Episode 90 Transcript

Grant: Welcome again to the Beatrice Institute podcast. It's been quite a while since I've recorded a new episode. I'm really excited to be back.

I'm especially excited to welcome to the podcast Dr. Francis Su. Dr. Su is the Benediktsson-Karwa Professor of Mathematics at Harvey Mudd College. He recently wrote a book that I've found very interesting called The Mathematics for Human Flourishing. As many listeners know, I'm very interested in how the promotion of human flourishing might be understood as the central animating principle of applied sciences, particularly health sciences and engineering. And Dr. Su’s trying to do something similar in the practice and teaching of mathematics. I'm excited to talk with him about many of the concepts that animate his book.

And I want to say at the outset, we both realized that we were overcoming colds, so our voices might be a little scratchy. We were excited enough to have this conversation that we thought we'd push through imperfect voices. It's good to be back.

Francis, it's great to have you. Welcome to the podcast.

Francis: Thank you. Thank you so much.

Grant: I want to start out first with a basic conversation about some definitions and concepts. What is math?

Francis: What is math? Yeah, it's a big question, but I guess I would briefly say, and a lot of people I know like to say, that mathematics is a science of patterns. Although I like to add that it's also in some ways an art. It's an art of making meaning from those patterns. It's one way to think about it. There's an art to it as well as a science.

Grant: Yeah. So how would you characterize the difference? You mentioned art and science, but one thing that came through in your book is how you understand mathematics to be a practice, right? Particularly as you draw on Alasdair MacIntyre. And that seems to be the animating theme of your book, is mathematics as a practice.

How do you characterize the difference between mathematics as a science versus mathematics as a practice?

Francis: Yeah, that's a good question. First of all, I guess I think of science as also a set of practices that people engage in. But, in the context that I like to think about mathematics as practice, it's a collection of virtues.

People often think of math as just a set of skills. A skill being something like knowing how to factor a quadratic, or knowing how to follow a certain procedure to get an answer. But I think math is actually much more broadly a set of virtues. I like to talk about it that way, virtues being habits of mind, dispositions, character attributes that shape the way you look at the world.

So, for instance, if I'm curious, that's an aspect of character and it leads to certain actions like, when I see an interesting question, I seek to learn more on.

So I think of virtue as a certain excellence of character that leads to excellence of conduct. And I think math builds – properly understood, properly taught, properly learned – builds in us certain kinds of virtues, like persistence in problem solving, like an ability to visualize a problem, an ability to quantify, to define, to strategize: these are all the various virtues that are part of the practice of doing mathematics.

Grant: And this is really the essence of flourishing, to the extent that we understand flourishing to be the perfection of virtues towards a particular set of goods that are proper to the human person.

I want to move to this question of human flourishing. What author or thinker has been really influential in your understanding of the concept of human flourishing, of what it means to live well?

Francis: Yeah, I think the first person that I heard talk about human flourishing in a way that was very compelling to me was N.T. Wright, the theologian. And I think Wright was talking about this in the context of animating: what does it mean to live in the world and to understand our purpose in the world as human beings who are spiritual? And, why are we here? And I found that very compelling and it in many ways began to be a frame that I began to understand how we should live in the world and also how I should approach my own discipline, and that being mathematics. And of course, it's been a big theme for me to try to understand deeper questions for why I do what I do. And so I found that very captivating.

Grant: So was it that listening to NT Wright that was the epiphanal moment or was there something else that sparked that connection between mathematics in the sense of a life well lived?

Francis: Yeah, I guess it began with Wright. And since then I've heard many other people talk about this idea of human flourishing. I guess if I'd taken a philosophy class in college, I might have encountered this a lot sooner. But I do like that frame for understanding the big question, why?

Grant: I'm going to return to that question of pedagogy, particularly math. And where I'm located is health sciences, particularly, in nursing school. And I think a lot about how we would think about pedagogy differently if we thought about both the flourishing student, the flourishing practitioner, but also making these disciplines oriented towards the flourishing of others. But we'll return to this question of pedagogy in a second.

How did you choose those – actually, I'll back up a little bit. Why did you choose to use the word “desires” rather than “goods”? Because those could arguably… the thirteen desires that you talk about could also be described as goods.

Francis: Oh, yeah. I don't have a philosophical background, so I guess I just began with the place that I started thinking about this question, which is why is it that some people gravitate towards mathematics and other people don't? Why is math a hobby for some people and not for others?

And when I think about the hobbies that I have or that somebody might have, you know, you wouldn't do something like a hobby unless it met some need that you have. And so I began to ask the question, what are the basic human desires that we might have, such as a desire for beauty or a desire for truth or desire for community, and how does math meet those desires?

And I guess I began thinking about this from that frame, thinking about desires. I hadn't thought about them as goods, but they are goods, they are things that because we desire them and they're basic. I guess I could think about it that way as well.

Grant: Yeah. And to the extent that every desire that's a good desire is oriented towards some natural, basic human good.

So how did you choose those thirteen? I can imagine there were probably other ones on the table that you excluded. How did you choose those thirteen and not others?

Francis: Yeah, that's a good question. And for those who haven't read the book, I've oriented every chapter around a basic human desire and talk about how math meets those desires. There were lots of potential ideas that came to my head. I guess I started just by picking the ones that I thought would resonate with people who… for instance, like myself as a professional mathematician, there are several there that would resonate with a potential mathematician, such as the desire for beauty. That's probably the number one reason mathematicians say they study mathematics.

But then I also tried to hit some basic human desires that were maybe not so obvious from the perspective of a mathematician, but maybe more obvious from the perspective of someone else who lives in the world and wonders why math is relevant, right?

So the book is aimed at multiple audiences, but one audience is those who know, who study mathematics, and know mathematics well, and there's another audience of those who don't.

And so, when you think about some of the things that human beings really gravitate towards, it's desire for community. But when I think about the best experiences that I have doing mathematics, it's often in community. And it's underrated, underappreciated, I think, when we talk about math, because people tend to talk about math is just a set of skills, right? And yet, if you talk to people who enjoy working and thinking mathematically, I think if you ask them to stop and reflect on it, they'd say, yeah, community is a big part of why I do what I do. And so I wanted to unpack that as well.

Grant: Right. As I look at these thirteen desires that are achieved through the excellent practices of mathematics, I see that many of them can be realized through other practices, right? There's obviously community that can be achieved elsewhere. Is there something uniquely effective about mathematics that helps us achieve these desires above and beyond other practices that might be oriented in the same way? Is there something unique about the practice of math?

Francis: Yeah, that's a great question. Yeah. I mean, certainly, many of the things I talk about, the desires and the virtues that are built by pursuing these desires, are not unique to mathematics. And in that sense, my book can be viewed as an argument for the liberal arts, right? How mathematics is a liberal art. It helps us to develop these broad perspectives on the world.

But I think there are some unique ways in which math cultivates, for instance, a desire for beauty. I think if you've ever encountered the beauty of reasoning, I think that is probably a kind of beauty that is maybe unique to mathematics in the sense that…I think there are very few other disciplines where the act of reasoning itself is an experience of beauty.

And if it appears anywhere else, I would call that mathematical, right? This is part of the way I think about mathematics is that mathematics is good thinking, right? It's not good calculating, it's good thinking. I like to say anything a calculator can do isn't really math. It's… arithmetic from my perspective is not really – okay, yes, it's math in the sense that there are certain rules that the universe obeys and arithmetic are some of them – but the joy and the wonder in calculation is to be found in the understanding, right? It's not just in the calculating.

Grant: Right. What would I learn particularly about math from pondering an M.C. Escher painting? You mentioned M.C. Escher a number of times in your book. What are the math principles that I could ponder?

Francis: Yeah. Well, yeah, that's a good example. If you think about how people respond to art, math is like that, right? Like the fact that what does M.C. Escher often do? There's often a shift of perspective when you're looking at an M.C. Escher print. There's the primacy of foreground and then you look at the background. You're doing this shift.

Or maybe there's an impossible waterfall that locally, if you look at part of the picture, it looks normal, but globally, it doesn't make any sense, right? M.C. Escher is famous for pieces like that.

And math is like that, right? Math is looking at an idea from many different perspectives. That's one of the amazing things that when people begin to appreciate math, they're like, whoa, I can solve this problem in so many different ways. And from this perspective, this principle holds right. And from that perspective, some other principle holds. That's very mathematical.

The shift between going local and global, I think, is a big theme in mathematics as well. So his pieces are very mathematical in that way. He often appeals to symmetry, often appeals to one sense of pattern and order, but then playfully distorts them in some way. And that's often what we find in math as well. You think there's some order, and then when you look at the big picture, something else unexpected happens. People often express beauty as a form of surprise, and that shows up in math all the time. Like, what does it mean for a set to have infinitely many things in it? That seems like a very simple concept, but when you look peel under the layers, you realize, whoa, there are many different sizes of infinity. That's an unexpected surprise.

Grant: What's your sort of arcane corner of mathematics that you study from a research perspective? And, in maybe three sentences, what's beautiful or compelling or unexpected about this very arcane corner that you study?

Francis: Yeah, let's see. I study… It depends on who I'm talking to, how I frame it. But if I'm talking to a mathematician, I would say I study geometric combinatorics, which is a study of a mixture of geometry, and combinatorics, which is the study of ways of counting things. If I'm at a cocktail party, I would probably talk about game theory, which is the mathematical modeling of decision making. And so what I do is I actually bring methods from geometric and topological combinatorics to the study of problems in the social sciences, namely the mathematics of decision making. And that's game theory. That's one way to think about game theory.

Grant: Tell me about a moment you've had with game theory where you had this breakthrough and you experienced some maybe transcendent moment that connected you with something particularly divine or beautiful.

Francis: Oh, interesting. Well, I'll give you an example of something that I think is a beautiful result. It's not my result. John Nash, who many will recognize as a Nobel prize winning – well, he's a mathematician, but he won a Nobel prize in economics for proving that every game has an equilibrium.

Now, a game is basically a set of players and strategies and when players play strategies, it achieves some outcome. And we call a set of players and strategies an equilibrium when everybody's response to everyone else are the best possible in some sense. It's called an equilibrium and it's called that because what Nash proposed is that games will naturally be found in equilibrium, when people play a game long enough, they begin to realize what the best responses are to other others.

Okay, so that's some fact, that that Nash proved, but the amazing thing is that Nash brought an idea from geometry and topology – topology is a study of continuity, it's one way to think about it – Nash brought this amazing idea; let's see if I can explain it. There's something called the Brouwer Fixed Point Theorem. Okay, and I'll just see for the podcast listeners, I'll have them imagine me holding up a cup of liquid. And the Fixed Point Theorem says that if I take my cup of liquid and I take a picture of it before I slosh it around, and then I slosh it around, and then I take a picture after, that there's going to be a point in the liquid, in the coffee, if you like, that is in the same point position in both pictures, even if I sloshed it around. It's kind of an amazing fact that there's always some point – maybe more than one, but always at least one point – that is fixed. It's called a fixed point. That is, it's in the same position in both pictures.

Now, that's a well-known result from this area called topology. And the beautiful, amazing, surprising thing is that that turns out to be the key to proving that every game has an equilibrium in this other area known as game theory. And, for that, Nash won a Nobel Prize, right? And why? Because it's sort of an unexpected result that comes from a surprising and beautiful theorem in another field of mathematics.

So that's another thing that's beautiful about it is it makes unexpected connections between things you thought were disconnected. And that connects to our feelings of transcendence, right? Like when you have an idea or you have an experience that's transcendent, it's often because you feel somehow strangely connected to the universe in some sense. It's seeing connections that you wouldn't have believed was possible.

Grant: I'm glad you bring up John Nash because I have another set of questions that have emerged. I've been reading this Chilean novelist named, I think it's Benjamín Labatut, I think his last name is. I'm not quite sure how to pronounce it; I've never heard it. Two books called, one's called THE MANIAC and the other is called When We Cease to Understand the World. And there are fictionalized accounts of many of the greatest mathematicians in the world. So, real events, real stories, he fictionalizes them. THE MANIAC’s about John von Neumann and When We Cease to Understand the World is about other similar types of mathematicians and physicists.

One central theme of the book is the co-location between genius and madness. Why is it that high level mathematicians are so prone to madness, or they at least seem to be?

Francis: Yeah. I mean, certainly the popular conception, people often think about examples like that, John Nash, maybe being one, who, well, was quirky and eccentric. I remember growing up watching – what was that dinosaur movie? – Jurassic Park, where they had the mathematician who was sort of a little bit… a little bit quirky, played by Jeff Goldblum.

I guess I would think about that in two different ways. I mean, one is that in some sense, you're just seeing a popular conception that – you know, I think there are lots of mathematicians who are not eccentric in that way, and it's probably precisely because we don't appreciate the full set of virtues that mathematics brings that we only recognize one dimension of it and that you see so many examples of quirky or mad mathematicians, right? If you only think that math is about solving calculations quickly in your head: well, there are lots of geniuses who may not have developed in other ways, but are very proficient at calculating in their heads, right? So then we see, “Ah, gosh, every mathematician must just be a quirky mathematician.” But I would say that, in one sense, I think we need to recognize more ways that people can be mathematical. And, when we do that, I think it would necessarily make that association.

That being said, mathematics more narrowly defined, when you see examples of quirky, mad mathematicians, there is something that is attractive, right? That even people who have not had a chance to fully develop in other ways, the fact that they would gravitate to mathematics is a form of comfort, solace – I'm not sure if that's the right way to put it – that they can see and appreciate aspects of the world that we can't see is a wonderful thing for them and for us.

Grant: Yeah. So this is a similar but a little bit different question. Are there particularly pernicious vices that mathematicians are especially prone to when they don't practice mathematics well? Because you talked a lot about the virtues that are associated with doing math well. Are there particular vices that – I don't want you to call out your colleagues or anything, but – are there any vices that maybe, maybe mathematicians are particularly prone to?

Francis: Yeah. I mean, you know, some of the devices that… Any vice can often be found in conjunction with some virtue. But maybe an overemphasis on certain practices could lead one astray. I mean, so an example might be the fact that math allows us to clearly define things, to talk about what we mean with precision, I think could in some ways be so overdeveloped that we're prone to try to over quantify things that shouldn't be quantified, right? Like, if to say that I love another person is an important, valuable, meaningful concept, but I wouldn't want to overdefine it in a sense that makes it more limiting.

Grant: Yeah. As I was watching – I don't know if you've seen Oppenheimer, the Christopher Nolan movie, but there's a sense in there in which I wonder if the folks at the Manhattan Project were overwhelmed by curiositas, this disordered curiosity that drove them to look into something so deeply that maybe they should not have been peering into, if that makes sense. And I wonder if that might be operative in that as well within very, very high level-mathematics.

Francis: Yeah, that's a great question. I haven't seen Oppenheimer and I should. But I mean, it also brings to mind a lot of the new technologies that are very mathematical, and are shaping our lives today: artificial intelligence to some extent, also social media, right? That's part of the whole mathematical computing revolution that's using algorithms to feed us things that we desire. But we've seen that if you only get fed things that you desire, that it leads to overindulgence, polarization. And that's not necessarily good for us.

Grant: So, if I wanted to be a math explorer, given the limitations of my time and in my station in life as a father and a husband, where would you suggest I start?

Francis: Yeah, that’s a great question. I can think of a few ways. There are probably many more. But one is if you're a parent, what parent doesn't enjoy watching their kids learn and pick up new ideas? I would say engaging in the playfulness that your kids have about life and the world and especially about reasoning is a good way to begin being a math explorer, especially when it comes to numbers and teaching our kids to flexibly use numbers, I think is a place where we can ourselves begin to see the delight and wonder of reasoning.

For others, it might be picking up popular math books – not textbooks, but books – about math and the popular imagination. That's another way, which I think – I think what we really need, actually, are people who have some idea of how math actually shapes the way we live, and could shape the way we live. And these popular accounts are a good way to do it without getting into the technical weeds. But once you see some of that, you might actually be motivated to learn more.

Grant: I'm struck in your book by the fact that mathematics is, in a very special way, a glimpse into the divine, right? Does studying math make it easier or harder to believe in God?

Francis: Well, there, there are many ways in which mathematics and the pursuit of the divine are similar. People often refer to math using language that sounds almost like spiritual language, right? Like the famous, mathematician, Paul Erdős, who was actually kind of anti-religious, still liked to talk about math as, the best mathematical proofs were written in a book, that God, only… let's see, how did he put it? That there's a book in which all the most beautiful proofs of mathematics are kept, right? God's book in some sense. He wasn't even a believer by any stretch of the imagination, but…

So people often have this the sense of reverence and awe. When you see a beautiful mathematical idea, you have this transcendent feeling, right? So there's lots of connections.

And so, from that perspective, I think mathematics can, it does, help people believe that there's some order, some rationale behind the universe. And it wouldn't surprise me if that evokes religious feelings. On the other hand, there's plenty of people who aren't religious in any traditional sense, but revere mathematics. And so if you don't have that inclination to believe in anything beyond that, it might lead you to worship mathematics itself.

Grant: One interesting thing about your book and hopefully folks will pick up and read it is the end of every chapter is a letter exchange between you and a man named Christopher Jackson, who's an inmate in a prison. And he discovered math late in life and is quite proficient at mathematics, that he discovers this while he's in prison. And I found those sections to be very interesting, very moving. How does someone like Chris get missed in the school system? Why didn't someone realize when he was twelve how proficient he was at math?

Francis: Yeah. Well, I think there are a lot of Christophers out there that we don't recognize the potential in because often when you think about mathematical potential, you're thinking about something that's very one-dimensional, that's not the full range of virtues.

But often in elementary school, it's all about speed, right? Like who computes quickly, right? And that's not even a thing when you get to the professional level, to be fast or slow. It's amazing to me that people get that idea. But why do they get that idea? Because our assessments, our homeworks, they're all worksheets, right? And they’re filled with computation problems, but maybe not filled with joy and wonder as much.

So I think it's easy to miss students who are proficient in other mathematical virtues, ones that might be, in fact, more important: being creative, being curious, being persistent in problem solving, being able to visualize, being able to abstract, right?

And these are the reason it's easy to miss these, is that they're hard to measure. It's easy to tell if you can solve twenty problems that are computational problems. It's hard to actually assess whether somebody's creative.

And so it's easy to miss the potential students and the potential, I would say, of every student, right? I think all of us have the capacity to grow in mathematical ways of being and to appreciate that. But when we say some people are math people and some people aren't, we're automatically prejudging people in a way that I think is unhealthy.

Grant: One thing I found interesting about your book is you definitely do talk quite a bit about the intersection between mathematics and social power and our social life together. Are the rules of math, the rules inherent to mathematics, socially constructed?

Francis: The rules. That's an interesting question. I mean, there's certain… So, mathematics is baked into the way the universe works, right? So in some sense, many of the things that you learn, like addition, multiplication, the distributive property, commutative property of integers, these are all baked into the universe, in some sense. So those are not socially constructed at all.

But I think what is socially constructed are what we consider to be interesting, right? The choices we make about what questions to study and what questions not to study. I think when you get to more abstract layers and levels of mathematics, it's maybe less clear what the important properties are to focus on. And so you might have one axiomatic system that chooses to focus on certain properties. And that's also a social construction. But on the other hand, over time, people begin to see ways of thinking about these things, that some are more beautiful than others. And you generally naturally gravitate towards the beautiful ones. And when you do that, often that's maybe less socially constructed. There's this idea throughout math and science that often the most beautiful ideas are the ones that are the truest ideas, and I think that's true in mathematics as well.

Grant: If you had to put a number on it, what proportion of the difference in mathematic’s achievement between men and women is socially generated as opposed to inherent differences in aptitude and interests?

Francis: Oh, there's a huge, big question, but… Once again, I'm going to point to the fact that there are many ways to be mathematical, and often these measures of math ability only focus on one dimension. And so it’s maybe missing the mark in terms of thinking about ability.

And of course, in terms of mathematical ability, proficiency, things like that, you just have a snapshot of what someone's able to do at a moment in time. And those are generally going to be normally distributed around some some shape, and you'll have often at the tails certain extremes. And I would say that, often because of the ways that society forms narratives around who does math and who doesn't do math, that you'll often see at the extremes of the tails, often more men than women. And that's not to say that I think that inherently women and men, have a difference in ability, because there's many different measures of mathematical virtue, and it's not been my experience that women are less capable than men at mathematics. So now of course I'm forgetting the actual question you asked.

Grant: I'm very interested in men in nursing, right? Particularly men in healthcare professions. And there's this real question about whether or not the differences in nursing are because men are just naturally oriented towards things; women are more naturally oriented towards people and there's obviously the bell curves that overlap where there's going to be some men in nursing and some women in engineering.

Francis: Yes. Right. I see. That's a very interesting question.

Grant: Yeah. And I find whenever I talk about this, most answers to this question collapse into: it's all social construction; men and women are exactly the same. Or: this is entirely differences in organic brain function.

So I'm wondering how you would think about this, maybe just in terms of differences in achieving PhDs in mathematics. Is there some inherent difference between men and women, or is it this a social sorting of some kind? That's the root of the question.

Francis: Yeah. I mean, I think in some ways there's a mixture here that's hard to untangle. And it is definitely true that women are discouraged from pursuing the highest levels in mathematics because of all sorts of reasons that are social, in fact. And it's also true that there are a wide range of people who exhibit proficiency in math according to whatever measure you choose, right? And so I'm not saying… there are people who pick up math quicker than others. I don't think that there's a… You know, if you look at the broad spectrum of what it means to be mathematical, I think that it certainly seems true that women gravitate towards community, right, and being communal in doing things, whether that be mathematics or other things. And I think that plays out sometimes in the way that who gets attracted to doing what they think math is, which often people think of as lone and solitary. Right?

So, I don't know. There's a lot of things to unpack here. But, if I had to look multidimensionally at the many different ways of being mathematical, I don't see overall sexual differences in math abilities.

Grant: I want to talk a little about pedagogy. You seem like you really like teaching, at least from, I got that from the book, that you seem to really have a passion for teaching. How has being located at Harvey Mudd influenced your understanding of flourishing in mathematics? It seems like a pretty special place.

Francis: Yeah. I mean, Harvey Mudd, it is a unique place because it's both a liberal arts college, which means students here come because they're looking for a broad background in not just the sciences, but also the humanities. But it's also a tech school that's focused on STEM. All our students here are studying math or science or engineering. And so that's one unique aspect.

Another unique aspect is that our mission statement explicitly talks about the need to train up scientists, engineers, mathematicians well-versed in the humanities and social sciences and the arts, who have an understanding of the impact of their work on society. So there's a very strong focus on not just learning science, but to use it wisely and use it well.

And I really resonate with that mission statement. I like the fact that I do want my students to see math as something that contributes to human flourishing for everybody. It has an aspect of wonder and joy. But when you see something that’s wonderful and joyful, you naturally want to share it with other people, right? You should naturally want to share it with other people. And so it's not just for my own sake that I should be doing math or science. It certainly should have some aspect of contributing to the flourishing of others. And I resonate with that.

Grant: So, in what ways do grades get in the way of excellence in mathematics?

Francis: Yes, that's a big question. And it's something that I thought a lot about. When we measure people's mathematical proficiency, whatever measure we use, there's maybe an undue emphasis or focus on that number as being a sign of somebody's self-worth, in either in mathematics or just in general, right? And often when somebody doesn't do well on some assessment, people look at that and say, “Oh, you're not a math person,” or “You are a math person,” some instant judgment. I like to say that grades are a possibly imperfect measure of your progress. It's a snapshot in time, but it doesn't reveal your trajectory.

And I guess because of the undue emphasis people have on grades, I guess, I hope that people would relax a little bit about what a grade actually measures, right? It's a snapshot. It's not a trajectory.

And I would say that even for students who are high performing, right? There are students who are high performing who sort of start seeing somehow that their performance in math means that they're better than other people. And somehow they start seeing that as a measure of their own self-worth. And so one of the dangers of that is that people begin to lose the sense of joy and wonder in math itself and begin to crave the performance aspects of mathematics. And I think there's something deeply lost there.

Grant: Last question. I know we're coming up on time. Do computers make us better or worse at math?

Francis: Ah, that's a good question too. Well, I mean, is watching TV good or bad for us? Well, certainly, you can have too much watching TV. And there's a way to watch TV that's mindless and numbing and deadens you to the world. But on the other hand, there's wonderful movies that enrich us and make us think, right?

I think that's very similar to the question about computation. Like, do we sit back and just automatically trust compute computers to do things for us? Or do we think carefully about how they're used and why we should use them, in what situations? I think it's a very similar question.

Grant: So is it better or worse?

Francis: What’s that? Oh, see, I didn’t answer the question. Are they better or worse for us?

I'd say on the whole, they're better in the sense that they can automate things that we don't necessarily need to do ourselves. But we have to think about them wisely and how we use them, right? And think critically about what we do with that information.

Grant: Yes, one potential argument that you have made in the book is that computation is not math. Math is the recognition of patterns to the extent that the computer can help with the computations, which isn't really math anyway. It can get you really focused on the business of the pattern recognition and the understanding, potentially.

Francis: Yeah. Yeah. And of course, computers can help us begin to identify patterns. But we're the ones who are going to have to say what's the meaning of those patterns? That's something that I don't think computation is going to effectively help us do.

Grant: Right. And maybe having your math students read more novels might actually help them figure out the meaning of the patterns as well.

Francis: Yeah. Yeah, that's right. That's right. I'm in favor of that.

Grant: All right, well, Francis, well, this is really a lot of fun. I'm so grateful that you took time out of your busy schedule to chat with me and I hope we can continue this conversation. We have very overlapping interests, so I hope we get to talk again sometime.

Francis: Yeah, me too. Thank you so much.

Grant: All right. Thank you. Take care.