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In the twenty-fourth episode of the second season of the “Saturdays at Seven” conversation series, Todd Ream talks with Satyan L. Devadoss, the Fletcher Jones Professor of Applied Mathematics and Professor of Engineering at the University of San Diego. Devadoss opens by exploring how mathematicians quite often find themselves encountering that which is mysterious. Acknowledging that mystery, persisting through it, and, at times, identifying results that may explain it yield intrinsic joy for mathematicians. Devadoss discusses that while he experienced that sense of joy in high school and college, he lost contact with it for the first few years of graduate school. Eventually, he found that playfulness—playfulness that offered little to no immediate use was one way to reconnect with that joy. Devadoss claims that eventually scholars in other disciplines may identify a use for what mathematicians offer but that process, while still with no guarantee, may take decades or even centuries. Devadoss’s own book-length projects concerning discrete and computational geometry as well as unsolved mathematical problems were designed as sites where that joy may be found. Devadoss then closes by discussing the rapid acquisition of power mathematicians are presently experiencing and how the greatest expression of their vocation during this season may be to “bend the knee” and share that power with scholars working in other fields—fields, according to Devadoss, that often demand persistence through far more complexity than mathematicians face.
- Satyan L. Devadoss and Matthew Harvey, Mage Merlin’s Unsolved Mathematical Mysteries (MIT Press, 2021)
- Satyan L. Devadoss and Joseph O’Rourke, Discrete and Computational Geometry (Princeton University Press, 2011)
Todd Ream: Welcome to Saturdays at Seven, Christian Scholar’s Review’s conversation series with thought leaders about the academic vocation and the relationship that vocation shares with the Church. My name is Todd Ream. I have the privilege of serving as the publisher for Christian Scholar’s Review and as the host for Saturdays at Seven. I also have the privilege of serving on the faculty and the administration at Indiana Wesleyan University.
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Our guest is Satyan L. Devadoss, the Fletcher Jones Professor of Applied Mathematics and Professor of Computer Science at the University of San Diego. Thank you for joining us.
Satyan Devadoss: Thank you so much. What a joy to be here, Todd.
Todd Ream: As with other guests who contributed a series concerning mathematics as a vocation, we’d like to open our conversation by asking about a 1960 article by Eugene Wigner. For individuals unfamiliar with him, he served on the faculty for a number of years at Princeton University, served as the Director of Research and Development at what became the Oak Ridge National Laboratory, and won the Nobel Prize in physics in 1963. In February 1960, he published “The Unreasonable Effectiveness of Mathematics and the Natural Sciences” in a periodical known as Communications in Pure and Applied Mathematics.
When reading this article, I found that he used some rather curious and inviting forms of language. For example, in one of his two points, he claims that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious, for which there is no rational explanation. In his second of two points, he then contends it’s just this uncanny usefulness of mathematical concepts that raises the question of our physical theories.
To begin, how do you respond to his curious use of language? Are mathematicians, for example, prone to using terms such as mysterious or uncanny? You know, how might you respond?
Satyan Devadoss: We are. We absolutely are. Most of our lives are surrounded by mystery. You know, every time you figure something out in this world, whether how to open a can of food, or how to open a door, or how a submarine works, whatever it is you figure it out, you realize, oh my gosh, there are 10 new questions I want to ask. And so, every time you get an answer, we are so curious as humans that we have 10 new questions that we’re trying to figure out.
So, if everything is solved, you just multiply it by 10, there’s 10 times more unsolved mathematics that any idiot would be thinking about, including me, than that is known. And so, we use, as mathematicians, we use mysterious all the time, because we don’t know about the 10, right? These things are alluding to some bigger stories out there, we have no idea.
On the other hand, what you’re talking about, Todd, is really interesting because that article is talking about mysteries of the connection between mathematics and the sciences in the natural world. And my response to that, I guess if you just want if I can just give you like a straight up response, I would say that I am foolish to be able to respond to that question directly. Because one of the joys of mathematics is actually to take ideas in the science world, bring it to the math world, and then completely ignore the natural world from that point on. Like, thank you for motivating me to think about this joyous question, and now I’m just going to be living in this abstract realm of glory.
And the scientists actually does almost just the opposite, right? They are trying to figure out the scientific world. And they’re using mathematical tools and structures and patterns to put scaffolding to make sense of what the heck is going on. Like, how does gravitational pull work? How does a cell, biological cell replicate, right? What does DNA look like? How does chemical bonds work? And so, they are thinking about mathematics as structures and patterns.
The thing about mathematics to me that I love is we’re about a hundred years, fifty to a hundred years ahead of what science is doing because we are just playing, right? We’re just thinking about these structures and patterns, the most abstract and I would truly say useless way. And 150 to 100 years later, those useless things eventually become useful to the scientists going, wait a minute, we want to think about atoms this way, but in fact, we don’t even know what these particles are. Let’s call them ABC. Oh my gosh, you guys already have a language for thinking about interactions of abstract things. We’ll borrow that.
So I would love to respond in a way that talks about this aha moment about this article. I think it’s a beautiful article. I’ve read it. It’s a really well written and quite a famous article. But in the mathematical world, I think this exact notion of useful, I think the quote is usefulness of math in the natural science, is exactly what I can’t say because I’m not a scientist. I’m not a, certainly I’m not a Nobel laureate in terms of physics, right? I’m completely in the opposite box. I’ve taken math ideas and just played with them.
Todd Ream: At what point then you know, in terms of this curve that occurs in history do the ideas that mathematicians work with then merge back with and become evident and perhaps in some cases useful to those in disciplines such as physics and chemistry? Is there a way to sort of chart what that curve might look like?
Satyan Devadoss: That’s a lovely question. I think the right person to ask would be a true math historian, but there’s some rough ideas that I have, which is clearly if you go back to the Greeks and the Romans and even go to the Egyptians and the Indians and all these great cultures, the notion of theory versus computation, we’re always hand in hand when you’re doing something, you’re calculating something, right?
You’re thinking of an abstract idea, wait, this is true for all triangles? And then you sit down and then you have to calculate something and figure it out. So this, this notion of abstract theory and what truth is about how something’s working versus let’s actually sit down, lift up the hood and see how the car is working and tinker with it. They always were together.
And so when we today talk about AI or computational mathematics versus pure mathematics and theoretical mathematics, that divide is quite artificial if you go back a couple thousand years ago. I would say the main split, and this is just my naive sense as a, just a teacher who’s been thinking about some ideas, would be Carl Friedrich Gauss. Gauss was basically one of the gods of mathematics who ruled the earth. And I think he was so proficient and prolific, and brilliant, that he actually knew the entire landscape of math.
He knew how numbers worked, how functions worked, how geometry worked, how algebra, he kind of knew everything and, but he was so prolific that afterwards, math became shattered into pieces, like nobody can see all of math anymore. You become specialized in these smaller things.
And of course, the Renaissance pushed the way to the Enlightenment, where you then become very specialized, right? You’re not, you’re not this holistic painter, biologist, artist, the way da Vinci is anymore, but you are now just a painter, who focuses on abstract surrealism from the 18 those kinds of things. And I think the same things happen in math.
Todd Ream: Would it be beneficial then to both mathematicians and say physicists and chemists if that curve in history, there were ways to draw the two closer together so that the time that lapses isn’t as great perhaps that it might be now or would that actually diminish the creativity that mathematicians seek to exhibit and artificially force ideas into say disciplines such as physics and chemistry prematurely?
Satyan Devadoss: I would not want the curve to touch at all, as, this is, this is my personal opinion, right? Like, for my life, not for the world of mathematics and the world of the sciences. And the main reason is, and this is a weird thing to say, as a human, I’m really interested in the natural world because I love ice cream. And I enjoy a glass of wine and pizza with my friends, right? So I love this embodied life that I live.
But as one who’s a mathematician, I want to play. So I don’t want to get into the realm of Oppenheimer. Where Oppenheimer’s playing and at the same time he’s going, oh my gosh. I am playing with ideas of God. Like, I can now, with my powers, destroy things.
If you come to my world, I’m talking about unfolding 300 dimensional boxes into 299 dimensional space. For example, you could take a 3D box and cut it open and fold it flat into 2D. And I’m doing this in higher dimensions. And an actual question is like, who cares? And the answer is nobody. Like I care. That’s why it’s glorious.
It has no ramifications to nuclear proliferation. It has no ramifications to the new COVID vaccine that I might be developing the antigens for, the new creation for. The disconnect is so there that I’m allowed to play without the dire notions of consequences that scientists deal with, that I don’t have to deal with. And so in one answer, selfishly, I want them to be disconnected.
Having said that, this physical world and applications that you talk about math and bringing that curve together, I think it’s actually important for many people to do that. So I think it shouldn’t be a field, like the entire mathematics shouldn’t arch towards holding hands with the sciences anymore. But there are many mathematicians and scientists who know how to connect those two things really clearly.
And we call them nowadays, I mean, applied mathematicians, right? People who work with scientists talk about the way birds fly in the sky, or the way bats evolve, or the sonar technology is like, oh my gosh, that reminds me of a math idea. I can use these modeling tools that predict what the bat might be doing at night and how it might be moving its wings. And so there’s this playfulness that’s going on.
But I think it’s kind of a discrete set of if you want to think about these two worlds, I can think of it as like a discrete set of tentacles that touch those two worlds. But I do not want the entire sheets to glue together, right? I just want these handful of things to say that this is joyous. There’s playfulness there as well, but in a general math context, we are empowered to live in this other world to just play.
Todd Ream: Thank you, thank you very much. I want to ask you now about your work own development as a mathematician. You earned an undergraduate degree in mathematics in North Central College and a doctorate in mathematics from John Hopkins University. At what point did you begin to sense mathematics would prove central to your sense of vocation?
Satyan Devadoss: That’s a great question. I think that really happened around my third year in graduate work. I didn’t enjoy math until that time. I guess to give you a sense of what’s going on behind the scenes, I loved Legos.
So I grew up in India. I was born and raised there. Never saw a Lego in my life. We had rocks and books and simple things, but no, none of these amazingly gorgeous mass manufactured stuff, right? India in the 1970s and eighties were just like beautifully handcrafted work because work was prevalent in terms of just people being there. But in terms of machines, that wealth did not come. Like a matchbox car would be more money than you could imagine for a kid to ever see.
And so when I came to the States in the 80s, I saw Legos and I couldn’t believe that it’s this toy that you could break apart and play with. And as I was playing with them, Todd, it was interesting, looking back in my life I realized two things about those Legos when I would play with them in high school and junior high and most likely even college if I remember. But the two things are I love to make things and then I love to talk about it.
The thing I usually end up, I’m just sitting in my room. I made some tractors, something like that. And I would just be talking truly to no one in the room and say, this tractor is amazing. Not only does it have a four by six clutch I would just be kind of trying to sell the tractor in my own mind to an audience that didn’t even exist.
And when I went to North Central, my dad was a professor there, so it was free for me to go there. It was great, you know it was a lovely experience. They had a 3-2 program. You go there for three years, and then you go to a place like Urbana Champaign one of the big universities in Illinois, and you go there two years. And you have this liberal arts broad background for the three years and you get this focused engineering in two years and you walk away with like a math and engineering degree. Perfect. That, gosh, I wanted to be a mechie to be able to play with gears and pistons and stuff. So I finished my undergrad degree and I realized, oh, I could just graduate in three years rather than going two more years to get any extra. So I took a lot of AP credits and I was a nerdy kid in high school. So I was able to graduate in a few years.
And I knew as a kid, it was absolutely clear in my life that my parents said, you have to get a PhD. Just like in your life, it was clear that you have to go to 4th grade. Like there was never a conversation about whether 4th grade is an option. It’s just if there’s a bully in 4th grade, you wouldn’t say, Mom it’s tough. I think I’m stopping right here. Like that just never would even occur to you, right? You just, you just get it done.
And you know, if you look back in the US in the 1950s, you could imagine high school might have been, but college was like a luxury to send kids there. And nowadays, college is almost the new high school, right? You might have to send your kids to college to get employed nowadays.
And so in my family, both my parents had PhDs. My aunt had a PhD. It was just kind of clear that that’s just—they came from India to get a PhD. So it was clear to do that. And so I went to Johns Hopkins, had to get a PhD, so I figured why go two more years and then get an engineering degree? Let’s just get off the train. How do I get off this? How do I get off this? I mean, out of true laziness, Todd. How do I get off this academic train as soon as I can?
So I was like, you know what? Instead of getting two more years of engineering undergrad and then doing a PhD, why don’t I just knock it out now? So I went, started with a math PhD. I met my wife at that time. She’s like, what are you doing? I mean, she was a kid. We’re both kids at that time. She’s like I said, I’m doing this PhD thing in math. She goes, I hate math. I hate it too.
And I really did not enjoy it because a place like North Central, a small college, you would know, a liberal arts school, you’re learning so many different disciplines. And math is one of it. And I love math, but gosh, I love philosophy. I love art history. I love the chance of econ just to play with all these different ideas. I took a class on bicycle maintenance, which I thought was phenomenal. And you go to grad school and all you’re doing in a PhD program is math a hundred percent. And that just drained me.
So to finally answer your question in this long winded approach in my life, I just want to say that it wasn’t until my third or fourth year in grad school when I finally got to play with my own ideas that I felt like I was playing with Legos again. I wasn’t following the instruction manual that would come with the Lego kit, what the first two years of grad school felt like, it’s like do this and do that. The third year I was like, oh my gosh, wait, I could just take these pieces and do what I want. And then I fell in love with them and I realized how glorious it was.
Todd Ream: Were any individuals along the way besides, perhaps your father helpful in terms of reconnecting with that love or connecting that love and then igniting that passion?
Satyan Devadoss: I don’t think my parents had any direct connection. The kind of work my dad did for math was, I find it archaic and boring, in all honesty. It’s like that part of math wasn’t exciting to me. So what they did do is clearly draw the line that this is going to happen, like you’re getting a PhD.
And the fact that they did it was listen, we’re scrubs. We came from India. We can do this. You’ve grown up in America to a certain degree, and you could certainly succeed here. So, it was made clear that it wasn’t that I couldn’t do it, or I was a brilliant kid. It was just, hey man, if you care about it, check the box and get it done. And so that was really instilling to me.
But in terms of other people regarding mathematics, I think the most important thing to me was actually things that weren’t related to mathematics. Like the community around me, like what it means to go through a hard time living alone and brothers and sisters around you that love you and take care of you. What does it mean to talk about issues of faithfulness?
Now I’ve left my home base; I’m alone in Baltimore. I’m out of the Chicagoland area and plugged into communities, plugged into churches. And you could say, oh, my gosh, there are people who are 30 years older than I am who are not my parents were speaking into my life in a really different way. So I think that really protected me and honored me throughout that process of grad school.
And there, and then just to close up, there are two young men who are a couple of years ahead of me, but all the three of us, they’re twins, the three of us all graduated from North Central together. And they’re still my best buddies. And I would say that those guys have been praying for me, fighting for me, calling me up, how’s your life going? And they all the three of us studied mathematics together. So that joy of kind of working together to launch was really, really lovely.
Todd Ream: Thank you. Are there any individuals in the history in your field or related to how you view mathematics, that have been helpful to read or instrumental in terms of how you think about your vocation?
Satyan Devadoss: Oh my gosh, that’s a tough question. The kind of math I study, I love to draw, so most of my math is very visual. I think the closest kind of touch point that people would have is geometry, which is just a study of shapes, like triangles and pentagons and stuff.
But I’m a topologist, which is if you just take the same shapes, but you just remove the skeleton out of it, right, so it just becomes like this wobbly thing. And instead of talking about a rigid, perfect sphere, you talk about deflated like a deflated ball, right. You can start talking about a rigid donut that you’d get a Dunkin Donuts. Now you just take the, take the filling out of the doughnut kind of becomes this floppy thing and that’s a topology. It’s a much more difficult thing to get a hold of sometime and understand what, how that shape is.
And so Henri Poincaré, who is one of the fathers of topology, the reason it’s hard for me to have kind of models in front of me are these are gods who walk the earth. And I feel like I truly am a little speck of an idiot who is just in this world and Poincaré can see things and connected ideas from topology to algebra to analysis to, and he’s like, you know what, why don’t we use, it’s almost like imagine you’re building a car and somebody goes, hey, let’s use that part in your house that goes that you flush the toilet. Why don’t we use that part for it? You’re like, wait, what? That’s not a car yet. That’s not a car part. He goes let’s try that.
You know the, radio transmitter, let’s just use the antenna and you could use that as a coil, like wait, that’s not a car part. You’re just you’re able to connect all of these different things that don’t belong together. That’s Poincaré’s power. That’s, that’s what the gods can see. They can see the landscape and connecting so it was, it was lovely to have him as somebody to aspire to, but very different worlds from my life.
Todd Ream: As a topologist, then perhaps it was that return to that experience, like with Legos, but can you say more about how you came to topology? And how it became a passion of yours and central also to your sense of vocation?
Satyan Devadoss: The most influential person in my mathematical career by far, and this is probably true for almost every mathematician, is your PhD advisor. That is your father or mother in some sense, like that is the person who you want to be like, who you shadow, you sit under, and you learn from. And my advisor, his name was Jack Morava, he’s a professor at Johns Hopkins. He was a superstar, he was like a mathematician’s mathematician.
And he could have I think he could have been at Princeton or Yale or Harvard any top place, but his wife was a superstar linguist at UVA in Virginia. And so these guys were kind of powerhouses. They don’t want to move anywhere else because they were kind of close to each other. And it was this kind of great collaboration where he would spend most of his sometimes at Johns Hopkins, but then go back on the weekends to be with his wife over there. And so they were there together.
And I learned so much about Jack, namely that he’s nuts. He’s nuts as in his brilliance is off the charts. And he would also, as an advisor, was a brilliant advisor and one of the worst advisors. And let me just explain this to you. He wouldn’t help me to know what the next step to do in a problem or an idea. So even when it was the, when the thought of topology was thinking let me, let me talk to Jack and he’s a topologist. He’s a, he’s a superstar algebraic topologist, he does very deep things. And I said I’d love to learn about things like knots, how to tie a knot, those kind of shapes, kind of the simpler topology in two dimensions and three, just, let’s get really visual. I love to draw. What can I play with? And Jack said, I don’t know anything about those things. So why don’t you learn it, teach it to me, and we can play around with that idea.
And so we started doing this and he would just be giving me ideas from four or five years into my future. So he would just be saying, hey, this is what I’m thinking about. This idea comes to me and this idea, and I’d be writing down things going, it’s going to take me two weeks to understand those words that you just said, Jack, you know. And I would, I would spend weeks just figuring out what those ideas were and I’d realize I don’t know how to connect that at all.
But 15 years later, Todd, he was right in everything. Like he was thinking about these really big intuitive notions of how things fit together, that I would just be unpacking over the years going, wait a minute, Jeff was right. Like that’s why he had said that. The reason he was a bad advisor is he said that to a grad student who’s an idiot. And he should have been saying it to like a full professor somewhere. But you know, but that’s how his mind works, right? He’s connecting these things. So he was incredibly influential on the way I personally think and try to connect ideas up also.
And my strengths and weaknesses I realize is echoing his. In fact, when most people ask in the math community, almost no one asks where you went to undergrad. No one asks almost where you went to grad school. The number one question everybody asks is, who’s your advisor? Because your advisors can move around also, right? Like, you can be a professor at Princeton a little bit, then go to Berkeley. And so, it’s not Berkeley or Princeton that’s exciting.
It’s, hey, who, where’s your DNA from? And the moment I say, oh, I’m up, I’m Jack Morava. They go, oh, you’re Morava’s kid. Okay, I know exactly how that brain’s working. You know, and how crazy that could be. And so, um-
Todd Ream: We know who we’re dealing with.
Satyan Devadoss: Exactly. You’re his kid, right? Like, that’s how, that’s how the children are working. I know what family you’re from.
Todd Ream: Yeah. I want to ask you a little bit more before we change topics again about one of your projects. So if you could talk with us about a recent project and in particular the sense of what is it you find intrinsically appealing or gratifying about that project and help us appreciate the joy that that, the pursuit of that project brought to you. What really gets you up in the morning?
Satyan Devadoss: Yeah. Oh, gosh. So my, I guess I’m wired in the way, going back to the Lego thing, I’m really wired to talk about things that I make and I have a hard time talking about things somebody else makes. And what I mean by that is I don’t find any flaws in anybody else’s work, but my joy is the kind of stuff I’m thinking about.
So I think what you had asked really resonates with me. I have a really hard time actually figuring out which of the ones I want to talk about right now, but I’m going to, I’m going to talk about one, one thing in particular, which I alluded to a little bit, which was unfolding boxes. Let’s just talk about this, the simple idea.
So imagine somebody gives you a box from Amazon, just a, just a rectangular box, or even you can even make it into a perfect cube if you want it to make life easier. And here are the rules to the game, Todd. The rules are, you want to break this down, break the box down so you can recycle it. Okay, so you just want to, it’s a 3D box, so you want to recycle it and save space, so you want to make it two dimensional.
You could take, you have, you have a knife, and the knife I’m allowing you to cut is, you can cut anywhere you want to, but that’s a lot of work to cut through the main material, why don’t you cut at the seams where they, where they kind of taped those edges together? So your rule are, your rule is, you have to cut along the edges. And if you cut along all the edges, you get, let’s think about it, if you have a cube, if you cut along all the edges, you have six squares in your hand. And it’s flat.
But now you have to like hold on to all the squares before they fall out of your hand. So you’d like to have one piece that you’d like to take and throw it in the recycling that’s flat. And so can you cut some of the edges, not all of them, so that it still stays connected, but lays flat? And if you think about this, if you actually cut the vertical lines and then the top part, it’ll actually unfold into a cross. It actually looks like a Latin cross where the four flaps of the cross go up and then the longer part goes to the top. Okay, that’s one way of unfolding it.
It turns out there is a really cool third grade problem. How many other ways can you unfold this box so it has that property? There are 11 ways total of doing it, that it unfolds and lays flat. Okay? Cool. And here is my favorite question in mathematics of all time which is, instead of a cube or a rectangular box, what if somebody gives you a weird shape like this one? Like an icosahedron, or think of any take a piece of clay and just chop off the sides so you get any polyhedra you could imagine, 300 sides made of triangles and pentagons and right?
The only rule is that this object has to be convex. And all that means is that if you’re inside the object, you can see anything else inside of that object. So like a horn of a Viking is not convex. If you’re one part of the horn, you can’t see the other part of the horn, staying inside. But basically, it’s basically saying if you want to summarize convexity, it’s almost like sphery, right? It has to kind of be spherish as many sides as you want to.
And here’s the question, Todd. If I give you one of those objects, some weird shaped thing, can you find a way to cut it, unfold it, lay it flat, just like I said. So it’s one connected piece. It turns out we can do that as mathematicians. We can prove that you can always do that. That there’s one added condition. The flaps can’t overlap themselves. When you’re unfolding, like, peeling it like an orange, it has to lay flat on the floor, but it can’t overlap themselves.
Like, if you think about the cube example that I told you about the cross, it doesn’t. The cross lays flat, none of the squares touch themselves. And the reason this is so important, and this alludes to the very first conversation we had, is for the physical world, if you’re manufacturing, if you want to recreate the box, then you could just laser cut that piece out and then refold it and get that box on your side. If it overlaps, then you got a problem because that one piece has to belong to two separate flaps and that’s not going to work.
And so this question, 500 years old from Albrecht Dürer, a Renaissance God, and the question was, given any kind of a box, can you always unfold it flat without overlap? And this lingo for folding it flat without overlap is called a net, N-E-T. And this is a question that is absolutely unsolved in mathematics. In fact, to the point that I’d say half the people working on this question think it’s true, that everything we’ve tried, all the computers running all the time try all these different examples, it’s always working. Half the people think, of course it’s going to work, we just don’t know why it’s working.
The other half the people think, you know what, we haven’t thought about the right thing yet. There is a box out there that no matter how you try it, you’re always going to have overlaps, no matter how you cut it. So half the people think it’s the other way. And I found this question so fascinating because we don’t even know how to bet on the horse. We don’t even know if the horse is going to win or lose. And so this to me is probably my favorite kind of prompt to fool around with.
I am, as I echoed before, not one of the gods who thinks about these things. So all I did was I switched gears and said instead of 3D boxes, let’s move on to 17 dimensional boxes. Let’s think about that. And you move into a space of mathematics that nobody else is playing at. And you’re just alone in that field, right? You’re lonely, but it’s lovely because you could think about it a little bit more. You could invite undergraduates to think about it. There’s not the gods of MIT and Stanford who are competing at that problem who want to solve this 500 year old problem. They really care about that 3D thing and we just move, move a little bit over, play in our own little area, but it’s still connected to it. And then we try our own, own hand at it. So that’s my favorite kind of stuff to do.
Todd Ream: Yeah, that’s fascinating. Wow. Fascinating. Thank you.
There’s only one problem that comes with this though. And you, you explaining and sharing this with me here today, is this evening I have to prepare the recycling to go out the curb. And my wife, Sara, may come out after about two or three hours and wonder what in the world her husband is doing out there in the garage when it usually takes 10 minutes for him to prepare the recycling. And he’s been out there for three, four hours, and he has this vexed look on his face as he’s trying to unfold and then refold this box.
Satyan Devadoss: Yeah, absolutely. My friend, it is a joyous thing to be, I mean, usually the great thing about it is we get most of the things we get are boxes, just rectangular boxes. So the solution’s done, but man, a cool question is, hey, how many ways can I unfold this thing? You know, I got this cross going, right? That’s kind of easy to do.
And sometimes your boxes have flaps on the middle, right? Even the top part is cut open. So now it gets, yeah, my gosh, I’m just getting chills thinking about now you just get to play. This is it. You just get to play. So I’m sorry for your wife.
Todd Ream: Yeah. Well, if I don’t get any sleep tonight, we’ll blame it on Jack, how’s that?
Satyan Devadoss: Perfect, that’s lovely.
Todd Ream: Yeah. Well, we’ll blame it on him and his mentorship of you as a graduate student.
I want to ask you about a couple of other different areas then and efforts that you’ve led over the course of your career. To begin in 2011. Princeton University Press published your Discrete and Computational Geometry, a book you co-authored with Joe O’Rourke. What led the two of you to pursue that project?
Satyan Devadoss: That’s a good question as I told you earlier, I’m a topologist. I don’t, I’m, I’m interested in geometry, these rigid shapes, and this echoing of unfolding boxes is an example of it, but I’ve been trained as a topologist; I really haven’t been trained as a geometer.
But I was a professor at Ohio State, and I was teaching graduate classes there, Todd, and computer scientists asked me if they can sit in, computer scientists faculty, asked me they can sit in on my first or second year grad class that I was teaching to students. And I said sure, okay so they came in and sat, took a bunch of notes.
And then later in the semester they said, we’re writing a big grant to the defense department. Can we just put your name on the grant? I said wait, what? We just need a mathematician. Here’s the kind of stuff we’re thinking about, is it, I was like, sure. I mean, like, let’s throw our hat in the ring. And at that time I went to Williams the next year Williams College in Massachusetts as faculty. And we got this big grant from the defense department. And I was trying to figure out what it was they were doing.
And I realized all the stuff they were doing is math that’s a hundred years old. The computer scientists are finally trying to figure out some of these ideas that are kind of classical, that have been done in the 1900s, early 1900s, that aren’t cool and hip anymore in the 2000s because we’ve moved on in the math community.
And so, let me just explain those words discrete and computational. Discrete just means if you have your car, it has a lot of smooth curves on it, right? You need wind resistance and it has these so, discretizing it is replacing the smooth car by a bunch of dots and lines and triangles so it looks like a triangular mesh. And so, the reason that’s done, is the same reason Pixar does it when they’re animating the Incredibles or Ratatouille or Bug’s Life is truly behind any of those images of Mr. Incredible, it’s just a bunch of polygonal mesh.
It’s just a bunch of polygons, but because a computer understands dots and lines really, really well. It knows how to move a line, it knows how to average a line, it knows how to turn a line, it knows how to add lines and dots. So that’s how computers think, it’s very linearly. So you approximate something curvy like Mr. Incredible with that stuff. You make it move as the skeletal structure, this discrete structure.
So you can compute it fast, and at the very end, Pixar does something called the rendering algorithm which smooths everything out so it looks like a human. And so it is very difficult for a computer to understand how shapes work unless they’re flat lines and triangles and those kind of objects. So you need to discretize things, make it into these dots and lines, discrete versions.
And the moment you discretize that, a whole of mathematics appears that is no longer trendy in 2000, but it was really trendy in the 1900s, which is, how does polygons work? How do polyhedra work? How do you unfold a box? Those kind of questions weren’t hip anymore. But computer scientists were realizing, oh my gosh, this is actually what we need to make our algorithms move really fast. And so they were sitting in my first year graduate class trying to learn some of the basic foundations in order to implement some of these ideas.
The problem was, this infuriated me. Because why is the computer science community getting credit for the math stuff that’s already happened? And so, most work in computational geometry was done by computer scientists for computer scientists. And I wanted to bring those ideas into the math world, make them hip and cool again, and introduce these new unsolved problems about boxes that were happening.
And so Joe O’Rourke is one of the fathers of the field of computational geometry. It’s about 30 years old, and he was at Johns Hopkins as a faculty there, then went to Smith College in Massachusetts and truly knew everything. He’s kind of like Gauss who kind of knew the landscape because it was a baby field. And now it’s massive. He wrote the Handbook of Discrete Computational Geometry, 1,400 pages, of just everything you can kind of understand, and even that handbook is now out of date because it’s growing so much.
So I asked Joe, how about you and I write a book, he already had written a textbook for undergrads, and I asked Joe, I think one of my, one of my greatest achievements was to convince this man who knew everything and who had already written a textbook to write another textbook with me just aimed at mathematics for mathematicians. Can we rip the computer science stuff out and just put in gorgeous images so students can fall in love with the ideas without the computational theory.
Todd Ream: I want to ask you now about your 2020 book that MIT University Press published, Mage Merlin’s Unsolved Mathematical Mysteries, a book you co-authored with Matt Harvey. What led the two of you to pursue that project?
Satyan Devadoss: One of the big struggles I have, I think probably my driving force, what, what really drives me is a sense of injustice or just anger. I don’t know how to, how to kind of frame it the right way. But I’m not saying this kind of, in a high end gosh, I want the world to be set right again in everything possibility, just in small things that I can actually address, right? How do you love somebody? How do you take care of family? How do you love your neighbors? And to me, this notion of injustice was, the first book was just, how do you bring those ideas that computer science was doing to math?
And the second thing that I really wanted to focus on was, when most people think about puzzles and unsolved things that exist in the world, things that bring you wonder and awe, in almost every discipline, you can ask somebody in your bus stop, in the subway station, in your coffee break, and say, hey, are there things in biology that we just don’t even know about? And people would say, yeah, of course, like cancer. Like, we don’t know how cancer works. We have no cure. There’s no pill for cancer. We don’t understand COVID. I mean, it’s multiplying these viruses. Like, what’s going on here? And so what, what about physics? Oh, you have a gravitational pull of space and time. What is it string theory? Like what’s going on behind physics? Almost every discipline you could think of, Todd, there are unsolved puzzles like paleontology, archaeology, anthropology, how do humans interact?
And if it comes to mathematics, somebody goes, hey, can you tell me something that is at the forefront of math? Like, where’s the wonder and amazement? And somebody goes, Pythagorean theorem, I don’t know, you’re just thinking of like, quadratic formula. And so the notion that a person off the street, just an average person, doesn’t know these wonderful things that people are playing with, the ideas that they’re in love with, that’s the bridge that I wanted to make.
So this book is basically a collection of 16 problems that nobody’s ever solved, unsolved puzzles in mathematics that mathematicians are struggling with, that a third grader can understand. So it’s just kind of pulled from there to here. And then it has a storyline where, I don’t know if you’ve ever seen MacGyver. I kind of grew up as a kid watching MacGyver, but he’s, he’s this person who can kind of figure out these things, right, with the simplest of tools. And so Merlin to us was MacGyver that King Arthur would send them out to, send them out to figure out stuff.
And he would knock out a whole bunch of things, but he’d keep a journal of things he couldn’t get. And this is the book that’s the collection of things, like I’m supposed to build a tent that does what? And he’s like, I just couldn’t figure it out. And then it turns out that’s a, that’s a math problem in disguise. That’s an unsolved puzzle in disguise.
And so Matt Harvey, he and I went to grad school together at Johns Hopkins. And I love to draw, but Matt is a person who’s an artist. And so very few people in my life speak into my art world. Like, when I’m drawing pictures for computational geometry, people would say, oh, that’s a beautiful drawing. That’s great. And Matt is somebody who’d say, well, I can make that way better. So I needed somebody who I can work with and play with, who’s a really good friend, who we could not only think about the problems, but actually paint the problems visually. So Matt and I started fooling around with that idea.
Todd Ream: Yeah, along those lines, then, in addition to being a mathematician, you’re also an artist who believes in the proper visualization of data from drawings to graphics to photography. And one of these things that these two books have in common is the rich use of illustrations. Would you please unpack then for us here how you see mathematics and art intersecting and relating to one another?
Satyan Devadoss: Yeah, thank you for that. I think in one sense from a disciplinary setting, I don’t know of any other discipline in a college, high school, grad work if you kind of break down the disciplines of the world, that is more in common with mathematics than art.
Going back to our conversation early on, Todd, you were talking about the sciences, right? You were talking about the usefulness of math and the natural sciences. But I think that’s the scientists using the hammer and the saw, right, and the nail in order to build a car. Like, they’re using mathematical tools. But the playfulness of mathematicians is, we’re just, what happens if you take a hammer and a saw and build it? Hu-saw! And a scientist is like, what the heck is a hu-saw? Like, I have no use for that thing. I got a hammer and a saw anyway. So we just play. We have no care about that use. And so this vast playfulness is really echoing what an artist does.
In the sense that you’re not saying to Da Vinci, well, make something that is really useful all the time. It’s like, no, play with these things. Of course, you might have a sponsor who’s sponsoring you to draw the person’s child, right, in this renaissance. But the way they drew that child was not obvious. It was playful. There’s a lot of hidden messages. There’s beautiful symmetry. And so that’s what mathematics echoes with me.
Having said that, I think there’s a deep divide between math and art. And what I mean by that is, it is, I still do not consider myself an artist. I think an artist is struggling with things and questions about what the world is about, that a mathematician does not struggle with. The goal of an artist is to evoke emotion, is to share a message, talk about some bigger, where words aren’t even enough, and you’re trying to use images to pull that through.
The goal of my life is not to evoke emotion. It’s just to share with you the glory of math. Like, I’m not really interested if you think it’s amazing or not, I just think this is, I think this is the coolest thing there is, and I’m giving you a truth statement that is going to stand the test of time.
So for example, like, I could tell you the sum of every triangle is 180 degrees. We’re not, in mathematics, we’re not going around and checking out, wait, does that work for this triangle? Did you find a new triangle, Todd, that I can check that out for? I don’t care, I know it’s true! That’s what we’ve done.
Whereas a scientist is walking around, is checking, wait a minute, is that true for this bacteria? Is that true for this enzyme? Is that true if I go to Mars, does the gravitational thing change? So they have theories and we have truths. So this, what an artist does and what a mathematician does is so different to me, and yet we’re playful in the same sense.
So I would say I am borrowing some of the principles and ideas and playfulness of art in my work, and the visual imagery, and I think it’s really motivating and driving me, but what I do is so different than what they do, and I don’t think I have the power to call myself an artist.
Todd Ream: Perhaps then some of those qualities then feed into your answer to this question because as our time unfortunately begins to become short here, I want to ask you about how you understand the academic vocation as you express it as a mathematician. What sort of qualities or characteristics define it and maybe chief amongst them is this playfulness but are there others that you might add?
Satyan Devadoss: Yeah, I mean, through the lens of the different disciplines, when we talk about history to economics to philosophy to psychology, I think one thing I really want to do is honor all of these disciplines. You know, I think they’re all approaches to these big questions of life, but I do want to say mathematics is in one sense the weakest of disciplines.
And the reason I say that is because we deal with things that are absolutely measurable. Like you can quantify, and that’s why it’s so powerful today, right? In the tech world, it’s so powerful today. In the STEM world, science and engineering, it’s so powerful today because we can quantify and measure things and that’s what we’re designed for. We can recognize patterns. We can talk about shapes, but we don’t measure complicated things. We measure triangles.
And so, because of that, we could do a lot of powerful measurement because the things we’re dealing with is simple. A biologist is also trying to measure things, but they’re dealing with a living organism. That’s a far more complicated playground. Historian is dealing with something that happened 500 years ago. That is a far more complicated playground to measure. I’m dealing with the images and the ideas that I have that I’ve created as a mathematician, not these things historically.
So, I want to kind of honor your question about academics. If you’re kind of pulling back to academics, I want to honor all of them. And they’re all trying to, I think, measure things, but the complexity level for these other disciplines increases. The way my heart breaks is when many people will say, oh my gosh, you’re a mathematician. You must be smart. If they found out I was a historian, they would say, oh, you’re a historian. You must be smart. They don’t say that in the world today. Why? Because math is so measurable. They think that is the only thing there is.
But they have forgotten this notion of complexity. People don’t understand that a historian deals with something a mathematician does not deal with, a far more complicated thing. And a biologist is dealing with something complicated, a psychologist, an anthropologist is dealing with mankind, humankind in general, that’s a far more complicated thing. And so, in terms of these disciplines, I think all of them should be honored in different ways.
Todd Ream: Thank you. As a mathematician, then what virtues do you find are most important to cultivate to express this understanding of the academic vocation?
Satyan Devadoss: I think there are things like beauty and playfulness and the notion of permanence that we have some truth that is always going to stand the test of time. That’s a beautiful thing in mathematics. We’re talking about these gorgeous ideas that you fall in love with, oh my gosh, you can do that with that, you can think about higher dimensions you fall in love with that. And the social playfulness, as we’ve talked about basically the whole time, is this, this playground of ideas you can have.
But I want to point to one thing that is usually not focused on in mathematics, which is this embodied life. So the thing I would, in the next hundred years, I think what we need to focus on is, what does it mean to value our bodies? And if you think about the math community, when you’re in third grade, you play with blocks. And you have these manipulative toys that a teacher might show you. This is what multiplication is like, and this is what addition is like, and they might move things around.
But by the time you get to junior high, it’s not cool to touch things anymore. You’re just in your mind, and certainly when you get to college, you’re in a disembodied life. This doesn’t make any sense to me because it’s not like my body is fading away and all of a sudden only my head exists in college. If that was true, if my arms fall off in junior high, and my legs fall off in college, I would say this is perfect because I don’t need that stuff, I just have my mind. But I still have my arms and legs.
So the question is, if mathematics has advanced so much with our mind alone, how much more can it advance if your body is also involved? That’s my favorite question that I’m struggling with. So we built a math studio, an 800 square foot playground of like popsicle sticks and toothpicks, chalkboard, display place. You could just come and play and make mathematics. Again, talking about playfulness, talking about beauty, talking about permanence, except use your body to try it.
And I think especially with AI and tech and our obsession with phones and technology, including this, I mean, it’s a beautiful thing because you and I get to talk and we get to hang out and people get to listen and share, but at the same time, it becomes this controlling or all powerful thing that we think is too important. I want an embodied life that also matters where pizza matters or a glass of wine matters where ice cream matters and we’re touching. So we can solve unsolved mathematics also now.
Todd Ream: Against what vices then do mathematicians, need to be vigilant in their vocation?
Satyan Devadoss: I think the biggest vice currently in the 21st century is the notion of ethics. If you go back to what we talked about, the fact that a biologist, Oppenheimer— a physicist has to deal with issues of ethics as they’re creating these ideas, a mathematician has not. Like unless you go back to the olden days when math the, the theory and computation were together. Nowadays, math is so abstract and higher dimensional stuff that I’m not thinking about the ethics of unfolding 900 dimensional boxes, so it doesn’t matter to me.
But with AI, Todd, the things that were in the corners of mathematics, the most abstract useless things, my advisor, Jack Morava, was working on things like homotopy theory, very high level abstract cobordism and K theory are now being pulled and the application to some of that is like within a month turnaround.
One of my friends is just received or has received, I think a 6 million grant from the Air Force to talk about the most abstract ideas of mathematics. Now, why would the Air Force do that? Because the Air Force wants a ROI. They have a high, this not like in 500 years, we want something to happen. He’s thinking about how the structure of ideas can give us a framing to measure what is true and false. How can you check an algorithm? How can you check something like Microsoft’s, Microsoft’s operating system, or Apple’s operating system, or maybe a missile guidance system to know what you programmed is what it’s gonna do? How can you check that using abstract mathematical ideas?
So now, all of a sudden, math is thrust in the world of reality. Going back to the very first conversation you and I had today, is that we’re just squished back into the world of reality again. And all of a sudden ethics matter. And I think mathematicians are just, just starting to figure this out. And so that to me is the, is the biggest danger coming up is we have to, we have to be prepared that our useless things are now becoming useful to weaponized for the world.
Todd Ream: Thank you. For our last question then today, in what ways can mathematicians and their contributions become more relevant to the Church? And can the Church be prepared to be more receptive to those contributions?
Satyan Devadoss: I think the greatest thing mathematics as a profession, and certainly believers in the math community can do is to give up their power for the other. We have, we are the most powerful people in the world right now. That’s true. If you think about the STEM world, science, technology, engineering, and math, that is the buzz in all of academia in America.
If you go to Stanford right now, if you go to the philosophy department or the English department, a history department, they’re a handful of brilliant faculty there, nobody’s hanging out with them. But if you go to the biotech world, my friend, you couldn’t stand in line for Starbucks because it’s out the door. There’s so many people in the STEM world that are obsessed because it will give you money. It will give you measurable outcomes, and it’s a powerful force for getting jobs right now and for making the world.
If you think about what AI is doing already, much less AI and mixed with everything in STEM and the people who empower all of them, we are the cocaine dealers because we make the stuff that empowers everything. And so if I walk into a room and somebody goes, oh, my gosh, you’re a math PhD, I’m honored compared to almost any other discipline there is.
But if you want to measure things in the world from a Christian perspective, if you want to talk about the resurrection of Christ, well, then you need somebody who could measure historical powers. If you want to talk about linguistic powers of Scripture, hey, is the book of first and second Samuel about David? Is that actually a work of art? Like, is it actually, or is it a bunch of random stories? Robert Alter, who’s a professor who’s a professor who retired from Berkeley, gives the greatest defense as to why the Old Testament scriptures, the Jewish Scriptures are a work of art as a work of writing, as a linguistic, as a literary critic.
And so, unless we in the math community give up our power and point that light to people who are struggling with incredibly equal but much more complicated ideas, then we will just absorb this wealth and not share it. And I think our time is to diminish a little bit more than we should. And there might be a time when the historian might need to diminish and point it back to math again. Now is not that time. But that’s the greatest calling I think we should have, is we should bend the knee for these greater disciplines to actually have a chance to play at the table.
Todd Ream: Thank you.
Our guest has been Satyan L. Devadoss, the Fletcher Jones Professor of Applied Mathematics and Professor of Computer Science at the University of San Diego. Thank you for taking the time to share your insights and wisdom with us.
Satyan Devadoss: Pleasure’s all mine, Todd. Thank you so much, my friend.
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Todd Ream: Thank you for joining us for Saturdays at Seven, Christian Scholar’s Review’s conversation series with thought leaders about the academic vocation and the relationship that vocation shares with the Church. We invite you to join us again next week for Saturdays at Seven.
