Professor David Bressoud (5:37)

Thank you. It really is my pleasure.

Mark Malloy (5:40)

Let me begin by stating up front that I am not a mathematician. I am, however, interested in the history of ideas and this interest has led me into the history of mathematics. As a general rule, non mathematical subjects, the history of philosophy, the history of ideas, even perhaps the history of science, do not necessarily do a very good job on the history of mathematics. This is understandable. Math is a complex and specialized field. Though this fact is understandable, it's also, as I'm sure you will agree, unfortunate for mathematics is an astonishingly influential and just as important a beautiful subject. Your book conveys this and so I'm hoping with our discussion that we can capture some of the fireworks of that history for our listeners.

Professor David Bressoud (6:31)

I hope so too. Yeah.

Mark Malloy (6:34)

Moreover, and I'm wrapping up the intro here. Moreover, for those interested in the history of ideas or the history of philosophy, many of the key figures in those fields were in fact influential mathematicians. Thales, Pythagoras, Plato, Descartes, Leibniz. I could go on. So I cannot stress enough. If you are interested in the history of ideas or the history of philosophy or history in general, listen to this episode and look more deeply into the history of mathematics. There are now many popular books and online courses oriented towards a lay reader. In conclusion, to any non mathematical listeners, if the conversation gets a bit technical at times, just Power through Mathematics is a technically complex subject, so are, in their own ways all other subjects. To learn a new subject, you take one step, then another, then another. If this is your first step in many years, we welcome you. One last thought. This episode will be cross posted to the New Books Network's Mathematics Channel. It is my belief, and I believe, Professor Brassu, I think you will concur, that even if we speak somewhat generally and broadly about the ideas under discussion, our discussion will be of interest and a value to mathematicians as well as with all subjects. Just about everyone could benefit from learning a bit more of the history.

Professor David Bressoud (8:00)

I think so, yes.

Mark Malloy (8:02)

I was wondering if you could begin by telling us a bit about yourself, both your training and the focus of your work.

Professor David Bressoud (8:11)

Sure. I was trained as an analytic number theorist. So this is analysis, basically advanced calculus applied to problems in number theory, the study of the structure of the integers. I spent the first part of my career as a research mathematician at Penn State, something that I was very good at and enjoyed. But in the early 90s, I began to realize that my real passion was in teaching and understanding the difficulties that undergraduates have in doing mathematics and learning mathematics. And I began to look for a more congenial place to pursue that passion. And in 1994 I made the move to McAlester College in St. Paul, Minnesota, a small liberal arts college which has been a wonderful home for me ever since then. I've written many textbooks, rather idiosyncratic textbooks, with my own take on how mathematics should be taught using the history in very large ways. I've gotten very much involved with the Advanced Placement Calculus program, and in fact, for six years I was on the committee that sets the syllabus for AP calculus and writes the exams, and I chaired that committee for three of those years. As you mentioned, I've served as president of the Mathematical association of America, and I'm currently director of the Conference Board of the Mathematical Sciences. So this is an umbrella organization that represents 19 of the professional societies in the mathematical sciences in the United States, and it helps to coordinate their efforts. And I find it particularly relevant to my own interests right now because it includes both the professional societies that represent pre K12 math teachers, such as the National Council of Teachers of Mathematics, and also the societies that represent college and university mathematicians, the Math association of America, the American Mathematical Society, Society for Industrial and Applied Mathematics. Right now, my focus is really on the transition between high school and college mathematics. Why do so many students fall at that transition and what can be done to help them?

Mark Malloy (10:48)

That's wonderful. I think for someone working in such a lofty field as number theory, it's wonderful that you are applying some of your talents to pedagogy as well. I would like to ask you about why you wrote this book. The book's title, Calculus Reordered, gives some indication, although only after one has read the book or learned about what the book is about. You are proposing a new pedagogical in addition, I should say, in addition to recounting the history, you are also proposing a new pedagogical approach to calculus, one that is informed by the history of its development. Could you recount the dilemma that you see mathematics educators facing between the rigorous Euclidean approach and the more organic, intuitive historical approach, and how you think reordering calculus education would help?

Professor David Bressoud (11:44)

Thank you. Yes. I've gotten very frustrated over the years observing calculus nationally, both as it's taught in high schools and also as it's taught in our colleges and universities, that so much of the emphasis is on the procedures, the techniques, a very rigid class of problems that students are taught how to solve without developing a real understanding of what calculus is about, where it comes from, how to how to understand it in a way that enables students to use it freely and use it in unfamiliar situations. As I've learned more about the teaching of mathematics, I've been very impressed by some of the work of Gershon Harel at the University of California, San Diego. And he has talked of the the importance of what he calls intellectual need that if you want to really ensure that a student learns material well, you need to create intellectual need within those those students. Now, it could be simply a problem that students believe is important and they want to be able to solve. But more often than a particular real life problem, it's an intellectual challenge that comes out there, a Pattern that grabs their curiosity and makes them want to learn more about something. I have found that in particular, the history of mathematics is very rich in supplying this intellectual need to understand why people came up with these mathematical ideas in the first place. Helps us provide students with questions that arouse their curiosity and help them to understand why we're teaching them, what we are teaching them. And so I've, I've always relied very heavily on the history of the subject as a way of communicating to students how well to, to back up a little bit. The, the general view of mathematics is it's a set of rigid guidelines that have mastered and a set of rigid problems that you've got to know how to solve without a sense of really getting absorbed in the material itself. And the history can help to break out of that. It can show students that mathematicians themselves often struggled, often were confused, often went down blind alleys, and that that's an important part of the process of learning mathematics. It's not just gaining mastery of a set of procedures, but it's really being able to, to fail to try things and fail and struggle with the mathematics.

Mark Malloy (14:42)

Fascinating. I think a close analogy, but a good analogy is to the history of science. For anyone who has not encountered the history of science, and I hope many of our listeners have, but it's a fascinating history. There's this, at least this is one way to tell it. There's this undiscovered world, this misunderstood world, and it's a process of feeling our way and working on problems which reveal new truths. And I, like I said before, I'm no specialist in mathematics, but I've come to the history of mathematics along those same lines. I was, I began reading some histories of it and I found it to be the same thing. There is this undiscovered world of mathematical objects and we could get to that. Of course, those, metaphysically speaking of these things, metaphysically it's complicated. But there is this beautiful mathematical world and when you approach it historically, it becomes a process of discovery with a cast of characters and with suspense and with revelation. So from my own perspective, I do think that introducing the history into the pedagogy of the subject would be enormously helpful.

Professor David Bressoud (15:55)

I've really found that, and overwhelmingly I found that my students really appreciate this. To understand where the ideas came from, how people struggled with them, that's an enormous assistance to the students who are trying to learn mathematics.

Mark Malloy (16:12)

I was hoping that you could quickly give us a top level overview. You, in the last answer, you gave us a brief definition but if you could give us a top level overview of mathematics, and I will request, if possible, that you kind of focus a little bit more on the pure side of mathematics versus the applied with. Without question, this is a false distinction. And most, if not all pure work is motivated by applied problems. That said, I think that the average person has some sense, through their education, however limited, of math as an applied science. And I think much that makes mathematics such a beautiful subject comes out of its pure aspects. So would you please begin by speaking briefly to us about what math is and what makes it a subject as rich and profound and creative as art, literature or philosophy?

Professor David Bressoud (17:12)

Sure, I'd be very happy to. I understand mathematics as the study of the deep patterns of the universe, the deep structures. Mathematics in many respects is like a game where you go into it where you don't know the rules, you're trying things out and trying to figure out the rules. And one of the beauties of mathematics, what makes it so attractive for the research mathematician, is as these rules get revealed, as you discover what can be done, what works, what doesn't work, you observe apparent patterns and try to see if they actually are there or not. The rules that are revealed are so beautiful that there's a feeling of inevitability about them. Once you see what those rules are, you know that they have to be true. It could not possibly have been any other way. So there's nothing arbitrary about mathematics. It's preordained and yet it's extremely subtle. So always as you go deeper and deeper into a subject, you discover that there are aspects of it that you never would have expected, reveal new realms to you that you wouldn't have imagined. If you're willing to let me go into a digression, I can talk a little bit about my own work.

Mark Malloy (18:35)

Please do so.

Professor David Bressoud (18:37)

Partition theory, as you mentioned, this is the simple question of asking how many ways can you write a positive integer as the sum of integers? It's a pure counting problem. It actually first arose in the 1700s in connection with the problem of finding roots of polynomials. But something important happened to this very simple sounding question. In the 1800s, it was discovered that there was a connection to something called elliptic functions. And elliptic functions are beautiful because of the symmetries that they exhibit, and that's one of the keys to the patterns. The beauty of mathematics is all of the symmetries that are out there. And a simple example of symmetries that mathematicians are concerned with are Islamic tilings. So these are examples of repeating patterns that can be used to cover a wall or a floor. And it's the study of these symmetries that gives rise actually to much of the work of M. C. Escher. Mathematicians love Escher and his work because so much of it involves very complex symmetries. The wallpaper patterns that you see in Islamic tilings are a simple example of that. But they begin. These symmetries begin to get much more complex. One example which can be found in Escher's work are the hyperbolic symmetries, where he has as an example a circle that's being tiled with fish that gets smaller and smaller as you get off toward the boundary of the circle, where they actually you have infinitely many fish that are touching that boundary. Well, the elliptic functions that came up in connection with partition theory, these are really generalizations of trigonometric functions. The sine and the cosine, they're periodic. But the elliptic functions which generalize the sine and cosine, they're doubly periodic. So they're functions of two variables that have periodicity in both of these variables and a lot of other structures. If you think of all the identities that can be constructed with trigonometric functions, you get a much richer set of identities that can be constructed with these elliptic functions. But the study of elliptic functions was just an entry into an even more magical realm of theta functions, really first developed by Jacob Jacobi in the early 19th century. And these both have the kind of tiling symmetries, but they also have a hyperbolic symmetry, this kind of symmetry that I referred to in the work of M.C. escher. And the kinds of ways you can combine these and the kinds of results that come out of them, they're a plaything. And one of the greatest examples of playing with all of this structure that's hidden inside the theta functions comes out in the work of the Indian mathematician Srinivasa Ramanujan, born in 1887. Actually, much of my work was based directly on the work of Ramanujan, great romantic figure. And in fact, recently a movie was done about the life of Ramanujan. He was played by Dev Patel, who's much better looking than Ramanujan actually was, with Jeremy Irons playing his mentor, G.H. hardy. But it's Ramanujan was a self taught mathematician. He had access to the library at the University of Madras. He read a lot of advanced mathematical books. But he then took mathematics off in an entirely new direction as a plaything, working with these theta functions and describing what could be done with them. One of the Things about beautiful mathematics. Mathematicians pursue it for its own sake. But the patterns are so essential that sooner or later, anything a mathematician discovers that's truly beautiful will be revealed in the world around us. And today, the work of Ramanujan and the work of others on these structures of theta functions that have been playthings for a century now have real applications. But nobody went after them to try to find the real applications. They went after them because of the beauty that they found inside them.

Mark Malloy (23:32)

I will just say quickly, as an aside to our listeners who did not follow everything that you said, this is just a stroll through the park, a tiny, tiny subsection of the world of mathematics. And the thing about it is that as you explore deeper and deeper and deeper, you find more and more and more beautiful subjects, more beautiful findings. And so it does, it does get more and more and more complex. And that does unfortunately become less and less accessible to the, to the non mathematician. But that shouldn't, that shouldn't turn people away who are interested in ideas and beauty and art, because it's is a beautiful subject and much of it is accessible to one degree or another.

Professor David Bressoud (24:21)

Absolutely. Yeah. Yeah. And the simple fact that the story of Ramanujan could be taught in novels, in books and in a movie, There are ways of visualizing that get across some aspect of what's happening there.

Mark Malloy (24:36)

And there are so many fascinating characters in the history. One of my favorites, maybe the, the most cliched character, I, I don't mean that he was a cliche. I mean having him as your favorite is a cliche. Is Galois, who was an incredible character. But we, we can leave that on. That's a separate discussion. We have covered mathematics in, in general, and I think you did a, a wonderful job. I really. That was exactly what I was getting at. I was hoping that you would show a little bit of your own enthusiasm and the way that you found your own way into it following the beautiful symmetries. So we've covered mathematics in general, but your book is about calculus. According to the Magazine Education Week, 15% of American high school students study calculus. And importantly, and leaving aside socioeconomic factors of privilege concerning who has access to higher levels of education, this 15% are, I would imagine, predominantly comprised of students who will go on to study STEM fields. So there are many humanities inclined students, intelligent people who will go on to positions of influence later in life, who are never exposed to these ideas. Some of these people will go on to write books, histories of philosophy, histories of ideas, without a solid understanding of what calculus is. And so people who are studying history may fail to appreciate what calculus is, just how impactful it has been on the development of science and technology and modernity as a whole. Can you give us quick, I know this is a large ask, a tall ask, but can you give us a quick overview as to what calculus is?

Professor David Bressoud (26:30)

Well, very simply, calculus is a set of tools that are used to understand, to study dynamical systems, systems that are subject to change. The first big application of calculus came in Newton's work on studying planetary motion. Today, the tools of calculus lie behind climate change modeling, they lie behind weather forecasting, they lie behind the study of epidemiological models. And anything that changes, whether you're looking at the stock market or whether you're looking at physical phenomena, trying to understand the physics of music, all of this comes back to the basic tools of calculus. I just, and I think I'll say a little bit more about this later, but in the 19th century, calculus got much more sophisticated, really began opening up windows into some deep structures, such as the elliptic functions and the theta functions that I mentioned. And at that point, it's no longer called calculus, it's now called analysis, but calculus itself, that one year course that's typically taught in the first year of college or the last year of high school, that's really about learning how to work with the tools that enable you to study phenomena that are changing.

Mark Malloy (28:06)

I was wondering if you could tell us quickly about Pierre de Fermat and the influence Pappas Collection and Diophantus Arithmetica, both ancient works, had on the development of modern mathematics. I believe, as I would guess you agree, that many people have a distorted view of mathematicians and mathematics in general. And I think the figure of Fermat is a wonderful means of showing people real mathematics as a living, breathing subject. I find this an especially wonderful moment in the history of mathematics, as I think it is a quintessential example of math as mystery, as an art form, even as a game. Fairmont, in his free time, joyfully working away at these puzzles and mysteries, utilizing the most extraordinary solutions and techniques, and ultimately arriving at the ecstatic eureka moment of discovery and proof. Fermat working away at Pappus and Diophantus, seems to me the farthest thing from the image so many people today have of the subject. Again, my hope here is to give people a glimpse of mathematics as an art form. Can you tell us briefly about Fermat and his relation to these classical texts?

Professor David Bressoud (29:34)

Sure. So Fermat himself was a lawyer working mostly for the Legislature in Toulouse, in France. He had studied law in Bordeaux, and we believe that it's in Bordeaux that he had first learned about algebra. And what most people don't realize is how in many respects, algebra was a new subject in the 16th century. Now, algebra has much older roots. It goes back to, to Greek study of properties of the numbers of the integers. And usually the origins of algebra are traced to the, the early Islamic scholars. In fact, the term algebra itself comes from the Islamic word al jabra. But in the, in the 16th century, tools that had been developed in the Islamic civilizations really got refined, especially in Italy, but also in, in France and in England and in Germany. And this is, it's the 1500s, when mathematicians were first able to find the exact formula for the roots of a cubic or quartic polynomial. These are polynomials of third or fourth degree. Now, up until the 1500s, algebra was written out in full sentences. You didn't say x squared plus 6 equals, equals 3x. You talked about the thing, writing out in full words, the thing squared plus six equals three times the thing. And you had to write it all out. And that's, as you can imagine, extremely cumbersome. So the 16th century saw a gradual refinement, a gradual development of what we now think of as algebra, which is all these symbols that are, that are floating around that people are learning how to manipulate. So Fermat, as he was studying for the law, also in Bordeaux, was learning about algebra. And he also at that same time began to learn of some of the classic texts from Greek mathematics. The two books that you mentioned, the Collection of Pappas and the Arithmetica of the Diophantus, these were written toward the end of the height of Hellenistic mathematics. So this is the third, the fourth century of the Common Era, beginning really right after the Dark Ages in Europe. Scholars in Europe began to rediscover these old Greek manuscripts, most of which had been preserved in the Islamic civilizations. They began rediscovering them, translating them into Latin so that it was easier to work with them. And among the last of the manuscripts to be translated, this is not until the late 1500s are both the collection and the Arithmetica. And the. The Arithmetica contains results on properties of integers. Good example of this. Take Pythagorean triple. So these are triples of integers that can be used as the sides of a right triangle. 3, 4, 5, 5, 12, 13. One of the problems that Diophantus looked at was how do you generate an arbitrarily long list of such triples of Integers that can be the sides of a right triangle. What Pappas did was to collect all of the results that were known about geometry. And there are almost no proofs in Pappas's collection. And so immediately, as soon as this was translated into Latin in the late 16th century, people who are interested in mathematics began looking at these results and asking, well, how did they derive that? How do I know that that's really true? And that's one of the challenges that Fermat took up, he and several other people. And Fermat had both this collection of geometric problems, and he also had these brand new tools of algebra that he had learned. And his incredible insight was to realize that he could take geometric problems and translate them into algebraic problems. And the method he used is what today we call the Cartesian coordinate system. It's named for Rene Descartes, who came up with the same idea actually at the same time, totally independently. But taking a geometric curve and translating it into an algebraic expression turned out to be a way of solving many of these problems that were in Pappas's collection. Obviously, that's not the way Pappas had originally solved them, because he didn't have access to, to this idea, but it did provide, provide proofs. Fermont's an interesting case, and I, I'd like to go a little further with him. As I said, he was in Toulouse, he wasn't near any other mathematical work, and he also never published anything. All of his mathematics was communicated through letters that he wrote, especially to people like in, in, in Paris, but also to other mathematicians in England, in Italy.

Mark Malloy (35:09)

And Germany, and then later even through other people who were looking through his books. Correct. The famous Fermat's theorem notation in the margin of one of his books.

Professor David Bressoud (35:25)

Right. So this was his copy of Diophantus's Arithmetica, and he had gone through it and he'd made marginal notes, including what is now called Fermat's Last Theorem. He didn't publish that. But what happened was his son, after he died, thought that his father's copy of the Arithmetica with all of its margin notes was so impressive that his son got it published with the margin notes published with the text from Diophantus. And so that's really what made Fermat's reputation was, was the publication of, of his copy of Diophantus together with all of his marginal notes and the story of Fermat's Last theorem. One of the things, as I mentioned, that Diophantus talked about was Pythagorean triples. So solutions to A squared plus B squared equals C squared in integers and. And Fermat had looked at this particular solution and said, gee, I wonder about A cubed plus B cubed equals C cubed. Can you find 3 positive inte.

Mark Malloy (36:34)

Just to clarify that point, you had said a few minutes ago that Diophantus had wondered if he could find arbitrarily many solutions.

Professor David Bressoud (36:43)

Not Diophantus, but. Oh, oh yeah, no. Diophantus had shown how to find arbitrarily many solutions.

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Mark Malloy (37:17)

Learn more@WhatsApp.com and okay, so he had showed how to find arbitrarily many. But. And again, correct me if I'm wrong, but sort of what, what Fermat's innovation was, was applying the new algebra to this type of question concerning the relations or the structure of the integers and asking the question, is it possible to prove that there are infinitely many or the lack thereof, or not infinitely many answers to certain questions we can ask about the integers.

Professor David Bressoud (37:54)

Exactly, yeah. So the question for squares A squared plus B squared equals C squared, that had been. Actually, that was solved long before Diophantus, but he reproduced those arguments. But what Fermat did was push it a little further. Can you do it with A cubed plus B cubed equals C cubed, can you do it with fourth powers or fifth powers or any higher powers?

Mark Malloy (38:21)

So just, just for the non specialists, when you say A cubed plus B cubed equals cubed, A is any integer, any whole number, B is any whole number and C is any whole number. Is there any. Can we find infinitely many integers that we could sub in where A A cubed any integer is A plus B cubed any integer for B equals C cubed any integer for C. Is it possible that you can find three integers for A, B and C where that equation works Exactly.

Professor David Bressoud (38:53)

So that's what he asked. He decided that you couldn't, you couldn't do it with third powers, you couldn't do it with fourth powers, you couldn't do it with any higher power. And then in the margin of his book, as he did with many of the other things he said, I have a remarkable proof that you can't do it once you get beyond squares, once you get beyond second powers. But I don't have enough room here to write out my justification. The years following the publication of Fermat's copy of the Arithmetica, together with all of his marginal notes, mathematicians latched onto this. Fermat had stated a lot of results inside this book. After all, these were just his personal notes to himself. He never had any intention that this would get published. Some of the things he said were correct and people were able to prove them later. Some of the things he said were wrong, and people were able to come up with counterexamples. This particular result about A cubed plus B cubed equals C cubed, or higher powers, that it could not be done. This was the last of the statements in Fermat's copy of Arithmetica to be decided. And that's why it came to be called Fermat's Last Theorem. Not that it's the last thing that he did or the last thing that he wrote, but just that was all that was left by about early to mid-1800s. Incidentally, the fact that it cannot be done wasn't proven until 1994 by Andrew Wiles, and that's extremely complex mathematics. And one of the fascinating things about the proof that Fermat was correct, it cannot be done for powers higher than the second power. There is absolutely no use for that theorem. It doesn't help us with anything else. But the methods that had to be developed in order to prove it are so powerful that they're very useful in many, many other areas of mathematics, which is another small window into the way that mathematics works. No matter what problem you're working on, no matter how eso. Esoteric it feels, if, if there's an intrinsic beauty and naturalness to it, if not the answer itself, then the process of trying to find an answer is worth the time that you put into it.

Mark Malloy (41:36)

And this, what we're talking about here is, is a field of math called number theory, the, the study of the properties of the integers. And that is kind of what makes that such, such a. An astonishing subfield of mathematics. Is that, in a sense, what mathematicians, including yourself, who are working in this field, are doing is really just trying to decode these puzzles, trying to find a means to figure out if this can be proven, and if so, what the answer is, are there infinitely many solutions to this problem? Are there any solutions to this problem or not? And yet centuries, I mean, from Fermat Writing down his, his supposed proof to that, to Andrew Weil's ultimate proof of it. Centuries transpire and immense tools of mathematics are developed in the service of this puzzle that they're working on, which then go into algorithms, go into. Have their part in reshaping the world that we live in.

Professor David Bressoud (42:42)

Exactly, yes.

Mark Malloy (42:45)

Let's drill down a bit into the heart of your book. Could you give us a quick history of what you call accumulation, which is the predecessor to integration in calculus, and ratios of change, which are the predecessor to differentiation in calculus? Maybe you could work your response up towards the fundamental theorem of calculus, which you prefer be referred to as the fundamental theorem of integral calculus to the degree possible. For the sake of clarity, I suggest we postpone discussing series and limits for the time being. I will ask about that, but just, just to keep things simple and progressing for our non mathematical listeners.

Professor David Bressoud (43:29)

Yeah. One of the things that really frustrates me about the teaching of calculus is how often I encounter people who've been through calculus and they think that the differential calculus is just a set of rules for turning one function into another function, and integration is just another set of rules for backing this up. When in fact the real purpose of calculus, of both integration and differentiation, are natural problems that arise. And we teach calculus as if differentiation comes first. In fact, it's the integration that's much, much older. And integration really comes out of problems of accumulation. And the right place to start is with the Greek challenge to find areas and volumes. And I'm going to digress a little bit here, but I think it's important to illustrate what's going on with accumulation or integration. Think of the formula for the area of a circle. How do you get that? How do you know that the area of a circle is PI times the square of the radius? If you take a circle, imagine a round pizza and you cut it, say into eight wedges each slices. And then you arrange these slices so that one of them has the point down and the one next to it has the point up. And they alternate in this way. So you'll have four slices with the point down and four slices with the point up. What you get is something that looks a little bit like a rectangle. Now, of course it's not. The top and the bottom aren't straight lines at all, they're scalloped. But if instead of cutting this pizza into eight pieces, we cut it into 32 pieces and we again alternate the wedges, the slices up and down, now it's looking a little bit more like a Rectangle. If we were to slice the pizza into a thousand pieces and rearrange them, now it really does look like a rectangle. If we could slice that pizza into an infinite number of pieces, it would actually be a rectangle. The length of it would be half of the circumference, which is PI times the radius. And the height of that rectangle would be the radius. And the area of that rectangle is PI times the radius multiplied by the radius, which is PI times the square of the radius. Now, this is something that was realized by very early Greek mathematicians. They knew about this formula for the. For the area of a circle. There's a basic problem here, because in getting it to actually be a rectangle, I had to take an infinite number of slices. And just as today, back then in ancient Greece, they were very uneasy about the idea of infinity. What does it mean to have an infinite number of pieces and in fact, infinity? That concept of infinity is so rife with paradoxes that Aristotle banned it. He said, you can't do mathematics with infinity. Just forget about it. Well, the ancient Greeks managed to find a workaround. They managed to show that the actual area of the circle, through some very complicated geometric and numerical arguments, they were able to show that the area could not be less than PI r squared and the area could not be greater than PI r squared. Well, if it can't be less than PI r squared and it can't be greater than PI r squared, it's exactly PI r squared.

Mark Malloy (47:27)

What you just said, you mean quite literally the, the, the lay reader might think that was a tautology, but their method was literally to prove that the, that it. We can prove that it cannot be less and we can prove that it cannot be more. And so we know what it is. And this is the type of thing I think I'd like to highlight that one of many methods that mathematicians use to do their work is to find ways to prove things. And in this case, you prove it can't be less, it can't be more. Therefore you know that it must be higher squared.

Professor David Bressoud (48:03)

Exactly.

Mark Malloy (48:03)

I apologize for interrupting.

Professor David Bressoud (48:05)

No, no, I'm glad you did that. To clarify that the next big step forward in the idea of accumulation takes place in the 14th century with the scholars in England, in France, and in Italy who began to study the concept of velocity. Velocity had never really been thought of as a number before. You get into the 14th century, and one of the questions that came up, you know that if you're walking at three miles an hour and you walk for two hours, you're going to travel six miles. Well, what if your pace isn't constant? What if your velocity is changing over this two hour period? How can you find out how far you go? And the idea that they came up with was, well, let's take that two hour period and let's cut it into tiny time increments and assume that the speed is constant on each of those tiny increments. Perhaps look at it at one minute intervals and treat the velocity or the speed as constant in each of those. If I've got a constant speed over each minute, I can figure out how far I've gone in a given minute and then I can add those up and get an approximation to how far I go over the two hours. Well, if instead of doing this over one minute intervals, we take 10 second intervals, we're going to get a closer approximation. As we take more and more of these intervals, we get closer and closer, until once we allow ourselves to talk about an infinite number of intervals, we'll have the exact distance that gets traveled over that time. And this is a prime example of an accumulation problem with that all calculus students learn about. If you know how velocity is changing, you can determine the distance that gets traveled. And integration was developed as a set of tools to solve these accumulation problems. And a lot of work done on accumulation problems, especially in the 1600s. Many lengths of curves, volumes. Johannes Kepler, the famous astronomer, did a lot of work early in the 1600s. He was inspired by the problem of finding the volume of a wine barrel. And so he was looking at finding volumes of various kinds of solids. That's integration. And it's really important for students to understand that integration is really about solving accumulation problems. The other big piece of calculus is what's called differentiation, and I refer to as ratios of change. What's happening in differentiation is that you have two connected quantities. As one of them changes, it forces the other to change. And what you're interested in is how changes in the one quantity of are reflected in changes in the other. And the first person to really work with this and produce a result that I consider to be differential calculus was an Indian astronomer around the year 500 of the common era, Aryabhata. And he was working with tables of trigonometric functions and especially tables of the sine function. And one of the difficulties with the sine function is that there is no way of getting an exact value for the sine of a particular angle. So what you have to do, there are a set of angles for which you can compute the Exact value. And so typically a table for the sine function would have these values worked out to precision. But then if you had an angle that was between two places in the table, you needed to interpolate, figure out where in between those values it actually lagged.

Mark Malloy (52:23)

A quick question on behalf of the listener.

Professor David Bressoud (52:26)

Sure.

Mark Malloy (52:26)

And. And also. And also on behalf of myself. So when we speak of the sine function, we're speaking about the sine wave, correct? It's.

Professor David Bressoud (52:35)

Yes.

Mark Malloy (52:37)

So it's, it's, it's undulating. Wave goes up, then goes down. Not in a V, not in a sharp. There's no jagged curves. It just goes up and down.

Professor David Bressoud (52:47)

Right.

Mark Malloy (52:47)

And I know that this is probably not historic. This might not be exactly rigorously correct. But just for the. On behalf of the listener, when we're talking about finding a value, is it sort of close to finding the, the, the tangent to the curve as that, as that sine wave undulates, is that sort of analogous to what we're talking about?

Professor David Bressoud (53:15)

Yeah, actually the image that we have of a sine wave today, something that repeats indefinitely, that's really an 18th century discovery. The origins for the sine function really have to do with chords looking at circles. If you've got an arc of a circle, the original question that was posed by astronomers that gave rise to the sine function is given a circle, if you know the length of an arc on that circle, what's the length of the chord that connects the two ends of that arc? And if you know, if you think about what you may have learned about trigonometry and the sine function, the sine function is actually half the length of that particular chord. And that's the way they were thinking of it. This is the way astronomers were, were using it. Well, from the time it was first developed, about the second century, before the Common Era, really up until the 1700s. But finding the lengths of these chords for certain arc lengths. So for example, if you've got exactly one quarter of a circle, so that's 90 degrees, you know exactly how long that cord chord that subtends the arc is. If the radius is one, then that particular chord is going to have length square root of 2. So you can get an exact value in that particular case. So there were exact values that could be worked out. The problem, though, was how to find good approximations to intermediate values. And that's where Aryabhata came up with the idea of looking at how the sine function changes as the angle or the arc length changes. If you're At a particular value, a small change in the arc length or a small change in the angle, how is that reflected in the output of that particular function? And that's really what differential calculus is all about. You've got two connected variables and you're interested in how change in one is reflected in change in others. Now, when we teach calculus, we usually do it in terms of. We introduce differentiation in terms of tangent lines and, well, the slope of a tangent line exactly prescribes how change in the input variable is reflected in change in the output variable. But I really worry because so many students think that differentiation, if it's telling us anything interesting, it's just telling us the slope of the tangent line. And we know from studies most students don't really understand what slope means. But I think it's important to get back to the more basic understanding that what the derivative is telling us is how change in one variable is reflected in change in another. And I hope I haven't gotten too technical there, but I think that's really important to convey, and I really wanted to make that point very strongly in my book.

Mark Malloy (56:42)

No, I think it's an important point. Let me put forward an example, and I may be wrong, so please correct me if I'm wrong, but we can speak of speed, how fast you're driving, we can speak of distance, how far you've driven, and the. Correct me if I'm wrong, but the first derivative of that function would be velocity, did I say that correctly?

Professor David Bressoud (57:06)

Of how far you've driven? Yes, the first derivative would give you the velocity.

Mark Malloy (57:10)

So the, so the ratio of change of. So we create a ratio of speed and distance, and then we take a derivative, we find the ratio of change of that, and that is the velocity.

Professor David Bressoud (57:24)

Right. So what the velocity is telling us is we're looking at the change in position divided by the elapsed time that it's taken place. And so we're looking at how time is, how changes in time are related to changes in position. And that's what the velocity is. It tells us a small change in time, how much is that going to change your position.

Mark Malloy (57:54)

And then if you could quickly tie it all together with the fundamental theorem of calculus or the fundamental theorem of integral calculus, I believe this is one of those moments in the history of math that it's sort of a transcendent moment, similar to when Descartes and Fermat independently discovered that geometry and algebra were two different languages. You could say with which to speak of the same underlying substance. And accumulation is being developed on its own, and ratios of change are being studied on their own. And then suddenly, I believe explicitly in the work of Leibniz and Newton, but perhaps hinted at in their predecessors, this relation is made. This relation is discovered where accumulation and ratios of change are in fact, different perspectives on a similar or a related or an identical underlying object. Is that correct?

Professor David Bressoud (58:59)

Yeah, exactly. As any student who is taking calculus knows, the techniques for differentiation are much easier than the techniques for integration. Integration is more difficult to learn. This is part of the reason why the standard calculus course teaches differentiation first and then integration later, because differentiation is easier. But most of the most interesting problems that people were tackling in the 17th century really involved accumulation problems. And the great insight that Newton had, and then later, independently, Leibniz had, was that this. This other problem that people had been working out throughout, working on, throughout the 17th century, this problem of ratios of change and calculating those ratios of change, that they could be used to give you an easy way of solving accumulation problems. That if you had a function or a representation of what's going on, that you needed to integrate or. Or find the accumulation of that particular quantity, if you knew that what you were looking at was the derivative of something that you'd found before, then you can reverse this process of differentiation. To get integration, we have Newton's notebooks and the notebooks that he wrote in the two years after he graduated from Cambridge, when he was back home in Lincolnshire. And we see the moment at which he discovers this. He has all of these derivative formulas that he's worked out, and now he's looking at integration problems. And it suddenly comes to him that he can solve his integration problems just by reversing the derivatives which he has already found. And it's just an explosion of integrals that he suddenly writes down in his notebook, because he's able to do these. And the importance of this, I've asked hundreds, if not thousands of students, what is the fundamental theorem of calculus and why is it fundamental? And if students say anything, they say, well, the fundamental theorem of calculus says that integration is the reverse process of differentiation. The problem with that is if you ask them what integration is, they say integration is the reverse process of differentiation. So it's a theorem without any meaning. In that case, what's important is that we have two very different ways of looking at integration. We can look at integration in its historical form as solving problems of accumulation. But what Newton and Leibniz realized is that we can actually accomplish this, find the solutions by reversing the process of Differentiation. And this is why I really want to call this the fundamental theorem of Integral calculus, because it says there are two aspects to integration, accumulation and reversing differentiation. And even though they seem very different, they actually, actually accomplish the same thing. And this is what's powerful. The. The term fundamental theorem referring to this result actually wasn't coined until the 1870s. I mean, people were using this, this connection, but they didn't actually refer to it as a theorem until the 1870s. And when they started referring to it as a theorem, they called it the Fundamental Theorem of Integral Calculus. This has been one little historical exploration that I've done. It didn't change its name into just the fundamental theorem of calculus until the mid-1960s, when calculus textbooks at that time started shortening it in order to make it easier, I guess, to make it easier to write and put in as the title of a chapter. But it really is about two very different ways of looking at integration, and that's the great insight of Newton and Leibniz. One of the messages I wanted to get across in this book is that Newton and Leibniz didn't sit down and create calculus out of whole cloth. Almost all of the results of calculus that a student learns in the first year were known before Newton or Leibniz started their work. But what Newton and Leibniz were able to do was to see how all of the pieces connected together and made a single coherent theory out of all these pieces that were floating around. And they made it into a powerful tool in that process, and that was their genius.

Mark Malloy (64:01)

Those pages of Newton's notebook where you see that explosion of integrals, that must be quite extraordinary to see that in person. The moment of insight and the enthusiasm of the following days.

Professor David Bressoud (64:20)

Oh, yes, yes.

Mark Malloy (64:22)

Okay, listeners, hold on to your hats. We're going to dive a little bit deeper. So far we have covered accumulation, again, what we today call integration, and ratios of change, what we today call differentiation. We have not yet covered two other essential aspects of calculus that Dr. Brasseux covers in his book. These include partial sums, which today are often called series, and the algebra of inequalities, which today is referred to as limits. For my part, separating these two questions has been intentional. I think that here the subject matter gets more complex and more difficult to convey without recourse to visual images, to being able to write down the mathematics on the page, especially in terms of series. That said, I do want to touch on these. Can you possibly touch on the part that sequences of partial sums or series had in the explosion of Mathematics that happened in the 18th century in the work of D', Alembert, Lagrange, and especially Euler, and then on the mathematics of inequalities or limits, and the part that they played on the 19th century in the rigorization of mathematics.

Professor David Bressoud (65:42)

Sure. So very early in the process of trying to study dynamical systems, applying the tools of calculus, it was realized that the standard repertoire of functions that people were using at that time just was inadequate to the, to the task. The solutions couldn't be explained in terms of the functions that were around at that time. And really before Newton, James Gregory, very, very prominent Scottish mathematician, was one of the first. But you also see this in Newton and others in the late 1600s, realizing that the answers, the solutions to the problems that they were posing in calculus, required polynomials of infinite degree. That may seem a little strange, but as you take polynomials of higher and higher degree, you're able to do more and more with them, until finally, if you allow yourself to go all the way out to infinity, you can find solutions to problems that didn't seem to have a solution. And so, especially during the 18th century, these polynomials of infinite degree, what we call power series, or Taylor series, they began to take on a very prominent role in enabling solutions to the calculus problems that people were working on.

Mark Malloy (67:11)

So, sorry, can we just very quickly slow down and remind us, because I just want to make sure that we have the visualization of this. So what exactly is a polynomial and to the degree possible? Could you just describe what, what of the, what the simplest, simplest Taylor series looks like? I'm thinking, for example, of the, and I don't think this qualifies, but the harmonic series, the summation of the harmonic series, you have infinitely many plus signs, some way to describe it so that the listener who is, is not familiar with what we're talking about, can begin to visualize what this looks like just on the page. Not even getting into the depths of the significance of it, just what it looks like.

Professor David Bressoud (67:54)

Yeah, so, so whenever you're adding up infinitely many things, you, you start to run into problems and apparent paradoxes. The harmonic series that you mentioned. So, so these are the, the basic harmonics. 1, 1/2, 1/3, 1/4, 1/5. What happens if you start to add these up and the, the number continues to grow? 1/ a half plus a third plus a quarter plus a fifth plus a sixth. As you add more and more terms, it gets larger and larger. And there actually, there's no limit to how large you can get it by adding more of these Terms, if you talk about going all the way out to infinity, adding up infinitely many of these reciprocals of integers, one over an integer, adding them up, the value is larger than any integer value. It's what we call an infinite value.

Mark Malloy (68:48)

And that is not always the case. There are series of fractions that you could add together that at infinity sum up to a finite value, say 2. But the harmonic series, 1/2 plus 1/3 plus 1/4 plus 1/5 that, if you add that up to infinity, the summation of that is infinite, is infinite.

Professor David Bressoud (69:13)

Right. If you took, on the other hand, one plus a half plus a quarter plus an eighth plus a 16th, where you're doing the reciprocals of the powers of two, that only adds up to two. As you take more and more terms, you get closer and closer to two, and you're never going to go beyond two.

Mark Malloy (69:32)

I think this is just worth touching on, because I do. I love this idea of proof, and I don't think that all listeners will be familiar with quite what that means. I loved earlier the proof that the area of the circle is not less and is not more. I know. I think this is Oresmi's proof, and it's quite elegant. It's one half is smaller than 1 3rd plus 1 4th, and 1 3rd plus 1 4th is smaller than 1 5th plus 1 6th plus 1 7th plus 1 8. And so essentially, with that alone, you can extrapolate that that will continue to happen as you continue this series. And so you get. It gets bigger and bigger and bigger. Not, I think, probably the 1/2 plus 14 that gets. There's no way to make that get bigger. It just. It just sums to less and less and less. Whereas the harmonic series, you're summing greater and greater and greater quantities. And that is what explains the difference between those two seemingly very closely related series.

Professor David Bressoud (70:37)

Exactly, exactly. And what a polynomial of infinite degree enables you to do is to take a wide variety of functions and represent them as one of these polynomials of infinite degree. For those who are familiar with a little bit of trigonometry. And the sine function, the sine function can actually be represented as a polynomial of infinite degree. If you take X and then subtract x cubed over 3 times 2 times 1, and then add back in x to the 4 fifth divided by 5, 4, 3, 2 1, and then subtract x to the seventh divided by 7 times 6 times 5 times 4 times 3 times 2 times 1. You keep doing this, you're getting closer and closer to the sine function. Until by the time you have infinitely many of these terms, you actually have something that is equal to the sine function. So the sine function can be represented as a polynomial of infinite degree, which is one way of approximating the sine function. It's one way of working with a sine function and understanding many of its properties. But these infinite sums of functions, these polynomials of infinite degree, they always are a bit tricky to work with. I like to think of infinity as a very explosive device that has to be handled with kid gloves. And the mathematicians, the great mathematicians of the 18th century knew when it was safe to use it and when it wasn't. But something happened in the early 19th century, and this was when Joseph Fourier was working on problems in heat transfer and he was doing the calculus of heat transfer. And he discovered that what he needed was not an infinite polynomial, but an infinite sum of trigonometric functions. And now suddenly, this infinite sum of trigonometric functions did not operate the way anyone thought any infinite sum of functions should operate. It was doing things that created paradoxes, such strong paradoxes that most of the great mathematicians of the time simply rejected his work. You see, they said it had to be wrong. There's no way you can add up infinitely many trigonometric functions. And the 19th century, really, it's an entire hundred year period of trying to understand what happens when you add up an infinite number of trigonometric functions.

Mark Malloy (73:26)

Sorry, just. Just to be clear on that, so the harmonic series is an infinite sum of fractions. And then the trigonometric function, a transition happens with Fourier where we're no longer summing fractions, we're summing functions. Is there a distinction there or is it.

Professor David Bressoud (73:51)

No, not really. Because the series, the power series, the functions that you're adding up are powers of X.

Mark Malloy (73:59)

Okay, so each, the terms of the, of the furrier series, each, each term, term in between the addition sign, those.

Professor David Bressoud (74:09)

Are, they're all trigonometric functions.

Mark Malloy (74:12)

Each one is a trigonometric. So it's not, it's not a polynomial X squared. Each term is its own function.

Professor David Bressoud (74:19)

Exactly, exactly. And, and what that forced was a reconsideration. At this point, I need to slide into the notion of infinity. Because people in the 17th century were very hesitant about using the idea of infinity. They were coming right out of the Greek tradition. They were basing a lot of their work on the Greek classics that avoided the talk of infinite, of infinity. You get somebody like Leibniz or Newton, and they're hesitant to actually use that as part of their argument. Now, they might talk, they might think in terms of infinity, but when they actually wanted to present their argument in full rigor, they avoided talking about infinity. But infinity is such a useful concept for understanding what's going on that their successors, people like the Bernoullis and Euler and Lagrange and so on, that you've got in the 18th century, they began using the idea of infinity with great abandon so much that it raised concerns among many people how free and loose they were with the idea of infinity. And then with Fourier, infinity blows up in their faces. It's creating things that they absolutely cannot explain. And so it takes 100 years to figure out what's really going on with this. And the person here who's really central to the story is a Frenchman by the name of Augustin Louis Cauchy, who really goes back to the ancient Greek way of showing that if you've got a statement that you want to make about something that involves an infinite sum or any other thing that involves infinity, you really have to go back to the Greek approach and show that the solution that you think exists can't possibly be any smaller than what you think it is, and it can't possibly be any larger than what you think it is. But Cauchy's brilliance was in building tools that made it easy to construct these kinds of arguments, but today is usually referred to as epsilon delta arguments. These are. These are tools for avoiding the use of infinity by actually showing that what you think is the solution is by verifying that this true solution couldn't be any smaller and it couldn't be any larger.

Mark Malloy (77:00)

And that is what today is referred to as the limit concept.

Professor David Bressoud (77:05)

Exactly. Exactly. When a mathematician talks about limits today, they're really using it in the ancient Greek sense, the way the Greeks justified the formula for the area of the circle. Now, mathematicians, physicists, anybody who works with calculus always thinks in terms of infinity. But whenever you run into difficulties, whenever things come out in ways that you don't expect them them to, or when you actually need a proof, then you have to go back to that Greek approach using the tools that the Cauchy developed, what referred to as the algebra of inequalities. Using inequalities to show that something can't be smaller and it can't be larger, and so it has to be exactly that value. Most times when you're trying to use the idea of infinity, your intuition is okay. But when intuition breaks down, the 19th century saw the development of tools that enable you to move forward and figure out what was going wrong and why it broke down.

Mark Malloy (78:16)

So to briefly summarize that point up to and to some degree, including mathematicians of Newton's age, for example, were hesitant to explicitly use infinity because it was counterintuitive that they would be able to do infinitely many sums, and also because perhaps they knew that it was not unproblematic. It could be tricky, and it could surprise you.

Professor David Bressoud (78:45)

Exactly.

Mark Malloy (78:47)

After Newton, for a century or so, that hesitancy is kind of abandoned in the enthusiasm to do real mathematics with the tools that Newton had developed. And this is the era, the age of Euler, who is another extraordinary character. And I think, I believe he has 80 volumes, a thousand pages each of papers, something incredible. They were more comfortable doing the math and not worrying too much about if whether or not infinity was problematic. Then in the 19th century, mathematicians decided, we really do need to sort this out. We need to be sure that the results we're getting are rigorously proven. And so Cauchy and others clarify the language and the mathematics around it with epsilon, delta and the concept of the limit, which really does found this and show us that the math that Euler and his contemporaries were doing was in fact, was in fact rigorously correct. Their intuition was true. But now we know that, whereas they. There was a bit of faith in when they were working.

Professor David Bressoud (80:05)

Oh, yes, yes. And there are wonderful stories in here. Weierstrass, late in the 19th century, finally was able to justify the idea of an infinite product that Euler had used very freely. But it took over 100 years before mathematicians could show rigorously that what he was doing was legitimate.

Mark Malloy (80:28)

So an infinite sum is adding infinitely many numbers, and an infinite product is multiplying infinitely many numbers. So Beierstrass showed that you are able to multiply infinitely many numbers and arrive at a single final result or product of that multiplication, Right?

Professor David Bressoud (80:50)

Yes. And actually not numbers, but functions, multiplying how many functions together.

Mark Malloy (80:56)

So that's why I always need to be checked, because my understanding of this is superficial but.

Professor David Bressoud (81:04)

Well, and it's fascinating because there's so many subtleties. And that's one of the things that's frustrating when we teach it as just learn this procedure and learn how to solve this particular kind of problem. But when there's so much beneath the surface that went into developing it and understanding it and all of the subtleties, all of the difficulties, and that's really what I've been trying to convey in this book, is to get that across for people who are teaching calculus, to help inform them about what they can communicate for people who are learning calculus so that they get a deeper understanding of, of what's going on in the subject. And the people who just are interested in the idea of calculus realize that it's important, but don't understand why it's important to give them some window into what's been going on as it's grown.

Mark Malloy (82:03)

From my non specialist perspective, I think that your book is on the right track. I think that it's wonderful. And I do think that your program or project of reordering calculus makes sense if you root it in the historical progression of the subject matter. That these subjects developed in this progression historically for a reason. I would argue it's, it's the, it's the intuitive way with which the human species came to understand them. And that same intuitive approach might work on the individual level.

Professor David Bressoud (82:40)

Exactly. Yes.

Mark Malloy (82:42)

Professor Brasseu, thank you so much. We've already taken up quite a bit of your time to wrap up. Could I just ask, is there anything you are working on now that you would be willing to share with us?

Professor David Bressoud (82:55)

Well, right now most of my work centers on students in the transition from high school to college. One project that I'm just starting to work on now is trying to understand how we came to the current situation with regard to calculus instruction, where most of the people who need calculus in college take it for a year in high school and then when they get to college, they take exactly the same course again when they get to college. And it's taught as if they have never seen it before. And it's an interesting story that really goes back to what happened with understanding the mathematics curriculum after the Second World War. It's a story that's unfolded over the past 70 years. And I'd like to tell that story of how we came to our current very dysfunctional system.

Mark Malloy (83:53)

And over the phone we spoke once previously preparing for this call. And I told you my own story, which was that I did quite well in high school math, but I, I declined. I, I, I believe that I did have the option to take calculus, but I, I did not take it. I, I reached pre calculus and then having done well in math, I thought, I thought maybe, who knows, maybe I, I will end up studying math. First semester freshman year of college, I took an analysis course and I was getting 20s on tests. It just absolutely clobbered me and that was the end of my mathematical career. No one's fault. But I do think it's not unrelated to what you're looking into in terms of the system as a whole and how we can map high school students onto transition them into the college more efficiently.

Professor David Bressoud (84:48)

Exactly. So many stories of students who have enjoyed mathematics through high school and get to college, and it's just an incredibly discouraging experience for them. And I hope that we can change that.

Mark Malloy (85:06)

Anything that we could do to encourage appreciation of the subject is a wonderful thing. Professor Brassu, your book is a wonderful introduction to this important and fascinating subject and I highly recommend it to all of our listeners. Thank you so much for writing it and for your time and insights today. It has been such a pleasure speaking with you.

Professor David Bressoud (85:33)

Thank you so much for this opportunity. I've enjoyed talking with you too.

Mark Malloy (85:47)

I hope you've enjoyed this conversation. I've been speaking with Professor Brasseux about his 2019 book, Calculus A History of the Big Ideas. It's a wonderful book. If you are interested in the subject matter, whether an expert or a lay reader, I highly recommend it. The theme music for this episode and for all my episodes is composed and performed by Dan Lurch. I'm Mark Malloy and you've been listening to the New Books and Intellectual History Channel of the New Books Network. See you next time.

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