*These transcripts may not be 100% accurate because guest interviews are automatically transcribed and sometimes voice over is altered in the edit
In 1903 a prodigy was born who grew up to be one of the most brilliant and influential figures of the 20th century.
Nobel prize winner Hans Bethe said this about him
“I have sometimes wondered whether a brain like his does not indicate a species superior to that of man”
The father of the hydrogen bomb, Edward Teller said
“He would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us.
Nobel Prize winner Enrico Fermi said to one of his PhD students.
You know, how much faster I am in thinking than you are. That is how much faster he is compared to me.
They were talking about John Von Neumann.
My name is Lowell Brillante and this is prodigy
John was born with a photographic memory common to prodigies. He grew up in an affluent hungarian household that contained a small library of literature.
He obtained a degree for chemical engineering at the same time as his PhD in mathematics. After studying under David Hilbert he became the youngest professor ever at the University of Berlin and published 12 papers in 12 months.
Then he accepted a position at Princeton and taught there for 3 years until he was offered a lifetime professorship at the Institute for Advanced Study where he worked alongside the likes of Einstein and Oppenheimer.
Although Von Neumann and Einstein were intellectual peers. The 2 could not have been more different.
Another member of the institute said quote: “Einstein’s mind was slow and contemplative. He would think about something for years. Johnny’s mind was just the opposite. It was lightning quick—stunningly fast. If you gave him a problem he either solved it right away or not at all. If he had to think about it a long time and it bored him, his interest would begin to wander. And Johnny’s mind would not shine unless whatever he was working on had his undivided attention.”
Von Neumann’s personality did not fit the stereotype of a quiet academic. He was a notorious practical joker and the life of the party with a mental library of jokes at his disposal. He was known as an occasional heavy drinker, whimsical, sarcastic and a terrible driver. Lastly he was described as a charming man prone to small acts of kindness who was highly admired by people who knew him.
Let’s look at some of his main contributions. Anyone of them alone is incredibly impressive. Combined it’s difficult to argue that there was a more brilliant person in the last century.
If you want to learn more about his life, Stuff you Missed in History Class has a great episode on him.
I needed an expert to help understand and explain these concepts so I reached out to Daniel Whiteson. He’s a professor at the University of California at Irvine and does experimental particle physics,
Daniel: Hi, I’m Daniel Watson. I’m a professor of particle physics at the university of California at Irvine. And I do experimental particle physics, which means that I smash protons together, the large Hadron Collider to try to figure out the fundamental nature of matter. And I’m the co-host of the podcast, Daniel and Jorge, explain the universe where we explain where we talk about all of the crazy bonkers things that we do and don’t know about our universe.
Daniel and Jorge Explain the Universe has been one of the top science podcasts in the world for years. It’s an audio library of interesting and complex information about how the universe works broken down in a way that is understandable and entertaining.
Von Neumann’s work in theoretical physics provided the foundation for Daniel’s field of research.
Back then physicists were seeing results in their experiments which couldn’t be explained using classical physics.
Daniel: And what Von Neumann did, which is really important is that he took all these crazy new ideas about how the universe might work. And he made them work mathematically. You like lay the mathematical underpinning so that we could talk about them and build on them and agree on what they meant.
He put together this idea of operator theory of quantum mechanics and the core idea there is that you think about the nature of a quantum particle. Not Hey, it’s a wave waving through space, but instead thinking about it in terms of an abstract space, like a quantum particle and its quantum state is like a point in some abstract space, an abstract space is like a space that’s not physical, a space that’s physical is like the space we live in has three dimensions forward and backwards up and down left and right.
He said, what if there’s Abstract dimensions to space that we can add on top of that. And they could have like complex numbers, right? They can be like three I, or, two minus one, I, or stuff like that. So he used this kind of space is called a Hilbert space. And he said, what if a quantum particles, quantum state is a point in that space.
So that specifies what the quantum state is. You have the three dimensions that you normally have. Plus you have these other. Weird abstract dimensions that you can add, and you can add as many dimensions as you can, as you like. So these Hilbert space is going to be like infinitely dimensional. And the whole point of operator theory is a quantum state is like a point in that space.
And then when you do something to the particle, when you like measure an electron. What you’re doing is you’re applying an operator to that point. It’s like when you take a vector, just like a normal vector, you have mathematically and you multiply it by a matrix. What do you get? You get another vector.
So matrix times a vector gives you another vector. And so the operator theory says, take a point in that Hilbert space, when you can make a measurement on it, you’re doing an operator on it. For example, You have an electron particle. So at a point in Hilbert space, you want to know it’s momentum. There’s a momentum operator that operator acts on the particle.
It gives you back the momentum, but then it also changes the particle. And so you often hear, for example, that in quantum mechanics, you can’t, you. That you often hear in quantum mechanics that you can’t get information out of a particle without interacting with it without changing it. This is the mathematical foundation of that.
He said, think of the particle as a point in this space, it’s already natural mathematically, you know that when you. Operate on a point in space when you operate on a vector which is a point in some high dimensional space with a matrix, you’re going to change that, just like if you take a vector and you apply a rotation matrix to it, you change the direction of that vector.
So this gives us like the mathematical foundation helps us understand why it is that when you measure something about a particle, you’re changing it because you’re performing an operation on the particle and he built the mathematical foundation for that. And that’s really powerful because in quantum mechanics, you got to have the mathematical foundations because we don’t have the intuition.
We can’t say this makes sense. And that doesn’t make sense because none of it makes sense. So you got to have the mathematical foundations to tell you what you can do and what things mean, and then to build up from there. So without that work, it would have been very hard to make really any progress in quantum mechanics.
And I love these stories when somebody mathematically invent some new tool and they don’t have an idea for like, why it’s useful or helpful, or even like relevant to our universe. And then somebody else comes along and says, Hey, this actually totally helps us with a problem in physics.
Another interesting concept from Von Neumann is Game Theory. Game theory is a mathematical model of strategic interaction among decision makers. It’s so universal that it has become an overall term for the science of logical decision making. So far no less than 11 game theorists have won the nobel prize.
Daniel: von Neumann took a mathematical view of basically everything. And I think he was watching people playing games and he was wondering, can you come up with an optimal algorithm for playing this game? And then he realized that’s like a. Bigger question, like in general, how do you find a strategy to play a game?
And what strategy are people using? Because people have been playing games long since long before we had game theory, but he invented this field, let’s analyze games mathematically and think about whether there are optimal strategies and how to find. The optimal strategy for a new game. And this is pretty awesome.
I love when people just create whole new field of inquiry, realizing that there’s a bunch of complicated decisions. People are making that. They’re not thinking about it mathematically and then bringing the mathematical tools to bear on that can really help you solve that problem. And so his big idea, there was this strategy called mini max, which when you say it sounds pretty obvious.
But it really was a big step forward. And some of these big ideas sound obvious in hindsight because now we’re used to them. But the concept of mini max was basically that you should make the moves, find the strategy that minimizes. The maximum possible score for your opponent. So say for example, you’re playing chess.
You want to play your moves to prevent your opponent from like capturing your queen or checking your King. That’s basically what you’re doing already. If you’re a good, intuitive chess player, but say you’re not a good, intuitive chess player say, or you’re a super powerful computer with no insight into chess.
You’ve never played a game before. Have you teach a super powerful. How do you teach a super powerful computer or like an alien to play chess? Will you tell them, make your moves so that your opponent can’t capture your, so make your moves so that your opponent can’t capture your queen right? Or protect your King.
That’s essentially minimizing the maximum score of your opponent. And that same basic strategy applies to almost anything which is pretty awesome.
Game theory in its most basic form describes optimal strategies for 2 players competing against each other in a closed environment, like chess. But it can also be applied to all sorts of fields that include decision making like economics. Here’s an example. You want to buy a TV and a business needs to sell TV’s. You want to buy it at the lowest possible price and the business wants to sell it at the highest possible price. If it’s priced too low the business won’t make a profit. If it’s priced too high you won’t buy it or you’ll buy it somewhere else. The business needs to set the optimal price to compete with you as well as other businesses.
Daniel: I think this led a lot of people to think about economics from a more mathematical point of view for a long time, economics was also qualitative, but over the last few decades or so there’s been a big. But over the last, half century or so, there’ve been a lot of folks thinking about economics quantitatively.
And to do that, you have to have like mathematical underpinning, which means you need a model for like how people are making economic decisions. Do you buy that pair of shoes when they’re a hundred dollars? Why did you buy it at a hundred and not buy it at 120? And to do that, you need an understanding of how they’re making those decisions.
And game theory is a basic foundation for that. Also, they’re weighing this against that. If I buy this, can I buy this other thing? And so it’s how people are strategizing using their money is very closely related to game theory. And folks like John Nash, for example, basically built on game theory to help understand a lot of things about economics.
We’re gonna get into the Manhattan project, mutually assured destruction, and computers right after a quick break.
Welcome back to prodigy.
Von Neumann also was a part of the Manhattan project which was responsible for the research and development of the atomic bomb. He was the leading expert on shaped charges and designed the explosive lenses needed to compress the plutonium core.
Daniel: The key thing here is that to get your bomb, to explode, you need your fuel to have a critical mass and a critical density. You don’t want it to have critical mass and density before it goes off, right? So you need to keep your fuel like separated or lower density while you’re like transporting it.
And you want your bomb to go off at a certain moment. And the idea is. And one of the ideas was to make the fuel, have the critical mass and density basically by surrounding it by an explosion. So you have a sphere of your plutonium or whatever, if you could have a sphere prickle implosion around it that compressed it, then suddenly it would have the density and the mass necessary.
To go critical to start that chain reaction and you get your nuclear explosion, that’s the goal. But then the question is how do you make like a perfectly spherical implosion around your fuel? You can create a little explosion, a little spherical explosion, just have a little mini bomb that gives you an explosion, but then you need to like, Have a bunch of these explosions scattered around your fuel in just the right way so that when the Wayfront hits the fuel, you get a nice spherical implosion.
And that’s not that easy. If you just put a bunch of tiny little explosions around it, you’ll get uneven surfaces when it hits the surface of your. You’ll get uneven pressure when it hits the surface of your fuel. So what he came up with were these ba sically explosive lenses, which shaped the direction of these wave fronts.
And you have like high speed and low speed explosions. So that went so that they add up. And when they get. So they add up in just the right way so that when it gets to the surface of your, so when it gets to the surface of your radioactive fuel, it’s applying equal pressure and you get like a consistent implosion, which is what triggers your nuclear explosion.
2 months after Germany surrendered, the Manhattan project produced 2 types of nuclear bombs. Japan still refused to surrender so the allied forces dropped bombs on Hiroshima and Nagasaki, killing hundreds of thousands of people, many of whom were civilians. Japan surrendered 6 days later.
Daniel: So I grew up actually in Los Alamos, New Mexico, where the Manhattan project took place. And when I grew up, all these stories about developing the nuclear weapons, there were heroic stories. People were told, they were told in the context of these men saved the world with their big ideas.
And it was only later that I realized like, Wow. These men invented weapons that could destroy everybody on the planet many times over and we’re terrifying and, also drop them on civilians and like basically kill the quarter of a million people at a time it’s horrifying. So there’s a lot of really interesting and difficult angles there.
And my parents worked at the labs and some of them were involved in. And sometimes they were involved in weapons questions. So I thought a lot about this kind of stuff. And it’s actually the reason that I ended up in particle physics rather than in a more applied area of physics, because I didn’t want any sort of moral question involved in my research.
Von Neumann knew that the Soviets were close behind us in weapons development so he fast tracked the development of inter continental ballistic missiles capable of delivering hydrogen bombs. Once the Soviets got them as well he pioneered the concept of MAD which stands for Mutually assured destruction. It’s a deterrence theory based on the concept that the threat of retaliation prevents an attack. So if the soviets launched 100 missiles at us we would fire 100 back before they hit.
Daniel: He didn’t want to do anything that people could turn into a weapon that could tear rise populations and slaughter people. But if I know him and thought about this a lot, because he helped develop these weapons. And of course he had thought a lot about game theory, right? And so if you had the weapons and the opponents have the weapons, you need to think about, what’s going to make our opponents fire, their weapons.
What’s going to keep them from marching, their weapons. And so he developed this idea of mutually assured destruction. And he gave it that name, I think on purpose to have a silly acronym, mad, he was a fan of silly acronyms. And the idea basically is if you have so many weapons and they have so many weapons that any war will end in everybody dead, then there’s no point in starting the war.
And so you can build all these weapons, point them at each other, and then nobody will die. And that’s the concept of mutually assured destruction, but it has a pretty big, basic flaw in it, which is that it only works if your opponent believes that you can launch your weapons after they have launched theirs, right?
If they can launch their weapons and just obliterate you, then they will. And they will win. So they need to, so to be prevented from being motivated to launching their weapons, they need to believe that once they launched there’s you still have a window to launch yours, right? Otherwise they have no incentive not to launch theirs, which means of course, that our weapons have to be on like a hair trigger.
The president needs to be able to launch weapons within two minutes of hearing that nuclear weapons are on their way, which means there’s no time for like careful review or congressional action or anything basically means that whoever’s the president has to have the button and has to be able to make a quick decision within minutes about whether to kill millions and millions of.
People, and this is terrifying and there’s been lots of times in our history when we’ve been like on the verge of starting a nuclear war, because we had misinformation that suggested that maybe the Russians had sent their nukes. And so this is just terrifying to me. And I think it’s a prank and this is just terrifying to me. And I think it’s unsustainable that we could have these weapons pointed at each other forever and not accidentally fire them at each other. So I think it’s bonkers.
Another societal staple Von Neumann influenced that you’re using to listen to this podcast right is the computer. He came up with the architecture for the cpu, memory and input/output process of modern computers.
Daniel: So von Neumann even was around when computers were basically invented and he had some of the really important foundational ideas just for like, how do you organize a computer? And these days, all the computers that we use. Use his idea. And his basic idea was that you have a computer program that has instructions.
It tells the computer what to do, like the instructions say, load this thing from memory. Add this piece of memory to the app, that other piece of memory store the result here in that memory, those are instructions. And also computer programs have data, right? You want to calculate the trajectory of your nuclear missile.
You have to. Input the angle and the velocity and all that kind of stuff. And the sort of idea at the time from some folks at Harvard, was it, you should keep the program, the instructions and the data totally separate. And the Harvard architecture for computers had those things totally separate. But Norman was like, now you should combine that stuff.
You should have all the program instructions and the data together, all inside the computer memory and that’s simpler. You don’t have to have two different kinds of memory program, memory, instruction, memory, and then. Also data memory. And it also is key because it allows the computer to basically reprogram itself.
If the computer can write to memory and its instructions for what to do are in memory, then it can modify its own instructions. All right. You can change what it’s been programmed to do. And so this flexibility, this power is really what led to the Von Neumann architecture for computing flourishing and dominating.
Basically all of computing. We have almost no. Okay. We have almost no computers these days that use the alternative architecture, the Harvard architecture. And so that’s a pretty foundational contribution. He’s basically the father of computing after Alan Turing.
He also applied that computer expertise to weather. He wrote the first climate modeling software and used it to compute the first numerical weather forecast.
He was one of the early people to figure out like how to take a mathematical question, that something you need to calculate and turn it into a computational one. These days computers are so powerful. You can almost just describe what you need to do.
And the computer will figure it out. You can write functions directly into your program, but back then, computers were simple tools you need to think about computers is like very. You need to think about computers back then as like very stupid. You need to think of computers back then as like very stupid assistants that never get tired.
So if you can take your problem and break it up into little pieces that you’re stupid, but never tiring assistant can do for you, then you can make some progress, but that translation is not always easy, turning a hard mathematical problem. Into a series of simple steps is not trivial. And so that’s what he was pretty good at.
And he definitely helped develop strategies for calculating whether on early computers, he helped on the, any actives in the fifties developing strategies for calculating. Developing strategies for solving differential equations that you needed to understand like how energy flows through the atmosphere on those early computers?
That was pretty cool. Although the first ones were so slow that like he could predict tomorrow’s weather, but it would take two days. And so his predictions were much more like post addictions in the beginning.
Von Neumann architecture has a bottleneck in the time and energy required to transfer data between the memory and processor. Recently there have been large investments made in an attempt to create an improved architecture.
We’re gonna get into cellular automata, self-replicating spacecrafts and singularity right after a quick break.
Welcome back to Prodigy.
Von Neumann and Stanislaw Ulam devised a model to calculate liquid motion by considering liquid a group of units with each unit being affected by its neighboring units. This is the basis of what is known as cellular automata.
Daniel: Cellular automata is a super duper cool idea. Basically it says, take a bunch of really simple objects that follow simple rules. And often these things are called finite state machines because these objects just have a few possible States, maybe they can be like move forward or turn left or turn right.
Something like that. And you take a bunch of these things following very simple rules and you put them together. And what are really complex patterns emerging, and this is something really deep about our universe. It mirrors like the way we think the whole universe. He’s put together. Cause you know, for example, every atom is only made out of a few kinds of particles, right?
Protons, neutrons, and electrons. And, but you put those together. You can make a hundred different elements, which are all pretty different and you mix those. It goes together and you can make an incredible variety of stuff out there in the universe, right? From stars to hamsters, to LA two ice cream, it’s all made out of the same basic.
Bits. So this concept that you can take a bunch of simple things that follow simple rules and from them, you can, and from them, you can see emerge really complex behavior is really deep and philosophically important in our universe. And he saw this sort of mathematically and computationally, and he built these little games where he would develop these cellular automata little things that like lived in a square.
Little things like lived in a pixel on an image and then followed rules and you might be familiar for example, with the game of life. It says, like a pixel is like a little organism and it lives if it has, if it’s surrounded by the right things and it dies under certain circumstances and you just let this play out and you get all sorts of really fascinating complex self-sustaining patterns that emerge.
It’s also a model used in epidemiology to calculate the spread of a virus like covid and estimate the benefit of social distancing and face masks.
Daniel: And it’s the foundation. Also, for example, Stephen Wolfram’s theory of everything, he thinks that the whole universe is built out of the, he thinks that the whole universe is built out of these little cellular automata follow a few basic rules, but it’s a really powerful idea, right?
It’s the powerful idea of emergence that all the complexity in the universe doesn’t have to be coded directly. Into the smallest level stuff. It can emerge later. For example, we can understand the physics of a baseball, which has 10 to the 27 particles in it without understanding what each of those particles are doing because simple properties emerge, even when you have a huge number of these things, all working together.
And it’s amazing that we can understand the universe at various levels that, that what emerges, isn’t just chaos and noise. But some sort of like simple, understandable things. And that’s what he saw in cellular automata that these basically little objects put together following basic rules result in complex behavior that you can understand at a higher level.
So to me, that’s one of the deepest contributions he made and something that really gives us insight about the nature of reality.
Cellular automata also parallels the transmission of genetic information. This was before the structure of DNA was even discovered.
Daniel: And the functions of the cell, right? Everything in the cell is like a tiny little machine and it’s just doing its job and together they do this incredible thing of being a cell and a DNA is similar to that because it’s like, it’s the language that you use to describe proteins. And so it’s a very simple language, there’s only four bases. And from that, you can just buy the organization of them just by the order of the basis. You can get basically any protein that exists inside life. And then those things come together to do obviously incredible things.
When you think of our first encounter with intelligent life, what do you see?
Will Seti receive a radio transmission in an alien language?
Will a massive spaceship fly into our solar system and destroy us?
Or would a civilization so advanced be friendly?
Would they suddenly appear on earth or are they already here?
Or maybe will we be the ones to discover them.
Space is vast and the distances between stars are massive. If an intelligent civilization wants to explore the billions of solar systems in the galaxy, they can’t feasibly send colony ships to each and everyone. We don’t have the resources.
A von Neumann probe is a self-replicating spacecraft and it’s a way we could explore the galaxy. We’d launch it to the nearest solar system. When the probe arrived it would mine materials then create copies of itself. Those copies would then go to other solar systems and repeat the process. This concept takes advantage of exponential growth and theoretically would be the fastest way to explore the galaxy.
Daniel: I love Von Norman probes. It’s fascinating. It’s wonderful. The thing that I love about them and this idea that you could build a probe, which goes out and makes more of itself, it’s very mathematical strategy because it relies on exponential growth, right? You only need to build a few of these probes and they go out there.
They make more of themselves. And pretty soon, even if your civilization, you could only afford to make five of these things, pretty soon you could have 5,005 million of them exploring the galaxy, because one of the frustrating things about the galaxy is that it’s huge. So if you want to find like another intelligent race of beings, there might only be a few of them out there.
It seems like it would take forever to explore all the billions of planets. But if you have a probe that grows exponentially, that builds more of itself, right? Each one builds two or five or 10 more of itself. Then very quickly you could have enough to explore the whole galaxy. There are calculations that suggest that it would only take half a million years to visit every single planet in the galaxy.
If you set a few of these probes off and that’s incredible because half a million years, while it seems like a long time, it’s actually. It’s actually a really short time compared to the age of the galaxy, which is in, 10, which is more than 10 billion years. And that immediately begs the question like if this is possible, then why hasn’t it happened?
This is the basis of the fermi paradox. Which asks the following question, considering that there are so many habitable planets in our galaxy, why haven’t we already seen signs of intelligent life?
Why have. It’s all right. Why hasn’t some alien species had this idea, build those probes and then sent them out there to visit us. Why haven’t we been visited by a Von Norman probe? So nobody knows the answer to that question.
Frank Tipler argued that based on the age of the galaxy and a moderate estimate of replication, these probes should be common. The absence of them solves the fermi paradox. The answer is that we are alone… But Carl Sagan disagreed.
Daniel: Carl Sagan actually said that if we could develop on Norman probes and we shouldn’t because they would run wild and pretty quickly they would consume all the matter in the galaxy in some after just a few million years, most of the matter in the galaxy would be Von Neumann probes. And so it’s definitely dangerous. He needs to build in some turnoff mechanisms. So they don’t just run wild and consume everything. But it’s a deep question. Like, why haven’t we been visited? It wouldn’t take that long.
And maybe we have been, and maybe there’s The wreckage or evidence for Ivano even probes somewhere on the earth, or if we wait long enough, we will be visited again. It’s a really deep and fun question. And one of my favorite things is a follow-up idea by a science fiction author called Bracewell.
He said, not only should you build one of these probes, you should equip it with artificial intelligence. So that when it arrives at some alien planet, you can have an intelligent conversation with those folks. You can answer their questions, they could ask reasonable and interesting questions before beaming the data back to earth, because you don’t want to get the data 50,000 years after it’s arrived.
And then have a follow-up question. That’s obvious that your probe didn’t ask or didn’t think to inquire about. So you build a little bit of AI into it and you can like really gather some useful information when it finds those aliens.
If you’re as fascinated with AI equipped von Neumann probes as I am, I highly recommend a recent science fiction book titled “We are Legion” by Dennis Taylor which explores the concept in a fun and interesting way. The probes were also the original reason for the monoliths in 2001 a space odyssey but Kubrick cut the scene.
Von Neumann also came up with the concept of technological singularity which theorizes that the accelerating progress of technology will reach a point that is beyond control which we cannot coexist with. Stephen Hawking was not alone in his concern that the development of true artificial intelligence could lead to our eventual demise. A poll of AI researchers suggested a 50% probability we’ll develop it by the year 2045. If the AI is capable of improving itself, it could lead to an intelligence explosion. The results of which are unknown.
This isn’t even all of what Von Neumann accomplished during his remarkable career but it’s too much to list.
In 1953 he was diagnosed with cancer and had trouble accepting his fate. He said, quote
“So long as there is the possibility of eternal damnation for nonbelievers it is more logical to be a believer at the end,”
He was referring to pascal’s wager which argues that people bet their souls on whether or not god exists. If you believe and are wrong the consequences are less than if you don’t believe and wrong. It’s an argument for religion based on decision theory.
2 years after being diagnosed with cancer he lay on his deathbed and entertained his brother by reciting passages from Faust. A German tragedy where a man trades his soul to the devil in exchange for knowledge.
I’ve studied now Philosophy
And Jurisprudence, Medicine,—
And even, alas! Theology,—
From end to end, with labor keen;
And here, poor fool! with all my lore
I stand, no wiser than before:
I do not pretend I could be a teacher
To help or convert a fellow-creature.
No dog would endure such a curst existence!
Wherefore, from Magic I seek assistance,
That I may detect the inmost force
Which binds the world, and guides its course;
John Von Neumann was under military guard at the time of his passing to ensure he didn’t disclose classified information while heavily medicated. He was only 53 at the time which surely deprived the world of many more brilliant insights.
Prodigy was created and produced by me, Lowell Brillante. Tyler von Klang came up with the theory of executive podcast producing. Professor Daniel Whiteson is a particle physicist and host of the podcast Daniel and Jorge Explain the Universe which is a fun and interesting way to learn all the things we know and don’t know about the universe.
Thank you so much for listening and please subscribe to the show, because I’ll be back next week, with another episode of Prodigy.