Is the Universe a Math Problem? With Terence Tao

Summary

Podcast Summary: StarTalk Radio

Episode: Is the Universe a Math Problem?
Host: Neil deGrasse Tyson
Guest: Terence Tao (Professor of Mathematics, UCLA)
Date: February 24, 2026
Co-Host: Paul Mercurio

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Overview

In this Cosmic Queries episode, Neil deGrasse Tyson and co-host Paul Mercurio are joined by world-renowned mathematician Terence Tao for an in-depth and lively exploration of the universe as a potential "math problem." The conversation traverses the distinction between pure and applied math, notorious mathematical conjectures, the unexpected applicability of abstract mathematics, the realities of math education, and the possibility that our universe is a simulation. The dialogue is peppered with humor, practical analogies, and illuminating anecdotes, making profound mathematical ideas accessible and engaging.

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Key Discussion Points & Insights

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1. What Mathematicians Do and the Power of Collaboration

[06:29–09:31]

• Terence Tao's Role at IPAM: As Director of Special Projects at the UCLA-based Institute for Pure and Applied Mathematics (IPAM), Tao describes how the institute unites mathematicians, scientists, and industry professionals to tackle cutting-edge interdisciplinary problems (e.g., AI, self-driving cars, faster MRI scans).

  • “Often industry people were working on these problems, but they were blocked by various mathematical obstacles and they needed mathematicians to talk to.” – Terence Tao [07:03]
  • Real-world impact: The collaboration led to modern, tenfold-faster MRI algorithms.

• Importance of Interdisciplinarity: Science is now too broad for any single expert; 21st-century progress depends on cross-disciplinary collaboration.

  • “The universe doesn't compartmentalize science as much as we want it to.” – Neil deGrasse Tyson [09:10]

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2. Pure vs. Applied Mathematics

[09:31–15:49]

• Definitions:

  • Pure math is curiosity-driven, often focused on pattern recognition in abstract systems, unconcerned with immediate application.
  • Applied math sits between pure math and practical engineering, providing methods and tools that can be used to solve real-world scientific problems.

• Example – The Digits of Pi:

  • Obsessing over pi’s digit patterns is a pure math pursuit, and while 99% of all numbers show no interesting patterns, proving this for pi remains unsolved.
  • “The fact that there’s not an interesting pattern is itself an interesting fact.” – Neil deGrasse Tyson [10:15]

• Toy Models and the Spherical Cow:

  • Math and physics often work with “toy models” (like the frictionless ‘spherical cow’) to simplify reality and understand the limits of certain systems.
  • Elegant abstraction helps set upper bounds and uncover universal principles.

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3. Can Applied Math Inform Pure Math?

[16:14–19:15]

• Bidirectional Influence: Sometimes, practical problems or anomalies from science feed back and inspire new lines of inquiry (and new conjectures) in pure mathematics—such as the universality of the bell curve (Gaussian) distribution.
  • “Mathematicians are in some ways exploring other universes, but just very abstract numerical universes.” – Terence Tao [18:20]
  • Quote from Vladimir Arnold: “Mathematics is the part of science where experiments are cheap.” – Terence Tao [19:07]

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4. Infamous Unsolved Problems: The Collatz Conjecture

[21:57–28:15]

• Collatz/Hailstone Conjecture Explained:

  • Take any number; if even, divide by two; if odd, multiply by three and add one. Repeat. Does every number eventually reach 1?
  • Computers have checked trillions of cases—no exception found, but no proof.
  • “You can get vast amounts of complexity from very simple operations… when you iterate even very simple operations over and over, you sometimes get enormous complexity.” – Terence Tao [25:23]

• Brute Force vs. Proof:

  • Checking endless cases with computers never suffices; mathematical proofs are necessary for infinite problems.
  • Tao’s progress: Shows most large numbers rapidly “shrink,” but the core conjecture remains unsolved.
  • “In mathematics, we very much value partial progress.” – Terence Tao [28:04]

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5. The Erdos Problems and Crowdsourced Mathematics

[29:53–35:55]

• Paul Erdos: Legendary, endlessly collaborative mathematician who famously proposed hundreds of small, seemingly simple, yet often difficult problems—some with cash prizes (invariably kept as souvenirs).
• Erdos Problem #1026:
  • Described as a game—arrange stacks of coins so an opponent, taking coins in strictly increasing or decreasing stacks, can’t win too much.
  • “There is order in chaos. That’s almost exactly how it’s described.” – Terence Tao [35:38]
  • Modern mathematical collaboration can be highly decentralized and spontaneous, leveraging the collective power of mathematicians and AIs worldwide.

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6. When Pure Math Finds Real-World Use

[36:03–40:32]

• “Unreasonable Effectiveness” of Mathematics:

  • Wigner’s phrase highlights how abstract math often finds applications decades later.
  • Most surprising example: Non-Euclidean geometry, invented for curiosity’s sake, turned out to be essential for Einstein’s general relativity.
  • “It was almost exactly what [Einstein] needed, almost word for word.” – Terence Tao [39:35]

• Math and Data Compression:

  • Both math and science seek to compress and explain vast data sets by discovering simple, universal laws.

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7. Would a Different Number Base Matter?

[41:14–43:54]

• Question: If humanity used a base other than 10 (e.g., base 12, 20, or even 2), would it fundamentally change our mathematical discoveries?
  • Tao notes different civilizations have used many bases (Babylonians: 60, French: 20, computers: 2).
  • While number representation might slow or speed up discoveries and computational efficiency, the underlying mathematical truths remain invariant:
    • “It’s always true that A plus B is B plus A, regardless of whether you use base 10 or base 20 or whatever.” – Terence Tao [43:04]

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8. The State and Emotion of Math Education

[45:07–48:17]

• Teaching Math as More Than Mechanics:
  • Modern instruction often fails to connect with students’ diverse learning styles—narrative, visual, symbolic, competitive.
  • “If the teacher doesn’t care, the students won’t either… good teachers are so precious and so rare.” – Terence Tao [48:17]
  • Engaged, passionate teaching—regardless of style or subject—is crucial to inspiring future generations.

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9. How Mathematicians Tackle Big Problems

[48:32–51:12]

• When Stuck on a Proof:
  • Mathematicians must find not only positive evidence but also negative proof (counterexamples). Mapping out “negative space” helps isolate viable solution paths.
  • Echoes Sherlock Holmes: Eliminate the impossible, and what remains is the truth.
  • “A lot of math is exploring the negative space of what doesn’t work… then you can see the very narrow path.” – Terence Tao [49:11]

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10. Future Math for a Strange Universe

[51:14–54:22]

• New Math for New Physics:
  • Our current math is highly successful at describing much of the universe, but when it comes to extreme conditions (black holes, the big bang), it breaks down.
  • The search is on for “quantum gravity,” possibly requiring revolutionary new mathematics:
    • “Nothing has really stuck as being convincingly the answer.” – Terence Tao [54:00]

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11. Are We Living in a Simulation?

[54:23–59:02]

• Mathematics and Simulation Hypothesis:
  • While Bayesian probability might, in principle, assign odds to the simulation hypothesis, the inability to enumerate all scenarios and assign prior probabilities defeats rigorous proof.
  • Any proof found within the simulation may itself be simulated, so absolute certainty is unachievable.
  • “Whoever designed [this universe] has great attention to detail… not like a cheaply made movie.” – Terence Tao [58:43]

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Notable Quotes & Memorable Moments

• “Science is just way too broad now… The 21st century is all about collaboration.” — Terence Tao [08:57]
• “Some problems you can only solve by brute force.” — Terence Tao [27:22]
• “String theory has very pretty math, but it doesn’t seem to be fitting reality as much as string theorists had hoped.” — Terence Tao [54:09]
• “Anything’s possible in the simulation.” — Terence Tao [60:17]

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Timestamps for Key Segments

• [06:29] — What is IPAM? & Power of Interdisciplinary Math
• [09:31] — Pure vs. Applied Math Explained; Pi Digits and Patterns
• [13:22] — The Spherical Cow Metaphor in Simplified Models
• [21:57] — The Collatz Conjecture Deep Dive
• [29:53] — Paul Erdos Problems & Collaborative Math in Action
• [36:03] — Pure Math’s Surprising Usefulness in the Real World
• [41:14] — Number Bases & Mathematical Discovery
• [45:07] — Math Education: Challenges and Emotional Connections
• [48:32] — How Mathematicians Know When They’re Stuck (or on the Wrong Track)
• [51:14] — Do We Need New Math for Black Holes and Quantum Gravity?
• [54:23] — Can We Mathematically Prove We’re Not in a Simulation?
• [60:17] — Tao on the Limits of Possibility: “Anything’s possible in the simulation.”

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Tone & Style

• The episode is marked by StarTalk’s signature blend of humor, curiosity, and approachability—making even the densest mathematical concepts feel relatable and engaging.
• Neil, Paul, and Terence Tao maintain a light, respectful, and sometimes playful banter, balancing technical detail with pop-culture analogies and audience engagement.

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In Summary

This episode of StarTalk masterfully weaves together deep mathematical ideas, pop culture, and cosmic curiosity. Terence Tao’s insights illuminate how mathematics both shapes and is shaped by our quest to understand the cosmos. Far from being “just a math problem,” the universe emerges as a profoundly interconnected, evolving puzzle—one in which pure abstraction, practical modeling, and the playful spirit of exploration all play their part. And if it’s all just a simulation? Well, at least the math is beautiful.

Transcript

[Paul Mercurio]

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[Terence Tao]

Cricket Wireless 5G got me moving different, man.

[Paul Mercurio]

Stop playing. You probably downloaded the trailer. Nah, for real.

[Terence Tao]

That's how fast this is.

[Paul Mercurio]

I'm streaming, gaming, scrolling, all the good stuff, man. How much you paying? It's Cricket, so you can count on great value.

[Neil deGrasse Tyson]

Cricket wireless lightning fast 5G speeds at prices that make sense. Switch today. Cricket 5G requires a compatible device and is not available everywhere.

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[Paul Mercurio]

But really drilled down and really applied ourselves.

[Neil deGrasse Tyson]

Cause 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 with right now. This is StarTalk. Neil DeGrasse Tyson, your personal astrophysicist. And we're going to have a cosmic queries edition on the subject of mathematics. Why would you laugh at me like that?

[Paul Mercurio]

Apparently you're going to scare people with this. Everybody thinks there's a math quiz coming up.

[Neil deGrasse Tyson]

That was a Halloween Laugh, wasn't it?

[Terence Tao]

It was.

[Neil deGrasse Tyson]

I got Paul Mercurio here. How you doing, man?

[Paul Mercurio]

I'm good, man. Good to see you.

[Neil deGrasse Tyson]

Yeah. You got your podcast. What was it called?

[Paul Mercurio]

Out with Paul Mercurio side Out.

[Neil deGrasse Tyson]

Did you get Disney permission for that?

[Paul Mercurio]

I did not. Well, thanks for bringing that up. I'm going to be getting sued right

[Neil deGrasse Tyson]

after this, and you are out of a job after May.

[Paul Mercurio]

Yes. Thanks for bringing that up as well. Anybody have any deadly diseases they want to talk about? Yes. The Late show gets. 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.

[Neil deGrasse Tyson]

And you've been with. With Stephen Colbert since the Colbert Report?

[Paul Mercurio]

Since the Daily Show.

[Neil deGrasse Tyson]

The Daily show, yes.

[Paul Mercurio]

I started at the Daily show as one of the original writers, performers there. He came in as a performer. We actually.

[Neil deGrasse Tyson]

You predate him on the Daily Show.

[Paul Mercurio]

I'm old school. I'm OG Wow. 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. Yeah. 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. Okay, so I'll be at your house cutting your lawn for two bucks.

[Neil deGrasse Tyson]

Okay.

[Paul Mercurio]

I hope our guest needs some help. I'll go to California. It doesn't snow out there, but I'll shovel anyway.

[Neil deGrasse Tyson]

All right. And by the way, Baron, you did

[Paul Mercurio]

get the title United Me Baron?

[Neil deGrasse Tyson]

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.

[Paul Mercurio]

Brilliant.

[Neil deGrasse Tyson]

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.

[Paul Mercurio]

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, non finality to it. It can take you everywhere, right?

[Neil deGrasse Tyson]

Yeah, yeah. Everywhere and anywhere and everywhere. Yes, exactly. Yes. So you know what we found? We found, like, a badass mathematician. Ooh, yeah.

[Paul Mercurio]

That sounds like a good movie title. Is he also an assassin?

[Neil deGrasse Tyson]

We cannot divulge that publicly. We have Terrence Tao. Terrence, did I pronounce your name correctly?

[Terence Tao]

That's correct. Yes. Please, please be here.

[Neil deGrasse Tyson]

Excellent.

[Paul Mercurio]

That's wrong. That's not how you pronounce your name.

[Neil deGrasse Tyson]

You are professor of mathematics at ucla. Ucla.

[Paul Mercurio]

Okay. That is not a community college.

[Neil deGrasse Tyson]

No, no. Yeah, yeah. UKLA 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.

[Paul Mercurio]

Can I just ask, Terrence, did they give you the title director of projects and you insisted the word special be put in there? That's what people are saying on social media. I'm just relaying information right now.

[Terence Tao]

No, it predates me, but certainly that title helped me accept the position.

[Neil deGrasse Tyson]

So we can say he has a special set of skills.

[Paul Mercurio]

Cool.

[Terence Tao]

There you go.

[Paul Mercurio]

This guy is definitely badass.

[Neil deGrasse Tyson]

He's got a special set of skills.

[Paul Mercurio]

Well, in a weird way, you could be like the accountant.

[Terence Tao]

Right?

[Paul Mercurio]

That's that movie.

[Neil deGrasse Tyson]

Yes.

[Paul Mercurio]

Where he's mathematically brilliant, but then also could just kill you with two fingers.

[Neil deGrasse Tyson]

Yes, yes. But that is not our guest today in love with the superhero. Right. Okay, so tell us about this center that you direct.

[Terence Tao]

Yeah. So I'm not the main director, but I'm the director of what's called Special projects. 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 electrical engineer and a 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.

[Paul Mercurio]

Can I just say, I recently had an mri. Still took a long time. Maybe you could have done a little better. I mean, I don't have all day, Terrence. I have stuff to do. I want in and I want out. I want it to be as fast as when I microwave soup. Right.

[Terence Tao]

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 your heart or 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 extract more data out of you.

[Neil deGrasse Tyson]

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 mechanical engineering, electrical engineering, Biology, chemistry, physics, body da de da dee da. And they have lunch together, 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.

[Paul Mercurio]

And I would think one of the benefits is that you look at something yourself or with your group, a new set of eyes comes in and brings a different perspective and sees things that you didn't see.

[Terence Tao]

Yeah, exactly. I mean, science is just way too broad now. Maybe 100 years ago it was possible to have pretty decent understanding of every corner of science, but that's basically impossible now. So you have to specialize.

[Neil deGrasse Tyson]

The universe doesn't compartmentalize science

[Paul Mercurio]

as much as we want it to.

[Neil deGrasse Tyson]

Yeah, it's all mixed together.

[Paul Mercurio]

You guys figure this out?

[Terence Tao]

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.

[Neil deGrasse Tyson]

This should be obvious, but let's just. So we're all on the same page, tell us the difference between pure and applied math.

[Terence Tao]

Pure math is curiosity driven math. We see patterns in very abstract things like numbers or shapes, and we just ask, 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.

[Neil deGrasse Tyson]

Would that include people who are obsessed with. With what's going on in the sequence of digits of PI Inference?

[Terence Tao]

Oh yes. That is part of pure mathematics. It turns out not to lead very far because there basically aren't any interesting patterns there.

[Neil deGrasse Tyson]

But did you prove that there are no interesting patterns?

[Terence Tao]

This is actually still an open question.

[Neil deGrasse Tyson]

So it could be that the fact that there's not an interesting pattern is itself an interesting fact.

[Terence Tao]

Right, except that we know that like 99% of all the numbers in the world have no interesting patterns. And the second, only a very small number fraction of numbers that actually have something detectable going on. So the fact that PI is part of the 99% is in retrospect, not exceptionally interesting.

[Paul Mercurio]

Do we have a theory why 99% don't have a pattern?

[Terence Tao]

Yeah, it comes from a law of probability. For the law of large numbers. If you pick a number randomly, just for every digit, you roll a die, a ten sided die, and you select digits at random. The low large number says that 99% or 99.99% of the time, you never see any patterns. 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 300 million Americans, but you can poll a thousand, 2,000. And the law of large numbers actually tells you that as long as you pol a very representative set of people, the outcome of that poll is actually pretty close 99% of the time to the general population.

[Neil deGrasse Tyson]

So I could ask a question that you might be able to answer. How many digits of PI would I have to roll out before I got 10 nines in a row?

[Terence Tao]

That would be about 10 to the 10. So I guess 10 billion.

[Neil deGrasse Tyson]

Okay, all right, we got an answer to that.

[Paul Mercurio]

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? No.

[Neil deGrasse Tyson]

Okay, so now applied math, the world other than mathematicians, care about the answer in applied math, right?

[Terence Tao]

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 those 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.

[Neil deGrasse Tyson]

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.

[Paul Mercurio]

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 tweet cheat that it's still.

[Neil deGrasse Tyson]

Do you mean approximation?

[Paul Mercurio]

Yeah. Well, you know the term is intentional simplified reality, right?

[Terence Tao]

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 mathematicians intentionally simplify reality. They work with what's called toy models. Physicists call them spherical cow type assumptions. Where you want to model a cow, it's 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 legs to the cow. But you start with the easiest cases to get an 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 build 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

[Paul Mercurio]

do, but mine has. That's why I have a bad ticker.

[Terence Tao]

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.

[Neil deGrasse Tyson]

Yeah, the patient doesn't die.

[Paul Mercurio]

Yeah, exactly.

[Neil deGrasse Tyson]

Yeah. 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 then to see how it influences 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.

[Paul Mercurio]

Okay. Where are the legs and where's the cow?

[Neil deGrasse Tyson]

So now if you add legs and a head and a tail, that reduces the number, but it means I have calculated the upper limit to how much milk the cow is gonna make.

[Paul Mercurio]

Because what does that sphere give the sphere?

[Neil deGrasse Tyson]

Exactly. So you can't come in later and say, here, have the cow produce five times.

[Paul Mercurio]

Wait. Having those details of legs, tail and head, it subtracts from the toe.

[Neil deGrasse Tyson]

Yes.

[Paul Mercurio]

That's the detail that you said. Doesn't matter matters.

[Neil deGrasse Tyson]

What I'm saying is it caps how much you can think about that problem. It keeps everybody in the same room.

[Paul Mercurio]

Got it?

[Neil deGrasse Tyson]

Yeah. It's not an unlimited place to guess outcome.

[Paul Mercurio]

Have you found that applied mathematics change a settled pure math theory, has that ever gone back and sort of.

[Neil deGrasse Tyson]

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.

[Paul Mercurio]

Is it Matt Damon writing on a board when he's a janitor? Exactly. Come on.

[Terence Tao]

Well, increasingly there's a conversation going in all directions. So traditionally, pure mathematicians just experiment with patterns. And based on analogies and intuition, they brought up, they propose that some phenomena that they see for one type of pattern also extends to some other setting. But sometimes it comes from the physical world. Physicists notice that something keeps happening. So, for example, there are these laws in physics that seem to be universal, that there are certain distributions. Like the bulk of distribution is an example of a universal distribution. If you plot, say, the high heights of people, or the size of cows, if you wish.

[Paul Mercurio]

Can I just tell you something right now? I want a milkshake in the worst way. I am so craving one anyway.

[Terence Tao]

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

[Neil deGrasse Tyson]

the Internet and we would in physics, I think in pure math. It's a Gaussian curve.

[Terence Tao]

Yes, that's the name for it. Yes.

[Neil deGrasse Tyson]

Named after Gauss. Brilliant mathematician, one of the most brilliant there ever was, I think.

[Terence Tao]

Yeah. So mathematicians did actually find an explanation for why this curve appears all the time, something called the central limit theorem in probability. But there are some other distributions that physicists have discovered that we haven't yet fully explained why they show up so often like gaps between spectral lines. And it's more technical to explain, but yeah, there are laws that physicists and other scientists have found are universal. And mathematicians. 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 alternate universes.

[Paul Mercurio]

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 you can. That's the applied part.

[Terence Tao]

Right.

[Paul Mercurio]

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?

[Terence Tao]

Yeah. So Vladimir Arnold, who is a famous mathematician, he once wrote that mathematics is the part of science where experiments are cheap.

[Neil deGrasse Tyson]

Oh, I like that.

[Terence Tao]

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 big F. But it tells you in theory what you can and what can't do and it sets good targets and then it allows you to allocate the more expensive resources more intelligently.

[Claude AI Announcer]

Startalk Radio is supported by Claude from Anthropic Science isn't about rushing to conclusions. It's about sitting with the uncertainty until it starts to make sense. Claude is the AI built for that kind of thinking and Anthropic committed to not running ads in Claude. So when you're chasing down an idea, there's nothing pulling you towards someone else's agenda. Try Claude for free at Claude AistaTalk and see why problem solvers choose Claude as their thinking partner. This episode is brought to you by Progressive where drivers who save by switching save nearly $750 on average. Plus auto customers qualify for an average of 7 discounts. Quote [e-mail] to see if you could save Progressive Casualty Insurance Company and affiliates national average 12 month savings of 700 by new customers surveyed who saved with Progressive between June 2022 and May 2023. Potential savings will vary. Discounts not available in all states and situations. Alienware is celebrating you during their annual Tech Days event with amazing deals on their newest gear. Save big on the latest gaming PCs like Area 51, their most powerful desktop and take performance to the with Intel Core Ultra processors. Plus don't miss great Tech Day savings on Alienware displays, accessories and more. Shop right [e-mail] deals.

[Terence Tao]

Hey, this is kevin the sommelier and I support startalk on patreon.

[Neil deGrasse Tyson]

You're listening to startalk with neil degrasse tyson. So I got here something called the Collatz Conjecture. What is that?

[Terence Tao]

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 solved us.

[Paul Mercurio]

Well, maybe if you tried a little harder.

[Claude AI Announcer]

We should try.

[Terence Tao]

I've worked on this, too.

[Paul Mercurio]

If you're just a little smarter, playing with spherical cows all day, I could describe it.

[Terence Tao]

I'm just worried. As I said, it could trap some audience members to work on it obsessively.

[Neil deGrasse Tyson]

That's good. We like trapping our people. So tell me about it. So these are the best of the unsolved problems, are the ones that can be described simply.

[Terence Tao]

So the Glass Conjecture is also called the Hailstone Conjecture for a reason. I'll explain later. So it says the following. So you give me a number, your favorite number, 37 or 69 or whatever, and then we do the following. So if your number is even, we divide by two, we make it smaller. So 16 becomes eight. But if it's odd, you multiply by three and you add one. So if you give me five, add. Multiply by five, it goes 15, add one, goes 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, 1, 1 is odd. You multiply 3 and add 1, it becomes 4 again, and you end up in a loop. 1, 42142, 1.

[Neil deGrasse Tyson]

It always takes you back down to there.

[Terence Tao]

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 tested ends up going down to 1, 421-421-4142. But we don't know if all of them do. So it's called the Hailstone Conjecture because there's this oversimplified model of hailstones, again, a spherical cow, that model of hailstone where hail forms because a little ice crystal forms in the clouds. Sometimes currents bring it up where it's colder, and then more ice forms, and then it goes down a little bit, and maybe it melts and the hailstone bounces up and down in the cloud, but eventually it lands to Earth. All the hailstones eventually hit the ground.

[Neil deGrasse Tyson]

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. Yeah. And the bigger the hailstones, the bigger the uplifting air was that kept in there for that long and the bigger

[Paul Mercurio]

insurance claims for your car.

[Terence Tao]

In principle, there could be some really unusually lucky hailstone 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. Occasionally you could defy the statistical laws of physics. In principle, there could be this very lucky number that just always keeps hitting the odd numbers and going up rather than hitting the even numbers to go. It would be like someone who's consistently winning at a casino at a game that's rigged. It's theoretically possible, but we don't know if you can actually do it with an actual number. And that's the collapse conjecture.

[Paul Mercurio]

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 starts with such a basic premise. That's the part that's confounding me and I guess obviously others.

[Neil deGrasse Tyson]

But. But you are now confounded.

[Paul Mercurio]

Thank you very much. Can I have my official spherical cow, please? Yeah, but what is that about?

[Terence Tao]

So it's part of a general phenomenon called chaos. So you can have very simple operations like halving a number if it's even or three, adding one if it's odd. As you say, if you just do it once or twice, it's a very easy operation. A kitten in third grade can. Can implement it. But when you iterate even very simple operations over and over again, you can get vast amounts of 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.

[Neil deGrasse Tyson]

We have computers, we have AI. Why hasn't the Collatz conjecture been affirmed?

[Terence Tao]

Well, because we have to check an infinite number of cases.

[Neil deGrasse Tyson]

We'll get to work on that.

[Paul Mercurio]

I was going to say there's a thing called a computer. Look at my finger just go like that.

[Neil deGrasse Tyson]

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. The SETI Institute did it, where you would upload software and they'd give you data. And while your computer was. You were not using your computer on screensaver, it was using your CPUs to calculate.

[Terence Tao]

Yeah. So there was a project, I think it was called Collats grid, which was exactly that, like city at home, but for collats. And it did extend the numbers so a couple quadrillion. I think 10 to the 18, 10 to the 19 we could do from this crowdsourcing. 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 to use mathematical laws that work for all numbers.

[Neil deGrasse Tyson]

Otherwise it's not elegant and it's not even interesting.

[Paul Mercurio]

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.

[Terence Tao]

Some problems you can only solve by brute force.

[Paul Mercurio]

Are we any closer to solving this conjecture? I mean, how close?

[Terence Tao]

So I worked on this a couple years ago. I proved a result that if you take a large number, like 10 to 15, 10 to 20, whatever, I could show statistically that 99% of all numbers that are very, very big would become very, very small, become much smaller than where they started. I couldn't Show they hit 1, 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 the 100, drops down to 100.

[Paul Mercurio]

You know what? I'm going to use a term from your world. Why don't you apply yourself?

[Terence Tao]

Yeah. In mathematics we very much value partial progress.

[Neil deGrasse Tyson]

They're very good.

[Terence Tao]

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.

[Neil deGrasse Tyson]

Do you like those math movies like the Beautiful Mind and the Matt Damon

[Paul Mercurio]

movie Good Will Hunter? You sit there and yell at the screen. That's not the way you do it. Are you one of those guys?

[Terence Tao]

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, was very cool. I think this is common. I have a brother who did some CGI back in the day. And every time he'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 Gifted 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 unsolved math problem. 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 it was actually some equations from one of my papers. Wow, okay. Two years before the movie was made I actually got an email from a director saying we were making a movie about a gift to kid and 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 from them for two years and then this movie came out so I was caught unawares, but it did actually come for me.

[Neil deGrasse Tyson]

So before we go to our question base, there's one more. Just tell me about the Erdos problems.

[Terence Tao]

Right, so Paul Erds was this Hungarian Methodist, he was rather extreme. So mathematicians have a reputation for being a little idiosyncratic, but he was rather extreme even among mathematicians. 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 two or three thousand papers. He's one of the most prolific mathematicians in history and he was famous for posing problems that he would attach little catch prizes to often. 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 Erdos 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 Erdish

[Neil deGrasse Tyson]

problem, I want to be that famous. I could pay people and they don't cash the checks.

[Paul Mercurio]

Maybe if Mr. Smarty Pants cash the checks he could buy himself a house and not have to sleep in other people's bed living. That's all I'm saying.

[Terence Tao]

There's a biography of Poerdish called the man who Loved Only Numbers and that is a pretty good description of it. I met him once and basically the entire conversation was about math. He was not one for small talk or anything.

[Neil deGrasse Tyson]

From my notes here we have problem 1026. Is that the correct way to say that? And was that an Erdos problem?

[Terence Tao]

Right, so recently there's been a systematic effort to actually. So there's a website which has collected over a thousand problems.

[Neil deGrasse Tyson]

Oh, so this is Just a number. This is rank. It's 1026. Right. That's all that is.

[Terence Tao]

Yeah. So yeah, we just gave each Erdish problem a number and this is the ticket that this number got, this problem got 1026. 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 lots of 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 threw out ideas, there was a discussion forum and people used the latest AIs to gather some numerical evidence. I was involved a little bit and

[Paul Mercurio]

I can ask a question and it's relevant and not off track. So when you're collaborating that extensively,

[Neil deGrasse Tyson]

who

[Paul Mercurio]

is curating all of that? Who is in charge of that? You need someone to sort of, of kind of manage that process. Right?

[Claude AI Announcer]

Yeah.

[Terence Tao]

So surprisingly, it's very decentralized. I mean, it's what there was about five, six people involved and there was just a chat room and we just all spoke, we contributed ideas. It was a very respectful environment. If we had to write, we haven't written a paper, the problem was solved. We haven't decided to actually formally make an 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 very spontaneous and very organic. I actually like this. There's more to collaboration than sort of a more directed top down thing where there's some principal investigator that sort of assigns tasks. We can do that too, but sometimes we get it from the crowd, which is great.

[Neil deGrasse Tyson]

Yeah. And 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, of my childhood math wizardry. So 1026, that number is divisible by both three and nine.

[Terence Tao]

Okay.

[Neil deGrasse Tyson]

Evenly divisible.

[Paul Mercurio]

Leave it to you.

[Neil deGrasse Tyson]

No, no, just add up the digits and what do you get?

[Paul Mercurio]

Okay, you get nine.

[Neil deGrasse Tyson]

Nine. So it's divisible by nine.

[Paul Mercurio]

There you go.

[Neil deGrasse Tyson]

And nine itself is divisible by three, so it's divisible by nine. Did I get that right?

[Terence Tao]

Yes, yes. That is a classic test for divisibility by.

[Neil deGrasse Tyson]

Exactly.

[Paul Mercurio]

But like a meaningful chunk value is forced to line up in some order. Right. Isn't that what this problem says on some level?

[Neil deGrasse Tyson]

Oh yeah. What was. We didn't say what the Problem was.

[Terence Tao]

Ah, okay, yeah. So you can phrase in terms of a game like suppose. 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, 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. 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 bounce up and down in such a way that it's hard for me to find a sequence of coins that go up or sequence of coins that go down.

[Paul Mercurio]

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.

[Neil deGrasse Tyson]

Right?

[Terence Tao]

Right. Yeah. So, yeah. So like, for example, if you.

[Paul Mercurio]

If.

[Terence Tao]

If you only allow 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. Of your coins. But the more columns, the most, 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. What is the. The max you can have from this game? So what is the fair price to charge from me to play this game?

[Paul Mercurio]

So it's like this chaos, in this order, in this chaos. Right. It's like a crazy group chat, but there's only always one guy making sense a little bit.

[Terence Tao]

Yeah, yeah, yeah, yeah, yeah. So it comes actually from this theorem of Erds and another mathematician, Zekevesh, that given any sequence of numbers that 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 and chaos. Actually, that's almost exactly how it's described.

[Neil deGrasse Tyson]

That's beautiful. That's a beautiful concept.

[Paul Mercurio]

Well, I have a beautiful mind.

[Neil deGrasse Tyson]

We should make a movie with that title. So you got questions?

[Paul Mercurio]

I have questions. Great questions, as always. Patreon membership, and they're always terrific. And we're going to jump right in here. This is JKW. Greetings, Professor Tao 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 all the time outside the realm of pure mathematics, but consequently subsequently rather find very useful applications in other fields. What is one of the most surprising examples of this?

[Neil deGrasse Tyson]

I love that.

[Terence Tao]

Yeah, this happens all the time. Eugene Wigner, 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 did, often decades later did scientists realize they were valuable for other things. The most famous example is that non euclidean geometries.

[Neil deGrasse Tyson]

There you go.

[Terence Tao]

People played with notions of curved space not because they thought that the actual world was not euclidean. But.

[Neil deGrasse Tyson]

Wait, wait, back up for a sec. There used to be just regular Euclidean geometry where everything is flat, boring. That's where you have squares with 90 degree angles and triangles that the angles add to 180. There's that. And that has direct applications. Geometry stands for earth measurement. Think about that. You're measuring surfaces. All right, so anyhow. So now.

[Terence Tao]

Yeah, so Euclidean geometry has all these amazing theorems. Like the sum of the angles of triangle is always 180 degrees. That's a classic theorem. But they were very, very hard to prove, very complicated to prove. And mathematicians were trying for a long time to see if there was any simpler way to prove these theorems. Like if you take away some of the axioms of geometry, could you still prove these theorems are at right angles and whatever. And then by doing so, they discovered these non Euclidean geometries where you're on a sphere or a hyperboloid instead of flat space. And now the sum of angles of triangle is not 180 degrees. The area of a circle is not PI r squared. And these are very weird geometries.

[Neil deGrasse Tyson]

And parallel lines converge or diverge. Right, right.

[Paul Mercurio]

Either or. Yeah, yeah.

[Neil deGrasse Tyson]

Depending on the curvature of the space. Yeah, 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 gonna do it anyway.

[Paul Mercurio]

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.

[Terence Tao]

Well, 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? He oh, yeah, there's this non Euclidean geometry stuff. There's this really bright guy, Riemann, who developed this wonderful Riemannian 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.

[Paul Mercurio]

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.

[Terence Tao]

Right.

[Paul Mercurio]

I would imagine.

[Neil deGrasse Tyson]

But that's not what motivates. They're not motivated by that.

[Paul Mercurio]

No, but I mean, once you get in it, you start to go, I know this doesn't apply now, but there's something, isn't it as intuitive at some point?

[Terence Tao]

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 mathematicians, it's mathematical data. In the case of, of scientists, it's physical data. And they want some nice theory or explanation to compress all that data into something that they can understand. And somehow when you compress either theoretical data or experimental data, it often tends to compress to similar looking theories, even if they come from completely different origins.

[Neil deGrasse Tyson]

Next question.

[Paul Mercurio]

Let's move on.

[Neil deGrasse Tyson]

These are good. That's a good one.

[Paul Mercurio]

Yeah, thank you.

[Neil deGrasse Tyson]

From Norfolk, England.

[Paul Mercurio]

England. This is actually from the Turkish Republic of North Cyprus. Whoa, whoa. Yeah, nice. Absolutely, yeah. They have a great Home Depot there. Greetings Jem here from Turkish Revolving North Central. 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, using another base number system all these years?

[Neil deGrasse Tyson]

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.

[Terence Tao]

Was it, well, I think like the Babylonians, I mean, every time you have to do either very big numbers or very small numbers, there's a Limit to. You can give every single number a different name, but that really doesn't scale

[Neil deGrasse Tyson]

every single numeral a different name. Right, right, right, right, right.

[Terence Tao]

So the Babylonians had a base 60 system, for example, the reason why an hour has 60 minutes and a minute has 60 seconds comes from the Babylonians.

[Neil deGrasse Tyson]

So you get to 60 and then you start counting again.

[Paul Mercurio]

Right. So you're back at zero.

[Neil deGrasse Tyson]

Yeah. So in base 10, when you run out of digits, you come back again, you start stapling the digits together to keep counting.

[Paul Mercurio]

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?

[Terence Tao]

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 had 1,000 minutes, you want to describe 1,000 minutes, you break it 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. They're still remnants of base 12. We talk about dozens and gross, although gross is very archaic now. We use base 20 score, 4 score, and 7 years ago it's a base 20 system. The French still use base 20 in their number system, and computers use base too. So binary 0 and 1.

[Neil deGrasse Tyson]

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.

[Terence Tao]

I think it can slow it down or speed it up a little bit, but still accessing the same numbers. So it's always 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. There was a base 3 system that the Soviets tried, didn't work very well. We found that binary works really, really well for computers. And so once we had to do computation at massive scale, we found the best base was base 2. But base 10 is completely fine for everyday purposes.

[Claude AI Announcer]

Startalk Radio is supported by Claude from Anthropic. Science isn't about rushing to conclusions, it's about sitting with the uncertainty until it starts to make sense. CLAUDE is the AI built for that kind of thinking and anthropic, committed to not running ads in Claude. So when you're chasing down an idea, there's nothing pulling you towards someone else's agenda. Try Claude for free at Claude AistaTalk and see why problem solvers choose Claude as their thinking partner.

[Kaeln Coleman]

Hi I'm Kaeln Coleman, winner of Target's HBCU design challenge. This challenge moved me closer to my dream of becoming a fashion designer through mentorship and support. You can find my design, along with creations from other black founders in Target's Black History Month collection.

[Neil deGrasse Tyson]

Quick, choose a meal deal with McValue, the $5 Mcle, the $6 McDouble meal deal or the new $7 Daily Double meal deal.

[Terence Tao]

Each with its own small fries, drink

[Claude AI Announcer]

and Four Piece McNuggets.

[Neil deGrasse Tyson]

There's actually no rush. I'm just excited for McDonald's for a limited time only. Prices of participation may vary.

[Terence Tao]

Not bell for McDelivery.

[Paul Mercurio]

Hello, Dr. Tyson, this is Raul. Hello, Dr. Tyson. Professor Tao, I am Ral. I have a few steps on the C train from the Natural History Museum. My question is to all of you, if you could, which means I'll handle this, guys. 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.

[Neil deGrasse Tyson]

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 teachers sucked, and now they have no appreciation for math the rest of their lives.

[Terence Tao]

Right. It is a tough problem, especially since there's a shortage of really qualified math teachers. It's not a professional profession that is very appealing. Often, that often 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 puzzles. 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's some way to have multiple pathways to learn the same material, maybe outside the classroom, some enrichment activities, I think that that would help.

[Paul Mercurio]

But you can get into the mechanics of what you, 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 gonna make a battery today. And he'd be. 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.

[Neil deGrasse Tyson]

You're saying mathematicians are generally unfun people?

[Paul Mercurio]

They're all dead to me, I gotta be honest. No, no, I don't mean that about. I mean about if any. Any presentation to human beings comes through best when the person delivering the information,

[Neil deGrasse Tyson]

whether or not it's math, whether it's

[Paul Mercurio]

math, science, if you're talking about English, if you're watching somebody interview somebody on tv, you're only compelled in that interview 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.

[Claude AI Announcer]

Word.

[Terence Tao]

Yeah. So, yeah, if the teacher doesn't care, the students won't either. But, yeah, good teachers are so precious and so rare. Hard to find.

[Neil deGrasse Tyson]

So are you a good teacher?

[Terence Tao]

I try, yeah.

[Neil deGrasse Tyson]

Time for a couple more questions?

[Paul Mercurio]

Sure. Absolutely. This is William Warren. William from Abingdon. Yeah, Abingdon, Maryland. Abingdon, Maryland. When you're working on a very, 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?

[Neil deGrasse Tyson]

I love that.

[Paul Mercurio]

Yeah.

[Neil deGrasse Tyson]

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. 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?

[Terence Tao]

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 counterexamples which satisfy all your hypotheses but don't satisfy your conclusion. Now, these hypotheses may not correspond to the real world problem that you were working with. But by comparing that counterexample with the real world problem, you can see what you're missing. A lot of math is actually exploring the negative space of what doesn't work and what you know doesn't work. And it's only after you sort of map out the negative space can you see kind of the very narrow path which dodges all the pitfalls and gets you to your goal.

[Paul Mercurio]

Do you ever just, like, say, I'm going to write it out cleaner and neater and then the answer will appear? Do you ever just kind of clarify it a little bit?

[Terence Tao]

Sometimes, yeah. You can just. What's the modern term? Like raw dog it or something? Yeah, but. Oh, yeah, that's what the young kids say. Okay. But the young.

[Neil deGrasse Tyson]

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 that's left is what is possible. Yeah. Or I mangled it, but that's the idea.

[Paul Mercurio]

Yeah.

[Neil deGrasse Tyson]

Yeah.

[Terence Tao]

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 or it's 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.

[Paul Mercurio]

It sounds like at some point you're not doing math, you're just seeing if the universe respects effort.

[Terence Tao]

Yeah, yeah.

[Paul Mercurio]

Come on, man, give it to me. I'm trying so hard here.

[Neil deGrasse Tyson]

I'll throw you a bone on this one.

[Paul Mercurio]

I got a headache. I'm supposed to take my wife to the ballet. I'm not going.

[Terence Tao]

Yeah, yeah. You can sometimes feel that these problems have agency and sometimes some malice in some cases. Malice.

[Paul Mercurio]

We have another one.

[Neil deGrasse Tyson]

Yeah, yeah. One more.

[Paul Mercurio]

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.

[Neil deGrasse Tyson]

Wow. Okay, so let me give a lead

[Paul Mercurio]

in, and I want a nice intersection.

[Neil deGrasse Tyson]

Let me tee him 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 in the universe. Right. So I have a physical model, and 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 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 had to modify my idea. Don't blame this gentleman in here just because your equation of your idea failed.

[Paul Mercurio]

No, no, no, I'd blame him.

[Neil deGrasse Tyson]

So in the broader context, could there be simply missing mathematics that in the same way the non euclidean geometry we are scratching our heads over dark matter, dark energy singularities and they're 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, us out?

[Terence Tao]

That I, 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 and extremely high temperatures or like a black hole like the rest of the universe, the math checks out. You know, we can make, we can make measurements of, of galaxies, you know, a billion light years away and, and, and, and all the measurements line up with, with what our, our current cosmological models give. The math works. But yeah, early universe and the Sino black holes, the current math is not giving us answers that make sense. And yeah, in physics I think that the biggest problem is that we don't have a theory of quantum gravity, which is the theory that would govern extremely strong gravitational fields at extremely small scales. And I think the current theory is that we have to abandon our notions of space and time. Even non Euclidean geometry will not be enough to understand what quantum spacetime looks like. And lippin proposal. String theory is the most famous. But nothing has really stuck as being convincingly the answer.

[Neil deGrasse Tyson]

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.

[Terence Tao]

Yeah, he's had a lot of ideas. Unfortunately there was some. String theory has very pretty math, but it doesn't seem to be fitting reality as much as string theory as I'd hoped. So sometimes even if the math is pretty, it's not the right answer.

[Neil deGrasse Tyson]

So we get time for one more question.

[Paul Mercurio]

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. Ah, I got you.

[Neil deGrasse Tyson]

Come on, you got to help us out. Because a lot of people are, Are. Have existential angst over that question.

[Terence Tao]

Right, right. It's a, It's a great question.

[Rosetta Stone Announcer]

I.

[Terence Tao]

We may not have enough. The bottleneck is not math. So there is a science, like if, if there's competing hypotheses explaining the world so, you know you're in a simulation, or you know the universe was, 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 up to update. So some data confirms some theories and makes their probability go up, and some data makes some theories less likely, and it probably goes down. And in principle, if you could compute everything you could end up with and say, oh, now I'm 20% confident I'm in simulation, or 80% that the universe is real or whatever. But the problem is that we don't know all the possible different types of universe that could happen, 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 world that looks like ours. So while people attempt to do these calculations, there's just so many gaps. And there's a lot of implicit biases in how you choose which universes. Maybe some hypothesis you have implicitly set up to fail or some that you're biased to make succeed, you can't do it.

[Paul Mercurio]

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 bunch of.

[Neil deGrasse Tyson]

But why would the proof be fake?

[Paul Mercurio]

Because you're within a simulation that's fake. So the proof within the simulation is fake.

[Neil deGrasse Tyson]

Right.

[Paul Mercurio]

Isn't that the argument? I mean, it's got the same credibility as like an email from a Nigerian prince at this point.

[Terence Tao]

Right.

[Neil deGrasse Tyson]

That's very dated. That's like, from 10 years ago.

[Paul Mercurio]

How about Amazon support? How about that?

[Neil deGrasse Tyson]

You're still getting Nigerian prince email. I am. That was, like, from 1994.

[Paul Mercurio]

He doesn't have any friends.

[Neil deGrasse Tyson]

Two years after email.

[Paul Mercurio]

If this were Fake. The proof would be fake, which means that reality's not reality. So we don't even know if we're in reality.

[Terence Tao]

Right. So you could never rule out a hypothesis with 100% certainty because whatever data you. 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, whoever's doing the simulation will have to keep faking more and more data to consistently lead to a completely different outcome. And at some point, it's just. Why would they go through so much effort?

[Neil deGrasse Tyson]

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. 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, we can measure, for example, the energy of cosmic rays. Just take that as a thing. Very, very high energy 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 the Truman Show. Oh, he's not gonna get to the edge. So let's just. So you get to the edge and. Oh, my gosh, we didn't think he could get. It's the edges of the programmer's parameters.

[Paul Mercurio]

So you see the flaw in the program, and then you know you're in a program.

[Neil deGrasse Tyson]

Oh, the limits of the program. What do you think of that puzzle? Possibility. Because you can't program infinity, Right.

[Terence Tao]

Yeah. So it seems like if the universe was a simulation, whoever designed it has great attention to detail.

[Paul Mercurio]

Yes.

[Terence Tao]

At really fine scales or something. You still see the same laws of physics that you see. You know, it's not like a cheaply made movie where once you're out of. Out of shot or something, everything's all made of cardboard. It wasn't made by a lazy simulator. It was made by a very obvious obsessive simulator. If there was, this is the first

[Neil deGrasse Tyson]

time I've ever heard the simulator complimented.

[Paul Mercurio]

There you go. Well, maybe we are in a simulator. Maybe you 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.

[Neil deGrasse Tyson]

A parallel universe with a Paul Mercurial in that, oh, my God, owes money.

[Paul Mercurio]

It's three times a day. I'm telling you, we're in a simulation controlled by the banks.

[Terence Tao]

The world is feeling less real these days, I think, just because there's so much simulated everything, unfortunately.

[Neil deGrasse Tyson]

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.

[Terence Tao]

Thank you. Yeah.

[Neil deGrasse Tyson]

And, Paul, always good to have you, man. Is your show still on the road?

[Paul Mercurio]

Permission to speak? Yeah. My show, directed by Frank Oz. We're touring with that. I'm doing my podcast Inside out with

[Neil deGrasse Tyson]

Paul McCarry, and I'd love to. And permission to speak. You engage the audience.

[Paul Mercurio]

I do. I basically.

[Neil deGrasse Tyson]

Which is a very important bit of improv.

[Paul Mercurio]

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 at the end of the show. Yeah, I would love to have Terry come on stage. Ladies and gentlemen, welcome to T man.

[Terence Tao]

Anything's possible in the simulation. There you go.

[Neil deGrasse Tyson]

That's the right answer to everything.

[Terence Tao]

Great.

[Neil deGrasse Tyson]

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.

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