This article is the transcript of an episode of Maja Malmcrona’s podcast, where we talked about abstraction, the purpose of school, the essence of things, and failed tattoos. You can listen to the podcast on Podbean or Spotify (and other platforms). You can also read the diverter article on abstraction, as a little warm-up.

And here we go with the transcript! I hope you will enjoy it.

Two individuals giggling and chatting about abstraction. The gorgeous imperfect art in the middle of my silly art is by Maja Malmcrona.

M: Maja, J: Julia (the diverter)

Giggles

M: I have depth, ok, I’m an interesting person.

J: Well, I have a butterfly tattoo on my back, so, you know…

M: Like a tramp stamp?

J: Oh no, no, no.

M: Dodged that bullet, at least.

J: I was young and I was like: “When I get older I want to remember that it was ok to be young, so I want to do something that’s about youth.”

M: That’s beautiful!

J: Yes! But now I have a butterfly on my back. And I think about younger Julia, and I think “Well I can’t remove it, because she was so excited about keeping a message for me in the future, I can’t betray her!”

M: For me it’s the opposite, because I was 14 when I got these and I was so horrified with myself, because I didn’t have my shit together, I was just a mess. And I don’t want to be reminded of this person, I want to believe I’m someone else now. But I got them in Turkey, and I paid 10 or 20 bucks for both of them. And to remove them, because you have to go back, you have to have at least 10 or 15 appointments, and it’s like 200 bucks each, or no, for me, I had a discount, so it was 150 bucks, so 150 times 10 I paid a thousand bucks to remove them.

J: And it’s quite small! If you want to remove a back tattoo, I don’t even know… Does it hurt?

M: It’s so painful, it’s insanely painful. And you put on this cream, this numbing cream before, thankfully. But it’s still painful… My sister is removing this very big one, and like you, when you got that tattoo, she’s like “Well you know nevermind, I want to keep it forever”, in the moment, because it’s so painful.

J: Yeah, I would just do a cover up, I would just tattoo my entire arm black.

M: Apparently, the further away from the heart, the harder it is to remove, because there’s less blood flow or something. And because mine was here, the furthest away on the hand, he was like “Yeah you picked the worst spot”. If it was something, maybe like what you have, on the collar bone, then that would be way faster. Regrets.

J: Brave, brave, brave. See, young Julia, I will respect your wishes! Not because I'm weak, but because I respect you.

M: That’s sweet! I like it.

M: Ok! Let’s get started. We aim for maybe 25ish minutes and then we see.

J: How does it work?

M: I just start rambling.

J: I’m so stressed out! We were just chatting normally, and now there’s this equipment, and I’m like “Heh, don’t know English no more.”

M: I like the unplannedness, because it’s more natural. But it also puts you on the spot.

J: Yeah haha, let me brain. It’s not working!

M: This has been in the works for a very long time. We talked about doing this recording a month ago. I think. But then we got distracted by things. And now, a few weeks ago, we were like, let’s set a date, in the future, and we carved it in stone and here we are.

J: Here we are! Success!

M: We can touch on a bunch of different things. But! I think mainly the reason why we… Maybe not the reason why we’re friends, but part of the big reason was because you wrote to me. I mean we also know each other from the gym, which is where I know all my friends from. But you wrote to me and said you were writing an article on abstraction and because I make abstract art that maybe I had an input. And then we just talked about abstract art, and then we had coffee talking about abstraction over and over again since then. Maybe we can start there and see what happens. How’s that?

J: Ok, that sounds good!

M: Ok! Also, wait, who are you? Maybe tell me who you are?

J: Ok! So, I’m french.

M: That’s all you need to know.

J: The disclaimer.

M: And I like baguettes.

J: And croissants! And I’m interested in abstraction, in the context of mathematics, and how we can teach abstraction, and should we even try to teach abstraction. And that’s how I got curious about the whole thing. I wanted to understand better what abstraction even means and what it means for different individuals. And that’s why I thought I would just ask interesting people who deal with abstraction about how they define it and why they find it interesting. Because that’s something that scares people off when we come to mathematics because they’re afraid of the abstract side of it, while some people are very excited about it. And I was wondering what’s the difference there.

M: Yes, that was the initial thing right, and I think you started the article with that. You wrote “When I think about abstraction it’s very exciting and other people are like ‘Oh god no no no!’”. And yeah, it’s interesting. It’s almost like it evokes strong feelings, abstraction, I know that with art at least, and I can imagine it’s similar in other fields.

J: Yes! And I think it’s very polarising. And then I thought: are we really talking about the same thing? And my intuition was that when those people said they don’t like maths because it’s too abstract, they meant abstract as it’s too vague, it’s too unfamiliar, they cannot connect to it. And for me, when I talked about abstraction in maths, I meant the magical super tool that I can reuse to solve one thousand problems. And, as a bit of a lazy person, I thought it’s great to have one tool to solve one thousand problems! And I was amazed by it, and I thought it was super powerful. And also to me, maths feels like something that we built, as human beings, and it’s a link that connects several generations, and we made together that very beautiful tool that we can apply and reapply to solve many difficult problems. I thought this was very powerful. And my feeling was that maybe, if I can show that to people, they will start liking maths as well.

M: So you talked to me, an artist. You talked to a philosopher, you talked to a linguist, a mathematician… Am I getting this right? Was there someone else?

J: A bunch of computer scientists.

M: Ah yes, because that is your field. So, you asked them to define abstraction. How did it come out?

J: I think I found many definitions that were close to what I was expecting or hoping for. But I was impressed by the connections between the different fields. In mathematics and in Computer Science, you can see how it’s about zooming out and seeing the bigger picture and say “This is a pattern that happens in many, many different situations and that we can use to solve all these different situations.” But then if you speak about linguistics, it was also the same thing. Abstraction was about taking away all the superficial details and going to the essence of things. But then, when we come to art, and in your context, then I feel a bit of a difference. Because in maths we want to remove all the superficial details, we want to remove colour, we want to remove texture, we want to remove those things. But in art, and in what you do specifically, there is a lot of texture, there’s a lot of depth, there’s a lot of tangible details to it. But that’s a tool that you use to express this essence that’s behind it. Or at least that's what I felt from your definition. And I thought that was interesting, because we are using something concrete to express something abstract. It’s something that I can touch, but still takes me to this more abstract place. And one of the feedback I got, because the people I interviewed also read the article…. Thank you! Very nice!

M: We were just being narcissists, like “”Wooh, look at my name in an article!”

J: Giggles The computer scientists were very touched by your definition in particular, because they were like “Oh, I had no idea…”

M: But wait, what was my definition, because I can’t remember. It was the removal of the superficial externals to convey the essence of something. And I was not talking about abstraction in general, I was thinking specifically about art. If you abstract in art, not in order to obscure, but in order to unveil something, unveil that essence, whatever that is, basically.

J: Yes exactly! And I thought that was a super interesting definition. Because if you look at the words that are being used to describe it, they are extremely similar across the fields, and that’s how I got computer scientists being like “Oh, I have something in common with an artist, or a philosopher!” and I was like “Yeah! Although you’re expressing it very differently, you’re on the same path in a way”. So I thought it was very interesting to connect those things. And now that brings me to something that maybe I would like your opinion on: I think that when you get it right, abstraction can look or be perceived as simple, in a way. That’s something I hear often about abstract art. People say “Oh, I could have done that!” or at least some specific parts of it. But it’s about the process of getting to something as simple to express this… In maths it’s the same, once you have abstracted properly, the result is extremely simple, it’s not so many symbols, it’s something very simple that can express all those things, it’s because you did it right. And so abstraction is not so much, in that context, about these simple forms, it’s about the whole process that tells you, ok, we can express this complexity like that. And I’m curious if you feel that relates to art in any way.

M: I have a sign in my studio that says “Paint a square”. And it’s basically a guide for when I’m stuck and I don’t know what I’m doing, rather than doing nothing, just paint a square, just paint a f*cking square. And yeah, again, on the one hand that’s just a way of getting over procrastination, but it’s also… Like a lot of the artists that I love, one in particular, Aleksandar Krajinović, I think he lives in Berlin, he’s essentially painting squares on white canvas. And Mark Rothko is also one of my favourites, he worked in colour field movements or genre, whatever it’s called, and he was just painting squares in different colours on a canvas. And you want to think that painting a square is very easy, we all did it in preschool, in order to just paint a square that is conveying what you want it to convey, not in terms of what it says to you, but in terms of what it feels like… And it’s so hard, I can look at artworks that someone else did and maybe it’s just a square and if I were to try and replicate that, there’s no way in hell I would be able to. Because it’s not about… Maybe it’s in terms of the language used: It’s not about simplifying something, it’s about distilling more, you make it more precise. You don’t make it vague, you make it more precise, by universalising it to some extent, by making something that a lot of people can relate to but on a very deep level. And that’s very difficult, I find. Does that make sense?

J: Yes, that totally makes sense. I agree with that as well. One thing we discussed in the article… You started by using the word vague as well, and that’s something we find a lot in maths as well. And something that came around in the article, especially from the philosopher point of view, was about: Do we need to abstract, and how far do we need to abstract? And if you think of abstraction in terms of zooming out, which was the mathematician definition, which is like: “Ok, you can look at the level of tables, and then rectangles, and polygones, and you can go as far away as possible”, but you need to also have the spectator of it to be able to connect all those things to some tangible experiences that they can relate to, and go through. And what the philosopher said was, do we really need to abstract, sometimes? Or are we losing something in abstraction? And I thought that was quite interesting, because it’s true that although sometimes we’re trying to find something that can encompass all those situations, all those experiences, sometimes we lose some people on they way and when they see it, they don’t relate to it at all and they don’t connect it to all these things we looked into in order to create that.

M: I think you and I have talked about that before as well. You mention texture in my work too. I do the same essentially, I just paint black squares over and over again. But they’re very dirty, and ragged, and just imperfect and ugly to some extent. Which I don’t mean in any bad way even, I like that, and it’s almost intentional to some extent. But it comes to abstraction and art, and maybe modernism and the modernist movement. Maybe in architecture it makes most sense. If you create, you build buildings, and they’re super ornamented, and it’s super fancy, and they’ve got little angels and stuff on them. Largely what the modernists were trying to do was to remove the context, because they wanted to get rid of the baggage that comes with sort of cultural context in terms of class and all of those things. Because all those ornaments, and all those symbols, mean something. And those meanings have implications, in terms of hierarchies in society and what not. So they wanted to get rid of that and just started building white boxes. And a lot of people hate those white boxes, but I like it and I also don’t. I think the intention is good, fundamentally. And in art it was sort of similar. I’m, I mean this is a simplification, but artists, maybe, didn’t wanna paint images of Jesus anymore, because that means something, that means something very specific, which excludes a lot of people. And just started painting sorts of symbols and geometries, that gets rid of those implications. But, like you said, if you remove all of that, you remove all of those implications, and all those hidden meanings. And if you take that very very far, you remove the human being altogether, and, again, there is something to that, but I think we shouldn’t get lost in that. And abstraction doesn’t mean anything for us in the end. It does, but we can never live there, we can never go there and visit that place, we will always be stuck in those worlds of this ugliness and dirtiness and what not. So I’m trying to find a balance between both. You have the abstraction, but you keep the human element. And maybe for me, that’s the texture and all the mistakes that are sort of incorporated onto the canvas, that I don’t erase, they’re always gonna be there.

J: Yeah, I like that a lot. You said that they wanted to remove those symbols that have strong meanings, and then this raises the question of what is left, and what is the meaning of what is left? Because before we had very clear ways of interpreting those symbols and the meaning was very obvious. But once you remove that, once you explore a new language, is there still a meaning? And can I still, as a person, understand that meaning? I wonder how you can process it. I feel like the meaning is a bit different and it’s not as explicit anymore, and it gives a bit more responsibility on the viewer, to be seeking for that meaning a bit on their own. So it shifts the perspective a bit from the creator who gives you a very clear meaning to interpret, to, ok, maybe you can build some of the meaning on your own.

M: Yeah, and not everyone has this sort of ability and energy and time to create that meaning from themselves. Off-air, we talked about religion. I’m not religious, whatsoever, I never have, never been. But I don’t shun it in the way that a lot of people do, as in it’s the root of all evil. I think it's probably the root of a lot of evil, but I can also see how it’s very valuable for a lot of people because it grounds you to some extent.

J: It’s quite interesting right, because in those terms, I feel we could say that religion gives you a concrete meaning, although if we think about God, it’s maybe not the most concrete thing we can think about. So I think it’s quite interesting how it’s still dealing with abstraction in a way, but still gives you very concrete ways of thinking about abstraction. Which I think is quite interesting. And I think we don’t have a choice. If you look at art, if you look at maths, we are using concrete tools to speak about abstraction, we use concrete symbols that you can draw with a pen, we use concrete texture that you can touch, that you can smell, that you can feel, and that’s the only way. If we want to capture abstraction, we need to represent it in so many different ways, to take oneself there. And I think that’s quite nice about human beings: that we’re even trying to do that. Everytime I think about it, I’m like wow, that’s quite brave!

M: Last week I talked to Adrian, who’s a mathematician, and we ran out of time so we didn’t dig into it too much, but I asked him what he thinks of the relationship between art and maths. And he said that, at first glance, a lot of complex mathematical graphs, and equations, and stuff, I don’t know anything about this by the way…

J: Giggles Numbers!

M: Ones, and zeros, and you know, those things! They can be very beautiful. And I’m wondering about beauty, because this is something I’ve thought about a lot. You and I went to a Jazz concert, which I also talked about in podcasts, and I ranted to Emma about this Jazz concert, and it was awesome. And it made me think about beauty. What would you think about abstraction and beauty, is there anything to be said about that? In maths, in computer science? Does that connect with you in any sense?

J: Yes, totally. Why I’m so attracted to those fields is because I find them extremely beautiful. It touches me deeply. And when I do maths, or when I do computer science - this sounds weird -, to me it’s a creative process. I’m always looking for ways of expressing myself and things that I have inside, and this is a way that I use to do that. I have times when I create something that I find beautiful, be it an algorithm or a theorem, and when all the pieces come together, and all the process comes and I reach this very simple expression of all those things that I was trying to express I find it extremely beautiful. I have a maybe more concrete example of beauty that I find interesting. I remember in maths class, when I was taking notes, and I was doing my proofs, the teacher came to me and he said “You are not doing maths anymore, you are doing calligraphy.” because maybe my handwriting was very good. And somehow he found it shameful to write maths beautifully and I thought no! I respect that field so much! You could take my notes and frame them because that’s the beauty of it. And I thought it’s ok to make it beautiful, I think it’s beautiful, and I think it’s ok to be creative in the way you represent it. It doesn’t have to be cold. And I think many people who follow the path of mathematics find it extremely beautiful, and it touches them. And I think that’s why it’s also such a polarising thing to talk about, because people who love maths, they love it so much that they’re gonna fight for it. If you say maths is not useful, they’re gonna say “WHAT?! Maths changed my life!” And that shows that it’s not as cold as it can seem because we’re so attached to it. And I feel the other half hates it which is also a strong emotional connection. To me that shows that there’s beauty in it and that’s the kind of connection that mathematicians have with maths. I saw a definition of maths that I thought was quite nice: It’s the longest connected human thoughts. As in, mathematicians keep building it and it’s the thought that has been going on for the longest time. And I thought this was a very beautiful way of thinking about it. That’s how I connect to it as well, maths is something that we are making together. And in that sense I find it immensely beautiful.

M: Again, it’s interesting that sort of polarisation. Maybe this is sort of simplifying, but I feel this has to do with, if you don’t understand something, it’s very easy to just instantly dismiss it. And your field is Learning… Sciences as well? Did I get that right?

J: Giggles Yes.

M: Huh, good, I’m sweating already. I can imagine that the way we teach subjects like that is a big problem. I am useless at maths, but I don’t necessarily think that I have to be. Because I remember when I was young, in highschool, going to maths class, we would go through these very simple things. I was in the lowest class, I was very bad, well, not very bad, but I was pretty bad. And the teacher is like “Now you have to do this, because… just do it.” And I was the person sitting there “but why? But why? Please tell me the bigger picture because otherwise I just can’t, it just doesn’t make sense to me.” And I had a lot of people say that. I listened to a podcast or an interview with a mathematician, who was like “I was really bad at maths! But then when I moved to higher level maths, it suddenly made sense, because now I know what I’m doing here!” And isn’t this the case with everything too? Back in the days, when you were asking your teacher in maths class “Why am I doing that? Why am I doing any of this, I don’t understand why?” and they would be like “Well you have to know this! Because when you’re in the store you have to know if you’re paying the right amount because you’re not gonna be walking with a calculator all day!” Which nowadays is not even true because we have iPhones! That’s sort of a way for them to connect it to the bigger picture, of why we are doing what we’re doing. Do you think that’s a problem in the way we talk about maths?

J: I think there are many problems. One being really concrete, at least in France, where it’s considered as an elite thing, and if you’re good in maths you’re smart, and if you’re not good in maths, you’re not smart. And I think this doesn’t make any sort of sense. But this doesn’t help. Then at a deeper level… Well maybe I’m gonna make some people angry.

M:: Yay! Controversy! This podcast will go viral!

J: The hot takes are happening! But… People are discussing… Should we keep maths, is it useful? But I don’t think school is about learning something that is useful. I went to a talk one day, and there was one researcher in theoretical computer science, which, to me, is very close to maths, and a student asked “Why are you researching this, what can it be used for?” And she said “I’m researching it for the beauty of it, I find it beautiful, and I want to contribute to this beautiful thing. And yeah, maybe it’s gonna turn out to be extremely useful one day, but right now I don’t know.” And I thought that was a very respectful thing to admit: I do it because I find it beautiful, because it touches me. And as human beings, that’s also a valid reason for doing something. Then about maths in particular, I think that when they try to give these sorts of concrete examples about where it can be applied… It’s like when you have these puzzles: Oh, I have seven candies, and I give three and a half candies to my brother, how many candies… This is not relatable, this is not concrete, in the sense I cannot connect to it. If I have to give candies to my siblings, we’re more likely to fight over it than to count!

M: Giggles Yes! I will give zero candy to my sibling, you think I’m gonna split that in half?!

J: Yes! It’s very simple maths! And I think this is fake concreteness that we’re trying to map over it. And something I got very lucky with is that my parents were studying when I was a kid, and I was seeing them solving equations, and I was like “Huh, it has numbers, and it has letters, and I like numbers, and I like letters, and I want to know how to do this!” And I saw an aspect of maths that I liked very much and then I went through school and highschool and I only found the beauty of maths again when I decided to study maths. But in highschool and before this I did not find this abstract content that I liked so much and I didn’t find the beauty of it. But I knew from my childhood that it existed so I kept pursuing it, but it took a while to come back. And I feel we might be losing people here, because we’re trying to do this more concrete maths. And even if we say “Yeah, you need to learn this theorem about triangles, you’re gonna use it everyday, because every time you’re gonna see a triangle in your life, you need to go and measure it.”

M: Come on! We know that’s true! Don’t you? Every time I see a triangle I have to measure it with my good old ruler.

J: Giggles Yeah, and it’s a bit embarrassing.

M: I just can’t help myself!

J: And I feel it’s a bit fake, and people are not stupid, they’re gonna see there’s a lack of meaning here, that’s not the meaning that maths is bringing, in my opinion. And that’s something we’re trying to explore. Should we start with concreteness, should we start with abstraction? And the results are not obvious. Some studies show that if you start with concreteness, people are going to be struggling more to get to this abstract level, which is so beautiful and important. So I think it’s a bit more complicated, and I think it’s not correct when we speak about “Is it useful?” Is art useful? We can debate about it for a very long time and I think it’s a similar debate when it comes to maths. There are many very concrete applications of it but I think many people who do maths don’t go around like “Yay! I’m gonna make bridges with my theorems!”

M: We are running out of time, but I asked in my Instagram stories “What is the meaning of poetry?” Which to me is the same question, than saying “What is the meaning of beauty?” or something. And the first answer was “It’s completely meaningless and that’s why it’s beautiful.” And I thought about that when you spoke about that woman who said “I do this because it’s beautiful and that’s it.” Maybe meaninglessness is not so meaningless, you know? Maybe there’s something to it that we need to value a bit more.

J: Yes I totally agree. And I think that’s also something that can come from your heart directly. As in, I’m attracted to this, I want to figure it out, I want to understand, I want to play with it. And I think that’s ok. I think we can be playful about things. I think we can be curious about things without having a clear plan, like “Ok, this is gonna take me there, this is gonna advance this and that.” And coming back to school, I think school should be more about this. Of course, it has to teach you some useful things for afterwards, but I think it’s such a valuable place to follow your curiosity and discover, and children are so curious. And if you start telling them “Yeah you need this to go grocery shopping!”, are we not forcing a narrative onto them that they don’t necessarily need to be curious about something? So yeah, I agree.

M: Yes, whoever this person was who wrote that comment, hats off to you. We both salute you. I can’t remember who it was! I should boost them!

J: Shout out!

M: Exactly, shout out to this random person! Ok, cool, we’re gonna wrap up! Thank you, this was fun!

J: Thank you very much!

M: And until next time! I still have no endings, I’m bad at this… So… Farewell!

J: Farewell to you all!

To go further:

Some beautiful definitions of Mathematics: (link)

Abstraction over concreteness in maths education: Kaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (2008). The advantage of abstract examples in learning math. SCIENCE-NEW YORK THEN WASHINGTON-, 320(5875), 454.

Or maybe not? Trninic, D., Kapur, M., & Sinha, T. (2020). The Disappearing “Advantage of Abstract Examples in Learning Math”. Cognitive Science, 44(7), e12851.