Our Mathematical Universe with Grant Sanderson (3Blue1Brown)

0:00Chuck, we finally got a math person as a guest on StarTalk. And one that I actually like. Sometimes it's very difficult to like math people because, you know, they know math.

0:17Coming up on StarTalk, we have an authentic math educator who's going to tell us what we're not thinking when we should be thinking anytime we think about math on StarTalk. Welcome to StarTalk, your place in the universe where science and pop culture collide. StarTalk begins right now.

0:46This is StarTalk. Neil deGrasse Tyson, you're a personal astrophysicist. Got Chuck Nice with me. Chuck, how you doing, man? Oh, man, I'm feeling great, Neil. Professional comedian and my co-host. That's correct. In that order. Sometimes. Sometimes in that order. So, Chuck, I think this, we're going to, parts of this is going to be a cosmic queries. Yes. But we have a lot to cover. So it's not going to be a full-up cosmic query. No. It's going to be, let's slip in what we can because the subject is math. What? The language of the universe? Learn math, kid. It's the language of the universe. You want to talk to the universe?

1:30You learn math. You learn math. And there's one of the funnest experts out there on this very subject. And his name is Grant Sanderson. Grant, welcome to StarTalk. It's lovely to be here. Thanks for having me. Yeah, you studied math in college. And you came out and you worked for the Khan Academy. Oh, my gosh. Oh, the Khan Academy. I know. If you've never logged into Khan Academy, you go there and you can learn everything about everything. And they care about how you learn. They don't just slap you with a textbook. And it's a completely different experience. I would call it even joyous, learning about practically every subject there is to learn.

2:16And if you fail to learn, you can look over your head while clenching your fists and going, Khan!

2:27it's a real opportunity there's a Star Trek reference there for those who are not geek enough to catch it right so Grant you have multiple math platforms like you have a podcast blog YouTube you have brilliant postings that millions of views and this is not trivial math that you're doing it's like real when I say real I mean it's the kind of math that most people would walk by and say, I don't need to know that. I don't want to know it. It's got nothing to do with me. Yet you somehow turned the tables on that. Now, before I ask you my first question, Chuck wants to ask you his first question. Yeah.

3:07And here's my first question, Grant. And it may sound a bit glib, but it's not. I'm very serious. What the hell is math? Okay. What the hell is math? Seriously. What the hell is math? Here's why I asked, just so that you'll get a bit more context. One, we started off by saying, hey, it's the language of the universe. You hear that all the time. But when I think of it, I'm like, all right, so we have these symbols, and we've attached values to these symbols, and then these values have concepts that are surrounding the values and the symbols. If I came from another galaxy, I wouldn't know your freaking math.

3:49So how is it the universal language? Well, okay, so it reminds me a little, this question came to me in another forum. I was at this wedding recently, and a person I was just meeting there who knew I did online math stuff said very sincerely, like, why is geometry math? Which is a different question from what you're asking, but it sort of gets into this, like, what do we even mean by math? And I initially answered saying it's something that we can know very precisely. So one of the oldest texts in math is Euclid's Elements. And one of the things he did there is he starts by laying out some axioms saying, assume these things are true.

4:24Assume that lines are a thing. Assume that points are a thing. Assume that if you have a point that's not on the line, it's possible to draw a line that's parallel to it. These sorts of things. And with just five of these assumptions, then rigorously proves things that have to be true beyond any doubt. So in every other field, there's a little bit of doubt. In every other field, you've got this little thing in the back of your mind that asks about experimental error, asks, maybe I didn't observe what there should be. And the thing that really separates math out from the other things is that, assuming you agree on the axioms, everything else, there is no more doubt.

4:58It's a truly rigorous proof. So we can say that aliens might use different symbols, but ultimately we're thinking math is out there to be grabbed by whoever has the capacity for logical thought. But it raises this question, what are the axioms? Where do you start? You can have all sorts of different maths. Once you're in an undergrad math major, you take all of these courses, like group theory or topology or whatnot, and they always start by saying, here's some definitions. I'm like, here's the definition of a group, and yada yada, it's a set with some added stuff. It looks like gobbledygook. And then you proceed forward and prove stuff.

5:37But I think a very reasonable question that the students could always have is like, why is this the object we're starting with? We could construct all sorts of other objects and move forward. And so there is this interplay with how it relates to the world and whether the math that you're developing is useful or not. Because I think if you take like a purely logical standpoint, you can develop a whole bunch of useless math. And in fact, you know, you could ask if some of the math being developed now, like maybe will never be useful. but history has this strange trajectory of things that seem to be useless coming back and being useful but it's a real deep question why there's some mathematicians who do math only because they're pretty sure it will never be useful well i mean there was this rather famous mathematician hardy who um this was around the time of world war ii he was writing this uh book called a mathematician's lament, or sorry, mathematician's apology.

6:31Hardy was one of the teachers of Ramanujan, if I remember. He was, yeah. He was the one who answered Ramanujan's letter and brought him in. Great mathematician. And he was played by, who's the actor in the film? Jeremy Irons, I think. Jeremy Irons. In fact, those who love our archives can dig up our Jeremy Irons interview, where we talk about that film where he played Hardy the mathematician and to get his reflections on what that whole story meant to him. So I'm sorry. I'll have to check that out. I'm just plugging our archive. One of my favorite actors combined with one of my favorite people in history.

7:10So it's just such fate for me. I can't believe I haven't listened to that. But Hardy wrote, he was sort of delighting in the fact that certain kinds of math, and he was highlighting the theory of prime numbers and relativity will never be used as instruments of war, which is this irony looking back because prime numbers have a lot to do with cryptography. Relativity has everything to do with GPS and everything with that. But the fact that he delighted in it is a little strange. It's something you would only expect from an academic maybe. But even when they're trying not to make it useful even when one of them is trying not to, it seems to kind of come back.

7:53And so to your question, Chuck, on what is math, if we just said it's the things that we can know for sure, that doesn't really feel satisfying because that somehow doesn't tie into this bizarre fact where it also describes the world. I would add that in the film Contact, not in the book, but in the film, the aliens first communicated with us using pulses that traced out prime numbers. right it's you know it's a special there are a lot of special sequences that's probably the simplest of the special sequences so that was another use of prime numbers if we don't know anything about prime numbers that's what i'm saying let's say for instance um we call them prime numbers but uh are those values universal so the i think you meet an alien species they come to earth um you say like do you know about prime numbers they don't know the language they're what do you mean by this word prime you start tapping out like two taps three taps five taps you do that the alien species has done like any thinking at all they'll be like i got you i got you i got you it's such a natural sequence it's it's like as soon as you have counting numbers and then you start to ask curious questions any intelligent entity is going to stumble on like primes being a thing just to be clear all this assumed that they count in base 10.

9:09No, it doesn't. It doesn't at all. Yes, it does. No, no, no, not. If you just tap twice. Well, just the number you were assigned to. That's how you represent it. Got it. Yeah, so how you write down the number. Yeah, what do you call it? Write down the number. You got it. Okay. So we would write one, three. Maybe they have a different base 10. But if you just do 13 taps, and then you do 17 taps, right? And you keep doing all the prime, like any, you know, any AI, any alien, all of them are going to be in agreement that like, this one's important. They're tapping out the glypsorpion sequence. Wow.

9:38Exactly, exactly. Chuck, know your prime number is just for this, just such an emergency. Just in case I run into an alien. Right. So I hate to ask you this question only because I don't like the question, but why not ask you? Is, I'll tell you why I don't like it in a minute, but is math invented or discovered? Well, it's okay. And forgive me for asking that question. No, no, it's a super common question. Let's do an example rather than like philosophizing. Okay. So one of the first substantive theorems that students learn in school is the Pythagorean theorem. So at some point, whether or not we've forgotten since then, many of us will have come across that if you have a right triangle and you label the two shorter sides, A and B, and then the longer side, C, no matter what right triangle you have, A squared plus B squared equals C squared.

10:29It's a cool fact. It's not obvious that it should be true. It's useful. It lets you make sorts of measurements. And it's the first one you meet that it doesn't just like fall out of definitions. It's interesting and it's all that. Now, if you ask, like, is that fact invented or discovered? Initially, it feels like, well, it's discovered for sure. You go out in the world, you see right triangles, you kind of, you can measure them. It's waiting there for you to see it. It's waiting for you to see it. But then you get a little further in math and then you start to ask questions like, what is space?

11:00What is, you know, three-dimensional space, two-dimensional space and things like this. and then one of the ways that mathematically you're defining two-dimensional space is rather than saying oh it's this pre-existing thing and we can put a coordinate system on it a lot of math will start from the coordinates themselves and saying it's all pairs of real numbers so rather than saying like oh there's this plane out in existence and then you put some like axes and you say this is your x coordinate this is your y coordinate to get to a point on space you say rather than all the tricky philosophical questions about what is a plane and whatnot we just say a plane is all pairs of real numbers.

11:33That's what it is. And then you can do math forward. You can make proofs and whatnot. But then when you say, what's the distance between two points, and you have to assign what they call a metric to the space, a way of telling distances, you sort of make the Pythagorean theorem true by writ. You define a metric that makes it true. And there's choices for other metrics that you could have put on it. And then it starts to become kind of sticky. Like, wait a minute, was this discovered anymore? Or if this is how we're doing all of our math where we say you know a plane is just pairs of numbers and that's mathematically what it is then then you don't have that same like oh it was waiting there to be discovered wait so grant how different is what you described from knowing that there's a variety of space where pythagorean theorem applies as we learned it but if you curve the space then triangles are different right the a triangle the angles sum to more than 180 degrees where a traditional triangle especially a right triangle it all comes to exactly 180 so this would be a different metric for the space yeah i mean so the idea that you can put different metrics on is sort of the superpower of being the mathematician right you're not constrained so and And non-Euclidean geometry was discovered before it was, well, I say before it was used in physics, but if you're doing things on a sphere, that's fine.

12:58But before all of the relativity stuff that really made it come to mind. Of course. Right, right. But this back and forth where you can have something that at one moment feels discovered and like this just kind of awe-inspiring truth, and at another moment feels like it was just baked into the definitions. This is a very long-winded way of answering the simple question that you asked, is math invented or discovered, for how I like to think about it. It is both. But more specifically, I think you go out and you discover stuff based on your existing, either your world or your existing math. You use those discoveries to inform the definitions of the next step.

13:31And then you make more discoveries in that next step. And then you use those discoveries to inform the definitions further on. So basically what you're saying is math is discovered the way Columbus discovered America. That's basically. What if we define it as the West Indies? We define it this way.

14:06I'm Jasmine Wilson, and I support StarTalk on Patreon. This is StarTalk with Neil deGrasse Tyson.

14:20Grant, you have a perspective on education that I highly respect. Even if I don't agree with every bit of it, I deeply respect it. And you part ways with many educators in that you don't necessarily prioritize. I don't want to put words in your mouth, so correct me if I misrepresent this. You don't put as a priority, you should learn math because it's relevant to your life. That's a typical way a math teacher would approach it. Worried that if they teach math and it's not relevant to people's life, they will just ignore it, get bored, go to sleep. So I have two questions for you. One, we probably agree that math more than any other subject is what people say is their hardest subject.

15:11So how do you get around that, if not by making it relevant to people's lives? So first of all, I think anytime there's a discussion of education, we should tear out the idea that explanation is different from education. And the idea of making a YouTube video or making an explainer is a very different task from being an educator in a classroom, eliciting something from a student. And that you're playing different games then, which is relevant to the answers here I think because if if relevance is on the table it's within you know a one hour from being able to explain why this piece of math like actually touches someone's life amazing run with that that's the most inspiring thing if someone can see how it's useful for them I have all sorts of friends who hated math but I lived in the Bay Area for a while lots of programmers there and they came to love math through programming because they were into computer science they saw that hey to do this graphics project I guess I really need to understand matrices.

16:07You also studied computer science as well. I did. In college. And so, yeah, so a lot of my like social circles are in those spheres. And, and so that's a story I've seen many times where here's this, this bit where usefulness is what, what made someone finally like it. That's great. But I think very often utility is just not on the table for a given piece of math all that soon. You, you, you want to teach some high schooler about the quadratic formula and they say when am i ever going to use the honest answer for that student is never times out of 10 never never like you might not just never never ever okay well i made a mistake once i was uh giving a talk to pixar and i use this as an example like the quadratic formula you'll never use it so we have to like motivate in these other ways and then afterwards all these graphics programmers come up they're like we use it every day they're like it was used approximately a trillion times to produce the movie Coco.

16:59And I'm like, okay, okay, fine. Like, yes, it can. Yeah. Yeah. To be careful what you say to Pixar people, there's some, there's some mathematically and scientifically literate folks working for Pixar. Yes. Um, but yeah, so given for most people, never, they're never going to use it. Then you say like, why learn it? And you could say it's simply not worth learning, right? You know, spend your time learning other things, but there's, there's something about the mental exercise of rigorous thinking that is so pure in a mathematical setting that it really flexes some muscles that are going to be useful elsewhere.

17:32Mathematician, go on. Okay. Yeah, no, it's self-serving, but like, I think the better analogy, if you want to tell a student, why should they learn this? It's like if you go to a football player and they're in the gym and they're doing some squats and you say, you're never going to be on the field with like, you know, 400 pounds on your shoulders that you're like lifting in this particular way. Like, no, no, no, it's building muscles. And so I think a lot of the math that we do in school, it's the same motivation. You're saying, no, no, no, you're not going to use like this particular thing in the same way that out on the field, you're not going to be doing like a literal squat, but the muscles built doing it like really do make you more powerful later.

18:09That also math can, and I think often should be taught as an art simply because it's available. And like, if you can get someone bought into the beauty of it, that is as inspiring, if not more so than the utility and often done right it's more on the table than the utility is given the constraint of say like a 20 minute online explainer or just to be clear i've seen some ugly math in my day put it out there i've seen some ugly paintings too right but that that doesn't mean we can't so so tell me is would you count information theory as mathematics 100 % yeah and what makes it so well part of it's what I'm saying it's something we can describe very exactly oh that's as we started this conversation the precision calculation is what you value here so you use information theory I've been told to help people in verbal yeah how's that even like how does that work well first of all for those of you who who don't know not not saying that it's me but what is information theory for everyone else yeah i mean you know for all the people out there that are listening right now like what do you mean information theory um so i will answer that question chuck by by way of talking about this project neil referenced where back in i think it was 2022 whenever wordle became incredibly viral on like the english-speaking internet this is one time when i got sucked in just as a programming project to say like how would you write like a bot that plays Wordle?

19:44Because it's so clearly something that a bot could play, but it's just a fun programming exercise to do it. And as I was doing it, I thought it was actually a really nice motivating example to explain information theory to the general public. Because here you've got this game that a lot of people like. The first thought that I had for how to go about it involved these ideas from information theory. I said, oh, this can help motivate. So to answer your question, Chuck, um often if you have a situation where like you're playing wordle and let's say you want to know is that is it better to play as a first word um so the nature of this game you guess some letters if those letters are part of the true word that you're like the secret word you're trying to get they'll show up yellow if they're there but in the wrong spot they show up green if they're there but in the right spot so it's this game of like using your previous guesses to get more information about what the true word might be.

20:35And there's that word information. And so the worst starting example that you could use is the word kayak. K-A-Y-A-K. In some very exact sense, this is the worst starting guess. Yeah, I'm already seeing it right now. That's a suck-ass word for what you just described. Those rules you just described, that is like the worst word. Because it's like, what are the odds there's a Y in the word? It's like not very likely. What are the odds that a K is in the word? Not very likely. You're not hitting that many vowels. So you have this sense that you're not learning that much about the word. By the way, the valuation of Scrabble letters tracks this very much.

21:15Yeah. Right. Because the rare letters have higher value. Makes sense. Okay. Okay, go ahead. I think we use this word information in a very heuristic sense, where it's the idea like, I learn a lot from doing something versus I don't learn a lot. But who made it actually rigorous was a very clever guy named Claude Shannon, who was working at Bell Labs. and this is around 1930s, 1940s, you know, you've got all of these games to try to send information across a telephone line, like if you're talking to someone, but sometimes it's noisy. And so you might ask, oh, should we like boost the signal? Should we add more lines?

21:45Like, how can you think more systematically about how well you can communicate? And so he wrote this paper that's like the seminal paper for what's now known as information theory. He was the first one to use the word bit in this paper. So this was the first time that bits as a concept ever came and the basic idea there was like if you let's say there's something where there's two ways that it might go you flip a coin it might be heads or it might be tails as soon as you see the results you've learned something about like which path the universe took and if there was only two possibilities you know that he said you've learned the the unit of information here is one bit but let's say there was like a hundred possibilities and you learn a specific one of those somehow it feels like you've gained more information and he came up with a formula for quantifying this And then he used that formula and like relatives of it to try to describe how much signal you can send across something that has a certain amount of noise.

22:39And it made it such that there was a theoretical limit where you could say this is how much information we could try to get across. And then clever and clever coding schemes could try to reach that limit. Now, in the context of Wordle, this means that you can put a number to basically how good a starting guess is based on the Shannon entropy is a fancy word for it. But basically, on average, how much information do you expect to get from it? Kayak has a terrible Shannon entropy in this context. A word like slate is actually very good. It's got that E, it's got that A, L and T are very common letters.

23:14S, even if it's not going to show up at the end, it's still a very common letter. And so I basically made this video that was giving a walkthrough of using information theory type concepts and trying to motivate what those concepts were as a way of pursuing this little hobbyist game of writing a plot to play Wordle. Here's one that you don't really hear much, but it's so important. And that would be the central limit theorem. So the central limb theorem is actually as important for why it's like overused and like perceived to be more useful than it is, than it is for its utility. But there's a lot of things in the world that seem to be distributed according to a certain normal curve.

23:55And here's what I mean by that. They don't seem to be, they are, right? What do you mean seem to be? Well, actually, no, this is a good fodder for discussion. Let's take heights, human heights, because I think this is a little bit in dispute. So if you go and measure a whole bunch of people's heights, you measure my height, your height, a thousand other people, a million other people, and you create a plot where you bucket it maybe by the inch or something. So everyone who's between five foot nine and five foot 10 falls in one bucket. Everyone between five foot 10, five 11, another bucket. You do this for people in the same demographic, let's say.

24:26So we'll separate men and women, well, same country, you notice that there's this shape that kind of bulges in the middle and it tapers out on either side. And you do this for all sorts of other statistics, and you see the same shape that seems to bulge in the middle and taper on either side. And it's not just that, you know, there's lots of shapes that could kind of bulge in the middle and taper on either sides, but there's a very precise function that you can use that describes this really quite well, even in all these different situations. Got this funny formula, It looks like e to the negative x squared with some constants thrown in.

24:56But the idea is that by having a formula to it, then you can make exact predictions like saying, hey, how many people, what percentage of people do you think would be taller than 6 '4"? Or what percentage of males would be shorter than 5 '2"? And all of these sorts of questions. So the central limit theorem explains why this curve and not any other curve, like why this curve comes up. And just as importantly, it explains when it's actually going to come up as opposed to maybe false cousins of it that seem to kind of look like it. Because the way that it'll happen is if you have a lot of different events that you're kind of adding together in some way.

25:36So the pristine platonic example for a normal distribution would be something like this. Let's say you and I stand outside and we're going to play this game where you flip a coin and when it's heads, you take a step forward. When it's tails, you take a step back. So you flip a coin, maybe take a step forward, flip a coin, take another step. And you play this game, not just with yourself, but with 10 ,000 of your friends. So you're all flipping your own coins. Each time, half of you take a step forward, half of you take a step back, roughly. And you flip, I don't know, 15 times. So you're all stepping.

26:06Some of you got very, very lucky and took 15 steps forward. A more typical person, you know, maybe they took like seven steps forward. and eight steps back. But then if you look at where everyone is, that will follow a normal distribution very, very perfectly or really, really closely. And it's because you have this idea that you took a certain kind of random event, flipping the coin, and you were adding the results many different times. And critically, all of those events were independent. Each coin flip didn't influence the next one. Whenever you have this, a bunch of independent things that you're adding together, it's not obvious at all that that will always tend towards the same universal shape.

26:43And it didn't have to be a coin flip. It could have been any other random process with its own little internal distribution. And you always get this idea that adding up a bunch of random things, which are all independent from each other, lands you on this universal shape. Now, the reason I was saying seems to and waffling a little bit there is I do think that there's a lot of times in statistics when people will assume something's normal when it's actually not or normally distributed yeah yeah like because i assume chuck is normal but i i question you're the only one that's why what do you say assume something that it would follow a normal distribution yeah especially when it comes to long tail events yeah i gotta ask you real quick before you go forward because when i'm thinking of this and picturing it it almost kind of makes sense the the distribution because you're only talking about a coin flip a coin flip but you said any random so let's take 10 white marbles and put them in a bag and one black marble and put them in a bag and these people play the exact same game with the same number of picks you're telling me that that same distribution is going to be represented the same way i mean if it's not a 50 50 right so um yeah this is good to be more precise so let's say you you pick out very often you're picking the white marble, 1 in 11 times you're picking that black marble.