Chapter 1: What is the main topic discussed in this episode?
Tetragrammaton.
I think that mathematics, hard mathematics isn't computational at all. It's almost like art. It's really about figuring out an insight, a non-obvious insight that unlocks a problem. Yeah, and I think the reason that I like it is because It's all about figuring things out. And I think that insight, there's a lot of it that's kind of mechanical.
It's like once you have it, you just turn the crank and you get the proof or you get the answer. Yeah, that special insight can be really, really magical. And to me, it's just about figuring things out. So the reason I actually got into mathematics, if you remember, it's because of physics.
When I was a kid, one of the first books that my dad sort of encouraged me to read was Stephen Hawking's Brief History of Time. And that book deals with kind of the big questions, right? what happened before the Big Bang? Why is the universe the way it is?
Chapter 2: How does Vlad Tenev view the role of mathematics in problem-solving?
You have this constant that measures the flatness of the universe, right? And if it's like, basically like too high the universe is going to expand at an increasing rate and everything is just gonna just die out in this like expansionary death if it's too low the universe is gonna re-collapse again we would have never had galaxies and stars And it turns out if we measure it, ours is exactly one.
It's like immeasurably close to one in this equilibrium between like death by expansion and death by contraction, which is crazy. If it's a little bit off, we would have never existed, right? If the laws of physics weren't exactly what they are, we probably would have never formed anything.
Yeah, because—and also, the universe has to be the exact age where one set of stars would have had to form and die and collapse, and we would have been—because our bodies are made of supernova debris, so there has to have been at least—we have to be second generation in a sense— So those were the questions that kind of motivated me. Like, where did we come from?
Why are the laws of physics the way they are? Is there one fundamental force? Like we know that electricity and magnetism, we found out that those were one force. Radioactive decay, then we figured out was part of that. Then we figured out the strong force.
Chapter 3: What inspired Vlad Tenev to pursue a career in mathematics and physics?
Then the next one is well if you follow that train all of the forces must have been one to begin with so there must have just been like one thing what is that that's what really i think motivated me just figuring that stuff out i think one of the things that i often think about from physics is the double slit experiment article or the wave yeah
Yeah, the double slit experiment is something that just sort of simple to understand, but I never really have been able to come up with, like actually the explanations that I can think of for it are kind of scary to me. Scary in what way? So the double slit experiment basically shows that by nature of us observing something or it even being possible to be observed. Its properties change.
So I think if you study quantum mechanics, which I did, they sort of say, well, everything's consistent, right? The math just sort of explains it. There's no spooky retro causality where you go back in time and change things. But it's just that the observer and... the thing being observed, the subject, are part of one system. And the observer can kind of impact the subject.
Yeah, and so what that makes me think of is Is this some kind of like lazy evaluation, right?
Chapter 4: What is the significance of the double slit experiment in understanding reality?
In programming, you have this concept of lazy evaluation. I don't know what that is. So if you have a computer program, that computer program has a bunch of memory that it can access. But to be efficient, you don't want to claim all the memory right away. So when the program gets to a particular point, Then it claims the memory, only when it's needed.
Let's say your program was simulation of the Earth, and you were just in the middle of the Pacific Ocean. It would only load sea. So you would only see blue ocean around you, and it wouldn't load America yet. But then maybe you get closer, and okay, now you're within 15 miles. You see some coastline on the horizon.
then america appears in the program and that memory is is claimed so that's kind of how people program computers because it's much more efficient i don't want to claim all the memory for the whole earth if you're only here and you only need all of this circle around you to experience the program so yeah what i think of in the back of my mind when i think of the double slit experiment is
does the universe work that way, right? And is it, okay, we're being efficient here. And if nobody's actually observing these particles, they're just a probabilistic haze that just looks like a big wave.
Chapter 5: How does quantum mechanics challenge our perception of observation?
And then only when you look at them, do they exist.
And I think now it's even changed to where you can predict in advance whether it's going to be a wave or a particle. It doesn't have to be in time. It can be
pre yeah yeah it's called the delayed choice quantum eraser yeah yeah where they actually create two particles right one of them goes through the standard path and the other one goes on a very very long journey where they can erase the history they've done this actually over in the canary islands a big multi-island quantum eraser Yeah, and they kind of end up at the same place.
If it's possible through data analysis after the fact to reverse engineer which way the thing went, then there will be no interference. But if you erase it and you make it impossible, then the interference shows up again. It's pretty crazy.
It's crazy. There's so much we don't understand. Do you think things like that are solvable or no?
I think it's worth trying. Yeah, I think so. I mean, that's really the point of physics and why I'm interested in AI models too.
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Chapter 6: What led to the GameStop trading frenzy and its implications?
I think they could be solvable. Are you into puzzles? I like to play around with puzzles from time to time, yeah. But math scratches the puzzle itch for you? I think so. I mean, puzzles... They're a good way to pass time for sure, but I don't think, I think it's different than true understanding. It's almost like a game. How mathematically interested are you?
I'm interested, but I don't know anything about it. I can tell you my favorite simple math problem. I think it illustrates this sort of difference between math being sort of like a computational thing and more artistic conceptual. So this is a triangle, right? We call it ABC, the three vertices. So let's say it's isosceles triangle.
Chapter 7: How did Robinhood navigate the challenges during the GameStop incident?
which means that these two sides have the same length. So one kind of interesting fact about isosceles triangles is that it turns out because these two sides have the same length, these angles are also the same. So what that means, the way you kind of write that is AB equals BC. implies that angle BAC equals angle BCA, right?
And if you look at this, I think the interesting thing is that this is kind of a self-contained fact. You've got these three points. The only thing you're using in this statement is those points. You have AB equals BC. So BAC, which is this angle, is equal to BCA. But then the question is, how do you prove it? Like, why is this true?
And the reason it's true requires you to do something that I can, this is like the, there's one creative step here. And you actually have to construct a new point from scratch. So you say, okay, I can create a point in between A and C, and I'll call it D.
Chapter 8: What future developments does Vlad Tenev foresee in AI and finance?
and this point is the midpoint, right? It's exactly, I can create a midpoint. If these two points aren't the same, there's a line connecting them that has some length. And if I have the midpoint, I can connect it to B and then I get two triangles, triangle ABD and triangle BDC. Now these triangles,
are actually the same triangle because by nature of this being the midpoint, this side is the same as this side, right? And then they're sharing this side. And so you have two triangles that are congruent. They have the same sides. And what that means is that all of the angles are the same too. This angle is the same as that angle. This angle is the same as that angle.
And there you get the fact that these two angles are the same. And this problem, by the way, is not just difficult for some humans, but it was somewhat difficult for AIs too. It's because the first versions of AI geometry solvers would work with the points. Let's say we have three points. You just have A, B, and B, C, and you're not using anything else for the statement.
Where does this point D come from? That's the genius of it. You have to know that by creating this new point, which is a midpoint. You can test. You can create this new additional structure that helps you unlock it, even though this point D doesn't actually appear in the statement anywhere. There's no D in here or D in there. That's called an auxiliary construction.
The first AI models that did really well at solving geometry problems, The only thing they did was figure out what the right auxiliary constructions to make would be. Then the rest of it is just all computational. Obviously, this is a pretty simple example.
In the abstract, the idea of adding a new piece of information that unlocks the picture, even though that new piece of information isn't really part of the equation.
Yeah. That's beautiful. Yeah. I think there's a certain elegance to it for sure. And I think there's crazy things in physics that still blow my mind that nobody can explain. Yeah. And if I had to start another company, I mean, I think harmonic scratches the itch.
There's like a very deep well of if you can just keep going and figuring out all of this stuff, I think you can just keep going forever. But there's also things in the real world that we can't understand and we can't explain. And I think that hearkens back to figuring out why the laws of the universe are the way they are. Yeah. Tell me about your spiritual life.
Yeah, I'm trying to figure that out. I think a lot of my day to day is sort of figuring things out. I mean, the way that this kind of, I think you asked math and physics, how do they influence what I do? I mean, a lot of what I do is really thinking of products, figuring out how people will use those products to have them have as much success as possible.
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