UK's Dr. Snezana Lawrence on the History of Mathematics

The Learning Curve Snezana Lawrence

Alisha Searcy: [00:00:00] Welcome back to the Learning Curve podcast. I’m your co-host, Alisha Thomas Searcy, and I have a guest co-host with me today. Jake Tawney, welcome to the Learning Curve podcast.

Jake Tawney: Thanks, Alisha. Just delighted to be here with you.

Alisha Searcy: Great. Well, why don’t you tell folks a little bit about yourself?

Jake Tawney: Again, my name’s Jake Tawney. I’m an educator by trade. Have been a educator really my, my entire career. Started off teaching math and computer science courses, of all things, in the great state of Ohio. And after doing that for about a decade, had an opportunity to move out to Phoenix, Arizona, where I joined a growing, and in fact still growing, wonderful group of charter schools [00:01:00] called Great Hearts Academies, that is dedicated to classical education.

Saw that organization grow from, I think, about 16 schools all the way up to 50 by the time I departed from them. That was a very difficult decision, but I was given a really great opportunity to work for the Institute for Catholic Liberal Education, which is, in some ways, the Catholic organization that’s helping a lot of schools around the country change their model of education over to much more of a liberal arts, or in some circles what we call a classical education model.

So I serve as the Director of Curriculum and Academic Resources, and the ICLE works with well over 300 schools through virtually every states in the United States.

Alisha Searcy: That is incredible. Well, we are happy to have you, and I have a feeling that you’re really going to enjoy today’s show. And so with that said, we usually start with articles, and so since you are guest co-hosting, I’m gonna put you on the spot and ask you to talk about your article first.

Jake Tawney: The article that struck me, or the news story that struck me in the last couple weeks, is the release of Pope Leo’s [00:02:00] new encyclical, Magnifica Humanitas, which is really an encyclical being billed about artificial intelligence. Of course, it’s quite lengthy. There’s a lot going on in this encyclical letter.

But it is true that the core of it’s about… As an educator, of course, what I was struck by was all the commentary that Pope Leo has on the nature of education. And of course, he builds this on w- what I would call a, a solid anthropology of what the human person is, right? And so he gives a lot of commentary on how AI can’t feel, it can’t experience, that really it’s fundamentally, as powerful as it is, it’s not human, right?

There’s something missing there. But then he builds on top of this sort of a, just a beautiful vision for education, where he says, y- you know, that education is a long journey requiring patience, and therefore needs time for developments and for engagement with reality beyond appearances. And then he quotes Plato.

“As Plato wrote, ‘The deepest and most important things are learned only after much time and effort, [00:03:00] by engaging with discussion with others, striking upon ideas, and experiencing together, like flint, until the spark of understanding is kindled within us.'” And I would just say, you know, one could hardly ask for a better defense of both the content and pedagogy contained in what my organization calls the Catholic liberal arts tradition, but what other schools and other organizations are calling classical education.

One could hardly ask for a better defense of that. This patient reading of the great texts, the art of inquiry that refuses easy answers. That slow formation of judgment that only happens through a real encounter with beauty and with goodness and with truth. I think this is a bold vision. It’s a bold vision that, that spoke to my heart as an educator.

And so in– again, in the middle of an encyclical that’s supposed to be about artificial intelligence, and it is, we just find this, this real gem of a vision laid out for what it means to educate students.

Alisha Searcy: I love that, Jake. I love the way that you have talked about this article and what [00:04:00] resonates with you.

I also love, as someone who is not Catholic, how the Pope has, I think, opined on some of the most pressing issues that we’re facing as a world, particularly around this issue of AI a- and education. And so thank you for bringing that forward and talking about it. And I think certainly our listeners should take a moment to read that piece as well.

So thank you for sharing. The article that I wanted to talk about really briefly this week is from The 74, entitled “Why Students Reach College Underprepared for Math and What To Do About It.” And I’m very interested to hear your thoughts on this, Jake. In this piece, the author is talking about how, you know, around the country, we are certainly talking about how students are unprepared when they get to college, particularly when it comes to math.

And of course, we’re talking about math today, and our listeners will find out in just a moment who our guest is. But as we talk about math, as we talk about certainly the NAEP scores that we’ve seen that have been on the decline both [00:05:00] in reading and in math, we’re also hearing often about how students are coming to college unprepared and having to take remedial courses.

And so this author is essentially making the point that it’s actually not the fault of the students that they are coming unprepared, and she gives an example of maybe some of these students are first-generation graduates who may be at the top of their class, but perhaps they attended under-resourced high schools in rural districts or other places where they may not have access to high-level math courses or even, I would insert in here the word quality of the math courses weren’t at the same level.

And so she’s saying that rather than blaming the students, perhaps the core issue is not what students lack, it’s instead what the institutions have failed to provide. So in some cases, I’m thinking people would say this is pretty provocative, but I think she actually makes a really good point, you know, thinking about who is responsible for making sure that students are prepared.

They can only demonstrate [00:06:00] proficiency based on what they’re provided, right, in terms of access, in terms of quality. And so what I appreciate, though, is that she has some solutions here, and I love the fact that the University of Georgia is actually one of the examples of things that are going well. And so she talks about how colleges should provide support alongside college-level instruction.

She mentions how the University of, uh, System of Georgia replaced traditional non-credit remedial math with a model that places students directly into college-level courses while providing just-in-time support through labs, tutoring, and aligned instruction. She says this approach has significantly improved outcomes, tripling completion rates, which is also really important, and gatework coursework And boosting pass rates from 20 to 66%, while offering more responsive, individualized help that keeps students on track.

And so again, I think a really important conversation for us to have nationally about, A, students being prepared, but even before they [00:07:00] get to college, making sure that the K-12 system is actually preparing them and ensuring that they are proficient in math. What say you, Jake?

Jake Tawney: Ah, I have lots to say about that, but I’ll- i’ll keep it short so that we can get to our exciting guests today. I was raised by a father who was not only a math teacher, but my math teacher in, in high school. And he often reminded me when I was a young teacher that it’s the wrong attitude for a teacher to show up at a classroom and say, “I’m here to teach, you’re here to learn, and if you don’t learn, then it’s your fault.”

That a teacher needs to very deeply in the core of their being feel that it is my job to form these students mathematically. It is my job to satisfy that unique thing in their human soul that can only be satisfied by learning mathematics and by being able to think mathematically. I just think a lot more attention needs to be given to how we go about teaching mathematics in our classrooms, how will we go about training teachers.

You know, I think it stan- starts with high-level curriculum. It moves into adequate training of our teachers. Mm-hmm. But I [00:08:00] also wouldn’t lose sight of building those systems of intervention, particularly at that middle school and high school level. Elementary schools tend to have a little bit better sense of systems of intervention, you know, in large part because of things like reading skills that are so important early on.

But as we move into middle school and high school, we need to have those built-out systems of intervention as well. And the final thing I’ll say is we have to kind of get beyond the, the typical debate that we’ve found, this swing in math curriculum from a curriculum that would just see us drill and kill, which maybe is how some of us learned math, and, and that seems to not work because students don’t understand why they’re doing what they’re doing, and it’s just a lot harder to memorize things when you don’t have context of meaning.

But then we’ve also seen it swing the other way to the point where it’s only about creative thinking, solving problems in your own way, but we’re not actually practicing the skills and facts necessary to access the higher things. And again, I think those effects compound as we move our way up through middle school and high school.

So I think we need to get back to, in some ways, the math model that was there, you know, 100 years ago or so, where it was [00:09:00] both and. We really want students to wrestle with the big questions in the discipline, but also to develop those facts and skills that are important so they can access the bigger ideas in, in future math courses.

Alisha Searcy: Well said. Thank you, Jake. I really, really appreciate that insight. I also think it’s interesting as you talk about elementary versus middle and high, you often hear teachers say that they feel more comfortable with the content at the elementary level, right? And the higher it gets, they’re a little bit intimidated.

So we’ve gotta address that as well, and I think you mentioned it in terms of training. So thank you for that. I like having you on here, Jake. You are good.

Jake Tawney: I appreciate that.

Alisha Searcy: Before we get to our upcoming guest, we wanna take a moment to note the passing of the scholar of Brown University, Gordon Wood.

He’s a Pulitzer Prize winner, renowned scholar, and we’ve had the privilege of having him on the Learning Curve podcast a few times. And so we are grateful for his work and his contributions, and he will greatly be missed. Coming up, we have Dr. Snezana Lawrence, who is [00:10:00] independent scholar affiliated with Middlesex University, so stay tuned.

Dr. Snezana Lawrence is an independent scholar affiliated with Middlesex University, London, and was the chair of the History and Pedagogy of Mathematics International Study Group between 2020 and 2024. She is the author and editor of several books, A Little History of Mathematics (2025); Mathematical Meditations (2025); A New Year’s Present From a Mathematician (2019); and Mathematicians and Their Gods (2015, co-edited with Mark McCartney). Since 2015, Dr. Lawrence has been a Visiting Professor at several universities, including Masaryk University in Brno, in the Czech Republic (2022-25); the University of Lorraine in Nancy (2014-15), France; and the University of Kragujevac in Serbia [00:11:00] (2022- ). She earned a B.Sc. in Architecture and Engineering from University of Belgrade and a Ph.D. in Mathematics from the Open University in the UK. Welcome to the show, Dr. Lawrence.

Dr. Snezana Lawrence: Thank you very much, Alisha.

Alisha Searcy: So you’ve authored several very interesting books about mathematics, including your 2025 volume, A Little History of Mathematics. Can you share with our listeners some of your background and offer a brief big-picture overview of why humans developed math to begin with?

Dr. Snezana Lawrence: Yeah, so let me just say, I’m really delighted that you invited me to, to talk about this little book here. So I’ll tell you how I started. I didn’t really… I, I always wanted to study mathematics, but my first degree was actually in architecture, and it was in old country, Yugoslavia, which doesn’t exist anymore, and I particularly enjoyed descriptive geometry.

Now, that’s an old kind of technique that was still taught there. And when I came to [00:12:00] England during the Yugoslavian War, I wanted to teach it because I was actually doing some teaching there back at the old home. So I started looking at where I can teach in England, and then I discovered that in England it didn’t actually even exist as a subject.

So that is how I started with the history of mathematics. That inspired me to learn why that wasn’t actually taught in England. So what is mathematics and why is it important and how did humans develop it is a very, very long story I suppose it is related to other kind of disciplines, like for example, visual arts or language, but obviously not the same.

Mathematics is particularly abstract, and it has been developed throughout the history of humankind as that kind of abstract way [00:13:00] of recording our thoughts and communicating and building on them. As someone who deals with mathematics, you start by looking at a pattern. You no- you notice something happening in a similar way that has happened somewhere else, and you start noticing that pattern, and then you record that pattern, and then you start actually then connecting that to other patterns.

That’s really how it works. That is, I suppose, my explanation of mathematics.

Jake Tawney: Dr. Lawrence, just so delighted to be here with you, and so delighted to have a copy of your book. The, the history of math is near and dear to my heart. There’s so much in the book, and at the risk of skipping through a couple things at the beginning, I would love to actually talk about Pythagoras a little bit.

Even if people know next to nothing about the history of math, there’s a pretty good chance they’ve probably heard of Pythagoras, in part probably because of the Pythagorean theorem. But his influence as a Greek philosopher, y- you know, on Plato, on Aristotle, and really kind of on the whole developments of Western [00:14:00] reasoning and math, is really astounding.

But people may not know this, he also had an influence on music. And so I wonder if you can talk just a little bit about Pythagoras, about his importance as a mathematician, and maybe even just dip a little bit into his influence on music.

Dr. Snezana Lawrence: Oh, yes. Sure, yes. He is a, a fantastic personality f- from the history of mathematics.

In fact, if you start talking, if… Quite often, if I, for example, talk to people and they ask me, “What do you do?” And I say, “I’m a historian of mathematics,” well, nine out of ten at least times people will say, “Oh, yes, yes, I know. I know about Pythagoras.” So he’s probably the first person who pops into people’s minds.

The strange thing is that we don’t actually know 100% that he actually ex- existed as a person. We definitely know the Pythagoreans as a sect, kind of mathematical sect existed. But what we have about him, we have from historians who lived hundreds of years after him. [00:15:00] So there are no writings directly that survive from Pythagoras, and that is the really the reason that quite a lot of historians of mathematics think that he might not have actually existed.

However, there are obviously some, as I said, historians who did think he did exist, and they note his influence on, for example, Plato and Aristotle in particular. So what was he known for? Well, he founded this group of kind of mathematicians, which quite a lot of people call a sect, and they call it a sect because they had very, very strict rules of life.

So they had the inner circle, uh, mathematically and acoustically, who– which was the outer circle, people who listened. So, as I said, they followed these strict rules of life. They had very stringent and probably a bit strange dietary regulations. For example, they prohibited [00:16:00] all eating of beans to all their members.

And they believed in transmigration of the soul, which is a kind of a reincarnation principle. So they were very, very deeply invested or, and interested in how mathematics can explain everything in the world. It wasn’t for them kind of a profession, it was part of their worldview. They looked at numbers.

The numbers for Pythagoreans signified various principles on which the world was built. They looked at the properties of numbers and studied and sometimes attributed certain beliefs to certain numbers. One of which was that certain numbers produced very beautiful ratios, and they produced ratios that can be translated into music.

I think the easiest thing you can imagine is if you have a string instrument and then you press that string in half, the [00:17:00] whole string will produce a certain sound, and then the string that you depress in half will produce half of that, and that will produce a very beautiful sound. So Plato was very influenced by Pythagoreans mainly in his theory of forms or theory of ideas.

And that theory talks about the real and ideal worlds. So in the real world, everything that we experience and we see is, for Plato, is a kind of copy or shadow of what perfect object or perfect experience would be in the ideal world. And if you translate that into mathematics, then a square in an ideal world is a, is a absolutely perfect square, but we can’t really recreate that perfectly in the real world.

So if you, if you draw a square, let’s say, it will never be absolutely perfect as it is in the ideal world. And there is this tension between [00:18:00] that ideal and real in this theory and in mathematics that really has influenced the development of mathematics really since then.

Alisha Searcy: Fascinating. Dr. Lawrence, in your chapter Unearthing Wisdom, you write the cuneiform script invented in Mesopotamia is in fact the earliest script we have evidence for in the history in the world.

It was in ancient Mesopotamia that the day was divided into 24 hours, each hour into 60 minutes, and each minute into 60 seconds. Can you tell us more about the cuneiform script, early measurement, and mathematics?

Dr. Snezana Lawrence: Yes, sure. This is a very sort of long-lasting culture, the Babylonians or Mesopotamians, or people call them in different ways, that really lasted for a very long time, but we don’t have very much from them.

We have these tablets. We have a huge number of tablets that survived, and they were clay tablets, and the cuneiform [00:19:00] script developed in the way it has because they used these clay tablets on which they made incisions with a stylus kind of as a pen. And then s- they, they were baked or they were dried, and that is why they survived.

So it is really astonishing to think we really didn’t know much about them until the 19th century when we started being able to learn about their civilization from all these tablets. A similar thing happened really, of course, with the Egyptians. When Egyptian civilization met its demise, we didn’t really for many centuries know how to decipher their writing.

You asked about the measurements. Well, Sumerians formed the first standardized metrological system, and these were then refined by the Babylonians. They used base 60, which is, as you said, just the way we count, we measure time now, and that really [00:20:00] remains from that metrological way of measurement. So they used base 60, and then they used body parts and everyday loads of grain and, and so forth to develop their other measures.

And every city and kingdom and city guild had their own standards until the formation of the Akkadian Empire when they standardized the units of measurement. So it was a very important civilization lasting for a very long time, and from which we know some things based on those clay tablets.

Mathematical clay tablets I described in the book were the tablet Plimpton three two two is a three thousand seven hundred-year-old Babylonian tablet which found its way to Columbia University, and I also described the Yale tablet, which describes mathematics in an interesting way because it relates to [00:21:00] Pythagoras’ theorem.

Plimpton three two two, which was bought from a dealer in then Constantinople by an American antiquary called Edgar J. Banks, and he sold it to a publisher, New York publisher, George Arthur Plimpton, and that is how that little tablet that I describe the mathematics of it found its way to Yale.

Jake Tawney: This is wonderful.

The whole history of writing numbers and how we write numbers is fascinating to me. And you mentioned that the, you know, the Babylonians had a base 60 system. W- we, of course, most of us have a base 10 system. But for all the differences in how we write numbers, there’s also been some discussion about what even constitutes a number itself.

And what I love in your book is how you, you talk about the nature of the number zero and how groundbreaking that is. It sounds like such a simple idea, you know, is zero a number? And yet we also know that people like Aristotle said that even one is not exactly a number. Like, one and number are opposites for Aristotle.

Of course, in, in [00:22:00] modern math, we have all kinds of numbers, irrational numbers, rational numbers, negative numbers, constructible numbers, complex imaginary numbers, transcendental numbers. But I wanna go back to zero because we all take for granted that zero is a number. Can you talk about the role of India as sort of the place that you say birthed the numeral zero as we know it, and why the number zero, as simple as it sounds, has become so important for mathematics as we move forward?

Dr. Snezana Lawrence: The origin of zero is something that fascinates mathematics and mathematicians and historians of mathematics, or has been fascinating for a very long time. In India, several really things point to India as the place that sort of gave birth to zero as we know it, close to about first millennium. And the symbol which stands for empty, void or vacant, had a purpose in calculations in, in Sanskrit’s numeral [00:23:00] system.

So that is why we sort of say that it draws origin to that. Nothingness in itself has really deep roots in the philosophy and religions of India, which may have encouraged the development of the symbol for it. So even now, for example, I recently spoke to someone who practices Taoism, and he spoke to me about how zero has a very important influence on meditation practices And we believe that zero was linked to a particular meditative practice in Hinduism and Buddhism.

And so a circle with a dot in the center symbolizes a meditation on concentrating on the nothingness between ourselves, a sort of empty space. And so this is not really very far from zero that we recognize. For a time, it was thought that the world’s oldest zero could be [00:24:00] found in the ninth century temple in central India and carved into a solid rock.

And so on the temple’s walls, the number 270 is written with a dot in the place where now we would write zero. So there are several Indian mathematicians who are credited with the invention of zero. And so Aryabhata is one from the fifth century. He was the first mathematician to use zero in calculations.

He himself is really an icon of Indian mathematics, so if you talk to an Indian mathematician, everyone will know his name. And his successor is Bhaskara, who was from the seventh century, so 600 to 680, Bhaskara one, uh, first. And he wrote around 629 a book based on Aryabhata’s mathematics. So here he also introduces zeros, and he [00:25:00] uses zero in writing equations.

So he introduces equations where both unknowns’ quantities are, are squared. That is very interesting, not only for that period, but for many mathematicians who came after him. For example, Fermat. And I suppose I should mention that one of the works in which zero appears, Indian works, is by Brahmagupta, uh, from

He, he was born in 598 and lives until 668, and he wrote two important manuscripts in mathematics and in astronomy, where he also uses zero. But he didn’t get everything right, and that is interesting, because in his work, he mentions that you can divide by zero, which, which we know now you can’t.

Mathematically, that is not possible.

Alisha Searcy: Thank you, Dr. Lawrence. You quote in your book saying, “Discovered in Thebes with some dusty ruins around [00:26:00] 1858, one papyrus would become probably the best known and most valuable of all artifacts telling the history of Egyptian mathematics,” end quote. Would you discuss why ancient Egyptian mathematics is often overlooked, as well as how it’s contributed to that civilization constructing such magnificent architecture, including the pyramids?

Dr. Snezana Lawrence: Yes, of course. I love Egyptian mathematics, don’t you? I think for every little child, when you first show them the pyramids, it is one of those things that give wings to imagination, I would, I would imagine. It certainly did for me. It is therefore really quite funny when you later on discover that we don’t know enough about Egyptian mathematics.

We don’t really know enough about Egyptian cultures yet, and that is because we, we lost quite a lot of Egyptian artifacts when the [00:27:00] Egyptian culture fell, and it wasn’t until the ’20s that we rediscovered how to read and how to understand what they were saying, what they were writing. So About this mathematical papyri, it is an interesting story because it is the first mathematical artifact of which we know a person who wrote it.

We know the name of the person who wrote it, and the name was Ahmes or Ahmose, and he wrote this around 150 BC. He says on the papyrus that he didn’t come up with all that mathematics, but he wanted everyone to know the secrets of mathematics. And so the papyrus was bought in Luxor in Egypt in 1858 by a Scottish antiquarian, Alexander [00:28:00] Henry Rhind, who collected various antiquities that he could find in Egypt.

And one of them was this Rhind papyrus. And then another one was Egyptian mathematical leather roll that he also bought. Unfortunately, this antiquary died quite young, but when he died, the British Museum acquired this particular artifact. There is also another piece of it in the Brooklyn Museum in New York.

So what does the papyrus say of mathematics? Well, it is made into what we have of it. We don’t have the full papyrus. It gives us some knowledge on arithmetic and, and sort of processes that they knew. Uh, there is 91 problems there about how to decompose unit fractions. And unit fractions are the fractions which have one in the numerator and some other number in the [00:29:00] denominator.

And so let’s say you have a number such as seven over 12, and you want to decompose that into unit fractions, then you can write it as one half and one over 12. So that process was useful because Egyptians were very practical people, and they looked at, for example, how to divide things and how to reestablish boundaries after the, the flooding of the Nile, and so on.

In book two of this papyrus, there are some things on the geometry, on volumes, on areas, and there is a little bit on the pyramids, but not enough, so we really don’t know how much they knew They must have known much more than what we have, particularly about the pyramids. So they were able to build these fantastic structures, but mathematics that we have of it wasn’t really, you know, wasn’t [00:30:00] sophisticated enough.

So there must be s- either something that has been forever lost or maybe that will be discovered at some point. I suppose the most interesting actual thing about it is their work on these unit fractions, and another interesting thing is that they were able to calculate pi to a very good approximation, and that approximation also appears on this papyrus.

Jake Tawney: Dr. Lawrence, it’s hard to talk about the history of mathematics without mentioning Euclid. Yeah. And I love that you titled your chapter on Euclid, The Greatest Mathematical Bestseller. This, of course, is referring to the 13 books contained in that collection that you could call the elements. And this, of course, to quote you, “Remains the most widely translated and distributed mathematical textbook ever created.”

And we, of course, know that even Abraham Lincoln used to carry a copy of the elements in his satchel, that he saw this [00:31:00] not just as a model of mathematics, but that kind of reasoning being a model of, of legal reasoning for himself. So I’m curious, why do you think the elements of Euclid and his geometry has had such a wide influence across the history of math?

Dr. Snezana Lawrence: I think Euclid is absolutely amazing figure, again, in the history of mathematics and, and his book, his, well, 13 books of Euclid’s elements even, even more so. Again, as in other very old stories from the history of mathematics, we are not 100% sure that Euclid himself wrote this. There are various theories, one of which emanated from a French mathematician from the 20th century who said that Euclid may have been a group of people, and maybe just their editor was called Euclid.

We don’t really know. But what we do know is that this fantastic structure of mathematical thinking really survived from 300 [00:32:00] BC to modern times. And why a figure like Lincoln would have a copy, I think it is because it teaches you really to think logically and to draw conclusions based on the premises that are already as clear and as simple as they possibly can be.

For example, the 13 books consist of definitions and then postulates and common notions and propositions. Everything that is described starts from these definitions and postulates and common notions. So I’ll give you some example. For example, definition one is a point is that which has no part. So you have 131 of these definitions, and everything you build upon those foundations, you know, you have to infer from these definitions.

There are five postulates, one of which is, [00:33:00] for example, stating all right angles are equal. And I think that kind of structure, if you are … Then if you actually build your arguments on things like that, you cannot really make conclusions which are illogical. There is only one kind of little problem, which is with the fifth postulate, which was a little bit more problematic, and that gave birth to the geometry called non-Euclidean geometry, but I explain that in another place in the book.

So let’s stick with the Euclid’s elements. What else can I tell you? For example, how Euclid came up with writing it. As y- as you know, Alexander the Great built Alexandria and in the very north of Egypt. And his successor, Ptolemy I, and his son built fantastic library there And apparently Ptolemy I wanted [00:34:00] to learn mathematics and found it too difficult because it was spread all over the different papyri, and he wanted one work from which to learn.

I think this is a … If it was a true story, which it may not be, maybe just a legend, it is a good kind of way of introducing people to mathematics. I think Euclid’s Elements is definitely still a very good starting point for someone who wants to learn what mathematics is all about.

Alisha Searcy: Dr. Lawrence, Nicolaus Copernicus was a Renaissance astronomer who first proposed the heliocentric or sun-centered model of the universe.

Johannes Kepler was a mathematician who built upon Copernicus’s work by discovering that planets move in elliptical or oval-shaped orbits rather than perfect circles. While Galileo Galilei was an Italian polymath who pioneered the scientific method to better understand the universe. Would you discuss some of the key relationships between [00:35:00] mathematics and modern science?

Dr. Snezana Lawrence: So Johannes Kepler is a, is a really beautiful example of how mathematics influences science, and that is because he started from a kind of wrong premise and came to a right conclusion. He was a, a late 16th, early 17th century figure. He studied at the University of Tübingen, and he was very surprised himself that he was good at mathematics, and he wrote a dissertation in defense of Copernicus’ heliocentric theory.

At that time, this was not really widely still accepted, and many church authorities really fiercely resisted the idea. When he finished his studies and he wanted to write a book about what he learned, because he had a theory, and so … Of how the world was built. So in 1596, he wrote this beautiful book, Mysterium Cosmographicum, [00:36:00] which is Cosmographic Mystery or Mystery of the Cosmos, where he describes what he thought the structure of the universe was like.

So in his book, he presented an image of a model of the universe, which you might have seen. It quite often appears in s- scientific publications as a sort of relic from the history of science. This model of the universe has five Platonic solids. I didn’t explain to you Platonic solids. These are the only five solids in three dimensions which are regular.

So each Platonic solid is built of regular polygons of the same size and type. So for example, cube is a platonic solid, and it is built of six equally sized squares. And so he, Kepler used these five platonic solids, which he encased each of these solids in a sphere, and then inscribed [00:37:00] in another solid.

So it was a nested kind of structure, and he suggested that the intervals between the spheres which encased these solids and objects, the solids themselves, corresponded to the sizes of each of the then known planets’ orbits around the sun. Later on, he didn’t, he, he realized this wasn’t actually the case because planets don’t move in circles, but they move in ellipses.

But nevertheless, this led him to actually start calculating and working on that. So later on, he wrote another book called Harmonices Mundi or Harmony of the World, where he, he looked at what we now call Kepler’s law, laws of planetary motion, and he worked out the speed at which the Earth goes around the sun changes as it comes closer to the sun and [00:38:00] slower when it goes further away from it.

He had a re- really rather beautiful kind of view of the world and the, of the universe. He connected that with the Pythagorean theory of music as well, and came up with this idea that there was this beautiful music, that’s why it’s called Harmony of the World, his book. There is this beautiful music that was played at the beginning of the world, and which can never be recreated again, and which is based on these beautiful ratios in science.

So it is interconnected, his vision, and I think quite a lot of scientists and mathematicians still have, in their own way, these visions of interconnectedness between science and mathematics.

Jake Tawney: So what I love about this story is how mathematics sort of comes out of the need to describe the world, right?

Either the shape of the universe or the shape of the solar system, or data that we’re finding. Of course, [00:39:00] you know, Kepler was assistant to Tycho Brahe, who spent decades, you know, observing the heavens to, just to try to c- collect data. Yes. And it’s from that, that the mathematics comes. But sometimes an entire branch of mathematics comes out of the need to describe physical phenomenon.

And here, I’m reminded of your chapter, The Lion and the Witch, where you discuss the calculus controversy between Newton and Leibniz, right, over who first invented calculus. Can you tell us about that debate, but also about that importance of a whole branch of mathematics coming out of the need to describe something like continuous change?

Dr. Snezana Lawrence: Yes. That is, as you say, a need that was really identified, and quite a lot of mathematicians from that time tried to work out how to describe this continuous change, not only for, i- in terms of mathematics, but because some of them had very practical applications in mind. For example, one [00:40:00] of the practical applications was the rebuilding of St.

Paul’s Cathedral’s dome by Christopher Wren, who knew that the most stable structure would be something, a curve. The most sta- stable dome would be based on a curve which kind of rotates, but couldn’t work out the exact shape of that curve, so he used approximation. But there, there were other interesting stories about why people wanted to do that.

That was a time when curves became really popular in mathematical investigations. And you’re right, this is about working out mathematics of continuous change. Much later, in the 20th century, there was another theory that studied the discontinued change, which is catastrophe theory. So mathematics does come up with these very practical applications that can be used in various fields in science or engineering or architecture even.[00:41:00]

So how did this start? Well, we know that Newton started studying in Cambridge, but very soon after he studied, well, his, he started his studies, the plague broke, and so he, he went back home to Woolsthorpe Manor, where he lived. And very early on during this period, he started … Because he couldn’t attend university, which in a way was a blessing, because he started actually reading from, directly from the masters.

So he looked at geometry of Descartes, among other things, and he looked at a Dutch mathematician, Van Schooten, who described these curves of continuous motion, and he also made some mechanical devices how to build these curves or how to draw the curves. And this is really the origin o- of that study by Newton, because he looked at how a curve is being generated by a movement of a point [00:42:00] So the point itself, as it moves to create this curve, leaves a trace as it moves.

So you can imagine, you know, that point just goes … There is some underlying law that moves that point. It leaves behind itself a trace. And Newton started thinking about how did he can mathematically describe this change between traces of that particular curve. And so he looked at the smallest possible quantities of change, and he s- called them, these changes, fluxions.

And what it led him to is being able to find derivative of a curve, which is really going to be equal then to a gradient of a straight line, which is a tangent to the curve at that particular point. So that was the Newton’s approach to inventing calculus. [00:43:00] On the other hand, Leibniz came with a similar

Well, he came to sort of same conclusion, but he started from a different point of view. So he looked at functions and how they have certain points where they achieve minimum or maximum, and then at these points, he noticed that the tangents at these points are going to be horizontal. They’re going to be flat, and that means the gradient of those tangents, of, of those straight lines, will be equal to zero.

And this is one of the most famous, again, stories from the history of mathematics, the issue of who invented calculus, Newton or Leibniz. And it divided really mathematical world in two camps, one for Newton, one for Leibniz. And the battle continued. The strange thing was that they both really did, they just started from a different point of view, so to speak.

Alisha Searcy: Very interesting. Dr. Lawrence, this has been [00:44:00] great to hear from you, to learn about some of the foundations and history of mathematics. Before we let you go, would you mind reading just a paragraph from your latest book?

Dr. Snezana Lawrence: Oh, yes, yes. I was recently writing something about this particular episode again. I’m going to read the beginning of chapter 28, which is on the unfolding of many dimensions.

Do you ever feel stuck in this world and wish you could escape into unexplored dimensions for a while, just for a little adventure? Lots of science fiction stories explore that. Traveling through time or visiting parallel universes. Perhaps some of the mathematical ideas we’ve already encountered have made our three-dimensional world, with all its homogenous space feel, so to say, a little flat.

We don’t have to rely on stories to experience different dimensions. Mathematics can offer it, [00:45:00] too. In fact, thinking about higher or more dimensions has been something that mathematicians have grappled with for many centuries. Let’s time travel some early mathematicians into our chapter now. And so on. I go on from the 17th century, and I go to modern times from here on until the rest of the book.

Alisha Searcy: Love it. You really brought math to life. Thank you so much for joining us. I know Jake and I both enjoyed this interview immensely.

Dr. Snezana Lawrence: Thank you very much. I’m really appreciative that you invited me, and I hope I explained a little bit more of what I wanted to present in this book. Thank you.

Alisha Searcy: Wow, Jake. That was incredible and very enlightening. And I have to say, I’ve learned much more about math today than I have in a very long time. What’d you think?

Jake Tawney: I thought it was great, Alisha, and I was just so delighted to be here. You know, obviously, math is a topic that’s near and dear to my heart, and yet every time I get on with a different guest, I learn something new, and so this was just, [00:46:00] just lovely.

It was so wonderful.

Alisha Searcy: Yes, it was. Well, before we go, we’ve gotta make sure we do the tweet of the week. This week it comes from Education Next. In a survey conducted last May, we found that 76% of voters said colleges and universities charge more than what the degree is worth. Hmm. I’m reading that as someone who’s paying college tuition right now.

So everybody make sure you check out that article in Education Next. Very, very interesting. Jake, it was wonderful to be with you. Thanks for co-hosting with me today.

Jake Tawney: Thanks so much, Alisha.

Alisha Searcy: Make sure to join us next week. We’ll have on with us Professor Suzanne Marz. She is professor emerita of English at Millsaps College and author of Eudora Welty: A Biography, the definitive story of an original Southern writer from Mississippi to international stature. See you next week.