The Mysterious Timelessness of Math

Josh Landy
Is math a realm of timeless universal truths?

Ray Briggs
Or are mathematicians just making it up as they go along?

Josh Landy
If equations are made up, why are they so useful?

Ray Briggs
Welcome to Philosophy Talk, the program that questions everything

Josh Landy
except your intelligence. I’m Josh Landy.

Ray Briggs
And I’m Ray Briggs. We’re coming to you via the studios of KALW San Francisco Bay Area,

Josh Landy
continuing conversations that begin at philosophers corner on the Stanford campus where Ray teaches philosophy, and I direct the Philosophy and Literature initiative.

Ray Briggs
Today, we’re thinking about the mysterious timelessness of math.

Josh Landy
You know, Ray, mathematics really is pretty mysterious.

Ray Briggs
Aw, I know you’re a literature guy, Josh. But if you just practice your times tables, enough, you’ll eventually get the hang of ’em.

Josh Landy
Hey, that’s not what I meant! I don’t mean, I’m bad at math. Actually, I had a great time doing math at high school. I nearly majored in it in college. But what I’m trying to say is, there’s something puzzling about the whole enterprise of mathematics like, why is it so useful?

Ray Briggs
Gee, I don’t know. Maybe it describes the fundamental structure of the universe.

Josh Landy
But how could anyone learn about the fundamental structure of the universe, just by scribbling symbols on a piece of paper? I mean, mathematicians don’t run experiments. They don’t even write down observations about the physical world.

Ray Briggs
Yeah, but math is good for so many things. We use it to build bridges and predict the motions of planets and explain why snowflakes have that funny six pointed shape.

Josh Landy
Absolutely. But that’s what’s so wild about it. Like how on earth can a bunch of doodles on a whiteboard help us build a bridge?

Ray Briggs
Huh, you make it sound like a coincidence, but that’s exactly what you would expect. If math was latching on to deep, important truths. Like, okay, for some shallow truth, like I’m wearing purple socks, you might have to stop and check whether it’s true, because it could easily have been false. But for a really deep truth like that two plus two is four, there’s no need to check because what could possibly make it false instead of true?

Josh Landy
Are you wearing purple socks?

Ray Briggs
Well, that’s the magic of radio, no one has to know.

Josh Landy
Okay, but I still, I don’t, I don’t know if I totally agree with what you’re saying here. Because, I mean, if something is a deep truth, doesn’t that make it even more important to verify it? Like, I don’t need to check whether you’re wearing purple socks. It doesn’t really matter either way. But surely, we all need to be certain that two plus two equals four.

Ray Briggs
Well look, we know that math works. Like that microphone you’re speaking into now, you didn’t build that. Math built that.

Josh Landy
No, engineering built it. If you want to make a microphone, you need to know about currents, and sound waves-

Ray Briggs
And equations and math! You can’t understand currents and sound waves without equations.

Josh Landy
Okay, but math doesn’t tell us anything new about the world. I mean, I think it’s more like a filing system, right, it helps you organize what you already know. But that information, that already had to be added by us.

Ray Briggs
Yeah, wait, Josh, what wouldn’t that make it kind of arbitrary? Like, okay, you could have a lot of different, equally good filing systems. Why not pick one where two plus two is five?

Josh Landy
Well, I mean, your system better at least be internally consistent. Like if your version of arithmetic says one plus one equals two, then it also needs to say two plus two equals four. But it’s still just a system for organizing and manipulating information.

Ray Briggs
No way. Math gets at deep truths about the universe. I mean, that’s why it’s so useful. Hey, I bet our guest is going to agree with me. She is Arezoo Islami, Professor of Philosophy at San Francisco State University, where she teaches the art of quantitative reasoning.

Josh Landy
She’s definitely gonna agree with me. One thing we’ll all agree on, is that math is really cool. And our Roving Philosophical Reporter Holly J. McDede went to talk to other people who love math, and find out just what it is they love about it. She files this report.

Holly McDede
Vi Hart is a YouTube star who makes videos you may not realize you want in your life. Hart has over 1 million subscribers and makes videos about math.

Vi Hart
Pineapples, unlike people, don’t have bilateral symmetry. You’ll never have that third spiral be not a spiral but just a straight line going up a pineapple.

Holly McDede
In one video, Hart attempts to determine whether the pineapple under the sea home to Spongebob Squarepants is actually a pineapple.

Vi Hart
When we look at SpongeBob’s supposed pineapple under the sea, it clearly has lines of pineapple things going straight up.

Holly McDede
Hart often begins these videos making fun of math class.

Vi Hart
Say you’re me in your math class and you’re supposed to be graphing functions as if there were some deep relationship between y and x that your teacher just won’t stop gossiping about. But like most gossip, you really don’t care about y’s unhealthy dependency on x. Really, y, get a life.

Edward Frenkel
I had no idea that real math is a vast and fascinating archipelago of knowledge.

Holly McDede
That’s Edward Frenkel, a Professor of Mathematics at UC Berkeley and the author of the book “Love and Math.” Growing up, he thought math was dull. Then he met a mathematician who taught him about the connections between math and quantum physics. He learned about how math was used to theorize about the existence of quarks, the elementary particles that protons and neutrons are built out of.

Edward Frenkel
it looked like a holy grail to me because now they were part of a coherent theory. And that’s when I knew that there was some- there was this magical, mysterious world that until that point was close to me. That was my initiation to that world.

Holly McDede
Frenkel grew up in the Soviet Union, where the government tried to control academic research. But math felt safe because it was too abstract for the government to understand.

Edward Frenkel
Scientists in general were given a sort of leeway. The government knew that they needed physicists, for instance, for their nuclear program and for development of various missiles and things like their space program.

Holly McDede
Mathematics, algorithms and technology determine a lot about our lives. But Frenkel wishes more than just a few elite scientists could see the beauty of math.

Edward Frenkel
Almost all mathematics that we study in school is more than 1000 years old. It’s really mindboggling if you think about, but how would anyone know because nobody ever talks about this?

Holly McDede
But some people are lucky.

Carlo Sequin
Clearly, somehow, geometry must have entered my bloodstream and has stayed there forever.

Holly McDede
Carlo Sequin grew up in Switzerland. He’s a professor of computer science at UC Berkeley. And from an early age he loved his numbers.

Carlo Sequin
Geometry really became this magic language for me that allowed to see pattern and understand things in a more deeper sense on them, not very formal, basic equations. But you know, I want to visualize things.

Holly McDede
The real breakthrough came when he saw a sculpture by Brent Collins, an artist whose work often represents mathematical equations. Sequin created computer tools to capture and enhance the shapes of Collins’ abstract sculptures. Like in the piece, “The Music of the Spheres,” it’s a curved bronze ribbon winding around a six foot diameter. The title comes from “The Harmony of the World” by Johannes Kepler.

Carlo Sequin
Referring to what Kepler actually thought, he thought it was the universe and all these planets and stuff and stars move in circles, and they play some kind of music. So for me, the first thing is, I want to have a little model of that.

Holly McDede
Every year, Sequin attends The Bridges Conference, a gathering to highlight the connections between math and art, music, architecture and culture. The event includes a family day where people work on puzzles.

Carlo Sequin
The children, some as young as four or five, they gets really, really excited, and then they are really captured by what they’re doing. Oh, wow, this is so great. You know what? That’s mathematics. You just did mathematics.

Holly McDede
Back on YouTube, Vi Hart shows us how math can be fun and beautiful. In one video made about a year into the pandemic, Hart talks about music and repetition. For a lot of people, the pandemic meant a lot of repetitive days.

Vi Hart
So if you feel like since last pi day, your world has turned upside down and you’re having trouble orienting yourself, there could be a mathematical explanation.

Holly McDede
Hart repeats a phrase on the piano in numeric terms, 31415. Hart says the digits of Pi are random notes when played in isolation.

Vi Hart
But if I’m playing sensitively, then you’ll hear something more structured. I like to connect the ascending 345, or maybe even with a little bit of a swell in the middle. And then the ones, instead of being part of the melody, become harmonic support for the melody like this.

Holly McDede
And maybe somewhere out there a math teacher is on the piano, trying something new, teaching a similar lesson about numbers and repetition. For Philosophy Talk, I’m Holly J. McDede.

Josh Landy
Thanks for that inspiring report. Holly, I will never think about pineapples the same way again. I’m Josh Landy, with me as my Stanford colleague Ray Briggs, and today we’re thinking about the mysterious timelessness of math.

Ray Briggs
We’re joined now by Arezoo Islami. She is a professor of philosophy at San Francisco State University, and co-author of “Marriages of Mathematics and Physics: A Challenge for Biology.” Arezoo, welcome to Philosophy Talk.

Arezoo Islami
Thanks for having me.

Josh Landy
So Arezoo, we just heard from a bunch of people about their love of mathematics. Where does your love of mathematics come from?

Arezoo Islami
Well, once upon a time, I was 16 years of age and there comes a new teacher to my school. He asked us to throw away our high school textbooks which we really huffily did, and get college books. He introduced us to a new form of beauty and mystery. And it was an ecstatic experience. He would often go over time, not by a minute, but by 45 minutes, not us noticing it. I fell in love with math then.

Ray Briggs
What a fantastic teacher. I have a question for you about math. Earlier, I was saying that math helps us discover the fundamental truths of the universe. And Josh was saying that it’s not a discovery, it’s an invention, and just helps us organize our other ideas. So who’s right?

Arezoo Islami
Well, you put me in a hard spot. My quick answer would be math is an invention on the basis of discovery.

Josh Landy
We’re both right.

Ray Briggs
What does that mean?

Arezoo Islami
Let me give you an example. Let’s think of geometry. At first geometry was a practical art of dividing lands, you can tell from its name. And at some point, it ended up being an axiomatic system in which people were no longer talking about lands, but they were talking about these geometrical shapes that are idealized mathematical objects.

Ray Briggs
Right, so there’s never been like a really perfect square in the world because all the lines in the world are like at least slightly curvy.

Josh Landy
And this point has no dimensions. There’s no point in the world.

Ray Briggs
Right? So okay, so geometry seems like it’s sort of about like, approximately square things and sort of not? Like how do I think about the relationship between the square that I imagined with perfectly thin straight lines and like a square of land that’s kind of a little bumpy and and not infinitely thin?

Arezoo Islami
So, this is fantastic Ray, what has been happening is we have started developing this tool if you like, to study lands, but as you’re realizing when we study line in geometry, it no longer has a thickness. That idealized line, which is something that we have invented on top of the, you know, edges of the buildings or that we are seeing, that allows us to abstract away from all these properties of lands or the color of your socks for that matter. And we can study them with exactness. Once we have developed this axiomatic system, which with idealized objects, then it becomes applicable in the study of the physical world. But that application always asks you to allow some corrections and errors and to add imperfections of the real world.

Ray Briggs
So I have more questions about geometry. I’ve heard that Euclidean geometry is not quite accurate, but if geometry is an idealization, anyway, why is Euclidean geometry less accurate than other kinds of geometry that people study?

Arezoo Islami
Again fantastic points. So different systems of geometry the question of which one of them is more useful in study in physics is a separate question from the accuracy of the system of geometry itself. So from the mathematician’s point of view, Euclidean geometry as well as different non Euclidean geometries, they all are very rigorous, they are axiomatic, you have proofs in them and so on. But when you want to study this particular universe, this particular earth, then we have the limitations that are coming from the particularities of this universe. So, once we have realized that everything curves, then we do not have actually straight lines, and we need a different geometry to study that.

Josh Landy
You’re listening to Philosophy Talk. Today we’re thinking about the mystery of math with Arezoo Islami from San Francisco State University.

Ray Briggs
What makes an equation true? If math is a product of our culture, does that make it culturally relative? Are numbers eternal fixtures of the universe or just figments of our imagination?

Josh Landy
The real truth about imaginary numbers, along with your comments and questions when Philosophy Talk continues.

Mathematics, magical and mysterious or merely mundane? I’m Josh Landy. And this is Philosophy Talk, the program that questions everything

Ray Briggs
except your intelligence. I’m Ray Briggs, and we’re thinking about the mysterious timelessness of math with Arezoo Islami from San Francisco State University.

Josh Landy
We were pre recording this episode. So unfortunately, we can’t take your phone calls. But if you’ve got comments or questions, please email them to comments@philosophytalk.org. Or you can comment on our website, where you can also become a subscriber and gain access to our library of more than 500 episodes.

Ray Briggs
So Arezoo, you’ve written about a fascinating controversy over imaginary numbers in the 1500s. Why were they so controversial?

Arezoo Islami
Before I start talking about imaginary numbers, I like to emphasize that this case is much more widespread in mathematics than we tend to believe. So let’s ask ourselves, what is an imaginary number? An imaginary number is a number that is like an imaginary i, that is the square root of negative one, the square roots of negative numbers. When we were little kids, they would tell us that the square root of negative numbers does not exist. So and then later, they corrected it. Something happened in the history of mathematics. Around early 16th century, Cardano, who was an algebraist and Italian algebraist, working on cubic equations, he realized he has found a method to solve cubic equations. But that method involves sometimes under the radical, under the square root, some negative numbers. So it was really puzzling. And at first, he tried to avoid using them because he said, well, a solution to an equation cannot be for them. Even a negative number, let alone [unintelligible] through square roots of negative numbers. But these weird entities, these kind of absurd, strange entities continue to lurk in the background.

Josh Landy
So why were they controversial? I mean, you know, so clearly, they’ve done a lot of work for us, right? I mean, it’s a very strange idea that you could have such a thing as a square root of minus one. But clearly, it’s paid off in mathematics. So why weren’t they just- why wasn’t that idea just embraced right away?

Arezoo Islami
So usually, when a new entity is introduced in mathematics, it is not a linear process by which people get there. So let’s put ourselves in the shoes of Cardano and other mathematicians of his time. All they were looking for was a solution to these equations where what today we call natural numbers, positive whole numbers. So when they had this square root of negatives, they thought, well, these are really weird, because first of all, where are they? If you need a geometric interpretation for these different numbers, can you show me where they are? So attempts began to find a geometric interpretation for these otherwise strange entities that they didn’t accept, strictly speaking as numbers at that time.

Ray Briggs
So what makes an interpretation count as geometric? Is it just like you can draw a picture of it like a triangle or like a graph of a function?

Arezoo Islami
Yes. So the attempts to find geometric interpretations led to proposing that maybe these imaginary numbers are vectors, or they were looking for them on the real number line, or on a one dimensional line, but they couldn’t find them. So the proposal was to look at them on the y axis, as opposed to the x axis. So they became a unit that had a 90 degree angle with the one unit that we knew before.

Josh Landy
That’s interesting, cuz it sounds like the hang up was, well, things in mathematics have to have a counterpart in the real world, right? So a square in mathematics has a counterpart in a square of land. But what’s the counterpart for the square root of negative one in the real world? Does that seemed like a reasonable way of phrasing it like it’s about the relationship between mathematics and the world?

Arezoo Islami
Good question, Josh. So it’s not so much about the relationship between math and the real world, but a relationship of mathematics with the rest of mathematics, namely, at that time, geometry had a really important place, and they were looking for geometric interpretations, so that they can give them a firmer ground to stand. But that wasn’t the end of the story, even after finding geometric interpretation for them as vectors or as units on the y axis, they still were considered to be suspicious. Well, if you think about zero or negative numbers, they had a similar history because of the metaphysical questions about them. But what ended up happening was that these imaginary units started spreading everywhere and such that they were used in the proofs of fundamental theorem of algebra. They were used as some kind of equivalent with trigonometric relationships, and they made a lot of mathematical relations so much easier.

Ray Briggs
So okay, I’m hearing kind of two different criteria for when we should believe that a thing that mathematicians talk about is real. So one is like, can you draw a picture of it? And another one is, is it useful for doing stuff? Are there other like standards that I should be thinking about, is one of those the right standard?

Arezoo Islami
I believe, after a pure mathematics of 20th century, we no longer are looking for geometric interpretations of things. So the case of imaginary numbers that I’m talking to you about is a case that is embedded in its historical context, namely, back in the time, they wanted a geometric interpretation, as well as its usefulness in other areas of mathematics. So maybe, let’s think about it this way, there is the discovery of new entities or the introduction of new entities in mathematics. Each era has its own standards for how they accept these entities into mathematics. And in our time, mathematicians really don’t care that much about whether this thing is really applicable in the study of nature. But remember, a lot of mathematics has its origin, in physics, in chemistry, in other areas.

Josh Landy
That’s really interesting, and it leads me to a potentially skeptical question, right? You’ve talked about how mathematical standards change. They’re local to their time in place. Could a skeptic say, well, gee, then maybe math, there’s something a little arbitrary about math. And you know, maybe we can’t entirely rely on math, because maths is constantly evolving and changing, the standards change. What would you say back to that?

Arezoo Islami
Good point, maybe there is a big difference between something being arbitrary, and something being maybe God given. So I wrote in one of my papers that I agree that God is a mathematician, but only if you think in this case, about humans, because humans as gods have created on the basis of their discoveries of this nature, some amazing discipline that has idealized objects in it. And it is very applicable, which we can talk more about applicability of mathematics. It is not arbitrary. But it doesn’t make it very close to the study of nature. It is still idealized. And mathematics has its own standard of rigor.

Ray Briggs
So I want to hear more about this idealization idea. So one thing I think sometimes is, look, if there had been no humans to study geometry, that wouldn’t make the conclusions we come to, in our geometry any less true or any less reasonable. But that suggests that there’s something independent of us in our culture, that makes the way we do geometry correct. And the same for like arithmetic or other branches of mathematics, you know, analysis, group theory. So how do I avoid thinking that like, well, you could make math different if you just thought about it differently? Cuz that doesn’t seem like it’s true.

Arezoo Islami
Right, so when we think about mathematics and its roots, that would give us a lot of insight, because, I mean, let’s think about a time that you really didn’t need to divide land somehow, like in a even you have humans, but the kinds of questions that they’re dealing with are completely different from our questions. In that case, I believe they could have made a different kind of mathematics. Now, that doesn’t make it completely a production of human mind. But we are embedded in this world and we are trying to solve certain problems in this world. It’s not armchair thinking, in the sense that we are just sitting and enjoying it, although it is a very enjoyable thing to do. But so its roots are coming from our study of nature. Think of it as a tool like the way hammer is a tool. It’s a conceptual tool that has given us all these possibilities to live in this world. One thing, the way I like to think about it is that every conceptual tool that we have, it almost extends our reach in the universe. You weren’t there at the origins of the universe, right? But what, how is it that we have come up with this idea that there was a big bang? Well, thank you to mathematics, and lots of work from theoretical physics that they have extended our reach into the nature.

Josh Landy
That’s absolutely fascinating. You’re listening to Philosophy Talk. Today we’re thinking about the mystery of mathematics. With Arezoo Islami from San Francisco State University. We’ve got an email from Tim in Portland, Oregon. Tim asks about the limits of our mathematical knowledge, he points out, Godel proved some true theorems cannot be proven, Turing show that some functions cannot be computed. So Arezoo, what does that tell us about the nature of mathematics?

Arezoo Islami
So mathematics, like other conceptual tools that we have, it is constantly evolving, and goes through changes and revolutions and so on. There are places that we do not, we do not have these two as strong as we like. And to me, that gives us hope, as humans, because we are in our knowledge, when we learn a lot, we realize that we don’t know a lot also. So it’s the beginning in a way of knowing more and more and more. So mathematics is not an exception to other areas of knowledge, like physics, like chemistry, like biology, there are limits to that, but we are very able creatures to overcome the very limits that are sitting before us.

Josh Landy
Okay, but I want to press you on this a little bit. And part of the reason I want to press you is because of this fantastic line in Borges. Borges says that, you know, basically we, humanity have created the world. But because we wanted to be aware that we created it, we deliberately left crevices of unreason, in its design, and he points to Zeno’s paradoxes. I think this connects also to Tim’s point about Godel, about Turing. But you think about Zeno’s paradoxes. You know, one of the paradoxes is look, an arrow is always stationary in any given moment, so it never moves. And another is, look, if you want to get from point A to point B, first, you have to go half the distance, but you have to go half of that distance you never get anywhere. And one way to think about the Zeno’s paradox, you take those two as a pair, you can solve one, if there’s a smallest unit of space. But if there’s a smallest unit of space, then the other one’s a problem. And so in fact, you can’t solve both at once: either there’s the smallest unit or space or there isn’t. Either way, you’re trapped in a paradox, Borges concludes that the world is unreal. Now, we might not conclude the world’s unreal. But shouldn’t we conclude that in a beautiful and fascinating way, sometimes what math does is shows us the limits of our knowledge or maybe the limits of mathematics, as opposed to showing us, you know, well, we’re on this road in the direction of knowing everything.

Arezoo Islami
This is a really good point. So one thing that I can say about Zeno’s paradoxes, it was a starting point for people to think about the concept of infinity. Infinity, the way it is understood in mathematics. And that infinity in a way, even when you think about the infinity of natural numbers, 12345, and so on, it seems that, by virtue of thinking about that infinity, we have somehow overcome our trappings in time, because to have a set like that, as an entity, you’re assuming somewhat that you have an infinite amount of time to do all the counting. So let’s recognize the power that mathematics gives us in overcoming our own human limitations. Another point that I want to raise is that in case of imaginary numbers that we already talked, and many other cases, mathematics has the power of turning a paradoxical case to be a very trivial case of mathematics in a way to build other things on the basis of it. That is not to say there are no limits to our knowledge. There are many, many, many limits to our knowledge, but our conceptual tools evolve in a way to turn these paradoxes into cases that we can actually solve and we can deal with. I’m aware that this is not a complete answer, but that’s the best I got.

Ray Briggs
Arezoo, I want to get back to the question of change over time. So one way in which math seems different from science is that every so often, scientists will have what the philosopher Kuhn called the scientific revolution. So they’ll they’ll just decide that their old theory was no good and they’ll switch to like a completely new framework. That doesn’t seem to happen in math.

Arezoo Islami
I have somewhat of a different view than this. To me, mathematics, is a science, is a branch of knowledge that constantly goes through revolutions. So what I called the case of imaginary numbers, or zero, or maybe even the move that happened, from arithmetic to geometry, to algebra to analytic geometry, all of these cases showed a break in mathematics at the time that they were happening. What do I mean by that? There were cases that they couldn’t solve. And for the time being, they thought that they are not solvable. But once they had found their solutions, they rewrote the entire mathematics such that these solutions, these paradoxical cases, were just the normal cases. So today, you and I, when we learned mathematics, to just say, here is the set of complex numbers, and here are its members without us hearing about all the breakthroughs and all the revolutions that had to go in these things coming about and being accepted as normal cases.

Josh Landy
You’re listening to Philosophy Talk. Today, we’re thinking about the mysterious timelessness of math with Arezoo Islami from San Francisco State University.

Ray Briggs
How can schools get better at teaching math? Why are so many students afraid of algebra? Is there such a thing as mathematical talent? Or do we all start from the same square one?

Josh Landy
Sine curves and learning curves, plus commentary from Ian Shoales, the Sixty-Second Philosopher when Philosophy Talk continues.

Will having a head for math give you greater access to the mysteries of the universe? I’m Josh Landy. And this is Philosophy Talk, the program that questions everything

Ray Briggs
except your intelligence. I’m Ray Briggs. Our guest is Arezoo Islami, from SF State University. And we’re thinking about the mystery of mathematics.

Josh Landy
So obviously, thinking about having a head for math, we have a question from Don in El Cerrito, California. Don says, I was an English teacher for a while and didn’t get very far in math. But it wasn’t that I couldn’t have been good at it. It was just circumstances. Now I’m dealing with my son who’s struggling in math. His mother says he’s not a math person. But Arezoo, Don doesn’t agree there’s such a thing as a math person. So what’s your view on that?

Arezoo Islami
I find myself really puzzled by this question of whether there is this natural tendency to learn certain things or a talent. I think whether it’s true or not, is irrelevant to whether we are giving people a chance to fall in love with this beautiful part of human culture. I have a sense that in the right situations, everybody can see the gorgeousness of this poetry we’re dealing with.

Ray Briggs
So this brings me to a question we like to ask on Philosophy Talk. We’d like to make people tsar of things. So if we were to make you tsar of mathematics education, what’s the first thing you’d change about the way math is taught in schools?

Arezoo Islami
Well, for one, I would put more philosophical minds to teach mathematics.

Josh Landy
Not self serving at all.

Arezoo Islami
And here’s why. Because a lot of students start disliking mathematics because they see it as this rigid system that they keep asking, why is it that two plus two is four? Why is it that the internal angles of triangle are 180 degrees? And they hear, that’s just the way it is. By a philosophical approach to mathematics, we can put our students in the place, in the shoes of the early mathematicians, and we encourage the question of why. Why is it that two plus two is four?

Josh Landy
And how do you get them into the poetry? I love what you said a moment ago about poetry, how math- I mean, I had wonderful teachers, and I’m endlessly grateful to them, because that’s what I felt. I always loved pure mathematics. Because I just found it beautiful. The proofs were beautiful. And they were something just gorgeous. I mean, there’s still to this day, branches of mathematics that have either no connection at all, with the physical world, or at least a very tenuous one. I mean, we were talking earlier about the mathematics of infinity. How do we get the, you know, high school kids to feel that poetry and that beauty?

Arezoo Islami
Thank you for raising that point. So sometimes I hear from colleagues that we need to tell our students what’s the application of this area so that they can like it. I tend to disagree, because it’s not for us. We don’t listen to music, because it has some kind of application. It’s because it arouses in us a certain aesthetic sense. So I play with my student this game, the first day that they come to the school. I say, well, suppose that math was a romantic partner of yours. How would you describe your current relationship? Some of them say it’s an arranged marriage, my parents wanted me to marry. And now we are divorced. Some of them they say, it’s kind of a relationship that mathematics beats on me. Some of them say, well, you know, we were kind of on a good terms until some events happen. And I asked them, what is your ideal situation, suppose you could have a magic wand and anything could happen? Almost all of them say I would like to have a really good relationship with mathematics. So I begin from the beginning with them and encourage the question of why and constantly gives them bits and pieces of history of mathematics. So they realize it is an alive field in which they can contribute. And it is beautiful, only if they are willing to spend some time.

Ray Briggs
I remember really hating to be told the practical applications of mathematics when I was a kid, because they were always things like, well, you can balance your checkbook, which sounded monumentally boring to me. But I’m curious, like, what’s the point at which your students get more excited about mathematics and start to see its beauty? How would you describe that?

Arezoo Islami
One point is when I introduced them to a notion of mathematical freedom, which is there are many mathematical problems that they have more than one solution, and you can show your creativity in getting to those solutions. That’s one point. And the second is that I realized, if I know what parts they are missing a step, so it’s this ladder, they’re gonna go up, and there are steps that are being missed. They’re trying to jump over it. And they always blame themselves for not being able to jump, I try to find the steps and help them walk over it. At some point in the middle of the course, inevitably, there are moments that somebody says, oh, I got it, I got it, and then their eyes start shining. So this is all I can ask for, because I feel they have been introduced to a new part of human culture and where they thought it’s a wall, it’s a window into an endless beauty.

Ray Briggs
I’m thinking more about sort of the history of mathematics and kind of the way that people across the world can connect over it and ways that sort of they cannot always connect over other languages. I know you just told us that there are mathematical revolutions, but it seems like they’re really old parts of mathematics that still are useful today and still get taught today. Like one of the proofs that that one of our listeners mentioned, Godel’s Incompleteness Theorem relies like on something that was first proved by like Chinese mathematicians centuries ago, like it’s a really old result. Like why does math last like that? And why does it seem to connect people across so many cultures?

Arezoo Islami
Absolutely. So the way I think about it is that the continuty of the discipline is in the continuty of practice. So first of all, when you and I taught were taught mathematics, we learned Euclidean geometry, which was taught like over 20 centuries ago, it is rewritten in a sense to be more relatable to us, but still, the results of mathematics remain the same, only maybe put in a different language. I think when you want to answer a question like that you should ask yourself, so why is it that painting or music or other forms of human culture have lasted for so long, and they have been in different cultures? It is a similar case.

Josh Landy
And that combination on the one hand of change over time, but also this mysterious timelessness at some things still seem to kick around is part of the fascination for me. But I wanted to come back to something we were talking about earlier, I wanted to actually throw a couple of quotations at you, see what you make them. So Bertrand Russell says, physics is mathematical not because we know so much about the physical world, but because we know so little. It’s only its mathematical properties that we could discover. And Einstein says, as far as the laws of mathematics refer to reality, they are not certain. And as far as they are certain they do not refer to reality. So these are both a couple of somewhat skeptical takes on the extent to which mathematics can capture the physical world. And of course, one objection, if it’s an objection, one thing, one consideration that people have raised is the question of time, because what mathematics at least seems to give us is a set of sort of, you know, timeless equations, right, kind of changeless states, descriptions of changeless states, how does time come in where you know, how can you get time into mathematics? In other words, can mathematics really be a full description or characterization of the physical world?

Arezoo Islami
There are many interesting questions in what you’re raising, and my current fascination and my philosophical research is on time. But let me say a couple of things. To me physics, more than any other discipline is co-constituted with mathematics. So when we are studying what we take to be concepts in physics, objects in physics, they are mathematical at their core. Let’s think about electron, let’s think about curvature of space-time. But what is happening is that when we are trying to use these idealized objects in the results of our experiments, there is always things that we need to add by hand. Because mathematics is the most general form that these patterns in nature take, of course, it has a rigor, and it has a perfection, that we do not see it in the world around us. We do use mathematics, but at the end of the day, we also put in other constants from nature, that need to correct for this.

Ray Briggs
So if there’s kind of one takeaway in the attitude with which our listeners approach the world that they can get from the study of mathematics, what would you say that is?

Arezoo Islami
Well, I would say, I fell in love with something that is so gorgeous, and so beautiful. And it expands human mind and expands human horizons, I would think it’s worth that everybody tries to give it a shot. And as you’re doing that, study some philosophy of mathematics along the way, that will make it more fun.

Josh Landy
That’s an excellent pitch and a beautiful place for us to stop. Thank you so much, Arezoo, for joining us today.

Arezoo Islami
Thank you so much. This was very pleasurable. Thank you.

Josh Landy
Our guests has been Arezoo Islami, professor of philosophy at San Francisco State University, where she teaches, among other things, the art of quantitative reasoning. So Ray, what are you thinking now?

Ray Briggs
Well, I’ve been thinking about some books by Sarah-Marie Belcastro and Carolyn Yackel on mathematics and the fiber arts, especially knitting, so they actually tell you how to make beautiful mathematical things using fiber arts methods. And I’m not very good at it, mostly because I’m not very good at the knitting part. But someday I will have a beautiful Mobius strip scarf, and it will be both mathematical and warm.

Josh Landy
That is fantastic. You’re making me want to get back to the Oulipo poets, who you know, the Fibonacci poems. I mean, the applications of mathematics to the arts are endless and fantastic. We’ll put links to everything we’ve mentioned today on our website, philosophytalk.org, where you can also become a subscriber and gain access to our library of more than 500 episodes.

Ray Briggs
And if you have a question that wasn’t addressed in today’s show, we’d love to hear from you. Send it to us at comments@philosophytalk.org, and we may feature it on the blog.

Josh Landy
Now a man whose speed defies all known constants. It’s Ian Shoales, the Sixty-Second Philosopher.

Ian Shoales
Ian Shoales. It is a known fact or a strong rumor or a modern trope that movie and television scripts are actually written by or with the help of algorithms. Traditionally, algorithms have been employed by math to figure out how to make a circle for example, out of those old squares you had laying around. An algorithm is a recipe, a plan, and much the same way that Darwinism slowly [unintelligible] its way over to Social Darwinism and kind of stayed there. Algorithms have [unintelligible] over from computer programming to Netflix programming, allegedly, and this new manifestation algorithms are production elements that ensure continued audience attention. What you do is you look at successful movies and reverse engineer them algorithmically, mash them all together. Then use that template to make sure the hero in your movie isn’t too angry, that his pals have a nice mix of gender and race, keep the plot really simple little blood is okay, next thing is a puppy and so forth. That’s how you get the hip suburban moms to stream your concoction, and also remember to never bad mouth Marvel movies. You will be canceled. Reading about all this stuff though reminds me at the folklore class I once took. There’s a term called the Stiff Thompson index after the professor put it together. It pretty much names any situation, theme, character etc. in fairytales. Examples: hero needs supernatural help with a task, like young man with Pus and Boots, cobbler with elves. The hero has to answer three questions. Hero gets lost in a forest. Be careful not to look back, which spells trouble for both Lot’s wife and Orpheus. These motifs have always been there. Do algorithms really show the story about elves helping a cobbler is more appealing than elves murdering the cobbler to steal the shoes, I don’t know. Still, these story elements are units of measurement, kind of the basic stuff of myths. And actual mythical numbers show up in real life as well kind of. Take distance: I would walk 500 miles. I can hear the whistle blow 500 miles. May not have trains anymore unless Joe Biden can get Republicans out of the way. But that distance will always be the same, that distance you and your sweet baby is always 500 miles. Meteor craters always measured in terms of football fields or car links. Before we had football and cars, nobody had any idea how big meteors were. Also in recent years any effort at reducing carbon dioxide emissions is measured in terms of how many cars taken off the road it will equal. That no cars are actually taken off any road kind of takes the edge off. I am still taken back when people talk about the four corners of the Earth. Globes don’t have corners. They don’t even have edges. Also, there is no 60 Minute Man and no Sixty-Second Philosopher, you caught me. Put me on the stopwatch and walk away. Some perspective: there are two cups in a pint. But what is a cup? Suddenly we’re talking ounces. It’s a forest tree situation. It’s a slippery slope. Who measured the first ounce? Are future ounces measured against the first ounce which is in an anodized aluminum climate control bank vault in Switzerland? We all know that civilization only invented clocks because we kept missing the train. Suddenly, we realize that these measures are based on more or less random traces made 1000 years ago so to speak, codified but wizards are experts and now we’re stuck with them. That’s why science loves the digital domain. Numbers don’t decay, get worn away by wind and sea; they count nothing but themselves. You can then apply them to everything from eggs to months of the year to disciples. All twelve, by the way. In conclusion then, the ounce itself is a trope or algorithm which combined a liquid form creates a cup or a TV script. Or we add more to make a court or a movie and double that, make a miniseries which is a gallon, which is why we have Bitcoin today and why half a loaf still is better than none. And what if we lose the cobbler? I got it. Reservoir elf, get a mathematician to write that script for you Orson. Elves are also standing by. Also, Quentin Tarantino’s lawyers. I gotta go.

Josh Landy
Philosophy Talk is a presentation of KALW local public radio San Francisco and the trustees of Leland Stanford Junior University. Copyright 2021.

Ray Briggs
Our executive producer is Tina Pamintuan.

Josh Landy
The senior producer is Devon Strolovitch, Laura Maguire is our Director of Research. Cindy Prince Baum is our Director of Marketing.

Ray Briggs
Thanks also to Merle Kessler and Angela Johnston.

Josh Landy
Support for Philosophy Talk comes from various groups at Stanford University and from the partners at our online community of thinkers.

Ray Briggs
The views expressed or misexpressed on this program do not necessarily represent the opinions of Stanford University or of our other funders,

Josh Landy
not even when they’re true and reasonable. The conversation continues on our website, philosophytalk.org, where you could become a subscriber and get access to our library of more than 500 episodes. I’m Josh Landy,

Ray Briggs
and I’m Ray Briggs. Thank you for listening.

Josh Landy
And thank you for thinking.