Our Mathematical Universe with Grant Sanderson (3Blue1Brown)

Is math discovered or invented? Neil deGrasse Tyson & Chuck Nice explore information theory, talking to aliens with prime numbers, Mandelbrot sets, and why math is often called the "language of the universe" with Grant Sanderson, the math educator behind YouTube channel 3Blue1Brown.

Speaker names marked ? are inferred from speaking time and the show’s known hosts — they are not voice-verified.

• 0:00Neil deGrasse Tyson?

  Chuck, we finally got a math person as a guest on StarTalk.

• 0:03Chuck Nice?

  And, and one that I actually like.

• 0:08Chuck Nice?

  Sometimes it's very difficult to like math people because, you know, they know math.

• 0:17Neil deGrasse Tyson?

  Coming up on StarTalk, we have an authentic math educator who's gonna tell us what we're not thinking when we should be thinking anytime we think about math on StarTalk.

• 0:32Neil deGrasse Tyson?

  Welcome to StarTalk, your place in the universe where science and pop culture collide.

• 0:41Neil deGrasse Tyson?

  StarTalk begins right now.

• 0:46Neil deGrasse Tyson?

  This is StarTalk. Neil deGrasse Tyson, you're a personal astrophysicist. Chuck Nice with me. Chuck, how you doing, man? Aw, man, I'm feeling great, Neil. Professional comedian and my co-host. That's correct. In that order.

• 1:02Neil deGrasse Tyson?

  Sometimes, sometimes in that order. So Chuck, I think this, we're gonna, parts of this is gonna be a cosmic queries. Yes. But we have a lot to cover, so it's not gonna be a full-up cosmic query. It's gonna 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 wanna talk to the universe, you learn math. Learn math. And there's one of the funnest experts out there. Okay.

• 1:35Neil deGrasse Tyson?

  On this very subject, and his name is Grant Sanderson. Grant, welcome to StarTalk. Lovely to be here. Thanks for having me. Yeah, you, you, you studied math in college and you came out and you worked for, for the Khan Academy. Oh, my gosh, if you've never seen.

• 1:54Neil deGrasse Tyson?

  Khan Academy. I know. If you've never like logged into Khan Academy, you go there and you can learn everything about everything. Yep. And they care about how you learn. They don't slap you with a textbook, and it's a completely different experience. It's, I would call it even joyous, learning about practically every subject there is to learn.

• 2:16Chuck Nice?

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

• 2:27Speaker 02

  It's a real opportunity.

• 2:28Neil deGrasse Tyson?

  Star Trek reference there for those who are not geek enough to catch it.

• 2:32Chuck Nice?

  Right.

• 2:32Neil deGrasse Tyson?

  So, so Grant, you have multiple math platforms. 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, ask you his first question.

• 3:07Chuck Nice?

  Yeah. And here's my first question, Grant. And it, it may sound a bit glib, but it's not. I'm very serious. What the hell is math? Okay.

• 3:15Speaker 02

  Okay.

• 3:16Chuck Nice?

  What the hell is math?

• 3:17Speaker 02

  What is math?

• 3:18Chuck Nice?

  Because.

• 3:19Neil deGrasse Tyson?

  No, what the hell is math?

• 3:20Chuck Nice?

  What the hell?

• 3:21Speaker 02

  What the hell is math? Okay.

• 3:23Chuck Nice?

  Here's why I ask, just so that you'll get a bit more context. Okay. 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 fricking math. So how is it the universe, the universal language?

• 3:52Speaker 02

  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's sort of kids in this, like, what do we even mean by math? And I, there was, 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. Assume 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 with just five of these assumptions then proves rigorously

• 4:37Speaker 02

  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. It's a truly rigorous proof.

• 5:00Neil deGrasse Tyson?

  So we can say that they might use 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.

• 5:13Speaker 02

  But it raises this question, like, what are the axioms? Like, where do you start? I mean, you can have all sorts of different maths. Like, 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 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 a wadiguk. And then you proceed forward and prove stuff. But 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 if you take like a purely logical standpoint, you can

• 5:55Speaker 02

  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 seeming seem to be useless coming back and being useful. But it's a real deep question, why?

• 6:13Neil deGrasse Tyson?

  There's some mathematicians who do math only because they're pretty sure it will never be useful.

• 6:19Speaker 02

  Well, I mean, there was this rather famous mathematician Hardy, who this was around the time of World War Two. He was writing this book called A Mathematician's Lament, or just sorry, Mathematician's Apology.

• 6:31Neil deGrasse Tyson?

  Hardy was one of the teachers of Ramunuson, if I remember.

• 6:35Speaker 02

  He was. Yeah, he was the one who who answered Ramanujan's letter and like brought him in. Great mathematician.

• 6:39Neil deGrasse Tyson?

  And he was played by, who's the actor in the film?

• 6:43Speaker 02

  Jeremy Irons, I think.

• 6:44Neil deGrasse Tyson?

  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, on, and what that whole story meant to him. So I'm sorry.

• 7:04Speaker 02

  I'll have to check that out.

• 7:05Neil deGrasse Tyson?

  I'm just plugging our archives.

• 7:06Speaker 02

  One of my favorite actors combined with one of my favorite people in history. So it's like, oh, it's just such bait for me. I can't believe I haven't listened to that. But I mean, Hardy wrote some, he was sort of delighting in the fact that math will never be, certain kinds of math. And he was highlighting, you know, the theory of prime numbers and relativity will never be used as instruments of war, which is, is this irony looking back because, you know, prime numbers have a lot to do with cryptography. Relativity has everything to do with, you know, GPS and, and everything with that.

• 7:35Chuck Nice?

  You idiot.

• 7:38Speaker 02

  But the fact that he delighted in it is, is a little strange. Right. And it's, it's something you would only expect from an academic, maybe. But it's like, even when, even when they're trying not to make it useful. Right. Even one of them is trying not to, it, it seems to kind of come back. And so to your, to your question, Chuck, on, you know, what, what is math? If, if, 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, this bizarre fact where it also describes the world.

• 8:05Neil deGrasse Tyson?

  I would add that in the film Contact, not in the book, but in the film, the aliens first communicated with us using pulses.

• 8:16Speaker 02

  Yeah.

• 8:16Neil deGrasse Tyson?

  That traced out prime numbers.

• 8:19Speaker 02

  Right.

• 8:20Neil deGrasse Tyson?

  It's, you know, suspicious, 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 didn't know anything about prime numbers.

• 8:29Chuck Nice?

  That's what I'm saying. Let, let's say, for instance, we call them prime numbers, but are those values universal?

• 8:37Speaker 02

  So the, I think you meet an alien species. They come to earth. You say like, do you know about prime numbers? They don't know the language. They're thinking about, what, 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.

• 9:05Neil deGrasse Tyson?

  Just to be clear, all of this assumed that they count in base 10.

• 9:09Speaker 02

  No, it doesn't. It doesn't at all.

• 9:11Neil deGrasse Tyson?

  Yes, it does.

• 9:12Speaker 02

  No, no, no, not if you just tap twice. Yeah, so how you write down the number? Yeah.

• 9:19Neil deGrasse Tyson?

  It's what you call it. Write down the number you got.

• 9:21Speaker 02

  Okay, so we would write one, three, maybe they have a different base ten. But if you just do thirteen taps and then you do seventeen 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.

• 9:35Neil deGrasse Tyson?

  They're tapping out the glypsorpien sequence.

• 9:38Speaker 02

  Exactly, exactly, yeah.

• 9:41Neil deGrasse Tyson?

  Chuck, know your prime number is just for this, just in case an emergency. I run into an alien, right?

• 9:47Neil deGrasse Tyson?

  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?

• 10:01Speaker 02

  Well, it's okay.

• 10:02Neil deGrasse Tyson?

  And forgive me for asking that question.

• 10:04Speaker 02

  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. It'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?

• 10:44Speaker 02

  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.

• 10:52Neil deGrasse Tyson?

  It's waiting there for you to see it and notice.

• 10:54Speaker 02

  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? What 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 preexisting 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 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. That's what it is. And then you can do math forward. You can like make proofs and whatnot.

• 11:37Speaker 02

  But then, when you say, what's the distance between two points? And you have to assign them what they call a metric to the space, a way of telling distances. You sort of make the Pythagorean theorem true by writ. You just, 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. Is this, was this discovered anymore? Or if this is how we're doing all of our math, where we say, you know, a plane is 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.

• 12:10Neil deGrasse Tyson?

  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.

• 12:42Speaker 02

  Yeah. I mean, so the idea that you can put different metrics on is sort of the, 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. But before all of the relativity stuff that really made it come to.

• 13:01Neil deGrasse Tyson?

  Of course. Right.

• 13:02Speaker 02

  But, 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, as math invented or discovered? For like, how I like to think about it. It is both, but more specifically, I think you, 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. And then you make more discoveries in that next step. And then you use those discoveries to inform the definitions further on.

• 13:36Chuck Nice?

  So basically what you're saying is math is discovered the way Columbus discovered America. That's basically.

• 13:47Speaker 02

  What if we define it as the West Indies? We define it this way.

• 14:06Speaker 03

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

• 14:21Neil deGrasse Tyson?

  Grant, 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. Right? 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 lives, they will just ignore it, get bored, go to sleep. So I have two questions for you.

• 15:02Neil deGrasse Tyson?

  One, we probably agree that math, more than any other subject, is what people say is their hardest subject. So how do you get around that if not by making it relevant to people's lives?

• 15:17Speaker 02

  So first of all, I think anytime there's a discussion of education, we should pair out the idea that explanation is different from education. And the idea of like 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 relevance is on the table, it's within, you know, one hour from being able to explain why this piece of math like actually touches someone else'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

• 16:00Speaker 02

  were into computer science. They saw that, hey, to do this graphics project, I guess I really need to understand matrices.

• 16:06Neil deGrasse Tyson?

  You also studied computer science as well.

• 16:08Speaker 02

  I did.

• 16:09Neil deGrasse Tyson?

  In college.

• 16:10Speaker 02

  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?

• 16:36Neil deGrasse Tyson?

  Is never.

• 16:37Speaker 02

  Nine times out of 10, never, never. Like you might.

• 16:39Neil deGrasse Tyson?

  Not just never, never, ever.

• 16:41Speaker 02

  Okay, well, I made a mistake once. I was 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. And I'm like, okay, okay, fine. Like, yes, it can.

• 17:02Neil deGrasse Tyson?

  Yeah, yeah. Be careful what you say to Pixar people. There's some, there's some mathematically and scientifically literate folks.

• 17:07Speaker 02

  Seriously.

• 17:08Neil deGrasse Tyson?

  For Pixar, yes.

• 17:09Speaker 02

  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.

• 17:31Neil deGrasse Tyson?

  Mathematician, go on.

• 17:34Speaker 02

  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, it's building muscles. And so I think a lot of the math that we do in school, it's, 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. That also.

• 18:11Speaker 02

  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.

• 18:29Neil deGrasse Tyson?

  Just to be clear, I've seen some ugly math in my day. Put it out there.

• 18:34Speaker 02

  I've seen some ugly paintings, too, right? But that doesn't mean we can't.

• 18:41Neil deGrasse Tyson?

  So, so tell me, is, would you count information theory as mathematics?

• 18:46Speaker 02

  Hundred percent, yeah.

• 18:47Neil deGrasse Tyson?

  And what makes it so?

• 18:50Speaker 02

  Well, part of it's what I'm saying, it's something we can describe very exactly.

• 18:53Neil deGrasse Tyson?

  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 Wordle.

• 19:07Speaker 02

  Yeah.

• 19:07Neil deGrasse Tyson?

  How's that even, like, how does that work?

• 19:10Chuck Nice?

  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?

• 19:19Neil deGrasse Tyson?

  For everyone else, Chuck.

• 19:20Chuck Nice?

  Yeah, I mean, you know, for all the people out there that are listening right now, like, what do you mean, information theory?

• 19:26Speaker 02

  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, hmm, how would you write like a bot that plays Wordle? Because it's, 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, 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 thought, oh, this, this can help motivate.

• 20:08Speaker 02

  To answer your question, Chuck, often if you have a situation where like you're playing Wordle and let's say you want to know, is it better to play as a first word? 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. And there's that word information. And so the worst, the worst starting example that you could use is the word kayak, K A Y A K. In some very like exact sense, this is the worst starting guess.

• 20:50Chuck Nice?

  Yeah, I'm already seeing it right now. That's a suck ass word for what you're just, what you're just described, that those rules you just described, that is like the worst word.

• 20:59Speaker 02

  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.

• 21:10Neil deGrasse Tyson?

  By the way, the valuation of Scrabble letters tracks this very much.

• 21:14Speaker 02

  Yeah.

• 21:15Neil deGrasse Tyson?

  Right? Because the rarer letters have higher value. Right?

• 21:18Chuck Nice?

  Okay.

• 21:19Neil deGrasse Tyson?

  Okay, go ahead.

• 21:20Speaker 02

  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 line? Like, 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 up. And the basic idea there was like, if you, let's say there's something where there's two ways

• 22:05Speaker 02

  that it might go. You flip a coin, it might be heads or it might be tails. As soon as you see the result, 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 unit of information here is one bit. But let's say there was like 100 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. And 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.

• 22:50Speaker 02

  Now, in the context of Wordle, this means that you can put a number to basically how good a starting guess is based on, like the Shannon entropy is like 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. S, even if it's not going to show up at the end, is 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 but to play Wordle.

• 23:32Neil deGrasse Tyson?

  Here's one that you don't really hear much, but it's so important. And that would be the central limit theorem.

• 23:40Speaker 02

  So the central limit 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. And here's what I mean by that.

• 23:56Neil deGrasse Tyson?

  They don't seem to be, they are, right? What do you mean seem to be?

• 23:59Speaker 02

  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, thousand other people, million other people, and you create a plot where, you know, 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, you know, five foot 10, five 11, another bucket. You do this for people in the same demographic, let's say. So we'll separate men and women, we'll, 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

• 24:43Speaker 02

  the middle and taper on either sides, but there's a very precise function that you can use that seems, 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, but the idea is that's 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, you know, 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, when it's actually going to come up as opposed

• 25:23Speaker 02

  to maybe the false cousins of it that seem to kind of look like it. Because the, 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. So the, the, the pristine platonic example for a, 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 play, 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.

• 26:05Speaker 02

  So you're all stepping, some 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 were, 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, it's not obvious at all that that will always tend towards the same universal shape. And it didn't have to be a coin flip. It could have been any other random process with its own little internal distribution.

• 26:48Speaker 02

  And you, you always get this idea that adding up a bunch of random things, which are all independent from each other, lands you on this, this universal shape. Now, the reason I was saying like 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.

• 27:07Neil deGrasse Tyson?

  Normally distributed.

• 27:09Speaker 02

  Yeah, yeah, like.

• 27:10Neil deGrasse Tyson?

  Because I assume Chuck is normal, but I question.

• 27:13Chuck Nice?

  You're the only one, that's why.

• 27:14Neil deGrasse Tyson?

  What do you say, assume that it would follow a normal distribution?

• 27:20Speaker 02

  Yeah, especially when it comes to long tail events. Yeah.

• 27:23Chuck Nice?

  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 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?

• 27:51Neil deGrasse Tyson?

  If it's not a 50/50.

• 27:53Speaker 02

  Right. So, yeah, this is good to be more precise. So, let's say you pick out, very often you're picking the white marble, one in 11 times you're picking that black marble. You play this game. They make 20 picks, they're all independent from each other. You will still see a normal distribution. What will be different? The mean will be different. So, everyone will have taken many more steps forward than back. So, they're definitely going to be shifted up the road.

• 28:14Neil deGrasse Tyson?

  To be clear, everyone, he's a mathematician. He used the word mean, he means average.

• 28:20Speaker 02

  Okay. The average, yeah, their average position will be different.

• 28:23Neil deGrasse Tyson?

  We have a mathematician in the house.

• 28:25Speaker 02

  Yeah, the standard deviation.

• 28:27Chuck Nice?

  It's not the Stern distribution, just like, get up, you take a step, I'm the mean distribution. Take a step, damn it, you loser.

• 28:38Speaker 02

  No, it's actually, it's a very insightful question. It's a very insightful question, because it gets at a thing that I didn't mention, which is how like, okay, it's not literally the same distribution where they're in the same point in space, but they'll be shifted up the road. They might be like spread out a little bit more. However, this curvy shape that you draw on top of it, that has a very specific function, it's e to the negative x squared, that's not an arbitrary function. That with a couple of parameters you tweak that just tells you like, where is it center? How much is it spread out? Just two different parameters. It'll, it'll still, it'll still match that. That's the real surprise of it. It's like, of course, some things are going to change, like where's the center? Of course, the spread is going to change, but the surprise is like, that's the only thing. Somehow every, everything else that's interesting about the specific, you know,

• 29:22Speaker 02

  ball and bag example you did, that gets washed away.

• 29:25Neil deGrasse Tyson?

  And by the way, the fluctuations in the cosmic microwave background in the early universe, between hot and cold around the average is exactly this distribution, which gives us information of what happened way back earlier at the big bang, the quantum fluctuations that would have led to it. Very powerful means of an L.

• 29:47Speaker 02

  Average number of prime factors? You take some giant number and you ask like, how many prime factors does it have? And you average that for a bunch of giant numbers normally distributed. It's just, it shows up in all these different.

• 29:57Neil deGrasse Tyson?

  So spooky, spooky.

• 29:58Speaker 02

  Yeah, yeah, it's spooky.

• 30:00Neil deGrasse Tyson?

  Yeah, but also I think there's the regression to the mean for things where you have two smart people and they're smart because of some combination of whatever happened in their life. And then they have babies. It's not likely that the smart, that their baby will be as smart as they are because there'll be regression back to the mean of everybody else. Where, so it works this way in biology is what I'm saying.

• 30:25Chuck Nice?

  Are you trying to, are you trying to create a depiction where I'm the baby here? What's happening?

• 30:34Neil deGrasse Tyson?

  One more thing before we go to Q and A. Let's, let's, why don't we give a shout out to fractals here? Tell me about a Mandelbrot set.

• 30:42Speaker 02

  Love me a good Mandelbrot set. So this is one of those images that really captures the.

• 30:48Neil deGrasse Tyson?

  Don't leave home without one.

• 30:50Chuck Nice?

  Let me just say, in all my years of StarTalk, that was the geekiest thing ever said.

• 30:59Chuck Nice?

  Love me a good Mandelbrot.

• 31:02Speaker 02

  Am I wrong? No. So go ahead. Oh, man, it depends on how much detail you want here. The one version of this is to simply tell people, go to YouTube and search like Mandelbrot set deep zoom and then simply look and gawk at the shape. And I think this is the way a lot of people engage with it. It's a really, really intricate shape that no matter how far you zoom in, there's always more detail to be found. And, and that's, that's cool in its own right. What's cool from the mathematician's perspective is that it has a really, really simple rule that was used to describe it, even though it's producing this mind-bogglingly complicated shape. And that rule is basic. Stop me again if this is too boring or like too much detail.

• 31:43Speaker 02

  So the, the heavy lift here is that you have to start by talking about complex numbers, which is one of those terms that like sounds more complicated than it is because they literally use the word complex. But in the same way that numbers we usually think of as being on a number line, there's a really natural way of having numbers that live in two dimensions. So you choose a point on two dimensional space and there's a number associated with it. And you can do things like multiply it by itself and, and add numbers to it. To normal, normal math operations. Normal math, normal math. And so you start with some point and you basically say, all right, I'm a, I'm going to call that point C as my like constant C. And then I'm going to play this game where I start at zero and I'm always going to take the square of what I have and add C.

• 32:27Speaker 02

  So for example, you start zero squared is zero and then you add C and you get to C. And you're like, great, I'm going to do that again. I'm going to square the number, which is C squared. So if C was one, you would have one squared and then you add C to it. So you have like two. And you play this simple game where it's almost like you're, you're at a calculator and you're just pressing the button, enter over and over and over, having done something like, what happens if I repeat this operation? Sometimes the process blows up to infinity. Sometimes it seems to stay bounded and bounce around. If you color each point on the plane based on which of those two things happens, you color it black. If it doesn't ever run to infinity, it's just kind of bounces around a whole bunch. And you color it some other color if it goes off to infinity. You maybe even give it a hue based on how quickly it runs to infinity.

• 33:10Speaker 02

  That simple set of rules, which you can write with like three lines in any programming language, that very simple set of rules produces this mind-bogglingly intricate shape, and it shows how complexity doesn't have to emerge from complex rules. It's possible for complex phenomena to emerge from very, very simple rules. And I think that's a powerful idea.

• 33:30Neil deGrasse Tyson?

  And by the way, Mandelbrot is a modern person, right?

• 33:32Speaker 02

  Yeah.

• 33:33Neil deGrasse Tyson?

  When was he born and died?

• 33:35Speaker 02

  When was he born?

• 33:36Neil deGrasse Tyson?

  I mean, 20th century.

• 33:36Speaker 02

  In my mind, yeah, 20th century. I think of Mandelbrot as like 70s is when I think of him.

• 33:41Neil deGrasse Tyson?

  Yeah, yeah.

• 33:42Speaker 02

  He worked at IBM, I think. I mean,

• 33:44Neil deGrasse Tyson?

  His first name was Ben.

• 33:46Speaker 02

  Benoit. Benoit.

• 33:47Neil deGrasse Tyson?

  Benoit Mandelbrot. So I just found out when Mandelbrot lived, born 1924, died 2010.

• 33:55Speaker 02

  Wow.

• 33:56Neil deGrasse Tyson?

  Yes, a man of our times there. Yes, indeed.

• 34:00Speaker 02

  Very cool.

• 34:00Neil deGrasse Tyson?

  And so, as you zoom in, you see new detail that is informed by the previous detail you just saw, but on a smaller and smaller, and it's that all the way down.

• 34:11Chuck Nice?

  Well, that's the very classic, famous depiction of a fractal.

• 34:16Neil deGrasse Tyson?

  Yes.

• 34:17Speaker 02

  It's the celebrity fractal. You know, this is just the, you know, the A-list actor fractal that everyone wants to call in if they're thinking of a fractal.

• 34:25Neil deGrasse Tyson?

  But since it can be defined so simply, the complexity is only what it looks like. But in terms of information theory, there's very little information associated with it. Because I remember, because I was active in this space briefly, and people wondered, can we use fractals to represent trees in an image from space, to model what we, forest versus land, you know, non-arable land? And then, some of that was abandoned, if not all of it, because nature actually has more detail than the fractal. So the fractal is

• 35:07Neil deGrasse Tyson?

  like the lazy way to look like, make it look like it has detail.

• 35:11Speaker 02

  Which, people do this in computer graphics, too. They'll use fractals and fractal ideas to give something, make it look like nature without having to use a lot of computer.

• 35:19Neil deGrasse Tyson?

  To actually to do it. Okay.

• 35:21Speaker 02

  Yeah.

• 35:22Neil deGrasse Tyson?

  Now, you have to hear my Mandelbrot joke.

• 35:25Speaker 02

  Okay, tell me your joke.

• 35:26Neil deGrasse Tyson?

  Okay, you probably know it, so just, okay. What is Benoit Mandelbrot's middle name?

• 35:34Speaker 02

  I have heard this one, so.

• 35:36Chuck Nice?

  All right, go ahead.

• 35:38Neil deGrasse Tyson?

  Benoit Mandelbrot.

• 35:40Chuck Nice?

  Of course. Okay, I have not heard that. Of course, I have not heard.

• 35:44Neil deGrasse Tyson?

  It is good. That's good.

• 35:46Speaker 02

  Can I correct? I'm gonna be that guy.

• 35:48Neil deGrasse Tyson?

  Oh, so be correct. Correct me. I misremember it. Go on, go on.

• 35:52Speaker 02

  Well, that's Avery. The version that I know that I think holds a little better, his middle initial is B, so everyone refers to him as Benoit B. Mandelbrot. And then you're answering this like, hey, what does that B stand for? Well, it stands for Benoit B. Mandelbrot.

• 36:04Neil deGrasse Tyson?

  Okay.

• 36:05Speaker 02

  So then you're constantly filling in the B.

• 36:07Chuck Nice?

  Of course.

• 36:08Neil deGrasse Tyson?

  Okay, you got it. That's the fault. Thank you. I shortened it because my memory didn't hold the full joke.

• 36:14Chuck Nice?

  I'm just gonna let you know, as the resident layperson here and a comedian, it's not funny. Okay, so.

• 36:31Chuck Nice?

  No, I'm joking. Joking.

• 36:49Neil deGrasse Tyson?

  So Chuck, you got some questions from our Patreon supporters.

• 36:53Chuck Nice?

  Absolutely.

• 36:54Neil deGrasse Tyson?

  Who have exclusive access to our question flow. So what do you have for our guest today? By the way, you might be our very first mathematician guest. I'm almost embarrassed to say that.

• 37:07Speaker 02

  I feel a little bit of hesitation with the word mathematician because I'm not a research mathematician. I often like hold that in a certain high esteem.

• 37:16Neil deGrasse Tyson?

  Okay, so I'll reword it. Out of respect for your concern to not oversell yourself, okay? You might be the first person we've had that truly loves math.

• 37:30Chuck Nice?

  Okay, yeah. You know, Brian Greene is gonna be very upset with you for that statement.

• 37:35Neil deGrasse Tyson?

  Oh, that's true. Okay.

• 37:36Chuck Nice?

  Yeah, man.

• 37:37Neil deGrasse Tyson?

  Yeah, but no, but he, I wouldn't call him a physicist there.

• 37:40Chuck Nice?

  Yeah, he is a physicist, so no matter what.

• 37:42Neil deGrasse Tyson?

  Okay.

• 37:42Chuck Nice?

  Okay. All right, let's go. And by the way, if you are a Patreon patron, we thank you very much.

• 37:47Neil deGrasse Tyson?

  Yes, thank you.

• 37:48Chuck Nice?

  Let's go to, um, hello, Dr. Tyson, Mr. Sanderson, Lord Nice. This is Hugo Dart from Rio de Janeiro, Brazil, with my six-year-old daughter Olivia. We are such huge fans of StarTalk. Uh, Mr. Sanderson, what new possibilities does the evolution of AI present to the visual representation of mathematical concepts? Is there a revolution in the area taking place or about to take place? What do you know? Thank you.

• 38:17Neil deGrasse Tyson?

  So is AI touching math? Is that basically that question?

• 38:20Chuck Nice?

  Yeah.

• 38:21Speaker 02

  It is, in, in, in some pretty interesting ways. So, couple different avenues. One of them is using AI to suss out conjectures. Things that, you, you look at a bunch of data about something mathematical. And a lot of times when, um, when humans come up with conjectures, which are guesses about what might be true that they have yet to prove, they just use their instincts. They say, hmm, it seems like this might be true, maybe I should try to prove it. And so there was some work with DeepMind, um, where they essentially had some computers try to generate, what's the interesting math?

• 38:50Neil deGrasse Tyson?

  DeepMind is, is Google, correct?

• 38:52Speaker 02

  That's Google, yeah.

• 38:53Neil deGrasse Tyson?

  Yeah.

• 38:53Speaker 02

  Yeah, and so that, that's one avenue. There's this other, other avenue where math lends itself to exactness, like we'd been describing, in a way that also lends itself to computers very well. And so this phenomenon emerging, really in kind of the last decade, is a notion of proof checking software, where when you write a theorem, rather than writing it in English, you write it in an actual programming language that can check for sure if the thing is true. And so, for, for example, I get a lot of, like, Both real mathematicians and me will get emails from people claiming to have proven some famous unsolved problem. And the normal thing to do with the emails is just ignore them because like this is not the normal way that like a real proof would would get surfaced. But proof checking software would mean that you can basically say, hey, if this is an

• 39:38Speaker 02

  actual proof, write it up in the software. We'll just run the software. If it checks out, then we know it's worth looking into. But this this interfaces with AI in an interesting way where if you want to teach an AI to do math, you can just generate a whole bunch of proofs automatically, whatever you write it in this special software. And whenever they it compiles, whenever the software checks out, you know it's valid. And then you have these AIs kind of reinforcement learn on the ones that are true, kind of like when they play chess against themselves or play a game against themselves to get better by doing it trillions and trillions of iterations. You can do this with math in a way you can't do with other fields. And so there's this thing called the International Math Olympiad, which is, I went to it this year. It's great fun. Every country sends six high school age students to basically.

• 40:22Neil deGrasse Tyson?

  But you went to it not as a contestant.

• 40:24Speaker 02

  No, no, no. It's for high school students.

• 40:26Neil deGrasse Tyson?

  I was about to say, you are old. You are old for the Math Olympiad.

• 40:29Speaker 02

  No, I was I was giving some lectures to the students. But the this was one of the first years that some companies had bots try to answer the questions. These are quite hard questions. They they're written proofs. They're they're not just like, enter a number kind of thing. And the the AI, again, DeepMind was doing it, that they developed, got a score that's equivalent to getting a silver medal at this International Math Olympiad. And that's very like, people didn't think this would be possible, like, because it's considered like such a, it requires creative thinking. It's not just rote mechanics. And so seeing this level of creativity, the reason that was possible is because they could do this game of letting a computer play against itself, in a sense, using these sorts of proof checking softwares.

• 41:14Speaker 02

  So short answer to our friend from Rio de Janeiro is like, yeah, there's there's lots of interesting interface with with AI and math.

• 41:20Chuck Nice?

  Corey He, who says, hello, Dr. Tyson and Mr. Sanderson. Sorry, Chuck, I don't like you, so I'm not mentioning you. I put that in there because he didn't mention me.

• 41:32Chuck Nice?

  This is Corey Hee from San Bruno, California. We use base 10 and it still leaves us with a lot of irrational numbers. Is there a number system that would make irrational numbers such as pi rational?

• 41:49Speaker 02

  So first of all, I used to live in San Bruno. So hello, Corey.

• 41:53Chuck Nice?

  Oh, really?

• 41:54Speaker 02

  And then, um, so one answer to this question is no, whether a number is rational or irrational has nothing to do with how you represent it. This kind of cuts to our alien discussion earlier. No matter what language we're using, what base system we're using, the idea of what makes a number irrational irrational, it's intrinsic to the number itself. So it's a property, not of the of the description, but of the of the number itself. But maybe.

• 42:19Neil deGrasse Tyson?

  Let's slip this in here just because you have to help me out. Pi, which is irrational and transcendental in base pi would be 10, 1, 0. So what's irrational about 1, 0?

• 42:37Speaker 02

  The word rational will mean you can represent as a fraction of whole numbers. So the the counting numbers, 1, 2, 3, 4, if you can represent something as a fraction of those. So in base pi, all of the counting numbers, which are a pretty natural object. You're just counting rocks out in the world would have these these wild descriptions that take a whole bunch of different, we wouldn't call them digits.

• 42:57Neil deGrasse Tyson?

  It'll force everything else into a weird, weird state.

• 43:01Speaker 02

  Also, it's it's not clearly well defined what it means to be base pi. Because like, what are the the symbols you use with base 10? You have 10 symbols that you're using to represent with base 4. You have four symbols that you're using, but base pi, like what? It's not entirely clear what a non whole number.

• 43:17Chuck Nice?

  It's one symbol. It's one symbol, but it goes on forever.

• 43:24Neil deGrasse Tyson?

  Okay, so you're right. If you made pi a nice even looking number, everything else relative to it has to, has to die. Give it, give of its smooth life for you to make pi look smooth.

• 43:38Speaker 02

  You can make it, yeah, it's the, the, the thing that's irrational, you could say, is the relationship between pi and the number one. So it's kind of like whichever one of those you make look as natural as you can, you're sort of, you have to, you have to own up to the fact that the relationship between the two has this, um, this property of not

• 43:56Neil deGrasse Tyson?

  Which is forced back. Okay. So the answer is no.

• 43:59Chuck Nice?

  No, the answer is no. But, you know, basically, you're not going to be able to make pi rational without making everything else, you know, some kind of weird representation of that.

• 44:11Speaker 02

  Can I, can I add on?

• 44:12Chuck Nice?

  Which is rational. But go ahead.

• 44:14Speaker 02

  I have a tendency to blather, so definitely stop me if, if I go too deep. Now, aside from talking about bases, you can talk about how do you represent irrational numbers? Where we, what we're used to do is writing them down with decimals, like pi is 3.1415, which effectively you're saying it's three and it's plus one tenth, plus four one hundredths, plus one one thousandth. You're kind of adding all these like powers of 100. But there's other ways that you could try to use numbers to represent irrational, like use whole numbers, counting numbers, to represent irrational values. And when I just want to throw into the ether for like the, the curious listener to go off and like land on the Wikipedia page if they want, is the idea of a continued fraction, where it's another way that you

• 44:55Speaker 02

  can use like a sequence of numbers, of counting numbers to describe these irrational things.

• 45:00Neil deGrasse Tyson?

  I've had nightmares about continued fractions, just, you know.

• 45:05Speaker 02

  That, what, what can maybe turn those nightmares into dreams is seeing how certain values, like the golden ratio, or e, when you look at their continued fraction, rather than just having a whole bunch of arbitrary seeming gobbledygook as the digits, you see these really, really natural sequences. And so somehow, if you want a different language with which to describe numbers, like continued fractions offers another language that exposes the regularity in certain irrational constants, even if the fact that they're irrational is something you can't.

• 45:34Chuck Nice?

  See, now, this kind of, this kind of strikes me as what we talked about in the beginning of the conversation, as useless math.

• 45:44Neil deGrasse Tyson?

  So Chuck, a continued fraction, the reason why it gave me nightmares is you write it. But you can't write the whole thing. It's this fraction that just spills off the page.

• 45:57Speaker 02

  But the same is true of um of decimal values. Like, I can't write pi, three point one four one five, still not pi, three point one four one five nine.

• 46:04Neil deGrasse Tyson?

  So you're saying if I'm okay with that, I should be okay with continued fractions? I don't know.

• 46:10Speaker 02

  I mean, they're awkward, like, the layout, because you have to use vertical space to kind of like.

• 46:14Neil deGrasse Tyson?

  Right, right. And it grows as you keep going.

• 46:17Speaker 02

  Yeah.

• 46:18Chuck Nice?

  So I I think we need a representation of irrational numbers where the numbers don't believe that the earth is round.

• 46:26Neil deGrasse Tyson?

  Okay.

• 46:28Speaker 02

  Yeah.

• 46:30Chuck Nice?

  All right.

• 46:31Neil deGrasse Tyson?

  Time for a few more questions.

• 46:32Chuck Nice?

  All right, here we go. This is Alyssa Feldhaus from Rocket City Huntsville.

• 46:38Neil deGrasse Tyson?

  Hmm.

• 46:39Chuck Nice?

  And she says, is there black hole math that we may be missing to see what the end result is beyond the event horizon?

• 46:52Neil deGrasse Tyson?

  So I I like that. So let me let me flesh that out even some more. The limits of our theories of the universe, so general relativity, has has limits that are known in advance. It cannot describe the singularity of the black hole or the singularity of the Big Bang. And these are gravitational singularities where it has been said, God divides by zero, which you're not supposed to do. So is and then with string theorists march to the rescue, except they haven't rescued anything yet. So at what point do we say, Grant, that we're missing physics? New physics to help us? And at what point might we say, come on, guys, give us another branch of mathematics,

• 47:37Neil deGrasse Tyson?

  and that will save us from our ignorance? Now, is that a fair retelling of this question? I think it might be.

• 47:44Speaker 02

  I like that retelling.

• 47:46Neil deGrasse Tyson?

  Okay.

• 47:46Speaker 02

  I'm gonna throw out a flag that feels like this one's a little above my pay grade in terms of, like, I know a lot of the specific insights around understanding what happens at the event horizon have been, you know, this has been one of the most fruitful areas for seeing relationships between quantum mechanics and general relativity. And the specifics of that, I'm not going to pretend to have my mind fully wrapped around to say like, ah, here's, here's what the insight will be that gets us one step further. But, but maybe Neil, you can provide a little bit of intuition on this where I think broadly, the idea of alternate, alternate mathematical tools to understand like that specific part

• 48:25Speaker 02

  of the universe feels like a ripe area for, for where progress can happen, where a lot of new physics happens at the boundary of what, of what we can observe and have observed. And, you know, singularities give something that's quite literally at this boundary. And so I think with, in history, when you have had new types of math come in to describe new physics, or it can even be old math that's describing new physics, right? Like when quantum mechanics was founded and people started using a bunch of matrix algebra to do this, it's not like matrices hadn't existed. It's just there was a utility for them that hadn't been known before. Same deal with imaginary numbers in quantum mechanics. They had existed, but there was a utility that was then found.

• 49:06Speaker 02

  Same, similar things might happen for whatever further steps one wants to make here where there might be some existing math out there that hasn't yet found a utility, but, but you try to use it.

• 49:18Neil deGrasse Tyson?

  Well, get back to work so you can.

• 49:20Speaker 02

  Yeah, yeah.

• 49:22Neil deGrasse Tyson?

  Chuck got time for just one more question.

• 49:24Chuck Nice?

  Just one more. All right. How about Frosty from Tennessee?

• 49:29Neil deGrasse Tyson?

  You don't know how he speaks.

• 49:30Chuck Nice?

  What's that?

• 49:31Neil deGrasse Tyson?

  You don't know if he's got that accent.

• 49:33Chuck Nice?

  Uh, no, I don't, but he does now. I'm Frosty from Tennessee.

• 49:41Neil deGrasse Tyson?

  I guess with a name like Frosty, he's got it coming.

• 49:44Chuck Nice?

  Yo, that's, yeah, man, Frosty's the cool, he's, never mind. Anyway, he says, many mathematicians speak of the beauty in mathematics. How would you explain this beauty to someone who might view math as purely functional or just plain difficult?

• 50:03Neil deGrasse Tyson?

  And I'd like to add to that, do you have any insights into why so many people find math to be this difficult? It's almost like. All of civilization, there are people who like math, and that's 1% of everybody. And then the 99% of people who hate math, was a worse subject, where they never did well in it, and then it was never intuitive to them. As an educator in that space, take it on.

• 50:28Speaker 02

  I think a lot of what's happening is that the things that we do in school are pretty distinct from what mathematicians find beautiful when they're referencing that. So it's not that they're both looking at the same thing, and one type of person finds it beautiful, and then 99% don't. We're looking at different things. Because in school, often what you're doing is implementing like a procedure. You learn the procedure for adding two-digit numbers when you're like in elementary school. You learn the procedure for solving for x. You learn the procedure for taking a derivative, all of these times along. The thing that people find beautiful, all these different levels, whether it's at the elementary, or you know, the usually calculus is like the highest one that a high school might teach. All of it, it's still very procedural. And that's, there's nothing wrong with that.

• 51:11Speaker 02

  Like, that's a necessary part of doing that. That's where you're doing your squats in the gym, right? This is building some muscles. But the thing that is beautiful will be unexpected connections. So seeing like, hey, I saw this bit of math show up off in this corner, and that same bit of math showed up in this completely different spot. I mean, pi is a really good example here, where pi shows up in all sorts of places you wouldn't expect it to. So I can throw down one very quick example, for something that maybe anyone can understand why it's at least surprising, even if maybe you don't see beauty in it right away. If I take 1 minus a third, you end up at somewhere on the number line, and if I add 1/5, you kind of go up a little bit.

• 51:55Speaker 02

  And if I subtract 1/7, and then I add 1/9, and I subtract 1/11, and I add 1/13. And I play this game of basically adding and subtracting in alternating fashion, all of these, like one over some odd number.

• 52:08Neil deGrasse Tyson?

  Ever smaller and ever smaller.

• 52:10Speaker 02

  Ever smaller. It kind of bounces back and forth, and it'll approach one particular point on the number line. That point is exactly pi divided by 4. When you look at that, like, pi is about circles. Why on earth would pi have anything to do with this game that I'm playing?

• 52:24Neil deGrasse Tyson?

  It's got no business showing up on your timeline.

• 52:26Speaker 02

  It's got no business.

• 52:27Neil deGrasse Tyson?

  On your number line.

• 52:28Speaker 02

  There's so many, the, the normal distribution we were talking about earlier that describes, you know, statistics all over, it has a pie in it when you describe the distribution properly. Why? Where's the circle? What, what on earth is going on there? So that, that side of it where there's a mystery, there's this, it's, you know, it's, it's the opening chapter of a mystery novel. You, you see this thing, you want to know like, who did it? Like, why did this happen? And then, and then the path to resolution. But I think the, if anyone enjoys a Sherlock Holmes story, for example, the part of your brain that that's tickling, where you have this mystery at the start, some path that feels like it could be discoverable, but takes a little bit of cleverness along the way. And then it's satisfying to see the knot tied at the end. Like, that is the feeling that people are describing when it's the beauty of math,

• 53:12Speaker 02

  which is different from the thing happening in most classrooms, which is the procedural side and, and understanding how to go through motions, which, which are useful, right? But it's just, it's a different, a different category.

• 53:24Neil deGrasse Tyson?

  Grant, you're giving us hope. Because people do like connections. I mean, there's that whole TV series and book series. Who's the host of that? The British guy? Yeah, yeah, James Burke. Right. It was about connections. And of course, there was, there were geopolitical, social, cultural connections. But nonetheless, I think we always like knowing that this over here is the same as that over there. And when you didn't otherwise realize that.

• 53:53Chuck Nice?

  But I think, I think the, the intrigue that you're speaking to is far more important from an educational standpoint, because the procedure is kind of the work, the nuts and bolts. Whereas what you are speaking to is the wonder and the mystery. And these are all concepts that show up in life, throughout life, and that always spark us to think and to ponder and to something bigger than ourselves. And if people were able to bring that into the classroom, I think you would find a lot more children being excited about math.

• 54:33Neil deGrasse Tyson?

  Not all educators are made the same, that's the problem.

• 54:36Chuck Nice?

  Absolutely. Yeah. I mean, honestly, I, I. I'm always fascinated when I listen to anyone talk about math from Neil to you, to you name it, to Brian Greene, to, you know, Brian Cox, you know, when he's, the way you talk about math makes it so much more interesting. But when I was in school, I had several teachers, especially in calculus, where I would get the right answer, but I got it the wrong way. And that was the end of it. They were like, yeah, you're wrong. You can't do it like that. And I'm like, okay, well, instead of, hey, let's go on this little journey to show you, you know, how you're thinking about this. I never got that. It was like, no, you got the wrong answer. And honestly, it only took a little, I was like, clearly, I don't think

• 55:19Chuck Nice?

  right. That's where I came away from calculus. My calculus experience was, I do not think right. And that was it.

• 55:25Neil deGrasse Tyson?

  It's your problem. It's all you.

• 55:29Neil deGrasse Tyson?

  So Grant, thank you for being a guest on StarTalk. And once again, we can find you on all platforms with three blue, one brown as the handle. And I don't think you had to compete with anyone to get that handle on all platforms, as you have successfully managed. And you have a podcast, YouTube channel, blog, you're active on X, and Instagram as well, correct? Yeah, there's some short videos up on there. Yeah. Your YouTube is chock full of all manner of videos, some short, some long, some courses you even give there. And they're so popular. And it gives us hope that not everyone in

• 56:13Neil deGrasse Tyson?

  the world thinks of math as the worst possible thing they could give their brain attention to. And you are living evidence of this. So, so Grant, thanks for being on StarTalk. Thanks for having me. All right, Chuck. Always good, man. Find you Chuck Nice Comic?

• 56:29Chuck Nice?

  Chuck Nice Comic everywhere. Thank you, Neil.

• 56:30Neil deGrasse Tyson?

  Everywhere. You got Comic C O M I C. This has been StarTalk, a hybridized cosmic queries all about math. Neil deGrasse Tyson here. As always, I bid you to keep looking up.