Episode 038: Category Theory for Normal Humans with Dr. Eugenia Cheng

Panelists:

Coraline Ada Ehmke | Rein Henrichs | Jessica Kerr

Guest Starring:

Dr. Eugenia Cheng: @DrEugeniaCheng | eugeniacheng.com

Show Notes:

00:16 – Welcome to “Shopping is Hard; Let’s Do Math!” …we mean, “Greater Than Code!”; Eugenia’s Introduction

Books:

How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics

Beyond Infinity: An Expedition to the Outer Limits of Mathematics

YouTube Channels:

TheCatsters

TheMathsters

Articles:

Eugenia Cheng Makes Math a Piece of Cake

Everyday Math

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01:54 – Getting Into Math: Is math useful? Is that the point?

A Mathematician’s Lament by Paul Lockhart

20:17 Category Theory

Textbooks:

Categories for the Working Mathematician

Category Theory (Oxford Logic Guides)

Conceptual Mathematics: A First Introduction to Categories   

38:17 – Changing the Terminology Around Gender to Focus on Character Traits Instead: Congressive and Ingressive Behavior

The Prisoner’s Dilemma

Are you Greater Than Code?
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Transcript:

REIN:  Hi, I’m Rein Henrichs. This is Episode 38 of ‘Shopping is Hard. Let’s Do Math.’ I am here with the wonderful Jessica Kerr.

JESSICA:  Good morning, Rein. Shopping is hard but this show is called Greater Than Code and I’m thrilled to be here today with Coraline Ada Ehmke.

CORALINE:  Good morning everybody. We have an amazing guest with us today and I know I say that every time but this time I really, really mean it. We have Dr Eugenia Cheng, who is a scientist in residence at the School of the Art Institute in Chicago. She won tenure in pure mathematics at the University of Sheffield in the UK, where she is now an honorary fellow. She previously taught at universities of Cambridge, Chicago and Nice and holds a PhD in pure mathematics in the University of Cambridge.

Alongside her research in category theory and undergraduate teaching, her aim is to rid the world of math phobia. Her first popular math book, How to Bake Pi was published by Basic Books in 2015, the widespread acclaim, including from New York Times, National Geographic, Scientific American and she was interviewed around the world, including on the BBC, NPR and The Late Show with Stephen Colbert and I have to confess I watched that clip today.

Eugenia was an early pioneer of math on YouTube and her videos been viewed over a million times to date. Her next popular math book, Beyond Infinity will be published this year. Eugenia is also a math columnist for The Wall Street Journal, a concert pianist and founder of the Liederstube. Eugenia, it sounds like you’re a big slacker.

EUGENIA:  Yeah. I’m really lazy. I never do anything.

CORALINE:  Your work with math has got some slight attention of some form, what got you interested in math at the beginning? Were you one of those kids who is just like really, really in a math your whole life?

EUGENIA:  I was really into lots of things my whole life and it wasn’t like I just sat around doing math the entire time. I was not the kind of kid who just read math books. But my mother really got me into thinking cool things and she showed me the coolest stuff about math, just as part of life. The whole time it wasn’t like, “Now, we’re going to sit down and do math and then we’re going to go and play.” It was always there so I never thought of math as something separate. I just always have this natural assumption that it was really fun and exciting and I had the most fantastic ideas in it.

Then I went to school and school wasn’t very interesting. I basically didn’t find anything in school, particularly interesting especially not the math. I understand why many people can’t stand math because of the lessons that they had at school. But because my mother showed me these really fun, interesting things, I held out hope all the way through that there was something better waiting for me at the end and indeed, there was. When I finally got to start doing research, I discovered the stuff I had been waiting for. A lot of it when I was going to graduate and there were little bits of it all day long but really, I found home when I got to make my own things up.

I think it’s sad that it took all those years through the drudgery of school and tests and homework before I was allowed to make my own things up. I think that that puts a lot of people off of math so people who like making their own things up, it sends them into other things like music or art or cooking or maybe coding or something where you do get to make your own things. That’s the message that I decided to try and give everyone else that it’s not necessarily just that thing you do at school, where you have to answer questions and get the right answer. You also get to play and experiment and create.

CORALINE:  I have to confess that I was one of those kids that struggled with math in school. To challenge myself in my senior year, I took an AP Calculus course and it was the only B I ever got, which was devastating and caused me the valedictorian position.

EUGENIA:  Oh, this is a really tragic story.

CORALINE:  But I did find that it got easier when I started taking physics classes because suddenly, I had a practical application for the math so the pure theory wasn’t doing it for me and it wasn’t sticking and I struggled with it a lot. But when I had something practical that I could apply it to, then everything started clicking a little bit more. Do you find that to be a common experience?

EUGENIA:  I have the opposite experience and I think that for many people, it’s about practical applications but the trouble with that is that many people aren’t interested in the types of practical applications that get presented. For people who are maybe more like artists and I see artists all the time now because I teach them at School of the Art Institute, they also aren’t interested in physics applications and they’re not really interested in engineering applications and they’re definitely not interested in business and finance applications. For those people there’s something else which I’m showing which is it’s about understanding how things work and it’s about showing what the deep structure is inside thought processes.

I want to reach those people because I feel that they’re the ones who are the most left out at the moment because it’s true that applications do help a lot of people but they don’t help me to like math. On the one hand, maybe I am an entirely unique person but I doubt it. I suspect, if that doesn’t help interest me, maybe there are a whole lot of other people who also aren’t necessarily interested by direct applications but rather in more broad thought-based ideological applications where it’s about understanding how to use your brain really well, rather than how to use the theory to solve this particular problem.

JESSICA:  Do you want to give an example of that?

EUGENIA:  One of the surprising example of that is how I understood my life and therefore, quit my job. That, I really felt was something that I thought about mathematically by focusing really carefully on things that are relevant and things that aren’t relevant and following chains of causation. I did an analysis of it, of my life. It was at New Year of 2013, at the beginning of 2013, I sat down and I wrote a list of everything that I think I’m good at and I wrote a list of everything that makes me happy and then I compare that with what my current life consisted of. I just didn’t feel like it used enough of the things that I think I’m good at.

I really believe in life, for me, it’s important to find all the things you’re good at and figure out how to use them in combination in the best way you can to contribute to the world in some way that you want you. I sort of axiomatized myself basically because I’m a mathematician, I like systems where you boil everything down to some very basic principles and you see how everything else follows from those principles.

I axiomatized myself, I found the things that I think I’m good at and I tried to figure out how I could get a life that used more of those things and I realized that a normal academic job as a professor in a normal academic university was just probably not going to do it. I had tried for a number of years because I really wanted to contribute to the education system in England that had given me everything that I have basically and it just wasn’t doing it so I thought about what I would do. I saw various constraints so I just try changing a few constraints to see what it have, which is a very mathematical thing to do, where you look at a situation and then you try generalizing it by saying, “What if I just relapsed this rule?” What kind of world would I create without that law, which doesn’t mean that you can because in real life, you can’t just get rid of laws. But it helps you to understand what’s making things function in the world that you have.

If you imagine getting rid of some law and saying, “What would happen?” then you can think about it that’s sometimes I go, “What would I do if I had tons of money?” I can’t just have tons of money but imagining a fantasy world, in which I have tons of money, helps me understand what my real desires are. I think that’s a mathematical process if the thought process comes from the idea of abstraction where in math, you take the real world, you forget some ideas, you move into the abstract world of ideals and then you think about what works in that world without the details and that helps you understand the world that’s got the details in it.

I realized that if there were no such thing as countries and borders and visas and passports, I realized I would simply move to Chicago and figure something out. I realized that the big hindrance was just that I needed a visa so I then had to think about how to get a visa. That’s how I started the process of quitting my tenure job and becoming essentially freelance and living in Chicago my best life.

JESSICA:  Wow. That’s beautiful. In the relaxation of constraints, you’re able to see over the wall of those barriers that you didn’t let yourself think past otherwise.

EUGENIA:  Yeah, I love that image because you see through the walls, you see what’s on the other side and then you can focus on how to get over that wall. Or you get an overview of the wall so you can see maybe where the lowest point is or where would be the best place to construct a way of getting over it.

REIN:  Yeah, I love that analogy and that reminds me of there was a time where I was reading a bunch of computer science papers to try to figure out what was going on in this world. I was really struggling to make my way through these papers. They were just in front of me and they were a huge challenge. I don’t have a formal background so they were this huge edifice that I was trying to climb and a friend of mine who comes from a similar background but has been very successful said, “You have to pick a fight with the paper. If you have to look at its assumptions and the chains of logic that follow them and say, ‘What if this one was different? Why is this one necessary for their result?'” Doing that gave me the motivation and sort of the path through this forest. What I realized later is that this applies to basically every challenge that I faced in my life.

EUGENIA:  And that is a great way of approaching math in a way that makes it more personal because the trouble with math is often that when you’re studying it, it feels like a bunch of rules that have simply been imposed on you and you have to follow them. But actually, if you think about why those rules are there, then you can just imagine one of them going away and seeing what happens and that helps you understand the system better. It also makes you feel like you have some agency in it.

REIN:  One of the things I love as you said, there are two possible outcomes, either you understand the paper or you have a new result.

EUGENIA:  There’s a kind of related thing to that, where sometimes I read a paper when I’m trying to go to sleep on the plane for example and I think, there were two possible outcomes, either I’ll understand the paper or I’ll get some sleep.

CORALINE:  Nice.

REIN:  I love the way you talked about mathematics as being a creative process and not just a rote. I reminded of a Paul Lockhart’s Mathematician’s Lament. One of the quotes I love from that is that mathematics is the music of reason, which really speaks to me. It seems like if you wanted to build a system that was designed to make people hate mathematics, it would look a lot like our current education system.

EUGENIA:  Yeah, I think you’re right. I think when you’ve done a really good job of building of really getting people to hate math and sometimes, I fantasize about the fact that if we simply stop teaching any math at all, we would probably achieve more than we’re doing now because at the moment, we’re achieving negative. If we just didn’t do it, we’ll achieve zero and because we know some math, we know that zero is more than a negative number.

CORALINE:  I thought I hated math until I read Gödel, Escher, Bach by Douglas Hofstadter and that spoke so much in me. I saw for the first time a relationship between math and music. I’m a musician. I saw for the first time the relationship between math and models of human intelligence, which sparked my work in AI, which I’ve been doing for 20 years now. It just opened my eyes so much and I wished that that book had been my textbook, instead of the boring calculus book that I actually had in my high school.

EUGENIA:  I know but calculus altogether is such a strange thing to teach people in high school. It’s definitely, dare I say, it’s very American. I don’t think other countries do that and it’s such a weird introduction to difficult mathematics because the structure of it is not, I think exactly illuminating so it puts off a lot of people who don’t just want answers, they won’t to understand. For me, there’s a real difference between a proof and an illumination because a proof is a step by step series of logical justification. But an illumination it sheds light on why something works. A calculus proofs, if you even get to them, don’t really shed light on what’s going on there. It’s kind of a trick to say, “I can’t shed light on its own. I’m just going to do this little trick and ping! I get to the other side. Huh?” I think that that is a strange introduction to the beautiful world of logical reasoning.

If you present it in another way, then it’s a really amazing introduction because it’s a way of getting around the problem of infinity that is really difficult to get around but it’s never presented like that, rarely anyway. When I taught it, I’ve presented it like that but usually, it’s some kind of weird set of manipulations and you just learned how to do tons of manipulations for no apparent reason. In the UK, where I’m from, there isn’t an obsession with teaching calculus. In fact, we don’t teach calculus until quite high up in an undergraduate math degree so I found it very surprising when I moved here for the first time and discovered that everybody does calculus and that’s the thing that you do, if you’re good at math, you do calculus. Then when you gets to university, the thing you have to do is you do calculus.

I was talking to another European person about this, “Do you understand why there’s all this focus on calculus?” And he said and he was just maybe a tiny bit cynical, “You know, what I’ve learned about this country, if you can’t understand why something is going, ask yourself where the money is coming from.” I thought, “Ah, yes. Calculus textbooks. They make people tons of money. They are really heavy and expensive and they make another edition every two years so everyone has to by a new one.

JESSICA:  Since math changes so quickly.

EUGENIA:  Right. Exactly and if you think about the number of people who are forced to buy those textbooks, I read an article about one of the calculus textbook authors who died recently and apparently, he’s very famous around here but I don’t know many calculus textbooks. He was so rich from calculus textbooks that he built himself something like an eight million dollar house with its own concert hall in it. The reason that I read an article about his house, it was ostensibly an architecture article and the house had calculus principles in it and he loved music so he built this concert hall. I just loved it and I thought, “Maybe, I should be writing calculus textbooks.” Then I remembered that I’m not primarily driven by money and that’s why I’m not writing calculus textbooks.

REIN:  There are obscure subjects that I’m interested in, where the only way to learn about them is to either read a bunch of papers, which is very undirected and most of them are behind paywalls or to buy an obscure textbook and they cost $400.

EUGENIA:  Yeah. I think that I love the idea of somehow trying to break the stranglehold that the publishing houses have on those textbooks but can you imagine the amount of pushback there would be if we tried to say actually, no, which is not going to teach calculus anymore.

CORALINE:  We get pushback in trying to put evolution in textbooks so I don’t think you’re going to change the textbook publishing world any time soon.

JESSICA:  It is changing. My 12 year-old doesn’t have a lot of textbooks and she does have a lot of online material that she uses.

EUGENIA:  Right and I love the fact that with the internet, there are all sorts of problems with the internet. But instead of focusing on the problems of the internet, let’s remind ourselves of the great things about the internet and the amount of information that can be shared without having to go through those old gatekeepers. Of course, the old gatekeepers want to keep things the way that benefits them. But guess what, we don’t have to go through them anymore and that’s why I started making videos on YouTube because suddenly, I had always wanted, I had always dreamed of making a textbook — for category theory, which is my research subject — and I think about category theory isn’t involved a lot of visual diagrammatic reasoning. It’s very difficult to show that statically in a book because you have to see how the diagrams grow as you’re thinking.

I always fantasize about writing a textbook that had a DVD at the back. That was like those language textbooks where you have a DVD of native speakers speaking it. I really wanted to just include a DVD so that everyone could watch the lectures with the text but to see the diagrams growing. But back in those days, millions of years ago, when we were young that it was really expensive to make DVD. You have to get a video team or something and then someone invented the YouTube and webcams and suddenly, it became possible to just point a webcam at a board and make a video and then just shove it on the internet and everyone could see it. I just had this amazing vision that everybody would have access to me teaching category theory.

The thing is at the time, hardly anyone was actually teaching it so as you say, loads of people were stuck with trying to learn it in from textbooks and the only textbook at the time was Categories for the Working Mathematician which is so dry and I don’t know how anyone ever managed to learn category theory.

REIN:  I could tell you what I did. I would read a paragraph of Mac Lane and then I would read five pages of Awodey. Then I would read the chapter of Conceptual Mathematics that would explain that to me. I would work paragraph to paragraph. Each paragraph in Mac Lane would be a forcing function that would require me to spend a week with other sources.

EUGENIA:  And the thing is that when I started making the YouTube videos, there weren’t that many other sources. I can’t remember when Conceptual Mathematics were in but Awodey’s book was years away from being written yet, I think anyway or maybe just a little bit away from it. But certainly, because I did my PhD in Cambridge, there was a category theory course there and then I went off into the world and met other people who had not had access to that. I saw that, because they learned category theory from Mac Lane, they had a, I want to say, slightly strange view but they could do lots of technicality without knowing quite what the point was.

JESSICA:  They have wrong proof but not the illumination?

EUGENIA:  Exactly. Thank you. They have the proof but the elimination. I think that is how so much of math is taught at all levels that people are trained to do manipulations and computations and to get answers that consist of a number here or a number there. But without the illumination, it doesn’t make that much sense to them plus it means that they’re not able to see why math is wonderful. They just see it as a series of manipulations and this happens at every level. It happens with high school math. It happens in elementary school math and everybody. It happens with math for scientists. Some scientists I’ve met, many of them actually, especially engineers, think of math as just something to help them answer their questions. They don’t see why research in it has any point.

But then it happens even inside math so category theory is like the math of math so it does for math what math does with the rest of the world, in a way. But then some mathematicians don’t see the point of that and they think of category theory as just some manipulations that don’t really have a point. Or they think that you only do it if you need to reach a particular answer, rather than doing it in order to illuminates what is going on in mathematics. The thing that I always want to do with my brain most is illuminate what is going on, just in anything: the world, math, my kitchen, music, anything.

JESSICA:  Eugenia, you said that category theory does for math what math does for everything else and you mentioned earlier that what math does for everything else has explained how the world works so just category theory explain how math works?

EUGENIA:  I think it does, at least at certain aspects of it. It’s important, I think to remember that math doesn’t explain how everything about the world works. What it does is it illuminate certain aspects of it. Anything that makes a connection between something and something else is a potential place where math can help because it’s really about looking at two different situations and saying what these two situations have in common and that part that they have in common is an abstract thing.

I think that there are times when, for example if we think about how we should treat minorities, then you can think about how we treat one kind of minority and then we can think about, whether that would be appropriate if we did something similar to another minority that maybe we’re more used to treating equally. For example, sometimes people say really stupid stuff about, for example gay people and then people sometimes people try to illuminate that how stupid it is what they said by replacing that with imagine that they’ve said that about African-Americans.

Then maybe they realize that that sounds really stupid if you said it about African-Americans and offensive and dumb. Then you go, “See, then why are you saying it about this other minority when you wouldn’t say about this minority,” and that process is a process of analogy between two different situations. You’ve moved into an abstract world, instead of talking about a specific minority, you’re talking about the concept of minorities and how we should treat them. That is kind of mathematical and that’s making an analogy between two different situations and finding what they have in common. That’s what category theory does for mathematics.

It says, “I’m looking at this situation where something seems to be happening, kind of freely without constraint and I’m looking at this other situation where there’s also something happening freely without constraint,” or the concept of things happening freely without constraint is therefore, something these two different mathematical situations have in common so that’s a more abstract piece of mathematics. Then category theory will produce a theory of things operating freely without constraint. Then we can put that back into algebras working freely without constraint or topological spaces working freely without constraint or logical systems working freely without constraint. Having made the theory, that’s at a different level.

CORALINE:  My interaction with category theory came from Hofstadter’s latest book, Analogy as the Fuel and Fire of Thinking and I tried to do some research in the category theory and found it impossible to find a starting point. But I did finally find a series of lectures that someone had done. One of the ideas that came, I think from the lectures in part and in part from my own realization and as an expert in category theory, I’m taking advantage of your expertise here, it seemed to me that from the beginning of rational scientific thought, the approach that people were taking starting early on probably with the great philosophers, was the idea of breaking things down into small problems, solving small problems and composing a larger solution from that.

EUGENIA:  Yes.

CORALINE:  It struck me that category theory does the opposite of that. It doesn’t look at the details. It blurs the details and looks at larger systems in the way that larger systems can be compared to each other so instead of the atomization that we saw in scientific research and I think in mathematics for a long time, it seems to be favoring the abstract over the concrete. Do you think that’s an accurate sort of perception of what’s going on with category theory?

EUGENIA:  I think that’s a really interesting observation but I think that that’s two different dimensions. Actually, I think there’s one thing which is splitting things up into small parts, decomposing them into small parts and fitting the parts together. There’s another which is abstraction, which is moving levels. I don’t think it’s about sticking things together to make bigger things. I think it’s about, as you say ignoring details so you get some kind of zoomed out picture. I might call it the bigger picture but I don’t mean that in the sense of sticking small pictures together.

I mean it in the sense of overview and you can do both at the same time so within category theory, there’s still a big focus on decomposing things into small parts, understanding the small parts and then working out how those small parts fit together. Then you can de-abstractify back to the small parts as well.

REIN:  I think, one of the good analogy for this is what happened in Google Maps when you zoom them out. They don’t show the streets anymore. They show the highways and things like that.

EUGENIA:  Oh, yeah. I think that’s a great analogy because when you zoom out on Google Maps, it still is some small pieces stuck together. You just see more things at once because you zoomed out but with less detail. You could still deconstruct it into small bits stuck together and then when you zoom in, it’s still also small bits stuck together. I think it’s two different points of view on how to understand a problem further. But I think that you’re right that if you start at the very zoomed-in level, then breaking into small parts and zooming out are two different things. I don’t think they’re opposites. I think they’re orthogonal.

CORALINE:  Someone who will go into category theory will do both under [inaudible] space?

EUGENIA:  Yes, I do that all the time. What I will do and what I do in my own search is that I zoom out to try and find the essential structure. But then after I’ve started thinking about the essential structure, I will break down the essential structure into small parts to try and understand the parts separately before understanding how they fit together and then understanding the relationship between the small parts, the fitting together and the zooming back in again.

CORALINE:  Is there a mechanism for detecting which details when you are at the zoomed in level matter and I’m thinking here of like work that I did with categorization using Random Forest algorithm to determine what traits decisions tree made on and which ones they could not be made on. Is there a mechanism like that that you [inaudible]?

EUGENIA:  That’s such an interesting question because thinking about it, that’s something that I have to do all the time. It’s really hard because it’s not an automatic process. It’s not an algorithmic process at all. When you’re figuring out what’s important, that’s one of the hardest parts. I think it’s something that is sometimes done quite badly, which is why sometimes people think of, they call it abstract nonsense because abstraction when it’s done just for the sake of abstraction, kind of is pointless. Whereas abstraction when it’s done for the purpose of eliminating what the point really is, is really amazing. But there isn’t a formula for how to figure out what’s really important and I think that —

JESSICA:  Oh, you mentioned one earlier.

EUGENIA:  I did?

JESSICA:  Yeah. You said change the rule and see what happens.

EUGENIA:  Yes. Thank you. I did and that is one way of doing it so you can try changing something and seeing if it affects it. When you’re making a good analogy and this happens all the time, when I’m having an argument with someone and I’m trying to be really logical, I will often try and find an analogy. If I think there’s something absurd about their argument, I will try to find an analogous situation that highlights how absurd their argument is by making it really absurd in another situation.

To find a good analogy, what I do is I retreat into the same part of my brain that does mathematics, I think. I play around and I trying to feel for what is really making the absurdity happen. I try ignoring some details of the situation or changing some details to see whether that absurdity still happens. If I find what I think is a really good analogy, usually the person will go, “But the different.” And I’ll go, “Yes, of course, it’s different. It’s an analogy. That’s the whole point of analogy. It’s different but the [inaudible] parts are the same.”

Then you have to convince someone that what’s crucial about the situation has remained the same, even though some surface details have changed. But then sometimes people make analogies because they think that they’re making a really good analogy but I think they’ve changed the crucial aspects of it so they haven’t actually eliminated the situation. There completely changed the situation but it’s very subtle and I think that is a fascinating aspect of how abstraction works and the nuances in our amazing human brains.

CORALINE:  It’s interesting you said you went to a mathematical place to from analogies. One of the things that Hofstadter talked about in his book is focuses on analogies and he talked about a mysterious process by which our brains are categorizing things in such a way that we can make analogies. One of the examples he gave was a family who took their son to the Grand Canyon and the mother and father are looking out at the sweeping vista of the Grand Canyon before them and they say to their son — I forget his name like, Chris, “Isn’t this really beautiful?” and looked over and Chris is fascinated with some ants crawling around on the ground.

Chris grows up and he ends up taking a vacation with his parents to Egypt and he discovers that in Egypt, glass soda bottles are prevalent and there are bottle caps everywhere so he decides to start a collection of bottle caps. They’re going to the temple at Karnak and he sees a bottle cap that he hasn’t seen before on the ground. He makes a big deal out of this bottle cap while he’s standing in front of the Temple of Karnak. His father is like, “That’s exactly like when –” It’s really mysterious how our brains can categorize things with enough abstraction that we can even say this situation is analogous to that other situations.

EUGENIA:  Yeah and I think that our brains are amazingly naturally wired or something to abstraction, even though a lot of people go, “I can’t do abstract math. I was fine with math until the numbers turn into letters. That’s too abstract,” and the thing is that numbers are already so abstract because numbers aren’t things. Numbers are analogies between groups of objects so you go, “Here’s three books and there’s three cookies and three bananas. What do those things have in common?” It’s the number three. It’s the concept of three. What is three?

When you’re teaching it to small children, you have to show them three of this thing and three of that thing and three of that thing and then you can’t tell them what three is. At some point, they have to make a leap and understand that three is a thing that those groups of objects have in common.

There’s a story I tell in my first book, which I love which is about a mother who I used to help in an elementary school in England. I did it for years. There’s one mother who also helped there and she was complaining. She had a three-year old time and his friends’ mothers they all say, “Oh, my child can count to 20. Oh, my child can count to 30.” She said, “My son can count to three but he knows what three is,” and I thought it was great because when you say that a child can count to 10, mostly you just mean they can recite the words one to ten, which is the start but that’s not what numbers are. Numbers are amazing concepts and somehow we have brains that can do that.

You know, language is already an abstraction from objects because when we say, ‘cat,’ we kind of know that cat is refined to some fluffy animal with four legs and things and we’re able to hold that abstract thing in our brain and know that it’s referring to various concrete things. The fact that we can do language, I think is already a sign that we have innate powers of abstraction inside us.

CORALINE:  And I can tell you that trying to marvel that in code is so difficult. I have my AI side project and trying to model things in such a way that you can concretely refer to them using linguistic terms but also understand how they relate to each other in a more abstract way is very, very difficult. When I learned that Leibniz was an alchemist, in addition to being a natural philosopher, that gave me some courage and I actually started exploring different classification systems from different areas of human endeavor, including Kabbalah where people have been working on the problem of classification and categorization for thousands and thousands of years and wasn’t all mathematicians and scientists doing it.

EUGENIA:  Yeah and I think classification is so interesting because we humans seemed to be desperate to classify things all the time and really, in order to groups things together so that we don’t have to process so many things in our head. I think that comes from, I’d like to say, laziness but in a way, it’s trying to be efficient, trying to have this so much data in the world around us that we have to process, that if we temporarily consider some of those things to be the same, then there’s just less data to process. I think that’s what analogy is doing in a way and I love the insight that we can get from trying to figure out how to get a computer to do a thing that we do.

Then we start realizing how weird it is, that our brains can do these things and there’s the classic example of how to get computers to recognize handwriting because there are all these things that we recognize as being the letter A, sort of spontaneously but they’re so different physically and the how that we can even say they’re different topologically and we can make an A with a gap in it and we can still know it’s a letter A, it’s just amazing when we realize that somehow the letter A is already abstract but even more abstract than that is the fact that all these different representations of it, we still recognize as being a letter A. Our brains are doing that all the time.

I think that if we couldn’t do that, we would be completely overwhelmed by the amount of information that we have to process all the time. I think the process of becoming an adult, one of the things that we have to learn to do is to do that more because we’re trying to process more and more information all the time so children can get really upset about the difference between that cookie and that cookie because they can see how different this cookie is from that cookie and if you give them up that one, instead of that one, they’ll scream.

Then we grow up and relax a bit and realize that they’re basically the same, those two cookies. They’re both chocolate chip cookies and we accept that some more things count as the same. In a way, what we’re doing all the time is learning how to ignore details that aren’t relevant but we have to make decisions about which details are relevant in the situation and which ones aren’t.

CORALINE:  I had an experience like that when I was young, when I was learning to count, apparently I had trouble saying that is two of the same thing because to me there are very distinct. When you said, “Three people,” I was like, “How can you have three of the same thing? It’s impossible.”

EUGENIA:  Yeah, that’s fantastic. For other children, they get confused if you say, “If you take one banana and one cookie, then can you add them together? What do you get?” You still get two things but you have to abstract further from the place that you started. Then when numbers turn into letters, you have to abstract even further so that one plus two can become part of the same concept as six plus seven. Then when you do category theory, you abstract even further from that. Every time you go further, some people kind of fall off or hit a ceiling because they’re not comfortable with forgetting even more details.

I sort of love forgetting more and more and more details. Sometimes, I think that there’s two directions that the world is going in, with the human interaction is going in, where on one direction, everyone is trying to show how unique they are and how different they are and how their identity is completely different from everyone else. But on the other hand, because of what I do as a mathematician at category theory, I love seeing how everyone has things in common and what we all have in common and if you get abstract enough, we’re all just people and we’re all really the same. There are different levels of difference that you can find so in a way, everyone is different. Or everyone is the same and everything else, every [inaudible] classification we make is just sort of some arbitrary decision in between those two things.

JESSICA:  For me, part of adulthood is being conscious of that, being conscious of what you’re grouping together and where you’re drawing your life with distinction.

EUGENIA:  Yes. That is so important. One of the things that category theory does is it says that you should always be clear what context you’re in right now and you can always change context but you shouldn’t mix up contexts. You shouldn’t be in one context but act like you’re being in another. If you’re always aware of what context you’re in, then it just enable you to think more clearly about things.

JESSICA:  Eugenia, when I met you at Empire Elixir the other day, you taught me some amazing words. Would you define ingressive and congressive for people?

EUGENIA:  I can, yes. It starts with the fact that I think that the words masculine and feminine are redundant and more than that, they’re actually abstractive. It doesn’t make any sense to talk about anything being masculine and feminine, except that a male person is if inherently, they call themselves male, they’re masculine because that’s all pertaining to male. I realized this because I had a narrative about my mathematical life that basically I went like this. When I started as a mathematician, I didn’t want to show any signs of femininity because I didn’t want people to have a chance to say that I wasn’t good at math because I was a woman.

But then when I became established in my career and I have tenure, I felt like it was okay to show some femininity but I only did it outside of work because I’m still afraid and then I gradually started letting in. Then I thought, “This doesn’t make any sense because I am a female,” so everything I do is feminine and why should some aspects of a career to be called feminine and some others not, as if there’s something wrong with me if I don’t show those aspects because I’m female but I’m showing masculine characteristics. I realized that what we really need is new words to talk about the actual character traits and separate them from the questions of gender because it’s really the character traits that may or may not cause people to think about things differently.

This is aside from the prejudice that people apply to people who present different genders. It’s just about character traits that are actually there that are different so I decided that we should have new words to replace masculine and feminine and I brainstorm with an amazing friend of mine for ages about this. We finally came up with ingressive and congressive. Ingressive is to replace masculine and congressive is to replace feminine.

The idea is that ingressive is about going into things and not being waylaid by what people think or by emotions and congressive is about bringing people with you and bringing people together and unifying and making connections between things. Maybe that ingressive behavior is traditionally associated with men and that congressive behavior is traditionally associated with women. But I think that when you make new words, then you can realize that both men and women and all genders of people can be ingressive and congressive at different times and in different situations. It doesn’t mean it’s right or wrong for one person to be one thing or another because of course, the trouble with masculinity and femininity as words is that they sound prescriptive so that it automatically sounds like women as supposed to be feminine and that therefore, if they’re described as showing masculine characteristics, then something is wrong. There’s a dissonance because they’re doing the thing that is associated with the other.

Similarly, if men show feminine characteristics and that this is problematic for all genders of people, there is the fact that male people then feel under pressure to do things like not show emotions and not cry and not worry about people’s feelings. Then female people, if they’re ambitious or powerful or they express opinions or they’re too strong, then people say that they’re being too masculine and there’s something wrong with it. Whereas if you think about ingressive and congressive instead, then you can actually think about which characteristics are good for different situations.

For example, I think that broadly speaking, I’ve come to think that congressive behavior is basically better for society but that ingressive behavior is rewarded more by society because society is based on competition. It’s based on how you present yourself so for things like competing for a job, you have to be ingressive to put yourself forward for promotion and to talk about how great you are. Whereas, when you’re actually doing some work with people, then it’s really helpful to be congressive because then you bring people together, you understand people.

I think that mathematics is usually presented in a very ingressive way so in high school, it’s about getting the right answer. If you’re good at it, then young people are put into competitions, where you have to solve as many problems as possible, as fast as possible and score as many points as possible. If you get it wrong, then you’re lost and you get a big red check mark against your work. Whereas, at research level, it’s congressive because it’s not about getting things right and wrong. It’s about understanding how things work and building structures. I think that sport is quite inherently ingressive because it’s about winning and beating the other people.

I realized this during last fall, I was teaching my art students, as usual and there was that thing that the Cubs won… Some kind of sports, I understand and the whole of Chicago was going nuts about it, except apparently me and all of my art students. We were talking about it and they just said that they were really not interested in it. I realized that I’m not interested in winning. I’m much more interested in learning and understanding and building things together. Not only I am not that interested in sports, I’m kind of put off by it because the whole idea of beating somebody else is something that I find really distasteful. I would much rather we all built something wonderful together.

That’s why I like music because in music, you’re not trying to beat someone else. You’ll all come together and you mix music together. Even in music, the process of getting to that point is often ingressive because you have to win competitions and you have to beat people in a competition in order to be selected to, say go to music school. We have all these ingressive hurdles that we put people through in order to reach things where congressive behavior would be much more beneficial. When I think about it like that, I can think differently about my interactions with other people and I can think differently about my interactions with myself actually and the fact that I am motivated by helping people, I’m not motivated by winning things.

If I try to motivate myself with reward, it doesn’t work. If I motivate myself instead with what I can do to help people, then it does work. I also realized that so many of my most frustrating interactions with other people come because they are being very ingressive. The behavior I have learned by being in the world and wanting to be successful is I have learned to basically fake ingressive characteristics because I know that and I see that that’s what’s rewarded. I have learned in the process of being a mathematician and being an academic, I have learned how to be ingressive right back at people when they’re being ingressive.

What happens when I do that is I’m very good at it and then I go home and I hate myself because I don’t want to be like that. That’s one of the reasons I quit my job because I discovered that, I didn’t have the terminology to say this at the time but now I do realized in retrospect what was happening was that I had to be ingressive in order to be successful. I felt I had to be ingressive to be successful and I didn’t like it. What I realize now is that you don’t have to be ingressive in order to be successful.

There are ways to be congressive and deal with ingressive behavior. It’s just hard and I’m figuring them out as I go along and what I hope is that we can gradually getting more in touch with our congressive side and find a way of diffusing ingressive behavior so that congressive behavior can be more valued because the trouble is that ingressive behavior is kind of louder. It’s like the fact that the loud people are always the loudest. They’re the ones who get heard. While ingressive people aren’t in charge of everything, they will continue to reward ingressive behavior as well. I think that is one of the big reasons why women are underrepresented in politics, in management, in academia, and dare I say it, also in crime because I suspect that ingressive behavior is also related to really, really being focused on winning and not really caring about losing. That’s why ingressive people, I think like playing sport because the idea of winning is really fantastic.

People say, sports teaches you to be a good loser and that’s a general assumption that that’s a good thing. But I think it’s kind of dangerous if taken too far because if you don’t mind losing, then you take risk, because you don’t care about the possibility of failure and people who take big risks will also do things like commit crimes because their idea of being arrested and going to prison doesn’t put them off. The idea of —

JESSICA:  Or start businesses.

EUGENIA:  Yeah, or start businesses. Or tell lies because the idea there is that they could succeed and they don’t really mind they fan out. But for me, this also includes the idea of being wrong because if somebody shows me that I’m wrong and gloats over it, that’s the crucial thing. I don’t mind if someone shows me I’m wrong and then we all learn something together. That’s a congressive thing. But if an argument is about someone trying to beat me by showing that I’m wrong and they’re right, I hate losing and that’s the only reason that I don’t like being wrong. I think that this is related to why people get put off at math because at a certain point, math becomes about being right and wrong and some people don’t like been wrong and they feel stupid so they go away to something that’s more obviously congressive.

But the ingressive people aren’t put off by being wrong. They’re really spurred on by the joy of being right and that, I think means that there is an accidental filter that puts off congressive people and keeps ingressive people in. I think this could lead to why women are still underrepresented in math. I think that we can change that by presenting math in a congressive way early on, instead of an ingressive way. There’s so many mathematical outreach activities that I look at them and go, “That’s really ingressive and there are ways that you could turn it into something congressive.” But you know, if you said that about boys and girls instead of ingressive and congressive, you would sound ridiculous because then you would end up saying, “This outreach activity appeals to boys and not girls.” Doesn’t that sound stupid? Whereas if you —

JESSICA:  It may be true.

EUGENIA:  — Ingressive people rather than congressive people. But if you talk about the activities that appeal to boys instead of girls, then you end up with stupid things like pink Lego.

[Laughter]

CORALINE:  I think you didn’t mentioned specifically. I’m thinking about things in terms of ingressive and congressive behavior, it makes a lot of sense to me. I just went through an interview process and interviews definitely rewards you for being ingressive.

EUGENIA:  Yes.

CORALINE:  Even the so-called pairing exercises, there is a power imbalance and you are not truly working together to solve a problem. What you’re actually trying to do is demonstrate your ability and that is competitive and you get a reward for being good at interviewing: you get a job. I remember one of the companies I worked for. I was one of two remote employees so I was already at a disadvantage because this company was not very remote-friendly and I was one of only two women on the team. The team had this approach of strong opinions they usually held which led to lots and lots and lots of argumentation. Argumentation was how decisions are made.

The two of us, first of all, we’re not able to get the same level of body language, cues and so on at the people physically present were able to do and we were not comfortable interrupting. We were not comfortable shouting and we were excluded from those conversations and I actually got dinged in a review for not participating in discussions enough and I was like, “That is not my communication style. I cannot possibly be successful in this role,” and I end up moving on.

EUGENIA:  Well, there are so many things to response to that one. One is about interrupting, which I think is really interesting. I heard a segment on NPR the other day. It was the usual thing about how men tend to interrupt women a lot. Someone called in and said, “If someone interrupts and I just say, ‘I’m still speaking, actually.'” I was thinking about it and I thought, I don’t have too much trouble with people interrupting me because if they do, then I can ask them to say that I’m still speaking but what I have noticed is that sometimes men won’t let me interrupt them but they will let men interrupt them. Then, if there’s a group of men who all want to speak, I can’t get a word in because they will only allow interruptions from men. If I wait for them to finish speaking, it never happens because they only stop speaking when they’re interrupted by a male person.

I haven’t thought of it quite like that before but I think in a way, interrupting is very ingressive and a more congressive approach is to hear everybody and to take everyone into account and not be trying to show how clever you are. I think that it is very ingressive to want to demonstrate how clever you are all the time. Whereas, I realized that what I really want to do is to help other people understand things.

There are two kinds of argument — well, there are lots of kind of argument — but there’s an ingressive argument where everyone is trying to win and the way you win is by showing that you’re clever than the other person and that they’re wrong. Whereas, I really like congressive arguments where the aim is to understand something. Then it doesn’t matter who is right and who’s wrong because whatever happens, you learn something and you’ve understood it.

I have realized that I am so much more congressive than I thought because I’ve been trying to emulate ingressive behavior to such a long time. I think a lot of the writing about how women can be successful is about teaching them how to be ingressive so that they can compete with men. That works for some people and I’ve met plenty of really ingressive women who become successful at that and who are proud of themselves for having done it. But I think there’s another way, which is to find ways of congressively dealing with ingressive behavior and I fantasize, and this is a kind of abstract dream again, where I fantasize about not an all-women institution. I’ve worked in all-women institutions and all that happens is there are ingressive and congressive women and the ingressive women take control and that’s that.

What I fantasize about is an all-congressive institution, where everyone is chosen because they’re congressive and everyone works together congressively and though, staff are working at congressive, I’m not quite sure what it would be like because it so far from things that I know. But I’ve tried to set some things up small microcosms like that for myself. For example, any class I teach is very, very congressive. I try to make it explicitly congressive so that it’s all about understanding things.

I say to my students, there are two types of question. There’s the type where it trying to show how clever you are and there’s a type where you’re trying to understand something and I will not accept the first type and I will accept all of the second type so it doesn’t matter. You don’t need to think the question is stupid. If you’re trying to understand something, it’s valid and if you’re trying to show how clever you are, it’s not valid and I won’t have that kind of question.

The other really congressive environment that I’ve set up is the Liederstube and my music saloon where we share music. I really like to think of it as sharing music, rather than performing it because performing is kind of ingressive and there’s a big risk of failure where it might go wrong. Whereas, when we just share it, all we’re trying to do is share our love of something with somebody else. We’re all doing it together. We’re not trying to be better than other people because there are so many performance situations where if you have five people performing in a row, then they’re all trying to be better than the others to show how good they are. No, we’re just all going it together to share something that we love and it’s not about judging how good someone is. It’s about doing something together.

Idealistically, I would love to do more to see more formal research into psychological aspects of this and what happens with these characteristics in education, in business, in politics, in human interaction. I think that I can see how self-help can run along these lines where if ingressive behavior is coming at you, you find ways of diffusing it. But also to think about when ingressive behavior is useful, I can think of a very, very small number of situations. It is, I think that ingressive behavior is better for people-selves. If you are ingressive, then it means that you will not be so upset by things going wrong or people being horrible to you and things like that. There’s that disgusting word, resilience which I really hate and I realized I hate the idea of being resilient because I think that resilient is ingressive. To me, it gives me the image of bad things happening to you and you don’t even care because you’re sort of oblivious.

I know that resilience is as in the common sense, for being really great but I prefer to think of transformation. I don’t want to be resilient to bad stuff. I want to be able to take bad stuff in and transform it into something good, which I think is completely different from resilience. I think it’s a congressive approach to dealing with things, as opposed to resilience, which is like learning how to be a good loser. I don’t want to learn how to be a good loser. I don’t want to just accept failure and defeat. I think first of all, we should get to a situation where there is no failure or defeat and everything is a process of transforming stuff for the better, the how idealistic do I sound.

REIN:  As you’ve been talking about this, I have figuratively and literally watched a series of light bulbs go off in my brain and this has been amazing. I thought I’d share a couple of them that may be relevant and may spark some discussion. One is that if you look at this from a game theoretical perspective, ingression wins. Ingression is about the success of the individual over the success of the group and conversely. But look at the frame that what was just presented, if what you care about is the success of the individual, then caring about the success of the individual wins.

Someone on Twitter told me, “Capitalism is math,” was the quote and I had no idea what this means but I’ve figured it out now. If you set up the system so that what you care about is individual success, i.e. Game theory, at least the common presentation of game theory, then the agents in the system that care about individual success will always beat the ones that care about group success. This is a sort of prisoner’s dilemma where the individuals do well at the expense of the group.

EUGENIA:  Yes, but then, the fact in the prisoner’s dilemma, the individuals don’t do well either.

REIN:  Well, the result, yes, is that the group as a whole does worse so they do worse. But some of them actually succeed so it’s not exactly in all cases. But then —

EUGENIA:  Right but the basic prisoner’s dilemma is that ingressive behavior actually causes individual loss as well as group loss and that its congressive behavior that succeed. That’s amazing.

REIN:  We’re all going to die because of climate change so in the end, we all lose. The other thing that I was thinking about when you talked about setting up a congressive environment is that the way that you mitigate ingressive behavior is having a power structure or relationship between participants such that you can stop the ingressive behavior. If that ingressive person was in a position of power in the group, they couldn’t be stopped. Or something them would be more difficult. But you are both in a position of power, sort of de facto as a leader, as the teacher and motivated to mitigate ingressive behavior.

JESSICA:  Or you set it up so that people can’t succeed individually. They need each other. I read somewhere that a team is defined as the people who can’t move forward without moving forward together. You can set that up in an organization.

REIN:  I agree but I also think that it requires maintenance. You can set up a team that way but that team could change to be something else.

JESSICA:  Oh, yeah just look at our government.

REIN:  Right so it requires both setting up, that framing and maintaining it over time and the people doing the maintaining are those who have the power to do the maintaining.

EUGENIA:  Well, the thing that I said about having an entirely congressive institution is that if all the people are congressive themselves, then that happens anyway. There are some tiny situations, I mean my friend Greg, who helps me come up with the words ingressive and congressive is the most spectacularly congressive person. Whenever I’m with him, I just feel like he’s one of the people I know who is explicitly more congressive than me. It’s so illuminating I would learn so much and I always feel so safe in that situation and that, if there were more people like Greg or if there were an entire environment, an institution that was entirely run by people like that, you almost wouldn’t need formal structures in place to make sure it carried on like that because everybody would be doing it anyway.

Of course, in order to try and really transform the world, we have to get more people to do it. But I have a suspicion that a lot of ingressive behavior is learned like mind was and that maybe people aren’t actually as naturally ingressive as that. People, especially little boys and especially in this country are taught to be that way because that’s how men are supposed to be and that —

JESSICA:  We’re taught that in math class.

EUGENIA:  Yeah, all of the structures of when we were growing up and educated lead towards that. Together with the emphasis on sports, the emphasis on right and wrong subjects. All of those things, children developed the things that are valued because that’s, like you say if you’re in a society that rewards ingressive behavior, then the ingressive behavior will succeed the most so people learn those behaviors. If we can gradually unlearn them and I have been on a really interesting journey of unlearning ingressive behavior in the last couple of years since I got this [inaudible]. I feel like the best things I’m doing in my life at the moment are because of that. I’m so much happier but like a prisoner’s dilemma, it’s not just individual benefit. I think I’m doing more good for the world at the same time, as also being happier. I think that all from unlearning the ingressive behavior and getting really in touch with my natural congressive roots.

I have another friend who I talked to a lot about this, who is an executive coach and she still really thinks that being a hybrid of ingressive and congressive is better but it’s interesting because she interprets some of the things I do as being ingressive. Although what I think she’s doing is she is ascribing ingressive personality to that behavior because most people who do that, do it ingressively. For example, the fact that I quit my job and just struck out by myself might appear to be ingressive because it might appear to be risk taking and it might appear to be doing something for myself. But it was very far from it because it came from a position of going deep inside myself and going, “How can I help more people and myself at the same time?” And it wasn’t a risk at all because I did it so safely. I didn’t just quit and go.

I took sabbatical. I had a one year job at University of Chicago. I wrote a book and it wasn’t until I had a promise from the School of the Art Institute that they wanted to keep me there, that I had my book was doing so well, that I felt secure, that I could earn money with that but I got myself a green card so I knew I could stay in this country. I spent three years transitioning from having a full time academic job to being freelance. I think that was an extremely congressive way of doing it.

She think that ingressive behavior is good because you step out of your comfort zone and that’s how you grow and achieve more things. I say, you can’t claim that I haven’t grown and achieve more things across my whole life but I have never stepped out of my comfort zone. What I have done is I have understood my comfort zone so that I can understand how to stretch it so that I’m always inside it. I never step outside it.

CORALINE:  It’s totally interesting that you talked about succeeding in parts by figuring out where your ingressive behaviors and tendencies were and kind of rooting them out, making a deliberate choice to be more congressive and I saw a parallel with that in my own life as a transgender woman before I transition full time. I was living kind of a double and I was trying very hard to present masculine and I tried very hard to be a man and failed spectacularly at it.

When I realized that I needed to transition, I decided I was pressing the big red reset button of my life and I could change whatever the hell I wanted to at the same time because the timing was good. Through the lens that you provided me now, I see that the people that I admired, they were not competitive, they were not people who cared about winning and losing. They were people who cared about other people. I made a deliberate effort to try to reshape myself into a person like that. For me personally, I have been exponentially more successful and more impactful since I took on or normalized the congressive behaviors and tendencies that I had in myself that I had repressed for all those years.

EUGENIA:  That is an amazing insight and I have tingles going on down my spine that you share that through the lens of these words that I’ve come up with because I really believe that when you have words for things, you can think about them differently and that opens us up to new ways of thinking. On the one hand, it’s just words but on the other hand, the words are a starting point for how we think. Just like in other languages, there are words or concepts that we don’t have and therefore, it’s hard for us to think about those emotions.

I remember the first time I heard the word, ‘schadenfreude,’ and I just thought, “Oh, my goodness. That’s fantastic,” and that’s not a word that we have so we just have to use the German word. I have always felt that once we have a word for something that we can think about it more clearly and I have not done anything as brave or dramatic as your transition but I have also gone through a small version of that where I wanted to get rid of the ingressive behavior. I realized while I was doing that, that many aspects of my previous life couldn’t continue, including my previous job.

REIN:  I just want to share one more thing if I can because I had another light bulb go off that is brighter than all of the other bulbs previously, cumulatively because you’re talking about having a new word, shapes, the thought that you can have. What I just now realized is that I’ve spent pretty much my entire adult life working towards this concept that I just now have the words to describe. This is why I have been interested in socialism because for me, the closest thing I’ve found to a philosophy in practice of building congressive organizations.

What I’ve discovered for myself just now is that’s actually the thing that I care about and socialism is a proxy for that for me. It’s a proxy for expressing in terms of how do we build congressive societies. But all of the work that I’ve done, consulting with teams in organization that trying to build good teams, has been trying to shift teams toward to being more congressive so that they work better together. I’ve just now realized that what I’ve been doing my whole career is that. Now, I have the word for it and I am so happy.

EUGENIA:  That’s amazing and it reminds me of one of the many amazing things that I heard Jessica say at the EmpEx Conference about if we think linearly, then we get to a point where we think about blame and we blame individuals for things. Whereas, on the way I interpreted it, was if we think ingressively, then we think about blame. But if we think congressively, then we see a system as a whole. Then we realize that it’s a system that we should be thinking about, not the individuals. I think that’s a congressive way of thinking.

REIN:  This is why I cared so much about systems thinking. I’ve made that a focus of my career. I’m going to be thinking about this, probably for years.

EUGENIA:  I think this is what category theory is. It’s about thinking about the whole systems as units, instead of individual things inside them and that’s what I think about society. I think there’s an ingressive view which is the one that look after themselves and there’s a congressive view which is that society is one whole, that we should always be thinking about that whole.

CORALINE:  I think that’s a great place to wrap up. This has been a stunning and transformative conversation. We’re actually going to skip reflections because we sort of arrived at consensus via little chat window while we are recording that we need more time. We need to internalize this more and think about this more so maybe we’ll blog our reflections this time around.

Eugenia, this has been an amazing conversation. You’ve really given me tools for thinking about things in a different way that I hadn’t thought of before. I love your perspective and thank you so much for being with us today.

EUGENIA:  Thank you for having me. I think it has been amazing and I have so many things that I want to think about as well and I hope you’ll continue the conversation too.

 

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