[MUSIC] In the previous part, we have been focusing on from the reality. From the complicated world we extract the laws of nature and we find that the laws of nature are surprisingly simple. As Einstein says, the most incomprehensible thing about our universe, is that it is comprehensible. But now we are going to ask a reversed question that considering that the laws of nature are simple, how the complicated reality emerges from the simple laws? Or alternatively, we can ask the question in another way that considering the laws of nature are simple then, is the reality simple? Or is the reality complicated? Or is the reality of any other kind? This is the question that we are asking. And to be more explicit, let me give you an example that, for example, the laws of nature, one of the simplest, the scientific law of nature, which is Newton's theory of mechanics and gravity. Think about Newton's mechanics, and the most well known example of Newton's mechanics is two massive bodies rotating around each other. And in this case this shows amazingly how the laws of nature the laws of gravity is so simple. So, this is the two body problem. And now, what about we proceed a little bit to ask a little bit slightly more complicated question, naively slightly more complicated question, which is what about I add one more body? How to study the gravitational problem with three bodies in it. In this case, is this their simple or it is complicated? Here, actually the law is very clear that the laws of nature here is very simple. Just write down the equation of motion of these three bodies extremely simple. But can you solve them? Can you find out what is the solution? What is the trajectory for this body? And can you find that out? Actually amazingly difficult I will show that to you. Amazingly difficult. And what about you consider further complicated problems, for example, exponentially many particles. And actually, once you are considering exponentially many particles, you'll find different cases. Sometimes simple, sometimes complicated depending on the details of the force, depending on the details of other conditions. So, to go from the laws of nature, simple laws of nature to our reality. This is a highly nontrivial problem and we have all kinds of possibilities. For example, we have the possibility from simple, too simple. For example, the Newton's two body problem. And we have from simple to more, okay? To more, I mean, for example, Newton's three body problem, for example, study 10 to the 23rd particles interacting with each other. Considering all the galaxies all the stars in our universe interacting with each other. And considering, for example, the atoms and molecules in solid matter for example, and in this case, different laws of nature. Different possibilities emerge. For example, there could be emergent simplicity, for example, thermodynamics. We know that the laws of thermodynamics they are simple, but actually the underlying dynamics. The underlying dynamics is the interaction of 10 to the for example, 23rd atoms and molecules interacting with each other. And emergent simplicity arises for thermaldynamics. And there are also relatively simple things. For example, condensed matter, not all condensed matter is simple. There are so many hard problems in condensed matter physics. But anyway, there are relatively simple laws in condensed matter physics. Again emerges from, for example, the crystal structure and how electrons move interact in these crystals. And how the oscillations of the crystal, the phonons interact with electrons etc. So simple laws emerge from extremely complicated situations. So this is emergent simplicity from simply we are considering more matter and this is very famous saying by understand that more is different. And here we'd like to proceed to another direction to ask another question which is more more different. Is there any other possibilities that if we study more matter that can arise and we do know some possibilities, for example, our universe. In our universe if you look at the large scale structure of our universe, that is pretty complicated. This is because gravity is attractive force. That pressure is usually not enough to prevent the matter from falling So eventually there are black holes etc, complicated objects form and gravity is unstable, gravity is complicated. However, this is kinds of matter governed by gravity. The problem appears intrinsically complicated. But surprisingly, I will show you that in some problems apparently extremely simple, for example, the three body problem of gravity. Just starting three bodies of gravity. And I can show you some even simple problems, that simple problems in that problem's complexity emerges. So surprisingly complex solutions emerges from surprisingly simple equations. I will show you that and that includes for example, look at these images. They are beautiful and naively, how are they made? They may be made of very complicated equations, are they? I will show you they are made by extremely simple equation actually as simple as possible. Nonlinear equations can make you this amazingly complicated, infinitely complicated images. And we can ask many other questions which appears to be simple but actually not simple at all. For example, how long is the coastline of Britain? And essentially we can ask this same question about any other country. How long is the coastline? Is it a simple question. Okay, maybe you think we just measure it, but depending on what kind of ruler you are using to measure it. If you measure with very long rulers or if you're measuring with very short rulers and eventually as short as possible rulers and not saying atomic scale. But classically as short as possible rulers, then you'll find that the lines of coastline of Britain is infinity. Amazing, because the coastline is very complicated shape. And actually you are not even asking the correct questions when you're asking how long is the coastline of Britain? You are making a grammar mistake. Why it is the case? Why complexity arises from apparently so simple question. And in these lectures, and also invite you to think deeper about determinism. Okay, from quantum mechanics, actually, we have already understood that the classical determinism is away. It's gone, it's died, that Laplace, originally, he believes that given initial condition, everything in the universe is determined. But now we know quantum mechanics has uncertainties. So this will destroy the Laplace's determinative. However, you can ask a weaker question in the sense that in some cases if we mseaure a quantum system, we got some uncertainties. But otherwise in usual cases, in usual cases, quantum uncertainties are small. Okay, that is very small. So for classical objects, can we still, in some sense, trust in Laplace's determinism? Just to put a few random number in when we are starting quantum mechanics, is that possible? Can we treat the classical objects with Laplace's determinism? But the quantum uncertainties always hiding under the table that we don't have to consider. Is that possible that determinism is more or less correct? Or if it is impossible that quantum uncertainties emerges and not always under the table, breaks the table and we totally have an uncertain universe. What is the case? This will be the content of complexity