One of my favorite professors, Gilbert Strang, comes by the lab to talk about his time at MIT since 1952. Anyone who knows professor Strang knows him to be one of the best mathematics teachers and a prolific writer with a heart of gold.
Professor Strang retired last spring (2023). His last 18.06 lecture was to a packed standing room of students, staff, faculty, friends, and family. We are privileged to have him visit the lab for this conversation.
...and Happy Birthday Professor!!
Podcast specific:
Podcast Instagram: @lockthequill
Comments or questions: lockthequill@mit.edu
The Lab at MIT:
Pappalardo Lab Instagram: @pappalardolab
Comments or questions: pappalardolab@mit.edu
Pappalardo Lab website
One of my favorite professors, Gilbert Strang, comes by the lab to talk about his time at MIT since 1952. Anyone who knows professor Strang knows him to be one of the best mathematics teachers and a prolific writer with a heart of gold.
Professor Strang retired last spring (2023). His last 18.06 lecture was to a packed standing room of students, staff, faculty, friends, and family. We are privileged to have him visit the lab for this conversation.
...and Happy Birthday Professor!!
Podcast specific:
Podcast Instagram: @lockthequill
Comments or questions: lockthequill@mit.edu
The Lab at MIT:
Pappalardo Lab Instagram: @pappalardolab
Comments or questions: pappalardolab@mit.edu
Pappalardo Lab website
[music]
Danny: We're kicking off season two with one of the world's most celebrated mathematics professors, Gilbert Strang. Professor Strang has been teaching at MIT since 1959, and in addition to being a beloved presence on campus, is one of the most viewed online educators in linear algebra, reaching millions of students around the world, and putting calculus in its rightful place.
[music]
Danny: Hey, Scott.
Scott: Hey, Danny.
Danny: All right. You know Professor Strang.
Scott: Oh, yes.
Danny: You've seen him around campus? the last 20, 30 years.
Scott: Oh, many times, yes. Been here forever.
Danny: He's going to be 89. He just retired this last spring.
Scott: Just a kid, right?
Danny: Just a kid. I had him sign my book.
Scott: Oh, that's so cool.
Danny: Totally cool. That is a treasure.
Scott: Yes, definitely.
Danny: Because I took the class with him over 30 years ago.
Scott: Yes, imagine that.
Danny: No. Everyone was a little younger then.
Scott: A lot younger.
Danny: Anyway, it's his birthday on Monday, and we wanted to wish him a happy birthday.
Scott: Monday, it's coming right up. Happy birthday, Professor Strang.
Danny: Happy birthday.
Scott: Yes, enjoy.
Danny: Here's the deal. We took all of our stuff on the road.
Scott: Oh, we did, yes. Yes, you and Steve.
Danny: Me and Steve, we loaded up a cart, and we went down the Infinite.
Scott: The cart that we made?
Danny: The little cart, not the big cart.
Scott: Oh, that's right. That was the little cart that time.
Danny: Yes. Stuff was falling off of it-
Scott: Oh, that's right.
Danny: -every bump we hit.
Scott: Now I'm with it.
Danny: We took it over to Building 26. We went ahead, and we crashed Tristan Ozuch's class. He's a new professor for 18.06, which is our undergraduate linear algebra class that was formally taught by Professor Strang. Have you gone to Building 26, 26-100?
Scott: You're talking about the auditorium.
Danny: Yes, the big auditorium.
Scott: I've been in the auditorium, and I've been through all the hallways underneath 26 many times.
Danny: That's how you get backstage.
Scott: I never knew that.
Danny: Yes. Steve and I couldn't find it. For all the years we've been here, that was a new discovery for us. We went upstairs.
Scott: The banana lounge?
Danny: Banana lounge. We didn't get bananas.
Scott: There must not have been good bananas, though.
Danny: No, we were just in a hurry.
Scott: Oh, yes, that'll do it, too.
Danny: Tristan was expecting us, so we were in a hurry.
Scott: No bananas.
Danny: No. We went upstairs, we went downstairs. We couldn't get in. We felt like total chumps. We've been here forever. We can't get in. We finally found it. We snuck all of our gear in during lecture.
Scott: Oh, no. , they were having lecture?
Danny: Yes, that's the whole thing. They were having lecture.
Scott: Oh, man.
Danny: This was their message.
18.06 Students: Happy birthday, Professor Strang!
[music]
Danny: Professor Strang.
Professor Strang: Yes.
Danny: Take us back to your undergraduate years.
Professor Strang: Well…you asked me to look way back to when I was an undergraduate, ‘52 to 5’5. MIT was a different place then. The number of students I remember would flunk out, and then they'd come back. There was much more of that. Now MIT has become highly selective. It's extremely difficult to get admitted. The students of today have no plans, whatsoever, to flunk out and come back… I'm just remembering stuff from years ago… Rush Schweikert was a classmate. We were all in the same-- it was groups of 30, and our names both started with S. Groups of 30 stuck together for the whole physics, chemistry, math. I had to take mechanical drawing. At that time, it was probably in Building 3 on the fourth floor or something.
Danny: Fourth floor. That's when the skylights were up.
Professor Strang: Yes. I mostly asked Schweikert to help me draw circles. Anyway, I would still need that help.
[music]
Danny: Professor Strang's circle-drawing tutor was Rusty Schweikert, an MIT alum and aero-astro who became a NASA astronaut. Rusty piloted the Lunar Excursion Module for the Apollo 9 mission in March 1969, a low-Earth orbit mission to test and prepare for the moon landing, which took place only four months later. Hey, if you need help drawing circles, I'd imagine astronauts are pretty good at it.
Professor Strang: Everybody, I hope, watching will understand that I'm not an engineer. I can't get my printer to work, much less a computer…. It was a nice place. I enjoyed it. There weren't very many math majors. Maybe one teacher that will be especially known to listeners about Course 2 was Steve Crandall. He became a mentor for me even though I was in math. I took, it must have been 2.001 or maybe it was then 2.01.
Danny: Mechanics and materials, stress-strain, elasticity, failure.
Professor Strang: We had to learn just very basic stuff. I remember free body-
Danny: Free body diagrams.
Professor Strang: -diagrams. That's right. Because Steve Crandall was pretty mathematical, there wasn't too much engineering that I couldn't deal with. It was math that I was happy with. Anyway, I was happy in that course. I can remember an exam on which-- this will reveal the truth, this exam. It was a quiz. It was a really tough quiz and some of the scores were low. My score was 99.
Danny: What happened?
Professor Strang: I'm bragging now about that except there was a one-point question and it was a true-false question or only two choices. I don't remember what the question is but I got it wrong. I realized at that point that I would never be an engineer. It was a question that an engineer would just see. It didn't involve a bunch of mathy abstract stuff. It was a practical question. Would this happen or would this happen? I missed it. That taught me.
Danny: That galvanized your decision to stay in math?
Professor Strang: I was already a math major, I guess. I realized that I was in the right place being a math major and just picking up the math part of mechanics.
Danny: It's unusual for someone from the math department to take these mechanical engineering courses.
Professor Strang: That's the only one I took, 2.001. It introduced me to Steve Crandall. At that time, we had to write a thesis and he was my advisor. The thesis actually got turned into a short paper. My very first paper was with Steve Crandall about the Rayleigh-Ritz method. Now, Rayleigh-Ritz, who were they?
Danny: Oh, there's a lot to unpack here. Part one: the paper. The paper is titled, An Improvement of the Holzer Table Based on a Suggestion of Rayleigh's. It was published in the Journal of Applied Mechanics in 1957 and describes adding an energy-based approach to the Holzer method to estimate the natural frequencies of a torsional system. Part two: the mentor. Steve Crandall, a legend in mechanics! Professor Crandall wrote the book on mechanics that we still use today in 2.001. Steve Crandall's mentor was J.P. Den Hartog of mechanical vibration fame, who was Steven Timoshenko's protégé at Westinghouse in the 1920s. Which brings us to part three: the secret engineer. Hearing this story, it's clear to me how Professor Strang was so effective teaching mathematics to mechanical engineers. His heritage is in mechanics, and furthermore, in the tradition of Timoshenko, Den Hartog, and Crandall. He may claim he's not an engineer, but I think otherwise.
Professor Strang: I've always felt Course 2 was a good one.
Danny: Told you!
Professor Strang: After MIT, I got a scholarship to Oxford in England. I was there two years, and then to UCLA because they were good in numerical analysis. My Ph.D. thesis was about finite difference methods for wave equations and heat equations and Laplace equations and so on. Finite differences, that was the lively topic with Peter Lax at NYU as our guru, as our leader. That was fine. I was a finite difference person for a while, theoretically. Then I thought-- Oh, it's funny looking back. My plan was, since I had zero money at that time, I thought I'd be an applied mathematician and work for IBM or something like that, but the MIT math department wrote to offer me a job as an instructor, and that solved the problem of what to do next after the degree, after the thesis. I came to MIT. Came back to MIT in ‘59. I was here ‘52 to ‘55, and two years in Oxford, two years at UCLA, and came back in ‘59, which seems like a long time ago and is a long time ago.
[music]
Some years later, along came something called the finite element method. I just would hear those words. I guess actually there were a couple of, of course 2 faculty, like Jurgen Bathe, who were key people in finite elements. Anyway, I thought, "Okay, I'll learn about this. What is it? Is it just finite differences in disguise?" No! it was different. Actually, it connects directly to Rayleigh-Ritz method. I just realized that as I was walking over. Finite elements is just a different way to set the equation. Finite differences- you start with the differential equation. You just take every term in that differential equation and you replace it by finite differences. Finite elements starts a little different way with things like energy and work. Closer to engineering, really.
That was a turning point for me to think about the finite element method because it was all done by engineers, mostly mechanical and civil engineers. It was a success numerically. It was exactly the Rayleigh-Ritz idea of having some trial functions and finding the best combination of trial functions to fit the equation. Involves some nice math, and those functions were piecewise linear, meaning just a bunch of line segments hooked together was the graph, or piecewise quadratic, piecewise parabolas. Oh…It was math heaven, applied math heaven. Finite elements, I'm sure, it grew and succeeded and displaced finite differences in many ways. That led me to a real big change in my life. One was knowing engineers and talking and reading the journals. The other thing was writing a book.
[music]
Danny: Professor Strang admits that the first draft of his first book about finite element analysis didn't quite hit the mark. The first draft applied only when regions were defined by a uniform square mesh, which he said “wasn't realistic”.
Professor Strang: That missed the main point of finite elements as I came to appreciate. I left that book aside. I turned it too much into a math problem from its real source as a great way to compute solutions to partial differential equations.
Danny: Talking to engineers brought Professor Strang to general geometries and irregular meshes, what he calls “the real thing”. The project was picked up again. An Analysis of the Finite Element Method was published in 1973.
Professor Strang: Anyway, then the book had general geometries, triangular meshes, and quadrilateral meshes. It's a happy memory. For me, a special part was to learn that I enjoyed writing a book. More or less, I kept going. I didn't stop because it just came naturally.
Danny: Professor Strang published more than a dozen books since Analysis of the Finite Element Method. Titles include Linear Algebra and its Applications, Introduction to Applied Mathematics, Calculus, Introduction to Linear Algebra, Computational Science and Engineering, and Differential Equations and Linear Algebra. There's no need to comparative shop for math textbooks. They are the bibles of the field.
Professor Strang: I don't want to fail to mention the course that came out of those books. We had the number 18 for math, 18.085. I'm hoping some people who listen to this podcast will have taken 18.085. That was absolutely my favorite course to teach. Partly because it was mostly taken by first-year grad students in Mechanical and in Aero were the biggest numbers in the class. They all came to do the job. It was a great class every year. I always said, as you mentioned, that I'm there to teach you guys and not to grade you, which is still my motto. You have to depend on the class to do their part. Those classes really did. They showed up for exams and took them and learned.
I could go back to 18.085, that was really special. I might comment that, MIT students are a little different now. Let me leave that for later.
Danny: I'm wondering if there was a particular time where you saw the most significant transitions.
Professor Strang: The big change, I see. Yes, that's a very good question. I don't know if I was conscious of it or it was just happening. Really, of course, I'm extremely proud of MIT. It's really tried to do things right and succeeded. We've had good presidents. For me, of course, the only president I would ever have called by his first name was Chuck Vest. He was just easy to know and he was the one responsible for OpenCourseWare. For me, if I had to think back about my time at MIT, it's pre-OpenCourseWare and since. [music]
Danny: MIT OpenCourseWare is an initiative to disseminate all of its undergraduate and graduate course content online for free. Professor Strang's undergraduate course is one of many examples with a complete set of over 30 lectures. I counted approximately 20 million views of these lectures on YouTube alone. Professor Strang was already a legend in the field by the time OCW came around, but he galvanized his role as the worldwide linear algebra professor with his online series. If anyone out there wants to learn about the existential pleasures of eigenvalues and the importance of singular value decomposition, Professor Strang's online lectures are the gold standard.
Professor Strang: It started in about the year 2000 that I thought, "Well, I could get my class videotaped." This is the linear algebra class, 18.06, because the physics guy who taught 8.01 was a superstar on the screen. In the big lecture room, 26-100, 8.01 was being filmed. My class happened to be in that room the following hour. I thought, "Well, maybe the camera could just stay in there for the next hour and film 18.06.” It wasn't that I thought it was going to go anywhere. The math department had some great teachers. Professor Rota was one of them, and a friend of mine. I just thought, "Okay, maybe if I set an example to have my class filmed, maybe Rota or the right guy would teach his class on film." I don't think that happened for Rota in particular, but of course, others have.
Anyway, well, maybe this is a part of my whole life, really, is to think that linear algebra deserves a bigger place, deserves a place alongside calculus in what students learn. We're approaching that now because data science and other new topics are becoming super popular, classes of 500 or something, taking data science.
Danny: We're dealing primarily with concepts in linear algebra.
Scott: That's right. With the matrix, the key data comes in a matrix and you have to understand how to get it out. Eigenvalues are part of it. Now 18.06 ends with singular values. They're the right replacement for eigenvalues. Eigenvalues are for square matrices. Data doesn't come in square matrices because the rows and columns are--
Danny: Always….generally different.
Professor Strang: Yes, they're measuring entirely different things. They might be patients and diseases or something.
Danny: Sure.
Professor Strang: Anyway, the most important math thing I could tell viewers is learn about singular values.
Danny: Beyond the eigenvalue.
Professor Strang: Beyond the eigenvalue. Singular values have really taken over. They're the powerhouse. That had to get into the class, into 18.06. Often it caught in on the last week so it couldn't be on the final exam and people weren't paying attention. Anyway, now it comes sooner.
Danny: As you added material, how hard is it to pull material out and how did you make those decisions?
Professor Strang: That's a problem, but I think night and day about how to organize linear algebra. Certainly, singular values have to be included.
If you had a lot of time, then you could go on with probability and deep learning, neural nets, but singular values. Actually, the singular values come from the matrix A transpose times A, or A times A transpose. Those amazingly have the same eigenvalues. You're back to eigenvalues of the symmetric matrix A transpose times A. As I say this, every person who computes is thinking, "Professor Strang, never multiply A transpose times A. It's a big operation. It makes a matrix numerically less reliable, more closer to risk!" In theory, the eigenvalues of A transpose times A, or their square roots, those are the singular values of the matrix. They're super important, and we have good algorithms. That's the main achievement, really, of numerical methods in these years is to find good algorithms. There's a seminar this afternoon at 4 o'clock in the math department about algorithms for finding eigenvalues.
Danny: Oh, no kidding.
Professor Strang: Yes, we can do that for matrices of order whatever, 1,000. When you get up to those numbers, you have to do sampling. For matrices of order 100, mathematicians would not have had any clue how to find the eigenvalues for a matrix of order 100. Now it's routine.
Danny: How about teaching in the classroom? How have things changed over the years for you?
Professor Strang: I was teaching here 60 years, so I saw a big changes in MIT. It's in the top-three worldwide. Sometimes it's in front of Harvard and Oxford and Cambridge, and other times one of those is number one, and we're two or three. Why? What does MIT offer especially? I think it's the more maybe down-to-earth-- it's a lab like the Pappalardo lab that MIT still has, which is a really good thing. Don't lose it. Don't become all theory.
Danny: Coming from a mathematician.
Professor Strang: Yes, coming from a mathematician who would get totally-- I'd be student number N with some large number.
Danny: You mentioned earlier that the student interactions have changed a little bit. I'm wondering if you can comment on that.
Professor Strang: I'd say the students are better, but we had some… You and I have had conversations with other faculty members. My last class, for example, was a big class in linear algebra. The number of student special requests, like they can't take the exam at the required time. Could I have permission to take it at a different time? Could I turn in the homework in late? Et cetera, et cetera. Students taking two courses that met at the same time. That doesn't make sense to me. It's become difficult, just mechanically, just to keep up with the number of special requests that are involved in teaching a big MIT class now. The students themselves, of course, are way better than we were. MIT students are super to teach.
Danny: What advice would you give?
Professor Strang: "Learn linear algebra," is my advice to everyone I meet. It's become recognized as maybe not quite as important as calculus, but that day is coming. It's calculus for--
Danny: Those are fighting words right there.
Professor Strang: Yes, sure. It's calculus for linear functions. The simplest functions are linear, first-degree. Those are ignored in calculus, but that's the heart of linear algebra.
Danny: As we wrap up, any reflections about MIT?
Professor Strang: Yes, I'm proud of MIT. We're doing this interview now when the world is in a difficult position because of the tragedies in Palestine and Israel, but MIT is really just …. When things go wrong, the presidents of MIT have made a big effort to help students get through them and see the good things that can come afterwards. Presidents have set a good example, and I hope the faculty has for doing their best. I think we've been fortunate in presidents and in alumni. The alumni think of MIT as a place that's trying to do its job right. They're correct. That's how I see MIT, too. It's been a fantastic place to work. [music]
Danny: Online, I looked up, there's, I'm sure you're aware of this, the Gilbert Strang Greatest Hits.
Professor Strang: I don't know.
Danny: You don't know it?
Professor Strang: No, I don't think I know about that.
Danny: Oh, boy. These are quotes that are attributed to you.
Professor Strang: Oh, the quotes, I see.
Danny: Why don't you just read a few and tell us what they mean?
Professor Strang: "Finite differences would have to scratch its head to figure out what to do, finite elements would just do it." I became a cheerleader for finite elements.
Danny: Who are you talking about in this one?
Professor Strang: "The French guy was a great mathematician, but not God, but the German guy, unbelievable."
Danny: Who were you talking about?
Professor Strang: I'm glad that made an impression. It's nice to think of mathematicians as actually, they were people. They walked around with other people, but they were amazing. I'm sure that the German in question is Gauss. Everybody thinks of Gauss as the greatest mathematician ever. I wonder who the French guy was. Laplace, maybe, or--
Danny: Fourier?
Professor Strang: Fourier. Oh, yes. Well, Fourier, he had a fantastic idea. Fourier is as modern as can be. If we think about calculus, that's Newton and Leibniz, but that's calculus.
Danny: You've got a real bone to pick with calculus, huh?
Professor Strang: I have to! Oh, yes! Here's one. "I think it's good to remember that MIT isn't a picnic all the time."
Danny: That's true. All right, last one. Here's a wholesome one.
Professor Strang: "Math is beautiful." I probably did say that. “You get some really nice answers. You can quote me on that.” Yes, math, it's just fun to see things work out right. Hopefully, that happens in class sometimes.
Danny: The way you're able to address the subject and make it tangible is one of the many ingredients to your magic sauce in teaching subjects.
Professor Strang: Oh, thank you. Thanks. Yes. I hope some of it comes through the books now that I'm not actually going to be in class anymore. That's been a big part of my recent life.
Danny: Sure.. You've touched millions of students around the world and you've affected my life in a positive way.
Professor Strang: Oh, that's great.
Danny: Thank you for everything that you've done and continue to do.
Professor Strang: Thanks for inviting me to have this conversation.
Danny: My pleasure.
Professor Strang: That's really nice to think that you took my favorite course, 18.085.
Danny: It's special to know it is was your favorite class.
[music]
Danny: Now a few greatest hits as read by our students.
Student 1: “We really only have two independent equations because this third equation turned out to be a fraud. Are they orthogonal? Life is going to end on Earth if they're not.”
Student 2: “Maybe Bessel himself remembered those stupid functions.”
Student 3: “There's a mechanism there. I'll stake my tenure on it.”
Student 4: “Oh, dear. Looks a bit messy.”
Speaker 5: “Taylor, he made a lot of trouble for millions of students.”
Speaker 6: “Math is beautiful.”
[music]
Danny: I want to thank Gil Strang for joining us in the lab. Congratulations on your retirement and happy birthday! Thanks to Tristan Ozuch for letting Steve and me crash his party, and thanks to our 18.06 students. Of course, thank you for listening! Next time, we'll be speaking with some former students, the youngest team to receive full FDA Class III PMA approval for their automated external defibrillator - Say that ten times over. Have a great weekend, everybody!
[00:29:43] [END OF AUDIO]