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.

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And now, here's my conversation with Grant Sanderson.

the universe, do you think their mathematics is different than ours?

There's an obvious sense, the notation is different.

I think notation can guide what the math itself is.

I think it has everything to do with the form of their existence.

Do you think they have basic arithmetics or are you interrupt?

Daishi Yeah, so I think they count, right?

I think notions like 1, 2, 3, the natural numbers, that's extremely, well, natural.

That's almost why we put that name to it.

As soon as you can count, you have a notion of repetition, right?

Because you can count by 2, 2 times or 3 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 that it captures the idea of continuity.

You could very well, you know, model the universe in motion of planets with that as the back end of your math, right?

And you still have kind of the same interface with the front end of what physical laws you're trying to, or 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 surface of what the different possibilities are if you have a completely different mode of thought, right?

Or a mode of interacting with the universe.

And you think notation is a key part of the journey that we've taken through math.

I think that's the most salient part that you'd notice at first.

I think the mode of thought is gonna influence things more than like the notation itself.

But 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, I would be comfortable saying.

Do you have a favorite or least favorite piece of notation in terms of its effectiveness?

Yeah, yeah, well, so least favorite, 1 that I've been thinking a lot about that will be a video, I don't know when, but we'll see.

We write the function e to the x, this general exponential function, with a notation e to the x that implies you should think about a particular number, this constant of nature, and you repeatedly multiply it by itself.

And then you say, oh what's e to the square root of 2?

And you're like, oh well we've extended the idea of repeated multiplication.

That's all nice, that's all nice and well.

But very famously you have like e to the pi I and you're like well we're extending the idea of repeated multiplication into the complex numbers yeah you can think about it that way in reality I think that it's just the wrong way of notationally representing this function, the exponential function, which itself could be represented a number of different ways.

You can think about it in terms of the problem it solves, a certain very simple differential equation, which often yields way more insight than trying to twist the idea of repeated multiplication, like take its arm and put it behind its back and throw it on the desk and be like, you will apply to complex numbers, right?

I don't think that's pedagogically helpful.

And so the repeated multiplication is actually missing the main point, the power of e to the x?

Yes, I mean what it addresses is things where the rate at which something changes depends on its own value, but more specifically it depends on it linearly.

So for example, if you have like a population that's growing and the rate at which it grows depends on how many members of the population are already there.

It looks like this nice exponential curve.

It makes sense to talk about repeated multiplication because you say how much is there after 1 year, 2 years, 3 years, you're multiplying by something.

The relationship can be a little bit different sometimes where let's say you've got a ball on a string, like a, like a game of tether ball going around a rope, right?

And you say, its velocity is always perpendicular to its position.

That's another way of describing its rate of change as being related to where it is, but it's a different operation.

That's what the whole idea of like complex exponentiation is trying to capture, but it's obfuscated in the notation.

When what it's actually saying, like if you really parse something like e to the pi I, what it's saying is choose an origin, always move perpendicular to the vector from that origin to you.

Then when you walk pi times that radius, you'll be halfway around.

It's kind of the, you turn 90 degrees and you walk, you'll be going in a circle.

That's the phenomenon that it's describing, but trying to twist the idea of repeatedly multiplying a constant into that.

Like I can't even think of the number of human hours of like intelligent human hours that have been wasted trying to parse that to their own liking and desire among like scientists or electrical engineers or students everywhere, which if the notation were a little different or the way that this whole function was introduced from the get-go were framed differently, I think could have been avoided, right?

And you're talking about the most beautiful equation in mathematics, but it's still pretty mysterious, isn't it?

Like you're making it seem like it's a notation, no.

I think the notation makes it mysterious.

I don't think it's, I think the fact that it represents, it's pretty, it's not like the most beautiful thing in the world, but it's quite pretty.

The idea that if you take the linear operation of a 90 degree rotation and then you do this general exponentiation thing to it, that what you get are all the other kinds of rotation, which is basically to say, if your velocity vector is perpendicular to your position vector, you walk in a circle, that's pretty.

It's not the most beautiful thing in the world, but it's quite pretty.

The beauty of it, I think, comes from perhaps the awkwardness of the notation somehow still nevertheless coming together nicely because you have like several disciplines coming together in a single equation.

Well, I think in a sense, like historically speaking.

You've got, so like the number E is significant.

Like it shows up in probability all the time, it like shows up in calculus all the time, it is significant.

You're seeing it sort of mated with pi, this geometric constant, and I like the imaginary number and such.

I think what's really happening there is the way that e shows up is when you have things like exponential growth and decay, right?

It's when this relation that something's rate of change has to itself is a simple scaling, right?

A similar law also describes circular motion.

Because we have bad notation, we use the residue of how it shows up in the context of self reinforcing growth, like a population growing or compound interest.

The constant associated with that is awkwardly placed into the context of how rotation comes about because they both come from pretty similar equations.

And so what we see is the E and the PI juxtaposed a little bit closer than they would be with a purely natural representation.

I would think, here's how I would describe the relation between the 2.

You've got a very important function we might call X that's like the exponential function.

When you plug in 1, you get this nice constant called e that shows up in like probability and calculus.

If you try to move in the imaginary direction, it's periodic and the period is tau.

So those are these 2 constants associated with the same central function, but for kind of unrelated reasons, right?

And not unrelated, but like orthogonal reasons.

1 of them is what happens when you're moving in the real direction.

One's what happens when you move in the imaginary direction.

They're not as related as the famous equation seems to think it is.

It's sort of putting all of the children in 1 bed and they'd kind of like to sleep in separate beds if they had the choice, but you see them all there and, you know, there is a family resemblance, but it's not that

So actually thinking of it as a function is, is the better idea.

The constant E sort of stands as this numerical representative of calculus, right?

So at the very least, there's a little cognitive dissonance using a constant to represent the science of change.

It makes sense why the notation came about that way.

Because this is the first way that we saw it in the context of things like population growth or compound interest, it is nicer to think about as repeated multiplication.

That's definitely nicer, but it's more that that's the first application of what turned out to be a much more general function that maybe the intelligent life your initial question asked about would have come to recognize as being much more significant than the single use case, which lends itself to repeated multiplication notation.

Let me jump back for a second to aliens and the nature of our universe.

Do you think math is discovered or invented?

So we're talking about the different kind of mathematics that could be developed by the alien species.

The implied question is, yeah, is math discovered or invented?

Is, you know, is fundamentally everybody going to discover the same principles of mathematics?

So the way I think about it, and everyone thinks about it differently, but here's my take.

I think there's a cycle at play where you discover things about the universe that tell you what math will be useful.

And that math itself is invented in a sense, but of all the possible maths that you could have invented, it's discoveries about the world that tell you which ones are.

So like a good example here is the Pythagorean theorem.

look at this, do you think of that as a definition or do you think of that as a discovery?

From the historical perspective, right, it's a discovery.

Because they were, that's probably because they were using physical object to build their intuition.

And from that intuition came the mathematics.

So the mathematics was in some abstract world detached from the physics, but I think more and more math has become detached from, you know, when you even look at modern physics from string theory to even general relativity, I mean, all math behind the 20th and 21st century physics kind of takes a brisk walk outside of what our mind can actually even comprehend.

Anything beyond 3 dimensions, maybe 4 dimensions.

No, no, no, higher dimensions can be highly, highly applicable.

I think this is a common misinterpretation that if you're asking questions about like a 5 dimensional manifold, that the only way that that's connected to the physical world is if the physical world is itself a five-dimensional manifold or includes them.

Well, wait, wait, wait a minute, wait a minute.

You're telling me you can imagine a five-dimensional manifold?

I would make the claim that it is useful to a 3 dimensional physical universe, despite itself not being 3 dimensional.

So it's useful, meaning to even understand a 3 dimensional world, it would be useful to have 5 dimensional manifolds.

Yes, Absolutely, because of state spaces.

But you're saying there in some deep way for us humans, it does always come back to that 3 dimensional world for the usefulness of the dimensional world.

And therefore it starts with a discovery, but then we invent the mathematics that helps us make sense of the discovery in a sense.

Yes, I mean, just to jump off of the Pythagorean theorem example, it feels like a discovery.

You've got these beautiful geometric proofs where you've got squares and you're modifying the areas.

If you look at how we formalize the idea of 2D space as being R2, right?

All pairs of real numbers and how we define a metric on it and define distance.

We've defined a distance so that the Pythagorean theorem is true.

So then suddenly it doesn't feel that great.

But I think what's going on is the thing that informed us what metric to put on R2, to put on our abstract representation of 2D space came from physical observations.

And the thing is, there's other metrics you could have put on it.

We could have consistent math with other notions of distance.

It's just that those pieces of math wouldn't be applicable to the physical world that we study because they're not the ones where the Pythagorean theorem holds.

So we have a discovery, a genuine bona fide discovery that informed the invention, the invention of an abstract representation of 2d space that we call R2 and things like that.

And then from there, you just study R2 as an abstract thing that brings about more ideas and inventions and mysteries, which themselves might yield discoveries.

Those discoveries might give you insight as to what else would be useful to invent and it kind of feeds on itself that way.

It's not that math is 1 of these or it's 1 of the others.

At different times it's playing a different role.

So then Let me ask the Richard Feynman question then along that thread.

Is what do you think is the difference between physics and math?

There's a kind of intuition that physicists have about the world that's perhaps outside of mathematics.

It's this mysterious art that they seem to possess, we humans generally possess.

And then there's the beautiful rigor of mathematics that allows you to, I mean, just like as we were saying, invent frameworks of understanding our physical world.

So what do you think is the difference there?

Well, I think of math as being the study of like abstractions over patterns and pure patterns in logic.

And then physics is obviously grounded in a desire to understand the world that we live in.

I think you're going to get very different answers when you talk to different mathematicians because there's a wide diversity in types of mathematicians.

There are some who are motivated very much by pure puzzles.

They might be turned on by things like combinatorics.

And they just love the idea of building up a set of problem-solving tools, applying to pure patterns, right?

There are some who are very physically motivated, who try to invent new math or discover math in veins that they know will have applications to physics or sometimes computer science.

Like chaos theory is a good example of something that's, it's pure math.

A lot of the statements being made, but it's heavily motivated by specific applications to largely physics.

And then you have a type of mathematician who just loves abstraction.

They just love pulling into the more and more abstract things, the things that feel powerful.

These are the ones that initially invented like topology and then later on get really into category theory and go on about like infinite categories and whatnot.

These are the ones that love to have a system that can describe truths about as many things as possible, right?

People from those 3 different veins of motivation into math are going to give you very different answers about what the relation at play here is.

Because someone like Vladimir Arnold, who is this, he's written a lot of great books, many about like differential equations and such.

He would say math is a branch of physics.

And of course he was studying like differential equations related things, because that is the motivator behind the study of PDEs and things like that.

But you'll have others who like, especially the category theorists who aren't really thinking about physics necessarily.

It's all about abstraction and the power of generality.

And it's more of a happy coincidence that that ends up being useful for understanding the world we live in.

And then you can get into like, why is that the case?

It's sort of surprising that that which is about pure puzzles and abstraction also happens to describe the very fundamentals of quarks and everything else.

So why do you think the fundamentals of quarks and the nature of reality is so compressible into clean, beautiful equations that are for the most part simple, relatively speaking.

So you have, we mentioned somebody like Stephen Wolfram who thinks that sort of there's incredibly simple rules underlying our reality, but it can create arbitrary complexity.

I'm asking a million questions that nobody knows the answer to.

I, it could be the case that there's like a filter iteration at play.

The only things that physicists find interesting are the ones that are simple enough.

They could describe it mathematically, but as soon as it's a sufficiently complex system, like, Oh, that's outside the realm of physics.

You know, maybe there's something where it's like, of course there will always be some thing that is simple when you wash away, the like non-important parts of whatever it is that you're studying.

Just from like an information theory standpoint, there might be some like, you get to the lowest information component of it.

But I don't know, maybe I'm just having a really hard time conceiving of what it would even mean for the fundamental laws to be intrinsically complicated.

Like some set of equations that you can't decouple from each other?

Well, no, it could be that, sort of we take for granted that the laws of physics, for example, are for the most part the same everywhere, or something like that, right?

As opposed to, sort of an alternative could be that the rules under which the world operates is different everywhere.

It's like a deeply distributed system where everything is just chaos.

Like not in a strict definition of chaos, but meaning like just it's impossible for equations to capture, for to explicitly model the world as cleanly as the physical does.

I mean, we almost take it for granted that we can describe, we can have an equation for gravity,

We can have equations for some of these basic ways the planet's moving.

Just the low level at the atomic scale, how the materials operate at the high scale, how black holes operate, but it doesn't, it seems like it could be, there's infinite other possibilities where none of it could be compressible into such equations.

It's also weird, probably to the point you were making, that it's very pleasant that this is true for our minds.

So it might be that our minds are biased to just be looking at the parts of the universe that are Compressible and then we can publish papers on and have nice e equals mt squared equations,

Well, I wonder would such a world with uncompressible laws allow for the kind of beings that can think about the kind of questions that you're asking?

Right, like an anthropic principle coming into play in some weird way here?

I don't know, I don't know what I'm talking about at all.

Or maybe the universe is actually not so compressible, but the way our brain evolved, we're only able to perceive the compressible parts.

I mean, we are, so this is a sort of Chomsky argument.

We're like really limited biological systems, So it totally makes sense that we're really limited little computers, calculators, that are able to perceive certain kinds of things and the actual world is much more complicated.

Well, but we can do pretty awesome things, right?

Like we can fly spaceships, and we have to have some connection of reality to be able to take our potentially oversimplified models of the world, but then actually twist the world to our will based on it.

So we have certain reality checks that physics isn't too far afield, simply based on what we can do.

Yeah, the fact that we can fly is pretty good.

It's a pretty good proof of concept that the laws we're working with are working well.

So I mentioned to the internet that I'm talking to you, and so the internet gave some questions, so I apologize for these, but do you think we're living in a simulation, that the universe is a computer, or the universe is a computation running on a computer?

What I don't buy is, you know, you'll have the argument that, well, let's say that it was the case that you can have simulations.

Then the simulated world would itself eventually get to a point where it's running simulations.

And then the, the second layer down would create a third layer down and on and on and on.

So probabilistically, you just throw a dart at 1 of those layers, we're probably in 1 of the simulated layers.

I think if there's some sort of limitations on like the information processing of whatever the physical world is, like it quickly becomes the case that you have a limit to the layers that could exist there, because the resources necessary to simulate a universe like ours clearly is a lot, just in terms of the number of bits at play.

And so then you can ask, well, what's more plausible?

That there's an unbounded capacity of information processing in whatever the like highest up level universe is, or that there's some bound to that capacity, which then limits like the number of levels available.

How do you play some kind of probability distribution on like what the information capacity is, I have no idea, but I don't, like people almost assume a certain uniform probability over all of those meta layers that could conceivably exist when it's a little bit like a Pascal's wager on like, you're not giving a low enough prior to the mere existence of that infinite set of layers.

But it's also very difficult to contextualize the amount.

So the amount of information processing power required to simulate like our universe seems like amazingly huge.

And we're easily humbled by basically everything around us.

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Grant Sanderson: 3Blue1Brown and the Beauty of Mathematics | Lex Fridman Podcast #64