously did finally , i will talk about how to end with what i believe is intelligence read so first of all, analogies. Another thing that we can say is that your mathematics is a theory of analogies and heres what i mean by that. That supposing we have two apples and two bananas and we can say oh, theres something they have in common and if we forget the details about them being apples and we say they are both two things this is fundamentallyhow we come up with the idea of numbers in the first place. Numbers are an abstract that the projects that have something in common and if you teach children how to count, i love helping children count and if youre trying to teach them how to count you have to wait until they make that affect move and you cant do it for them, they just have to sort of see whats going on and every time we make another abstraction lead in the process of math education, some people dont quite make it and there are various reasons for that and i think one is it can seem pointless if its not well motivated and another is cyou have to wait until you can do it. Nobody can do it for you. So the other thing is there are different ways to do it. Theres not a completely Automatic Process so if for example we wanted to this is very exciting. If instead of saying 2 things we send what these have in common with b 2 groups, that would be an abstraction but in that case we would not be able to include or example 2 chairs in that situation. That is not an example of 2 fruits we have to go up one level further to 2 things in which case we can encompass those examples and what im going to argue, one of the things im going to argue is that abstraction seems to take us further away from real life enables us to bring in far more examples and we could before so heres a more mathematical example where if we say, if we look at one 2 and 2 3 they are both examples of a the people often say to me i was fine with math until the numbers became letters. Im going to show you what the point is of numbers becoming letters. We can also say look at one times to 2c and they are examples of eight times the now, a b and eight times the , oh. Thank you very much. Thank you. So a b and eight times the are examples of a something b that is against further level of abstraction and these levels of abstraction are bottom levels of what you might do in Elementary School when youre doing arithmetic upand this is what happens when you startmeet algebra and nothen this level is part of what happens if you go to be a math major in university and uk abstract algebra , maybe group theory. Its about finding operations and none of these levels are right or wrong, thats not the point that sometimes people think math is all about what like your shedding on a particular situation and one of the things my phd advisor Martin Hyland taught me was that jamison to find the most abstract possible approach, the aim is to find a good level of abstraction for what you are trying to do. And what happens in normal life typically is that we talk about things being analogous to each other but we dont focus on what is making them analogous and that ambiguity that we leave leaves open the possibility of disagreements just based on using Different Levels of abstraction where not making clear whereas in math we are very explicit about which level were using so that we remove that particular ambiguity. So y. Heres an example of how that ambiguity comes in. On its own. So if we talk about straight marriage and samesex marriage people say theres no difference really between those some people say its terrible and so whats 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 and indeed samesex marriage is not part of that picture, however some of us believe that its actually really about two unrelated adults in which samesex marriage is part of that and people disagree because theyre using Different Levels of abstraction and the next thing that happens because were not being precise about which level where using so people who disagree about the upperlevel can hallucinate er that weve gone further than we have and they get upset and say the next thing we know we will be allowing any 2 adults even if they are unrelated. Ive redacted whats going on but in case there were children in the audience was then we can go up further and have lets say 2 humans read or we could say 2 living creatures. For we could say to creatures. And the point is that just because some of us have decided that we want to go to hear is not automatically mean we have to shot all the way up to the top but when were not being precise about what level of abstraction where using it can open us up to those arguments about saying this is the same and this is the same. Im not saying this approach solves that problembut im 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 become can be seen as being interconnected my favorite diagram of interconnectedness. And it is an abstraction of the London Underground system we are not many details about where things are and in fact is not geographically accurate at all but its very useful for seeing how stations are connected by which lines but because its not geographically accurate you can end up slightly hapless tourists trying to take the train from say less garden even though their two minutes walk away because you cant tell from this picture so here is the geographically accurate picture of the London Underground which is a different abstraction and its not better or worse, its probably less useful if youre trying to train somewhere is quite interesting being where everything is so the point is that these are two different abstractions that eliminate different aspects of the situation and this is what math does read an abstract from the situation to see what we can learn so heres an abstraction that i find interesting to do with relationship breakdown so this is what often happens when relationships break down is that one person, ill call them alex feels disrespected and when alex feels disrespected alex is unable to show love and his partner stan on love as a result of which is unable to show respect alex feels disrespected and we have a vicious circle thatcan escalate. I can further abstracts and label these arrows as action arrows and these arrows as feelings. And this doesnt solve the problem but it makes the start because we can think about how we can break at least one of these arrows because you only have to break one to break the circle you can say is easier to t break an action arrow or a feelings arrow area maybe we can control our feelings but maybe we can decide to act on them. Even when alex feels disrespected they can s concentrate on showing love regardless of that and then then the situation will spiral out of control so we revisit to the two action arrows and we can 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 as 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 but police feel threatened by black people so they defend themselves against black people which makes black people feel threatened by the police eyso they defend themselves against the police so the police feel threatened and i would like to point out im not saying this is what happens if this overview of what happens but it has been shown even when black people dont do anything to defend themselves and everything are psupposed to do there is violence against them so we can again say that we break the action arrows or the ceiling arrows and we might argue that maybe the police, why do they feel threatened . There the police, excavated trade , they feel less threatened by people and black people so maybe we can train them but we can also train them to take action differently the escalating rather than immediately escalating things and we can say take responsibility, some people yell and say thats the law. I would argue that its really the police who have the power in the situation so i think they should be the ones taking responsibility for changing it and if they dont that doesnt help the situationbut maybe we can find some more clarity about what is going on and this is something i think abstraction can help us with. Another way i use interconnectedness to help me is when there are very many factors prosecuting the same thing in an interconnected way so there will be a gracio