Po-Shen Loh: Mathematics, Math Olympiad, Combinatorics & Contact Tracing | Lex Fridman Podcast #183

1. 0:00 – 1:43

   Introduction

   1. LFLex Fridman0:00

      The following is a conversation with Po-Shen Loh, a professor of mathematics at Carnegie Mellon University, national coach of the USA International Math Olympiad Team, and founder of Expi that does online education of basic math and science. He's also the founder of Novid, an app that takes a really interesting approach to contact tracing, making sure you stay completely anonymous, and it gives you statistical information about COVID cases in your physical network of interactions so you can maintain privacy, very important, and make informed decisions. In my opinion, we desperately needed solutions like this in early 2020, and unfortunately, I think, we will again need it for the next pandemic. To me, solutions that require large-scale distributed coordination of human beings need ideas that emphasize freedom and knowledge. Quick mention of our sponsors: Jordan Harbinger Show, Onnit, BetterHelp, Eight Sleep, and LMNT. Check them out in the description to support this podcast. As a side note, let me say that Po and I filmed a few short videos about simple, beautiful math concepts that I will release soon. It was really fun. I really enjoyed Po sharing his passion for math with me in those videos. I'm hoping to do a few more short videos in the coming months that are educational in nature on AI, robotics, math, science, philosophy, or if all else fails, just fun snippets into my life on music, books, martial arts, and other random things if that's of interest to anyone at all. This is the Lex Fridman Podcast, and here's my conversation with Po-Shen Loh.

2. 1:43 – 5:21

   Planes and bridges

   1. LFLex Fridman1:43

      You know, you mentioned you really enjoy flying and experiencing different people and different places. There's something about flying for me, I don't know if you have the same experience, that every time I get on an airplane, it's incredible to me that human beings have actually been able to achieve this. (laughs) And, and when I look at, like, um, what's happening now with humans traveling out into space, I see it as all the same thing. It's incredible that humans are able to get into a box and fly in the air, and, and safely, and land. And the same, it seems like, and everybody's taking it for granted, so when I observe them. It's, it's quite fascinating because I s- see that cleanly mapping to the world where we're now on, on, uh, in rockets and traveling to the moon, traveling to Mars. And at the same kind of way, I can already see the future wh- where, where we will all take it for granted. (laughs) So I don't know, I don't know if you have, uh, you personally when you fly have the same kind of magical experience of like, how the heck did humans actually accomplish this?

   2. PLPo-Shen Loh2:49

      So I do, especially when there's turbulence-

   3. LFLex Fridman2:51

      (laughs)

   4. PLPo-Shen Loh2:52

      ... which is, you know, like on the way here-

   5. LFLex Fridman2:54

      Yeah.

   6. PLPo-Shen Loh2:54

      ... uh, there was turbulence and it, the, the plane jiggled. Even the flight attendant had to hold onto the side.

   7. LFLex Fridman2:59

      Yes.

   8. PLPo-Shen Loh3:00

      And I was just thinking to myself, it's amazing that this happens all the time and the wings don't fall off.

   9. LFLex Fridman3:04

      Yeah. (laughs)

   10. PLPo-Shen Loh3:04

       You know, like given how many planes are flying.

   11. LFLex Fridman3:06

       Yeah.

   12. PLPo-Shen Loh3:06

       But then I often think about it and I'm like, you know, a long time ago, I think people didn't trust elevators-

   13. LFLex Fridman3:12

       Yeah.

   14. PLPo-Shen Loh3:12

       ... in a 40-story building in New York City. And now we just take it completely for granted that you can step into this shaft, which is-

   15. LFLex Fridman3:19

       Yes.

   16. PLPo-Shen Loh3:19

       ... 40 floors up and down and it will just not fail. (laughs)

   17. LFLex Fridman3:24

       (laughs) Yeah. Again, I'm the same way with elevators, but also buildings when I'll, I'll stand on the 40th floor-

   18. PLPo-Shen Loh3:30

       Oh, yeah.

   19. LFLex Fridman3:30

       ... and wonder, "How the heck are we not falling right now?" Like-

   20. PLPo-Shen Loh3:35

       Yeah. (laughs)

   21. LFLex Fridman3:36

       ... how, how amazing it is with the high winds, like structurally just the earthquakes and the vibrations. I mean, g- natural vibrations in the ground. Like, how is this, how are all of these, you go to like New York City, all of these buildings standing? I mean, to me, one of the most beautiful things, actually mathematically too, is, uh, bridges. I used to build bridges in high school from like toothpicks, just like out of the pure joy of, like, physics making some structure really strong. U- understanding, like from a civil engineering perspective, what kind of structure will be stronger than another kind of structure, like suspension bridges? And then you see that at scale, humans being able to span a body of water with a giant bridge. And, and it's, I don't know, it, it, it's so humbling. It makes you realize how, (laughs) how dependent we are on each other, sort of. I talk about level up, but it, there is, there's a certain element in which we little ants have just a small amount of knowledge about our particular thing, and then we're depending on a network of knowledge that other experts hold. And then most of our lives, most of the quality of life we have has to do with the, the richness of that network of, of knowledge, of that collaboration and then sort of the ability to build on top of it l- levels of abstractions. You start from, like, bits in a computer, then you can have assembly, then you can have C++, or you have an operating system, then you can have C++ and Python, finally some machine learning on top. All of these, they're abstractions. And eventually we'll have AI that runs all of us humans.

   22. PLPo-Shen Loh5:17

       (laughs)

   23. LFLex Fridman5:17

       But anyway, uh, but speaking of abstractions and programming,

3. 5:21 – 7:46

   Writing a computer game from scratch

   1. LFLex Fridman5:21

      in high school, you wrote some impressive games-

   2. PLPo-Shen Loh5:24

      (laughs)

   3. LFLex Fridman5:24

      ... for MS-DOS. I got a chance to, in browser somehow, it's magic. I g- got a chance to play them. Alien Attack 1, 2, 3, and 4. What's the hardest part about programming those games? And maybe can you tell the story about, uh, about building those games?

   4. PLPo-Shen Loh5:41

      Sure. I actually tried to do those in high school because I was just curious if I could.

   5. LFLex Fridman5:46

      (laughs)

   6. PLPo-Shen Loh5:47

      And the, uh, yeah, and, and the-

   7. LFLex Fridman5:47

      It's a good starting point for anything, right?

   8. PLPo-Shen Loh5:49

      Yeah, yeah, yeah. It's like, could you? But the, the appealing thing was also, it was a soup to nuts kind of thing. So something that has always attracted me is, I like beautiful ideas. I like seeing beautiful ideas, but I actually also like seeing execution of an idea all the way from beginning to end in something that works.

   9. LFLex Fridman6:06

      Mm-hmm.

   10. PLPo-Shen Loh6:06

       So, for example, in high school, I was lucky enough to grow up in the late '90s when even...... a high school student could hope to make something sort of comparable to the shareware games that were out there. Not, I, I say the words sort of, like, still quite far away.

   11. LFLex Fridman6:21

       Mm-hmm.

   12. PLPo-Shen Loh6:21

       But at least I didn't need to hire a 3D CG artist.

   13. LFLex Fridman6:24

       Mm-hmm.

   14. PLPo-Shen Loh6:25

       There, there weren't enough pixels to draw anyway.

   15. LFLex Fridman6:26

       (laughs)

   16. PLPo-Shen Loh6:27

       Even I can draw, right? (laughs) Uh, uh, bad art, of course. But the, the point is, I wanted to know, is it possible for me to try to do those things? Uh, where back in those days, you didn't even have an easy way to draw letters on the screen-

   17. LFLex Fridman6:40

       Mm-hmm.

   18. PLPo-Shen Loh6:40

       ... in a particular font. Y- you couldn't just say, "Import a font."

   19. LFLex Fridman6:42

       Mm-hmm.

   20. PLPo-Shen Loh6:42

       It wasn't like Python. So, for example, back then, if you, if you played those games in, in the, in the web browser, which is emulating, um, the, the old school computer, um, those ... even the letters you see, those are made by individual calls to draw pixels on the screen.

   21. LFLex Fridman6:56

       So, you built that from scratch, though almost building a computer graphics library from scratch?

   22. PLPo-Shen Loh7:01

       Yes. The primitive that I got to use was some code I copied off of a book in assembly of how to put a pixel on a screen in a particular color.

   23. LFLex Fridman7:08

       And the program- programming language was Pascal?

   24. PLPo-Shen Loh7:11

       Ah, yeah. The first one was in Pascal.

   25. LFLex Fridman7:14

       Okay.

   26. PLPo-Shen Loh7:14

       But then the other ones were in C++ after that.

   27. LFLex Fridman7:18

       How does the emulation in the browser work, by the way? Is that, is that trivial? (laughs) 'Cause it's pretty cool you get to play these games that have a very much '90s feeling to them.

   28. PLPo-Shen Loh7:26

       Ah, so it's literally making an MS-DOS environment-

   29. LFLex Fridman7:29

       Yeah.

   30. PLPo-Shen Loh7:29

       ... which is literally running the old .EXE file.
4. 7:46 – 11:21

   Programming competitions

   1. LFLex Fridman7:46

      you did a bunch of programming competitions. What was your interest, your love for programming? Um, what did you learn through that experience, especially now that as much of your work has taken a long journey through mathematics?

   2. PLPo-Shen Loh8:03

      I think I always was amazed by how computers could do things fast. If I wanted to make a- an abstract analysis of why it is that I saw some power in the computer, because if the computer can do things so many times faster than humans, where the hard part is telling the computer what to do and how to do it, if you can master that asking the computer what to do, then you could conceivably achieve more things. And those contests I was in, those were the opposite in some sense of making a complete product.

   3. LFLex Fridman8:34

      Mm-hmm.

   4. PLPo-Shen Loh8:34

      Like, a game is a product. Uh, those contests were effectively write a function to do something extremely efficiently. And if you are able to do that, then you can unlock more of the power of the computer.

   5. LFLex Fridman8:45

      But also doing it quickly. There's a time element from the human perspective to, to be able to program quickly. There's something nice. So, there isn't, like, almost, like, an athletics component to where you're almost like, uh, an athlete seeking optimal performance as a human being trying to write these programs? And at the same time, it's kind of art because you're ... the best way to write a program quickly is to write a simple program, is to have a damn good solution. So, it's not necessarily you have to type fast. You have to think through a really clean, beautiful, uh, solution. I mean, what do you think is the use of those programming competitions? Do you think they're ultimately something you would recommend for students, for people interested in programming, or people interested in building stuff?

   6. PLPo-Shen Loh9:34

      Yes. I think so because especially with the work that I've been doing nowadays, even trying to control COVID, something that was very helpful from day one was understanding that the kinds of computations we would want to do, we could conceivably do on like a four-core cloud machine on Amazon Web Services out to a population which might have hundreds of thousands or millions of people.

   7. LFLex Fridman9:55

      Mm-hmm.

   8. PLPo-Shen Loh9:56

      The reason why that was important to have that back of the envelope calculation with efficient algorithms is because if we couldn't do that, then we would bankrupt ourselves before we could get to a big enough scale. Uh, if you think about how you grow anything from small to big, if ... in order to grow it from small to big, you also already need 10,000 cloud servers, you'll never get to big.

   9. LFLex Fridman10:19

      And also, the nice thing about programming competitions is that you actually build a thing that, uh, works. So, you, uh, you finish it. There's a completion thing, and you realize ... I think there's a magic to it where you realize that it do- it's not so hard to build something that works, to have a system that, uh, well, successfully takes in inputs and produces outputs and solves a difficult problem. And that directly transfers to building a startup essentially that can help some aspect of this world as long as it's mostly be, um, based on software engineering. Things get really tricky when you have to manufacture stuff.

   10. PLPo-Shen Loh10:56

       (laughs)

   11. LFLex Fridman10:56

       (laughs) That, that's why people like Elon Musk are so impressive that they ... it's not just software. Tesla autopilot is just, not just software. It's-

   12. PLPo-Shen Loh11:05

       Yes.

   13. LFLex Fridman11:05

       It's like you have to actually, like, have factories that build the cars, and there's like a million components in- involved in, in the machinery required to assemble those cars and so on. But in software, one person can change the world, which is, uh, uh, incredible.

5. 11:21 – 16:52

   Math is hard

   1. LFLex Fridman11:21

      But on the mathematics side, what, uh, if you look back or maybe today, what made you fall in love with mathematics?

   2. PLPo-Shen Loh11:29

      For me, I think I've always been very attracted to challenge as, as I already indicated with the writing the program. I guess if I see something that's hard or supposed to be impossible, it cert- so- sometimes I say, "Maybe, maybe I wanna see if I can pull that off." And with the mathematics, the math competitions presented problems that were hard that I didn't know how to start but for which I could conceivably try to learn how to solve them.

   3. LFLex Fridman11:58

      Mm-hmm.

   4. PLPo-Shen Loh11:59

      So, I mean, there are other things that are hard called, like, get something to Mars, get people to Mars.

   5. LFLex Fridman12:03

      Yeah.

   6. PLPo-Shen Loh12:03

      And I didn't, and I still don't think that I am able to solve that problem. On the other hand, the math problems struck me as things which are hard and with significant amount of extra work, I could figure it out, and maybe they would actually even be useful. Like, that, that mathematical skill is the core of lots of other things.

   7. LFLex Fridman12:22

      That's really interesting. Maybe you could speak to that because a lot of people say that math is hard as a kind of negative statement. It always seemed to me a little bit like that's kind of a positive statement, that all things that are worth having in this world are hard. I mean, everything that people think about that they would love to do, whether it's sports, whether it's art, music, and all the sciences, it, they're going to be hard if you want to do something special. So, is there something you could say to that idea that math is hard? Should it be made easy or should it be hard?

   8. PLPo-Shen Loh13:04

      Ah. So I think maybe I wanna dig in a little bit onto this hard part and, and say, uh, I think the interesting thing about the math is that you can see a question that you didn't know how to start doing it before, and over a course of thinking about it, you can come up with a way to, to solve it.

   9. LFLex Fridman13:24

      Mm-hmm.

   10. PLPo-Shen Loh13:24

       And so you can move from a state of not being able to do something to a state of being able to do something where you help to take yourself through that-

   11. LFLex Fridman13:33

       Mm-hmm.

   12. PLPo-Shen Loh13:33

       ... instead of somebody else spoonfeeding you-

   13. LFLex Fridman13:36

       Yes.

   14. PLPo-Shen Loh13:36

       ... that technique. So actually here, I'm already digging into maybe part of my teaching philosophy also-

   15. LFLex Fridman13:42

       Yes.

   16. PLPo-Shen Loh13:42

       ... which is that I actually don't want to ever just tell somebody, "Here's how you do something." I actually prefer to say, "Here's an interesting question. I know you don't quite know how to do it. Do you have any ideas?" I, this is, I'm actually c- coming up with, uh, I'm, I'm actually explaining another way that you could try to do teaching.

   17. LFLex Fridman14:00

       Mm-hmm.

   18. PLPo-Shen Loh14:00

       And I'm contrasting this to a method of, "Watch me do this. Now practice it 20 times." I'm trying to say a lot of people consider math to be hard because maybe they can't remember all of the methods that were taught. But for me, I look at the hardness and I don't think of it as a memory hardness. I think of it as a, can you invent something hardness. And I think that if we can teach more people how to do that art of invention, uh, in a pure cognitive way, not as hard as the actual hardware stuff, right?

   19. LFLex Fridman14:31

       Mm-hmm.

   20. PLPo-Shen Loh14:31

       But like in terms of the concepts and the thoughts and the mathematics, teaching people how to invent, then suddenly actually they might not even find math to be that tiresomeness hard anymore-

   21. LFLex Fridman14:42

       Right.

   22. PLPo-Shen Loh14:42

       ... but that rewardingness hard of I have the capability of looking at something which I don't know what to do and coming up with how to do it. I actually think we should be doing that, giving, giving people that capability.

   23. LFLex Fridman14:55

       So hard in the same way that invention is hard, that is ultimately rewarding. So, maybe you can dig in that a little bit longer, which is, um, do you see basically the way to teach math is to present a problem and to give a person a chance to try to invent a solution without, with minimal amount of information first? Is that, is, is that basically how do you build that muscle of invention in a student?

   24. PLPo-Shen Loh15:26

       Yes. So the way that... I, I, I guess I have two different sort of ways that I try to teach. Actually one of them is in fact this semester because all my classes were, uh, remotely delivered. I even threw 'em all onto my YouTube channel.

   25. LFLex Fridman15:37

       Awesome.

   26. PLPo-Shen Loh15:37

       So you can see (laughs) you can see how I teach at Carnegie Mellon. But I'd often say, "Hey, everyone, let's try to do this. Any ideas?" And that actually changes my role as a professor from a person who shows up for class with a script of what I wanna talk through. I actually d- I don't have a script. The way I show up for class is there's something that we want to learn how to do and we're gonna do it by improv. Uh, I'm talking about the same method as improv comedy-

   27. LFLex Fridman16:02

       Mm-hmm.

   28. PLPo-Shen Loh16:02

       ... which is where you tell me some ideas and I'll try to yes and them.

   29. LFLex Fridman16:07

       (laughs)

   30. PLPo-Shen Loh16:08

       (laughs) Uh, you, you know what I mean?
6. 16:52 – 54:09

   Contact tracing that preserves privacy

   1. LFLex Fridman16:52

      But you did mention, um, disease and COVID and you've been doing some very interesting stuff from a mathematical but also software engineering angle of coming up with ideas. It's back to the, "I can... I see a problem. I think I can help." Uh, so you stepped into this world. Can you tell me about your work there under the flag of NOVID and, uh, both the, the software and the technical details of how the thing works?

   2. PLPo-Shen Loh17:21

      Sure, sure. So first I want to make sure that I say this is actually a team effort. I happen to be the one speaking, but there's no way this would exist without an incredible team of people who inspire me every day to work on this.

   3. LFLex Fridman17:32

      Mm-hmm.

   4. PLPo-Shen Loh17:33

      But I'll speak on behalf of them. So, um, the idea was indeed that we stepped forward, uh, in March of last year when the world started to become, our part of the world started to become, our part meaning the United States started to become paralyzed by, uh, COVID. The shutdown started to happen. And at that time it started as a figment of an idea which was network theory, which is the area of math that I work in, could potentially be combined with smartphones and some kind of health information anonymized. Exactly how, we didn't know yet. We tried to crystallize it. And many months into this work, we ended up accidentally discovering a new way to control diseases, which is now what is the main impetus of all of this work is to take this idea......and polish it and hopefully have it be useful, not only now, but for future pandemics. The idea is really simple to describe. Um, actually my main thing in the world is I come up with obvious observations.

   5. LFLex Fridman18:33

      (laughs)

   6. PLPo-Shen Loh18:33

      (laughs) That's, that's, so I'll explain it now.

   7. LFLex Fridman18:35

      Einstein did the same thing.

   8. PLPo-Shen Loh18:36

      (laughs)

   9. LFLex Fridman18:36

      And he wrote a few short papers, so...

   10. PLPo-Shen Loh18:38

       (laughs) But, but, but so, uh, the idea is like this. If I, if we describe how usually people control disease, for a lot of history, it was that you'd find out who was sick, you'd find out who they've been around, and you try to remove all of those people from society against their will.

   11. LFLex Fridman18:57

       Yes.

   12. PLPo-Shen Loh18:57

       Now, that's the problem. The, the against their will part gives you the wrong kind of a feedback loop, which makes it hard to control the disease, because then the people you're trying to control keep getting other people sick. You can see already how I'm thinking and talking about this, feedback loops. This is actually related to something you said earlier about even, like, how skyscrapers stay in the air.

   13. LFLex Fridman19:16

       Yeah.

   14. PLPo-Shen Loh19:17

       Uh, the whole point is control theory. Uh, y- you actually want to... or even how an airplane stays. You, you need to have control loops which are feedbacking in the right way.

   15. LFLex Fridman19:26

       Mm-hmm.

   16. PLPo-Shen Loh19:27

       And what we observed was that the feedback control loop for controlling disease by asking people to be removed from society against their will-

   17. LFLex Fridman19:34

       Mm-hmm.

   18. PLPo-Shen Loh19:34

       ...was not working. It was running against human incentives, and you suddenly are trying to control seven billion, eight billion people in ways that they don't individually want to necessarily do. So, here's the idea, uh, and this is inspired by the fact that at the core of our team were user experience designers. That's actually the f- in fact, the first thing I knew we needed when we started, was to bring user experience at the core. Okay.

   19. LFLex Fridman19:58

       Beautiful.

   20. PLPo-Shen Loh19:58

       But so, um, the idea was, suppose there was a panthe- suppose hypothetically there was a pandemic. What would you want? You would want a way to be able to live your life as much as possible and avoid getting sick. Can we make an app to help you avoid getting sick? Notice how I've just articulated the problem. It is not, can we make an app so that after you are around somebody who's sick you can be removed from society? It's can we make an app so that you can avoid getting sick?

   21. LFLex Fridman20:30

       Yes.

   22. PLPo-Shen Loh20:30

       That would run a positive fee- uh, what, how do I w- I don't know if I want to call it positive or negative, but it would run a good feedback loop.

   23. LFLex Fridman20:36

       Yes.

   24. PLPo-Shen Loh20:36

       Okay? So then how would you do this? The only problem is that you don't know who's sick because the w- especially with this disease, if I see somebody who looks perfectly healthy, the disease spreads two days before you have any symptoms. And so it's actually not possible. That's where the network theory comes in. You caught it from someone. What if we changed the paradigm and we said, "Whenever there's a sickness tell everybody how many physical relationships separate them from the sickness." That is the trivial idea we added.

   25. LFLex Fridman21:09

       Mm-hmm.

   26. PLPo-Shen Loh21:09

       The trivial idea was the distance between you and a disease is not measured in feet or seconds, it's measured in terms of how many close physical relationships separate you, like these six degrees of separation, like LinkedIn. Simple idea. What if we told everyone that? It turns out that actually unlocks some interesting behavioral feedback loops, which for example, let me, let me now jump to a non-COVID example to show why this maybe could be useful, actually we think it could be quite useful. Imagine there was Ebola or some hemorrhagic fever, imagine it spread through contact, through the air in fact. Pretend, pretend. Um, that's a di- that's a disastrous disease. It has high fatality rate and, uh, as you die you're bleeding out of every orifice. Okay? So...

   27. LFLex Fridman22:01

       Yeah, not, not pleasant.

   28. PLPo-Shen Loh22:02

       Not pleasant. So, so the question is suppose that such a disease broke, who would want to install an app that would tell them how many relationships away from them this disease had struck?

   29. LFLex Fridman22:14

       Hmm, like a lot of people.

   30. PLPo-Shen Loh22:16

       A lot of people. In fact, almost... I don't want to say almost everyone, that's a very strong statement, but a very large number of people.
7. 54:09 – 1:09:49

   Math Olympiad

   1. LFLex Fridman54:09

      But if we look at the cleanest formulation of that, o- of looking at a problem from different perspective, you're also involved with the International Mathematics Olympiad, which takes, uh, small, clean, uh, problems that are really hard, but once you look at them differently, can become easy. But that little jump of innovation is, is, uh, is, is the entire trick. So maybe at the high level...Can you say what is the International Mathematical Olympiad?

   2. PLPo-Shen Loh54:41

      Sure. So this is, uh, a, the competition for people who aren't yet in college, a math competition, which is the most prestigious one in the entire world. It's the Olympics of mathematics, uh, but only for people who aren't yet in college. Now, the kinds of questions that they ask you to do are not computational. Usually, you're not supposed to find that the answer is 42. (laughs)

   3. LFLex Fridman55:03

      Right. Yeah.

   4. PLPo-Shen Loh55:03

      Right? Instead, you're supposed to explain why something is true.

   5. LFLex Fridman55:06

      Yes.

   6. PLPo-Shen Loh55:07

      And, uh, the, the problem is that at the beginning when you look at each of the questions, first of all, you have four and a half hours to solve three questions, and this is one day, and then you have a second day, which is four and a half hours, three questions. But when you look at the questions, they're all asking you, "Explain why the following thing is true," which you've never seen before. And by the way, even though there are six questions, if you solve any one of them, you're a genius-

   7. LFLex Fridman55:29

      Yeah.

   8. PLPo-Shen Loh55:29

      ... and you get an honorable mention. So this is, (laughs) this is hard to

   9. LFLex Fridman55:31

      Really hard problem.

   10. PLPo-Shen Loh55:32

       Really hard.

   11. LFLex Fridman55:33

       So, what about, is it one person? Is it a team?

   12. PLPo-Shen Loh55:35

       Ah. So it's, each country can send six people, and the score of the country is actually unofficial. There's not an official country versus country system, although everyone just adds up the point scores of the six people and they say, "Well, now which country, uh, stacked up where?"

   13. LFLex Fridman55:51

       Yeah. So may- maybe as a side comment, I should say that, um, there's a bunch of countries, including the former Soviet Union and Russia, uh, where I grew up, where this is one of the most important competitions that the country participates in. Like, it was a source of pride for a lot of the country. You, you look at the Olympic sports like, uh, wrestling, weightlifting, there's certain sports, and hockey, that Russia and the Soviet Union truly took pride in. And actually, the Mathematical Olympiad, it was one of them for many years. It's still one of them. And that's kind of fascinating. We don't think about it this way i- in the United States. Maybe you can correct me if I'm wrong, but it's not nearly as popular in the United States in terms of its integration into the culture, into, uh, just basic conversation, into the pride. Like, you know, if you win an Olympic gold medal or if you win the Super Bowl, you can walk around proud. I think that was the case with the Mathematical Olympiad in Russia, not n- not as much the case in the United States, I think. So I just want to give that a little aside because beating anybody from Russia, from the Eastern Republic, or from China is very, very difficult. There ... Like, if I remember correctly, you know, there's people, this was a multi-year training process. They train hard, and this is, this is everything that they're focused on. My, uh, my dad was, was, uh, w- was a participant in this and it's ... I mean, it's, uh, it's as serious as Olympic sports. You think about like gymnastics, like young athletes participating in gymnastics, it's just as serious as that if not more serious. So, I just wanted to give that a little bit of context 'cause we're talking about serious high level math (laughs) a- athletics almost here.

   14. PLPo-Shen Loh57:46

       Yeah. Uh, actually, I also think that it made sense, uh, from the Soviet Union's perspective because-

   15. LFLex Fridman57:52

       Mm.

   16. PLPo-Shen Loh57:52

       ... if you look at what these people do eventually, even though, uh, uh, let's look at the USSR's-

   17. LFLex Fridman57:58

       Yeah.

   18. PLPo-Shen Loh57:58

       ... Olympiad, uh, International Math Olympiad record. Even though they, I say even though they won a lot of awards at the high school thing, many of them went on to do incredible things in research mathematics or research other things, and that's showing the generalization, generalizability of what-

   19. LFLex Fridman58:14

       Interesting.

   20. PLPo-Shen Loh58:14

       ... they were working on. Because ultimately, we're just playing with ideas of how to prove things.

   21. LFLex Fridman58:22

       Mm-hmm.

   22. PLPo-Shen Loh58:22

       And if you get pretty good at inventing creative ways to turn problems apart, split them apart, observe neat ways to, um, turn messy things into simple crystals, well, if you're gonna try to solve any real problem in the real world, that could be a really handy tool too. So I don't think it was a bad investment. I, I think it clearly worked well for Soviet Union.

   23. LFLex Fridman58:45

       Yeah. So this, this is interesting. People sometimes ask me, you know, "You grow up in, in, under communism, you know, was there anything good about communism?" (laughs) And it's difficult for me to talk about it because it's not, uh, communism is one of those things that's looked down on, like, without in absolutist terms currently. But you can still, in my perspective, talk about the actual, forget communism or whatever the actual term is, but, you know, certain ways that the society function that we can learn lessons from. And one of the things in the Soviet Union that was highly prized is knowledge, not even knowledge, it's wisdom, and the skill of invention, of innovation at a young age. So we're, we're not talking about a selection process where you pick the best students in the school to do the mathematics or to read literature. It's like everybody did it. Everybody ... It was almost treated as if anyone could be the next Einstein, anybody could be the next, uh, I don't know, Hemingway, James Joyce. And so you're forcing an education on the populous and a rigorous deep education. Like, as opposed to kind of like, "Oh, we wanna make sure we, um, we teach to the weakest student in the class," which American systems can sometimes do because we don't want to leave anyone behind. The m- the, the Russian system was anyone can be the strongest student and we're gonna teach you to the strongest student and we're going to f- just to pretend or force everybody, even the weakest student, to be strong. And what that results in, it's obviously this is what people talk about is a huge amount of pressure. Like, it, it's psychologically very difficult. This is why people struggle when they go to MIT, it is a very competitive environment, it can be very psychologically difficult. But at the same time, it's bringing out the best out of people, and that mathematics was certainly one of those things. And exactly what you're saying, which kind of clicked with me just now, as opposed to kind of a spelling bee...... in the United States, which I guess you spell... I'm horrible at this. But it's a competition about spelling, which I'm not sure, but you could argue doesn't generalize well to future skills. Mathematics, especially this kind of mathematics, is essentially formalized competition of invention, of, uh, of creating a new ideas, and that generalizes really, really well. So, that's, that's quite brilliantly put. I didn't really think about that. So this is not just about the competition. This is about developing minds that, uh, will come-

   24. PLPo-Shen Loh1:01:34

       Yeah.

   25. LFLex Fridman1:01:34

       ... come to do some incredible stuff in the future.

   26. PLPo-Shen Loh1:01:37

       Yeah. Actually, I want to respond to a couple of things there. The first one is one, um, which is this notion of whether or not that is possible in a non-authoritarian regime. I think it is. And that's actually why I spent some of my efforts before the COVID thing, uh, actually trying to, uh, work towards there. The reason is because if you think about it, let's say in America, lots of people are pretty serious about training very hard for football-

   27. LFLex Fridman1:02:01

       Yes.

   28. PLPo-Shen Loh1:02:01

       ... or baseball, or basketball. Basketball is very, very accessible, but lots of people are doing that. Why? Well, actually, I think that what is, what- what- what- what was going on with the authoritarian thing was at least the message that was universally sent was being a good thinker and a creator of ideas is a good thing.

   29. LFLex Fridman1:02:21

       Yes, exactly.

   30. PLPo-Shen Loh1:02:23

       There's no reason why that message can't be sent-
8. 1:09:49 – 1:17:06

   Hard math problem

   1. PLPo-Shen Loh1:09:49

   2. LFLex Fridman1:09:49

      So I don't know if you can, uh, say something insightful to this question, but what do you think makes a really hard math problem on, on this olympiad, maybe in the courses you teach, or in general? What makes for a hard problem? You've seen, I'm sure, a lot of really difficult problems. What makes a hard problem?

   3. PLPo-Shen Loh1:10:08

      So I could quantify it by the number of leaps of insight-

   4. LFLex Fridman1:10:13

      (laughs)

   5. PLPo-Shen Loh1:10:13

      ... of changes of perspective that are along the way. And here's why.

   6. LFLex Fridman1:10:16

      Yeah.

   7. PLPo-Shen Loh1:10:16

      This is like a very theoretical computer science way of looking at it, okay? It's the, it's that each reframing of the problem and using of some tool, I actually call that a leap of insight. When you say, "Oh, wow, now I see I should kind of put these plugs into those sockets-"

   8. LFLex Fridman1:10:31

      Mm-hmm.

   9. PLPo-Shen Loh1:10:31

      "... like so."

   10. LFLex Fridman1:10:32

       Yes.

   11. PLPo-Shen Loh1:10:32

       "And suddenly I get to use that machine. Oh, but I'm not done yet. Now I need to do it again." Each such step is a large possible, large fan out in the search space.

   12. LFLex Fridman1:10:41

       Mm-hmm.

   13. PLPo-Shen Loh1:10:42

       The number of these tells you the exponent. The base of the exponent is like how big, how many different possibilities you could try.

   14. LFLex Fridman1:10:50

       (laughs)

   15. PLPo-Shen Loh1:10:50

       And that's, it's, that's- that's actually why, like, if you have a three-insight problem, that is not three times as hard as a one-insight problem, because after you've made the one insight, it's not clear that that was the right track necessarily.

   16. LFLex Fridman1:11:03

       Right.

   17. PLPo-Shen Loh1:11:03

       Well, unless you're very good at-

   18. LFLex Fridman1:11:04

       There's still a branching o- of possibility. Yeah. (laughs) Uh. Uh, you're saying there's problems, like on the Math Olympiad, that requires more than one insight?

   19. PLPo-Shen Loh1:11:14

       Yes. Those are the hard ones. And- and also I can tell you how y- how you can tell.

   20. LFLex Fridman1:11:18

       Uh-huh.

   21. PLPo-Shen Loh1:11:18

       So this is how I also taught myself math when I was in college.

   22. LFLex Fridman1:11:21

       Yeah.

   23. PLPo-Shen Loh1:11:21

       So in... if you are taking a... Not taught myself, I was taking classes, of course. But I was trying to read the textbook, and I found out I was very bad at reading math textbooks. A math textbook has a long page of stuff that is all true, which after you read the page, you have no idea what you just read.

   24. LFLex Fridman1:11:36

       Yeah.

   25. PLPo-Shen Loh1:11:37

       (laughs) This is just-

   26. LFLex Fridman1:11:37

       A good summary of a math textbook.

   27. PLPo-Shen Loh1:11:39

       (laughs) Okay. Yeah, because it's- it's-

   28. LFLex Fridman1:11:41

       Yeah.

   29. PLPo-Shen Loh1:11:41

       ... it's not clear why anything was done that way.

   30. LFLex Fridman1:11:44

       Yeah.
9. 1:17:06 – 1:22:02

   Is math discovered or invented?

   1. PLPo-Shen Loh1:17:06

   2. LFLex Fridman1:17:06

      You know, there, there's this funny effect where, uh, j- just looking at different fields, where people discover parallels. They'll, they'll prove something, it'll be a totally new result, and then somebody later realizes this was already done 30 years ago in another discipline, in another way. And it's really interesting. Now, we did this offline, another illustration you showed to me. It's interesting to see the different perspectives on a problem. It, it, it kind of points like, there's just, like, very few novel ideas, that everything else, uh, that most of us are just looking at different perspective on the same idea, and it makes you wonder this, this, uh, this old silly question that I have to ask you, is, uh, do you think mathematics is, uh, discovered or invented? Do you think we're creating new idea, we're building a, a set of knowledge that's, that's distinct from reality, or are we actually like, it's... Or is math almost like a shovel where we're digging to like this core set of truths that are... oh, that, that were always there all along?

   3. PLPo-Shen Loh1:18:20

      So, I personally feel like it's discovered. Uh, but that's also because I guess the way that I like to choose what questions to work on are questions that maybe we'll get to learn something about, why is this hard? I mean, I'm, I'm often attracted to questions that look simple but are hard.

   4. LFLex Fridman1:18:36

      Mm-hmm.

   5. PLPo-Shen Loh1:18:36

      Right? And what could you possibly learn from that? Sort of like probably the attraction of Fermat's Last Theorem, uh, as you mentioned.

   6. LFLex Fridman1:18:42

      Mm-hmm.

   7. PLPo-Shen Loh1:18:42

      Simple statement. Why is it so hard? So, I'm more on the discovered side. And I also feel like if we ever ran into an intelligent other species in the universe, uh, probably if we compared notes, there might be some similarities between both of us realizing that pi is important. Because it's... Uh, you might say, "Why? Why humans? Do humans like circles more than others?"

Episode duration: 2:20:15