StarTalk Radio - Is the Universe a Math Problem? With Terence Tao

Episode Date: February 24, 2026

Do we need new math to explain dark matter? Neil deGrasse Tyson and comedian Paul Mecurio explore unsolved problems in math, simulation theory, base systems and more with mathematician Terence Tao.NOTE: StarTalk+ Patrons can listen to this entire episode commercial-free here: https://startalkmedia.com/show/is-the-universe-a-math-problem-with-terence-tao/Thanks to our Patrons Drake Mccammon, Lourosa Thomas, Christopher, WireRider, Kevin Walter, Erik Majsai, Zoe Briskey, Roger W. Cavallo Jr., Tibi Chiorean, Maggie, Guy Cummins, Sean E, Sebastian, Ty Powell, Iceo Bergins, Anissa Aponte, Keith Autry, Guy_%, Kyle Kern, Scott Morris, Francisco Cueto, Psychoquark, Joe Rivera, Chris Coyle, Brian White, Michael Boring, Adrian Mihai, Rowdy Payton, Ben Huang, Jerred Cook, KennyS, Gonzalo Oria, Herb Tobias, Kevin Revels, Sean Taylor, AndersonRobotics, Faruk Arslan, Antonio Alcala, Rudolfo Munguia, Tim Winter, B Boyette, Pilar Rodriguez, Justin Brackenbury, Kosta Papageorgiou, Mickey Mouse, Charles Garcia, Sultan Bin Zaal, Brian Rall, Mike, Dave, Rodolphe Phelouzat, Joshua Fisher, Xan Kriegor, Michael L. Price, Wendy Welch, RichInDC, Justin Lebron, Brad Cook, Auralen Isara Maelis, Obi Wan Kenobi, Avinav Abraham, Stanko, Colton Murphy, Patrick Quinn, Haze, John Stamm, Roberto Delgado, Hans Gab, Richard Nolan, Tony R, Trisha Hadden, Stephen Flenley, Robby Vasquez, Abraham, Dr. Bebop, Damon, Richard Thompson, Prince N., David Rivanis, Daniel Slack, Guy Bergeson, Shawna Couplin, MrFish, Lisette Ramos-Voigt, Scott Mccoy, Steffen Thomas, Cassio Magellan Campos, Rodger Reinhardt, Michael Becker, Billie Lyons-Super, Todd Chambers, Mo Elzeinab, Talal, Joseph Glynn, John Hermanns, TheNaden, Mark Parker, Seth Davison, Anon328648926, and David Bentley for supporting us this week. Subscribe to SiriusXM Podcasts+ to listen to new episodes of StarTalk Radio ad-free and a whole week early.Start a free trial now on Apple Podcasts or by visiting siriusxm.com/podcastsplus. Hosted by Simplecast, an AdsWizz company. See pcm.adswizz.com for information about our collection and use of personal data for advertising.

Paul, we have revisited the universe of math. Yeah, but really drilled down and really applied ourselves. Because that's how we roll. Yeah. Coming up on StarTalk. Welcome to StarTalk. Your place in the universe where science and pop culture collide. StarTalk begins right now.

This is StarTalk. Neil deGrasse Tyson, you're a personal astrophysicist. And we're going to have... have a Cosmic Queries edition on the subject of mathematics. Who are laughing me like that? Apparently you're going to scare people with this? Everybody thinks there's a math quiz coming up. That was a Halloween laugh, wasn't it?

It was. I got Paul Mercurio here. How you doing, man? I'm good, man. Good to see. Yeah, you got your podcast. What was it called again? Inside Out with Paul Mercurio?

Inside Out. Did you get Disney permission for that? I did not. Well, thanks for bringing that up. I'm going to be getting sued right after this. And you are out of a job after May. Yes, thanks for bringing that up as well. Anybody have any deadly diseases they want to talk about?

Yes, the late show got canceled. So I work on the late show. You've been on the late show a bunch. We love you there. Yeah. Well, not everybody. No. And you've been with Stephen.

With Stephen Colbert since the Colbert report. Since The Daily Show. The Daily Show, yes. I started at The Daily Show as one of the original writer's performers there. he came in as a performer. You predate him on the daily show. I'm old school. I'm OG.

And we actually shared an office together. We'd write a lot together. And then he had the Colbert Report. I worked on that. So he and I have been together a long time. And it's weird and sad, you know, because it's like it's not a lot of change over in the 10 years that we've been on the late show.

So it's like a family breaking up. You know, it's really kind of, yeah. So I'll be at your house, cutting your lawn for two bucks. Okay. I hope our guest needs some help. I'll go to California. Doesn't snow out there, but I'll shovel anyway. All right.

And by the way, Barron? Barron. You did get the title of Barron. United me, Barron. Only if the people asking questions, remember that. That's where that comes from. You'll find out. So who do we have today?

I love me some mathematics. Ever since high school. Brilliant. I've been a big fan of mathematics. Even the obscure math that doesn't relate to anything ever, but it's still fun. Yeah. But of course...

Well, initially you like math because there's a finality to it, but then when you really get into it, you realize there's a whole bunch of non-finality to it. It can take you everywhere, right? Yeah, yeah, everywhere, anywhere and anywhere. Yes, exactly. Yeah, so you know what we found? We found, like, a badass mathematician.

Ooh. Yeah. That sounds like a good movie title. Is he also an assassin? We cannot divulge that publicly. We have Terence Tao. Terrence, did I pronounce your name correctly? That's correct, yes.

Please, please be here. No, excellent. That's wrong. That's not how you pronounce your name. You are professor of mathematics at UCLA. UCLA. Okay. That is not a community college in Oxnard?

Yeah, uh-uh, uh-huh, community college. Your professor of mathematics at UCL, that's officially University of California, Los Angeles. Yes, it is. And here's the best part. director of special projects at IPAM Institute for Pure and Applied Mathematics. Can I just ask, Terrence,

did they give you the title of the director of projects and you insisted the word special would be put in there? That's what people are saying on social media. I'm just relaying information right now. No, it predates me,

but certainly that title helped me accept the position. So we can say, we can say, he has a special set of skills. Oh. There you go. This guy is definitely badass.

He's got a special set of skills. In a weird way, you could be like the accountant, right? That's that movie. Yes. Where he's mathematically brilliant, but then also could just kill you with two fingers. Yes, yes. But that is not our guest today.

Okay. In love with the superhero. Right. So tell us about this center that you direct. Yeah, so I'm not the main director, but I'm the director of what's called Special Projects. Yeah. So IPAM is,

an institute that brings together pure mathematicians and applied mathematicians and scientists and people from industry to work on topics where it's time for mathematics to get involved. So, for example, we hold very early workshops and conferences on AI, deep fakes, self-driving cars many years before they became a reality. And often industry people were working on these problems, but they were blocked by various mathematical obstacles and they needed mathematicians to talk to. So we create these programs where we just have lots of experts in different fields in the same room, listening to talks and just socializing and good things happen.

I once talked with an electrician engineer and statistician 20 years ago in one of these programs, and we ended up finding a new way to do MRI scans. That's actually 10 times faster than traditional scans. In fact, now all the modern MRI machines use our algorithms. Can I just say I recently had an MRI? Still took a lot, long time. Maybe you could have been a little better. Right.

I mean, I don't have all day, Terrence. I have stuff to do. I want in and I want out. I wanted to be as fast as when I microwave soup. Right. I think when they make them faster, they trade it off for more detail. So they can get a more detailed image of you. The cutting edge now is.

they can get videos of like your heart, whatever, in real time. So since you're there anyway, I think they will keep you for the same amount of time, but they can expect more data out of you. At the risk of stating the obvious, one of the great things when you establish institutes or these umbrella gatherings of people from different professions that wouldn't otherwise ever talk to one another. I mean, think about it.

You have departments of, you know, mechanical engineering, electrical engineering, biology, chemistry, physics, by-di-da-di-da-di-da. And they have lunch together, you know, they have coffee together, and they don't know what anybody else is thinking or what anybody else is doing. And now you put this umbrella over an institute, and you bring them all together and, oh, my gosh. And I would think one of the benefits is that, you know, you look at something yourself or with your groove,

a new set of eyes comes in and brings a different perspective and sees things that you didn't see. Yeah, exactly. I mean, science is just way too broad now. Maybe 100 years ago was possible to have a pretty decent understanding of every corner of science, but that's basically impossible now. So you have to specialize. The universe doesn't compartmentalize science. As much as we wanted to.

Yeah, it's all mixed together at all times. You guys figure this out. Yeah. All the modern problems in the world are really interdisciplinary. I think the specialist problems we've kind of solved in the 20th century. And the 21st century is all about collaboration. This should be obvious, but just so we're all in the same page, tell us the difference between pure and applied math.

Pure math is curiosity-driven math. You know, we see patterns in very abstract things like numbers or shapes, and we just ask, you know, does this pattern keep continuing? And it doesn't necessarily, it's not necessarily motivated by any practical application. Or maybe it started out that way, but people just sort of just kept asking questions. Would that include people who are obsessed with what's going on? on in the sequence of digits of pie? In principle, yes, that is part of pure mathematics. It turns out not to lead very far because there basically aren't interesting patterns there. But did you prove

that there are no interesting patterns? This is actually still an open question. So it could be that the fact that there's not an interesting pattern is itself an interesting fact. Right, Except that we know that like 99% of all the numbers in the world have no interesting patterns. And it's like only a very small fraction of numbers that actually have something detectable going on. So the fact that pie is part of the 99% is in retrospect not exceptionally interesting. Do we have a theory why 99% don't have a pattern? Yeah, no, it comes from a low of probability for the low of large numbers. Like if you pick a number randomly, for every digit you roll a die, a 10-sided die.

and you select digits at random, the law of large numbers says that 99% or 99% of the time, you never see any patents. You get as many zeros as ones, as many ones as twos. There's only a very small fraction of numbers that exhibit bias. It's a very useful law. This is why, for example, we can do polling. Like if you want to poll 300 million Americans and see what they're thinking, you can't poll 3,300 million Americans, but you can poll 1,000, 2,000.

And the low of large numbers actually tells you that as long as you poll a very representative set of people, the outcome of the poll is actually pretty close 99% of the time to the general population. So I could ask a question that you might be able to answer. How many digits of pie would I have to roll out before I got 10 nines in a row? That would be about 10 to the 10. So I guess 10 billion. Okay. All right. We got an answer to me that.

You're both wrong. It's 11 billion. I'm the guy that knows nothing and you're both wrong. Why are we doing it now? Okay, so now applied math, the world other than mathematicians, care about the answer in applied math. Right, yeah. So applied math, it's still not quite directly applying to the real world.

That's what scientists and engineers do. But it's about the mathematics that is of practical value to the scientists. So, for example, an atmospheric scientist might care about predicting climate or the weather. And applied mathematicians might develop the software or the equations to actually model the atmosphere or oceans. They may not actually work directly with real data. They may only work with sort of test data or something. And it still has a lot of theory to it. But it's so intermediate between the pure mathematicians and scientists and engineers.

We face that a lot in astrophysics where you can generate data that you can then apply your ideas to to see if that works, and then you check later to see if the data you generate matches what you actually end up observing. So just the methods, tools, and tactics. So when does applied mathematics sort of do a cheat, if you will, right? Sort of simplify things intentionally, drawing on pure mathematics. And where's that line where it's too much of a cheat or it's just enough of a cheat, that it's still...

Do you mean approximation? Yeah, well, you know, the term is intentional simplified reality. right? Yeah, yeah. So there's a spectrum. So the real world is messy. And if you try to incorporate every single aspect of it, it's just too much to model. And you can't see the forest for the trees. So yeah, so methodicians intentionally simplify reality. They work what's called toy models. Physicists call them spherical cow type assumptions where you want to model a cow is actually easier for the physics. If the cow is a complete sphere and frictionless, that's not realistic. but it's a good starting point.

And then over time, you decide to add some friction, add some leg to the cow. But you start with the easiest cases to get some initial idea. I guess the difference between math and the other science and basically anything else is that we can change all our hypotheses and work with these toy cases. If you want to build a bridge and you're an engineer, you can't just build a toy bridge over a creek first and then float your way up to a real bridge. If you're a heart surgeon, you can't do some experimental surgery on rats or something first. And then, well, I guess in training, maybe you do.

Mine has. That's why I have a bad ticker. Okay. Well, my condolences. But in most professions, you don't get to play your toy models because you're not allowed to fail. But in math, math failure is very, very cheap. You know, you try a problem. You don't solve it.

Fine. You just toss the paper away and you try again. No one gets hurt. No one dies. Yeah, the patient doesn't die. Yeah. Exactly. Yeah.

And let me add something because you just blew by the spherical cow. It's worthy of another comment. Okay. So in physics, everything is about simplification. Because often the details, while they may be interesting, don't actually influence the outcome in any important way. Okay. And if they do, you would have solved the case than to see how it influences the outcome. the outcome. So here's the point. You're tasked with, I want to design a cow that can maximize

milk production. Okay. So then you say, consider a spherical cow. So a sphere holds the maximum amount of milk for the area, okay? The surface area. Okay, where are the legs? So now, if you add legs and a head and a tail and, okay, that reduces the number, but it means I have calculated the upper limit to how much milk the cow is going to make. Because what does that sphere? The sphere, exactly. So you can't come in later say, here, have the cow produce five times as much. But wait, having those details of legs, tail, and head.

It subtracts from the total. Yes. That's the detail that you said doesn't matter. It matters. What I'm saying is, it caps how much you can think about that problem. It keeps everybody in the same room. Got it. Yeah.

It's not an unlimited place to guess outcomes. Have you found that applied mathematics can change a settled pure math theory? Has that ever gone back and sort of... Well, let's lead into that by asking, tell us about unsolved problems in mathematics. And when they're solved, who solves them? Is it the computer person? Is it the pure math person? Is it something else?

Is it Matt Damon writing on a board when he's a janitor? Exactly. Come on. Increasingly, there's a conversation going in all directions. So traditionally, few methoditions just experimenter patterns, and they, based on analogies and intuition they brought up, they propose that some phenomenon that they see for one type of pattern also extends to some other setting. But sometimes it comes from the physical world. This is noticed that something keeps happening. So, for example, there are these laws in physics that are simply universal, that there are certain distributions.

Like the bulk of distribution is an example of a universal distribution. Many, if you plot, say, the heights of people or, you know, or the size of cows, if you wish. Can I just tell you something right now? I want a milkshake in the worst way. I am so craving one. Anyway. Yeah, but lots of distributions in nature. have the same shape.

In this case, a bell curve. There's even a meme about it on the internet. And we would, in physics, I think in pure math, is a Gaussian curve. Yes, that's the name's right. Yes. Named after Gauss, brilliant mathematician. One of the most brilliant there ever was, I think. Yeah.

So mathematicians did actually find an explanation for why this curve appears all the time. It's something called a central limit theory in probability. But there are some other distributions that physicists have discovered that we haven't yet fully explain why they show up so often, like gaps between spectral lines and it's more technical to explain. But, yeah, there are those that physicists and other scientists have found are universal, and mathematicians are, and they're a good source of conjectures for mathematicians to work out. Yeah, well, mathematicians are in some ways exploring other universes, but just very abstract

numerical universes, not as interesting as the sci-fi outer universes. So, I mean, to put this in sort of practical terms, is pure. math versus applied. It's like you're painting a room and you exactly calculate the number of gallons of paint. And then two days later, you've made three trips to Home Depot because you need three more gallons and you're going to have a nervous breakdown because that's the applied part, right? Once you get into it, you sort of go, oh, wait a minute, I didn't factor this in. And that's sort of what happens with applied math. And it would then correct some of your pure math calculations, right? So it's sort of this, it's this symbiotic relationship.

between the two, right? Yeah, so Vladimir Arnold, who is a famous mathematician, he once wrote that mathematics is the part of science where experiments are cheap. Oh, I like that. Before you invest billions of dollars in a new telescope or a collider or something, you do the math. And you see what is theoretically possible,

maybe assuming spherical cows and things. But it tells you in theory what you can or what can do, and it sets good targets. And then it allows you to allocate the more expensive resources more intelligent. Hey, this is Kevin the Somelier, and I support StarTalk on Patreon. You're listening to StarTalk with Neil deGrasse Tyson. So I got here something called the Colatz conjecture.

What is that? Yeah, so this is a dangerous conjecture. It has trapped many mathematicians and amateurs because it feels like something so simple that we should be able to solve it. But it's been around for at least 100 years, and we haven't sold us. Well, maybe if you try it a little harder. we try I've worked on this too if you were just a little smarter

playing with spherical cows all day I could describe it I'm just worried as I said it could trap some some audience members to work on it obsessively people That's good we like trapping our people So tell me tell me about So these are the best of the unsolved problems

Are the ones that can be described simply So the glass conjecture It's also called the Hailstone Conjecture For a reason I'll explain later So it's satisfying So you give me a number, your favorite, number 37 or 69 or whatever. Okay. And then we do the following.

So if your number is even, we divide by two. We make it smaller. So 16 becomes 8. But if it's odd, you multiply up by 3 and you add 1. So you can be 5, add melt up by 5 because 15 add 1 to 16. So odd numbers become bigger, even numbers become smaller. So now you just repeat this process. So 5 becomes 16, but then 16 becomes 8 because it's even,

8 becomes 4, 4 becomes 2, 2 becomes 1, 1 is odd, you mark up by 3 and add 1, it becomes 4 again, and you end up in a loop, 1, 4, 2, 1, 4, 2, 1, 4, 2. It always takes you back down to there. Well, that's the conjecture. So we've used computers. If you take any number up to say a trillion, every single number that we've tested ends up going down to 142, 142, 142, 142. But we don't know if all of them do.

So it's called the hairstone conjecture because there's this oversimplified. model of hailstones. Again, a spherical cow type model of hailstones, where hail forms because a little ice crystal forms in the clouds, sometimes currents begin up where it's colder and then more ice forms and then it goes down a little bit and maybe it melts. And the hellstone bounces up and down in the cloud, but eventually it lands to Earth. All the hellsons eventually hit the ground. But just to be clear, they hit the ground because they reach a mass that the upward currents can no longer sustain them and then they drop out. It's too heavy.

Yeah. And the bigger the hailstone, the bigger the uplifting air was that kept them there for that long. And the bigger insurance claims for your car. In principle, there could be some really unusually lucky hellstone that somehow always hit the currents that go up and don't hit the ones that go down and just keep bouncing up and up forever. I mean, occasionally you could so defy the statistical laws of physics.

And so in principle, there could be. this very lucky number that just sort of always keeps hitting the odd numbers and going up rather hitting the even numbers to go down. It would be like someone who's consistently winning at a casino at a game that's rig. It's theoretically possible, but we don't know if you can actually do it with any given, with an actual number. And that's the collapse conjecture. But this is so basic. It's sort of a yes. It's a yes, no moment, right? It's even odd. It should seem like why does it devolve into chaos? Because it's starts with such a basic premise. That's the part that's confounding me and I guess obviously others, but...

You are now confounded. Thank you very much. Can I have my official spherical catap, please? Yeah, but what is that about? So it's part of the general phenomenon called chaos. So you can have very simple operations like halving a number of it's even or mongonger 3901. It was odd. As you say, if you just do it once or twice, it's a very easy operation.

a kid in third grade can implement it. But when you iterate even very simple operations over and over again, you can get vast much complexity. And so sometimes you don't. Sometimes you get these universal laws, like these bell curves and things settle down. But sometimes you just get this enormous complexity. The act of reproduction and splitting DNA is fairly simple. But it leads to immense biodiversity.

We have computers. we have AI. Why hasn't the COLA's conjecture been affirmed? Well, because we have to check an infinite number of cases. We'll get to work on that. You're gonna say. There's a thing called a computer. Look at my finger. Just go like that. Why don't we crowdsource it?

All right, it's still not an infinite number of people, but there's quite a bit of computing power out there. You know, the SETI Institute did it, where you'd upload software and they'd give you data. And while you were not using your computer on screensaver, it was using your CPUs to calculate. Yeah. So there was a project. I think it was called Collapse Grid, which was exactly that.

Like City at home, but for Collapse. And it did extend the numbers. So a couple quadrillion, I think, or 10 to 18, 10 to the 19, we could do from this crowd sourcing. But no matter how much you do, there's still an infinite number of numbers left to go. So if you want to roll out all the numbers, you need proofs. You need mathematical, you need to use mathematical laws over all numbers.

Otherwise, it's not elegant and it's not even interesting. Well, it's not elegant. It's like you're putting a bunch of crazy ingredients in a blender. You always get, and it always outputs oatmeal, like every time. Some problems you can only solve by brute force. Are we any closer to solving this conjection? I mean, how close? So I worked on this a couple of years ago.

I proved a result that if you take a really large number, like 10 to 15, 20, whatever, I could show statistically that 9% of all numbers that are very, very big, would become very small, and become much smaller than where they started. I couldn't show they hit one, but I could show that 99% of all numbers become as small as, say, the logarithm of their number. So like 10 to the 20, I could show drops down to 20, 10 to 100, drops down to 100. You know what? I'm going to use a term from your world.

Why don't you apply yourself? Yeah. So in mathematics, we very much value partial progress. They're very good. Yeah, we can't solve completely, but we are happy with half a loaf or 99% of a loaf, because someone else can build upon that. Do you like those math movies, like the beautiful mind and the,

the, the, the, the, the, the, the, the, the, the, the, goodwill hunting. Do you sit there and yell at the screen? That's not the way you do it. Are you one of those guys? I enjoy those movies mostly for the non-math part of it. Now, maybe if there's an expert in schizophrenia, they would be complaining at that, but saying that the math was very cool. I think this is common.

I have a brother who did some CGI back in the day. And like every time it's animated movie, he cannot enjoy the special effects because he knows how they were made. So I remember watching one of these movies. A movie called Gipter stars Chris Evans and this amazing young child actor. who is supposed to be this math genius, and her mother was a math genius, and she was working on an answer of math problem. And it actually was a problem that I worked on.

And at some point, they were going to review some of her notes on how she was making progress was the problem. And I was actually quite interested to see what they would do. And they actually showed a little snippet of her notebook. And there was actually some equations from one of my papers. Wow. Two years before the movie was made, I actually got an email from the directors saying we were making a movie about a gift to get in math.

Could you supply some samples of some math computations that would look good? And I actually supplied some from my own work, some from others. And they said, thank you very much. And I didn't hear of them for two years. And then this movie came out. So I was called unawares. But it did actually come from me. So before we go to our question base, there's one more.

Just tell me about the aerod. most problems. Right. So Paul Erdisch was this Hungarian mathematician. He was rather extreme. So mathematicians have a reputation for being a little idiosyncratic, but he was rather extreme even among methodicians. He didn't own a home.

He would travel the world constantly and crash on other mathematicians' couches, basically, for his whole life. But while doing so, he would talk math with them and they would often write papers. He has like 2,000 or 3,000 papers. He is one of the most prolific methoditions in history. and he was famous for posing problems that he would attach little cash prizes to often. Like here's a little problem I just came up with. You get $25 or something if you can solve this problem.

And in fact, many of these problems did get solved and Erdus would send them a check with that amount of money. But these checks were almost never cashed because they were more valuable framed on the wall as someone who had solved an Erdush problem. I want to be that famous. I can pay people and they don't cast a check. Well, maybe if Mr. Smarty Pants cashed the checks, he could be. buy himself a house and not have to sleep in other people's bed. That's all I'm saying. There's a biography of Poetters called The Man Who Loved Only Numbers.

And that was, that is a pretty good description of it. I bet him once. And he basically, basically all the entire conversation was about math. He was not one for small talk or anything. From my notes here, we have Problem 1026. Is that, is that the correct way to say that? And was that an Erdoche problem? Right.

So, yeah, so recently there's been a systematic effort to actually, so there's a website which has collected over a thousand urdish problems. Oh, so this is just a number. This is right. It's 1,026. Right. That's all that is. Yeah, so, yeah, we just gave each oudish problem a number.

This is the, and this is the ticket that this number got, this problem got 1,026. And there's been a systematic effort in recent years to solve these problems by any means possible. So some people use pen and paper. Some people use computers old school with also computations, and some people are using AI tools. And, yeah, there was a recent problem, 1026, that got solved by a combination of all of these. Lots of people throughout ideas. There was a discussion forum, and people used the latest AIs to gather some numerical evidence. I was involved a little bit.

And I can ask a question, and it's relevant and not off track. So when you're collaborating that extensively, who is sort of, who is curating all of that? Who is in charge of that? You need someone to sort of kind of manage that process, right? Yeah. So surprisingly, it's very decentralized. I mean, there was about five, six people involved, and there was just a chat room. And we just all spoke with contribute ideas.

It was a very respectful environment. If we had to write it, we haven't written a paper. The form was solved. we haven't decided to actually formally make it official paper. If we did, then we'd have to organize it a bit better. But a lot of these crowdsourced solutions, they're just, actually, very spontaneous and very organic. I actually like this more to collaboration than sort of a more directed top-down thing where there's some principal investigator that's sort of a science task.

We can do that too. But sometimes we get it from the crowd, which is great. Yeah, the PI won't necessarily always be the cleverest person. someone can come in from somewhere else and jump right in on it. I just want to show off a little bit of my childhood math wizardry. So 1026, that number is divisible by both three and nine. Okay. Evenly divisible.

Leave it to you. No, no, just add up the digits. And what do you get? Okay. You get nine. Nine, so it's divisible by nine. There you go. And nine itself is divisible by three.

So it's divided by nine. Did I get that right? Yes, yes, that is a classic. Classic test for divisibility. Exactly. But the meaningful chunk of its value is forced to line up in some order, right? Isn't that what this problem says on some level? Oh, yeah.

What was? We didn't say what the problem was. Ah, okay, yeah. So you can play it, you can phrase in terms of a game. Like, suppose you have a pile of coins, like 100 coins, and you arrange them in stacks. Like maybe you put 30 coins here and then 50 coins here and then 20 coins here. So you arrange them in stacks. stacks and some stacks are taller than others.

And then I get to pick some of the stacks and claim those coins for myself. But I can only pick a sequence of stacks that's in increasing order or in decreasing order. I can't pick a sequence of stacks that goes up and down. So your aim is to lose as little money as possible. My aim is to get as much money from you as possible. So you want to arrange your stacks to sort of bounce up and down in such a way. It's hard for me to find a sequence as a sequence. of coins that go up or sequence of coins that go down.

And it's no matter how scrambled the list is, you can pick this up decreasing and increasing order, no matter how scrambled it is, right? Right. Yeah. So, yeah. So, like, for example, if you're only allowed to divide your coins into three piles, I can always pick the two biggest ones, because the two biggest ones will either be an increasing or decreasing sequence. So I can always grab at least two thirds of your coins.

But the more columns, the more stacks that you're allowed to make, the less and less I can win from you. And the question was exactly how much, what is the maximum loss you can, you can have from this game? So what is the fair price to charge from me to pay, to play this game? So it's like this chaos in this order in this chaos, right?

It's like a crazy group chat, but there's always one guy making sense a little bit. Yeah, yeah, yeah, yeah. So it comes actually from this film of Erdush and other mathematician Zekevesh, yeah, that given any sequence of numbers, as it goes up and down, you can find within it, some sequence that goes up for a long time or some sequence that goes down for a long time. Yeah, there is order in chaos, actually. That's

almost exactly how it's described. That's beautiful. That's a beautiful concept. Well, I have a beautiful mind. Make a movie with that title. So you got questions. I have questions. And great questions, as always. Patreon membership. Yes, and they're always terrific. And I'm going to jump right in here. This is JKW. Greetings Professor Tau and Dr. Tyson. James from Norfolk, England here, my question is rather simple. How often are discoveries made in pure mathematics, which would appear to have no practical application, not all the time, outside the realm of pure mathematics, but consequently, subsequently,

subsequently, rather, find very useful applications in other fields. What is one of the most surprising examples of this? I love that. Yeah, no, this happens all the time. Eugene Vigner, a physicist, once called this the unreasonable effectiveness of mathematics. in the physical sciences. That there are lots of concepts that mathematicians played with for their own sake. And only later, often decades later, did scientists realize they were valuable for other things.

The most famous example is that non-Euclidean geometries. There you go. People played with notions of curve space, not because they thought that the actual world was not Euclidean. Well, wait, back up for a sec. There used to be just regular Euclidean geometry. geometry, okay, where everything is flat. Boring. That's where you have squares with 90 degree angles and triangles have, the angles add to 180.

There's that. And that has direct application that, you know, geometry stands for Earth measurement. Oh. Think about that. You're measuring surfaces inside. Yeah. So anyhow, so now. Yeah, so you can't geography has all these amazing films, like the some of the angles

or triangles always 10080 degrees. It's a classic theorem. But they were very, very hard to prove, very complicated to prove. And mathematicians are trying for a long time to see if there was any simple way to prove these, to prove these theorems. Like if you take away some of the axioms of geometry, could you still prove these films at right angles and whatever? And by doing so, they discovered these non-Euclidean geometries, like where you're on a sphere

or some hyperboloid instead of flat space. And now the sum of angles of triangle is not 180 degrees. The area of a circle is not p.R. squared. And these are very weird geometries. And parallel lines converge or diverge. Right, right. Either or, yeah. Yeah, depending on the curvature of the space.

Yeah, it's completely wacky stuff. And what were they thinking at the time? Were they thinking, oh, this is just pure math. No one will ever care, but I'm going to do it anyway? I mean, it's really like, it's like you're creating, discovering things that you don't think you'll ever use. It's like either you're a genius or super anal retentive.

It's like you keep throwing stuff in your garage for 40 years, then you clean it out and you're like, that's why I kept that hat box. See, I knew, right? Isn't it like that on some level? Just say yes, Darren. Fiction and art is like that too. You explore worlds that aren't real,

and there's value to that. So, yeah, so these were kind of geometries that weren't real in some sense until Einstein, when he was developing general relativity, realized he needed a theory of curved space, and he asked a mathematician friend, I think Marcel Grossman, hey, do you know of any mathematics that deals with curved space? It's, oh, yeah, there's this non-declin geometry stuff.

There's this British bright guy Riemann who developed this wonderful Romanian geometry, and he took a look, and it was almost exactly what he needed, almost word for word, the language he needed to express the theory of relativity. So it's almost like when you're doing pure mathematics, somewhere in your gut there's an instinct that there's something this can be used for or will be used for, right? I'm going to imagine. But it's not what motivates.

They're not motivated by that. No, but I mean, once you get in it, you start to go with this. I know this doesn't apply now, but there's something, isn't it as intuitive at some point? Yeah. So my theory is that both pure math and science are motivated by compressing the world around them. So they have all this data in the case of pure math, It's mathematical data.

In the case of scientists, it's physical data. And they want some nice theory or explanation to compress all that data into something they can understand. And somehow when you compress either theoretical data or experimental data, it often tends to compress to a similar-looking theories, even if they come from complete different origins. Next question. Let's move on. That's a good one. That's a good one.

Yeah. Thank you. From Norfolk, England. England. This is actually from the Turkish Republic of North Cyprus. Whoa. Yeah. Nice.

Absolutely. Yeah. They have a great Home Depot there. Greetings, a gem here from Turkish Revolution and North Center. I hope you're all having a great day. Here's my question. If due to some reason we were to adopt another base system instead of 10 and spent all these years doing math in that system instead of base 10, do you think there would be theories that

we would fail to develop? Alternatively, do you think we would have maybe developed even better or more successful theories if humanity had been using another base number system all these years. I love that. In fact, let's start out by asking you, when did the concept of base as accounting system reach maturity that can have been from ancient times, was it? Well, I think like the Babylonians, I mean, every time you have to do either very big numbers or very small numbers, you know, there's a limit to, you can give every single number a different name, but that, that, that, that, that doesn't scale.

Every single numeral, a different name. Right, right, right, right. So the Babylonians had a base 60 system, for example. You know, the reason why an hour has 60 minutes and a minute has 60 seconds comes from the Babylonians. So you get to 60 and then you start counting again. Right. So you back a zero. Yeah.

So in base 10, when you run out of digits, you come back again. You start stapling the digits together to keep counting. But why do you just say, Terry, that you either have to start with a big number or a small number? Can you elaborate on that a little bit? The way base systems work is that if you have a really, really big number, you try to break it up into tens or in the case of Babylonians, 60s. So if you have a thousand minutes, you would describe a thousand minutes, you break up into hours and it would be like, what, 16 hours or something plus some change. So actually, humans have used multiple base systems. The Babylonians had base 60.

Base 12, there's still remnants of base 12. We talk about dozens and gross. Although gross is very archaic now. We use base 20 score, four score and seven years ago. It's their base 20 system. The French still use base 20 in the number system. Yeah. And computers use base too.

So, you know, binary, zero and one. But does using one base or another hide certain access to discovery or reveal access to discovery? Because that's really what the question is getting at. I think it can slow it down or speed up a little. little bit, but you're still accessing the same numbers. So, you know, like, it's all true that A plus B is B plus A, regardless of whether you use base 10 or base 20 or whatever. So once computers came along, people did experiment with, like, there was a base three system

that the Soviets tried, it didn't work very well. We found that binary works really, really well for computers. And so, yeah, once we had to do computation at massive scale, we found the best base was base two. But base 10 is completely fine for everyday purposes. Hello, Dr. Tyson, this is Rawl. Hello, Dr. Tyson, Professor Tao. I am Rawl. I have a few steps on the sea train from the Natural History Museum. Oh, nice. My question is to all of you, if you could, which means I'll handle this guy. If you could, how would you change the pedagogical landscape for mathematics education? I recall reading an old math text where

question at the end of the chapter had me write a short essay on the behavior of a function rather than do something more mechanical. This exercise is really quite beautiful and left me with far more intuition than if I had been asked to do something else. Yeah, so what is the state of math education? Because there's this whole set, there's a whole demographic of the mathematically walking wounded. In their math class, they got bad grades, the teacher sucked, and now they have no appreciation for math the rest of their lives. Right. It is a tough problem, especially since there's a shortage of really qualified math teachers. It's not a profession that is very appealing, often that they don't often get enough respect or salary for what they do. My theory is that different

people have a different kind of math language. So some people are very visual learners and they like to see lots of pictures. Some people are very narrative driven and they want to see a story. Some actually like working with symbols and solving public. some like playing games, some like being competitive. So there's many, many different ways to access math. But when you teach a class of 30, 40 kids, you can only teach one way. And inevitably, many of the students in that class will not click with them, the style. So if there was some way to have multiple pathways to learn the same material,

maybe outside the classroom, some enrichment activities, I think that would help. Well, you can get into the mechanics of what you need to do to make it more interesting. But at the core of it, and I mean this seriously, and I've said this about Neil, it really is about the emotionality of the person delivering the information, right? If that person is engaged, I had a terrible science teacher in middle school. He smoked cigarettes and he'd be like, all right, we're going to make a battery today. And I always say if I had Neil as a science teacher, it's the only compliment I can give him, is I would probably be in science today because he emanates passion,

and enthusiasm and love and fun. And at that point, you can come up with all the sort of mechanical mechanisms through which you teach math, but if it's being delivered in a dry way by someone who's indifferent or disconnected, it is never going to land on the students. You're saying mathematicians are generally unfun people.

They're all dead to me. I got to be honest. No, no, I don't mean that about, I mean about if any presentation to human beings comes through best, when the person delivering the information, Whether or not it's math, whether or not it's math, science, if you're talking about English, if you're watching somebody interview somebody on TV, you're only compelled in that interview

because by that interview because you're seeing a real relationship between two people, which emanates initially from the host. It's all about emotionality. And then the information comes and is absorbed. Yeah. So, yeah, if the teacher doesn't care, the students won't either. But good teachers are so precious and so rare hard to find. So are you a good teacher?

teacher? I try. Yeah. Time for a couple more questions? Sure, absolutely. This is William Warren. William from Abingdon, yeah, Abingdon, Maryland. Abingdon, Maryland.

When you're working on a very difficult proof, how do you decide whether you're missing a key idea versus simply not pushing far enough with the tools you already have? I love that. Yeah. Because in physics, when we approach a problem, you first lay down all the parameters. And if you're missing a parameter and don't know it, you're not solving the problem. Right. So if you're solving a problem that's never been solved before, how do you gain the confidence

that you have everything necessary at your disposal to actually solve it? So it's really important in math to not just prove positive results, but negative results. So results where you know you cannot prove the thing you want with the hypotheses you have because you can find some counter examples, which satisfy all your hypotheses, but don't satisfy your conclusions. Now, these hypotheses may not be, um, um, correspond to the real world problem that, that you were working with, but by comparing that count example with the reward problem, you can see what, what you're missing. Um, so actually, a lot of math is actually exploring the negative space of what doesn't work

and what you know doesn't work. Um, and it's only after, uh, you, you sort of map out of the negative space. Can you see kind of the, the very narrow path, you could, which dodges all the pitfalls and, and gets you to, to, to your goal. just like say I'm going to write it out cleaner and neater and then the answer will appear. Did you ever just like kind of clarified a little bit? Sometimes, yeah, you can just, what's the modern thing? Like raw dog it or something?

Yeah. Oh, yeah. That's what the young kids say. Okay. But what you did describe was very Sherlock Holmesian because Sherlock Holmes is once you've removed all, everything that's not possible, then all it's left is what is possible. Yeah, I mangled it, but that's the idea. Yeah.

Right. Yeah, so it's good to not be emotionally invested in one outcome that this has to be true or this has to be false. But to actually actively work to prove both the one conclusion, all is opposite. And sometimes you're surprised. Sometimes your initial guess is wrong. And actually the answer is the opposite of what you thought it is. It sounds like at some point you're not doing math. You're just seeing if the universe respects effort.

Yeah. Come on, man. Give it to me. I've tried so hard here. I'll throw your bone on this one. I got a headache. I'm supposed to take my wife to the ballet. I'm not going.

Yeah, you can sometimes feel that these problems have agency and sometimes some malice in some cases. Malice, ooh. We have another one? Yeah, yeah, good. One more. Joel. Hello, Dr. Tyson, Professor Tao.

Joel here from Chambersburg, Pennsylvania. Will a new math system have to be invented the more we explore space, it seems as if there are a lot of places in the universe where our math simply just breaks down. Wow. Okay, so let me give a lead-in, and I want to... Nice intersection of... Let me tee them up here.

So it's not where our math breaks down, which, in fact, it does, but you don't blame the math for that. The math is just a representation of the physical model that we have created for the universe. Right? So I have a physical model. I say I want to represent it using math. And then I do that and I calculate with it,

hey, I'm getting some good answers here. And like, as Einstein did, you get some good answers. And then you reach up the center of a black hole and you end up dividing by zero. And that's no-no. And so the math blows up. But I'm not blaming the math because that was just the math I used

to represent my idea and I need to modify my idea. Don't blame this gentleman here just because your equation of your idea failed. No, no, no, I'd blame him. So in the broader context, could there be simply missing mathematics that in the same way the non-Euclidean geometry, we're scratching our heads over dark matter, dark energy, singularities, and the limits of our physical theories, is there a new kind of math that we're just waiting for you guys to invent that can help us out? That, I believe so. Yeah, so the math we have has become extremely good at

explaining most of the universe. So as long as you're not at extremely very tiny scales, or extremely high temperatures or like a black hole, like the rest of the universe, the math checks out. You know, we can make measurements of galaxies, you know, a billion light years away and all the measurements line up with what our current cross-political models give the math works. But yeah, there are some, the early universe and the center of black holes. The current math is not giving us answers that makes sense. And in physics, I think the biggest problem is that we don't have a theory of quantum gravity, which is the theory that would govern extremely strong gravitational fuels at extremely small scales.

And I think the current theory is that we have to abandon our notions of space and time. Even non-uclidean geometry will not be enough to understand what quantum space time looks like. And Lipin proposal, string theory is the most famous. But nothing has really stuck as being convincingly the answer. One of your own people was right in the middle of this, Ed Witten, right? He's a mathematician who lent his efforts to string theorists. Yeah, he's had a lot of ideas. Unfortunately, there was some string theory has a very pretty math,

but it doesn't seem to be fitting reality as much as the string theorists had hoped. So sometimes even if the math is pretty, it's not the right answer. So we get time for one more question. One more question. This is a good one. This is Hayden from Hawaii. Like Dr. Tyson, my favorite movie is The Matrix. Is there a mathematical way to prove or disprove?

We are not in a simulation. Is anyone working on this? Now, if you answer no, that's exactly what the simulation would want you to say, which means we're in a simulation. Aha! I got you! Come on. You've got to help us out.

Because a lot of people have existential angst over that question. Right, right. It's a great question. We may not have enough. The botanic is not math. So there is a science, like if there's competing hypotheses explaining the world. So you're in a simulation or the universe was created by a deity or whatever. There's a branch of statistics called Bayesian probability,

which can help you guide which of these outcomes is more possible. You have to lay out all the possible scenarios that could be true, assign some prior probability to each being true, and then take all the data that you have. And there's a formula that allows you to take for every observation that you have, update, so some data confirms some theories and makes their probably go up. And some, some, um, some data makes some theories less likely and it probably goes down. And in principle, if you are very, if you could compute everything, you could, you could end up with and say, oh, now I'm

20% confident I'm in simulation or or 80% that the universe is real or whatever. Um, but the problem is that we don't know all the possible different types of universe that could happen. Um, and we don't know what their prior probabilities are. And there's just so many of them, we can't compute how many of them would replicate a war that looks like ours. So while people attempt to do these calculations, there's just so many gaps and the sort of implicit biases in how you choose which universities. Maybe some hypotheses you have implicitly set up to fail or some that you're biased to make succeed. You can't do it because if you're trying to prove that it's fake,

the proof that you have is fake. So you're caught in this loop. So you have a- Why would the proof be fake? Because you're within a simulation that's fake, so the proof within the simulation is fake, right? Isn't that the argument? I mean, it's got the same credibility as like an email from a Nigerian prince at this point, right? That's very dated.

Thank you. How about 10 years ago? How about that? You're still getting Nigerian Prince email? I am. That was like from 1994. He doesn't have any friends. Like two years after email.

If this were fake, the proof would be fake, which means that reality is not reality. So we don't even know if we're in reality. Right. So you could never rule out a hypothesis with 100% certainty because whatever data you have collected could itself be faked, as you said. But it just may take enormous effort. Like if you collect more and more data and it keeps pointing to a different hypothesis that the universe is real, you know, whoever's doing the simulation will have to keep faking more and more data to consistently do a completely different outcome. And at some point, why would they go through so much effort? So there's another pathway into this, which is when you program a world, there's a part of the program where you set up the basic parameters for it. You know, how big is it? How old is it? What's the passage of time? Population.

You just set it up and you take it from there. Well, in our world, there are, we can measure, for example, the energy of cosmic rays. Just take that as a thing. Very, very high energies. You can imagine, suppose the energy distribution has an abrupt cutoff for no obvious reason. Maybe that's the programmer's limit because they didn't think we'd ever get there. It's like a Truman show. Oh, yes.

He's not going to get to the edge. Yeah. So you get to the edge and, oh, my gosh. We didn't think you. The edges of the programmer's parameters. So you see the flaw in the program and then you know you're in a program. Well, the limits of the program. What do you think of that possibility?

Because you can't program infinity, right? Yeah, so it seems like if the universe was a simulation, whoever desired it has great attention to detail. Yes. At really fine scales or something, you still see the same laws of physics that you see. It's not like a cheaply made movie where once you're out of a shot or something, everything's all made in a cardboard.

It wasn't made by a lazy simulator. It was made by a very obsessive simulator, if it was. This is the first time I've been. ever heard the simulator complimented. There you go. Well, maybe we are in a simulator. Maybe we are in a program. I mean, I'm getting endless spam calls about a loan that I supposedly asked about and took

out or whatever. That's a parallel universe. With a Paul McCurio in that universe. Oh, my God. It's three times a day. I'm telling you, we're in a simulation controlled by the banks. The world is feeling less real these days. I think just because there's so much simulated.

everything, unfortunately. Yes, it is. And thank you. And good luck. I think sometimes you need a little bit of that as you explore that moving frontier. Thank you. Yeah. And Paul, always good to have you, man.

Is your show still on the road? Permission to speak? Yeah, my boy show, directed by Frank Oz. We're touring with that. I'm doing my podcast inside out with all the curing. And I love that. And permission to speak, you engage the audience. I do.

I basically... Which is a very important bit of improv. Yeah. So we basically found that, you know, everybody's got a story. and we're in a world where people want and need to tell their stories and people share. And you can add some math to your... Yeah, we'd love to have Terry come on stage. Ladies and gentlemen, welcome the T-Man.

Anything is possible in the simulation. There you go. That's the right answer to everything. All right. This has been StarTalk Cosmic Queries edition on mathematics of all things. Until next time, Neil deGrasse Tyson, bidding you. to keep looking up.