Grant Sanderson: 3Blue1Brown and the Beauty of Mathematics | Lex Fridman Podcast #64
Lex Fridman (host), Grant Sanderson (guest), Lex Fridman (host)
In this episode of Lex Fridman Podcast, featuring Lex Fridman and Grant Sanderson, Grant Sanderson: 3Blue1Brown and the Beauty of Mathematics | Lex Fridman Podcast #64 explores grant Sanderson Reveals How Notation Shapes Our Mathematical Reality Lex Fridman and Grant Sanderson (3Blue1Brown) explore the nature of mathematics, debating whether it is discovered or invented and how alien civilizations might develop very different, yet overlapping, math. Sanderson argues that mathematical notation and chosen abstractions profoundly shape how we think, teach, and even misinterpret core ideas like the exponential function and Euler’s formula. They contrast mathematics and physics, discuss why the universe appears compressible into simple equations, and touch on topics like the simulation hypothesis and infinity. The conversation also dives into Sanderson’s creative process, the pedagogy of problem-solving and visualization, and the deep aesthetic beauty—and lingering mystery—of objects like the Riemann zeta function.
Grant Sanderson Reveals How Notation Shapes Our Mathematical Reality
Lex Fridman and Grant Sanderson (3Blue1Brown) explore the nature of mathematics, debating whether it is discovered or invented and how alien civilizations might develop very different, yet overlapping, math. Sanderson argues that mathematical notation and chosen abstractions profoundly shape how we think, teach, and even misinterpret core ideas like the exponential function and Euler’s formula. They contrast mathematics and physics, discuss why the universe appears compressible into simple equations, and touch on topics like the simulation hypothesis and infinity. The conversation also dives into Sanderson’s creative process, the pedagogy of problem-solving and visualization, and the deep aesthetic beauty—and lingering mystery—of objects like the Riemann zeta function.
Key Takeaways
Notation can mislead as much as it illuminates.
Sanderson argues that writing the exponential function as e^x over-emphasizes repeated multiplication and obscures its true nature as the solution to a simple differential equation governing processes where rate of change is proportional to value, especially in the complex plane.
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Math is both discovered and invented in a feedback loop.
Physical observations (e. ...
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Understanding emerges from concrete examples before abstractions.
Effective teaching, Sanderson says, starts with specific, visual, low-level instances (like actual arrows for vectors) and only then introduces general definitions, letting the learner’s brain infer patterns rather than beginning with formalism.
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Problem-solving and teaching dramatically deepen learning.
Reading or watching lectures is not enough; working through exercises and trying to explain or teach concepts (even via code or videos) solidifies understanding far more than passive exposure.
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The apparent simplicity of physical laws may be partly a selection effect.
They suggest that physicists focus on phenomena that admit compressible, elegant descriptions, and that minds like ours may only be capable of perceiving and modeling the ‘simple’ slice of a possibly more complex reality.
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Infinity and high-dimensional objects are manageable as abstractions, not mental pictures.
Sanderson frames infinity as the property of ‘always being able to add one more’ rather than an actually completed totality, and says higher-dimensional spaces are useful via state spaces, even if we cannot directly visualize them.
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Mathematical beauty often lies in deep structure plus lingering mystery.
Sanderson finds the Riemann zeta function especially beautiful because it tightly links simple entities (natural numbers, primes) in non-arbitrary ways yet still contains vast unsolved mysteries, unlike Euler’s formula which he feels he now fully ‘sees through.’
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Notable Quotes
“Calculus is the study of change, so there’s a little cognitive dissonance using a constant to represent the science of change.”
— Grant Sanderson
“I think notation actually carries a lot of weight when it comes to how we think about things, more so than we usually give it credit for.”
— Grant Sanderson
“It’s not an either/or. It’s not that math is one of these or it’s one of the others. At different times, it’s playing a different role.”
— Grant Sanderson on math being discovered vs. invented
“Things that are too arbitrary, it’s just hard for those to feel beautiful… you feel like you’re speaking to patterns themselves or nature itself.”
— Grant Sanderson
“Explanation is great… I remember about 10% of what I read and about 90% of what I teach.”
— Grant Sanderson
Questions Answered in This Episode
How would our understanding of core concepts like exponentials or complex numbers change if we completely redesigned mathematical notation from scratch?
Lex Fridman and Grant Sanderson (3Blue1Brown) explore the nature of mathematics, debating whether it is discovered or invented and how alien civilizations might develop very different, yet overlapping, math. ...
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If aliens had a fundamentally different sensory and cognitive apparatus, which parts of our mathematics do you think they would still inevitably discover?
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To what extent are we missing entire domains of useful mathematics because our current physical theories and technologies don’t yet demand them?
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How can mainstream math education be restructured to prioritize problem-solving, visualization, and example-first explanations over formal definitions?
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What current mathematical objects or conjectures (like the Riemann zeta function) do you expect future generations to view as ‘obvious’ in hindsight, and why?
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Transcript Preview
The following is a conversation with Grant Sanderson. He's a math educator and creator of 3Blue1Brown, a popular YouTube channel that uses programmatically animated visualizations to explain concepts in linear algebra, calculus, and other fields of mathematics. This is the Artificial Intelligence Podcast. If you enjoy it, subscribe on YouTube, give us five stars on Apple Podcast, follow on Spotify, support on Patreon, or simply connect with me on Twitter @lexfridman, spelled F-R-I-D-M-A-N. I recently started doing ads at the end of the introduction. I'll do one or two minutes after introducing the episode, and never any ads in the middle that can break the flow of the conversation. I hope that works for you and doesn't hurt the listening experience. This show is presented by Cash App, the number one finance app in the App Store. I personally use Cash App to send money to friends, but you can also use it to buy, sell, and deposit bitcoin in just seconds. Cash App also has an investing feature. You can buy fractions of a stock, say $1 worth, no matter what the stock price is. Brokerage services are provided by Cash App Investing, a subsidiary of Square and member SIPC. I'm excited to be working with Cash App to support one of my favorite organizations called FIRST, best known for their FIRST Robotics & Lego competitions. They educate and inspire hundreds of thousands of students in over 110 countries, and have a perfect rating on Charity Navigator, which means the donated money is used to maximum effectiveness. When you get Cash App from the App Store or Google Play and use code LEXPODCAST, you'll get $10, and Cash App will also donate $10 to FIRST, which, again, is an organization that I've personally seen inspire girls and boys to dream of engineering a better world. And now, here's my conversation with Grant Sanderson. If there's intelligent life out there in the universe, do you think their mathematics is different than ours?
(laughs) Jumping right in. I think it's probably very different. There's an obvious sense. The notation is different, right? I think notation can guide what the math itself is. Uh, I think it has everything to do with the form of their existence, right?
Do you think they have basic arithmetics? Sorry to interrupt.
Yeah. So I think they count, right? I think notions like one, two, three, the natural numbers, that's extremely, well, natural. That's almost why we put that, uh, name to it. Um, as soon as you can count, you have a notion of repetition, right? 'Cause you can count by two two times or three times. And so you have this notion of repeating the idea of counting, which brings you addition and multiplication. I think the way that we extend to the real numbers, there's a little bit of choice in that. So there's this funny number system called the surreal numbers-
Mm-hmm.
... that, um, it captures the idea of continuity. It's a distinct mathematical object. You could very well, you know, model u- the universe and motion of planets with that as the backend of your math, right? (laughs) And you still have kind of the same interface with the front end of what physical laws you're trying to... or what, what physical phenomena you're trying to describe with math. And I wonder if the little glimpses that we have of what choices you can make along the way based on what different mathematicians have brought to the table is just scratching the service, surface of what the different possibilities are if you have a completely different mode of thought, right? Or mode of interacting with the universe.
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