I'll end this unit by talking a little bit about some ideas from the history of philosophy and science, focusing on Newton and Laplace. This isn't a definitive or comprehensive history by any means. I just want to highlight a few ideas that will help set the stage for the next unit, when we encounter chaos and the butterfly effect. My main goal is not so much to get a historical account, but to look at some historical ideas to help maybe elucidate, or bring to the forefront, some ideas about science, and how science goes about explaining the world, and some assumptions about the world, that I think are still with us today. And these are some ideas that the discovery of chaos and various related phenomena cause us, I don't think to throw out completely, but maybe to re-examine or reconsider a little bit. In any event, our story begins with Isaac Newton. In 1687, Newton published the Principia Mathematica, in many ways the crowning achievement of the Scientific Revolution. So, there's a lot that Newton did in this, but one of the main things he did was, he laid out a theory of motion. He explained why things move. And this is most clearly embodied in what's now known as Newton's Second Law of Motion. You've likely seen this before if you've taken a physics or physical science class. So the equation says – On the left-hand side we have: F⃗_{net}. F is the forces, and net means total. So we're looking at the total forces that are acting on an object. The arrow on top of F means that force is a vector quantity. When considering the total forces, we need to take into account not just how strong a push or pull it is, but the direction of that push or pull. And vectors are a mathematical way for doing that. The right-hand side of the equation is: ma⃗. a is acceleration. It's the rate of change, or the rate of change of the position, or the rate of change of velocity – how fast something's slowing down or speeding up. Like forces, acceleration is a vector quantity. In order to describe motion I need to say not only the magnitude of acceleration but also the direction. And m is an object's mass. So what Newton's second law says is that if I know the total forces acting on an object, and I know its object's mass, I can figure out its acceleration, and from its acceleration I can figure out how it will move. The basic idea is that forces determine motion in the same sense that the functions, iterative functions and differential equations, that we've been studying so far, are deterministic. So Newton's second law is a rule; it tells us how things move if you know the forces that act on them. So the second law is – I think of it as a theory of motion, it says why things move. Another key part of Newton's Principia was that he put forth the idea that the laws of physics are universal. And in many ways I think this is the most revolutionary part of Newton's work. So he said that the laws of physics that describe how an object falls in Cambridge, England, those laws are at the same in Brussels or Beijing or Bogota, anywhere. Then not only anywhere on the Earth, anywhere in the Universe. So the same law that describes how an apple falls in England describes how the Moon orbits the Earth or how the Earth orbits the Sun. So, as I said before, Newton's Principia is in many ways a crowning achievement of the Scientific Revolution. And to summarize in very brief strokes, the idea of the Scientific Revolution is that knowledge can be generated through careful observation and repeated experiment, building on the work of others, and that that knowledge is expressed in a coherent and consistent logical framework that's often mathematical in nature. So Newton in the Scientific Revolution gives way to a period of intellectual history called the Enlightenment, that I think of as spanning most of the1700's. The Enlightenment is associated with expanding individual rights and democracy, continued scientific advances, a belief in the power of logic and reason over authority and doctrine. So during the Enlightenment a scientific method and scientific worldview starts to solidify. And I sort of think of this as a Newtonian framework. So, some ideas about the world and about science that aren't directly contained in Newton's Laws but sort of flow naturally from them. So, in the Newtonian framework, the world is mathematical, mechanistic and material. It's mechanistic because things happen for a reason, objects move because forces act on them. So motion can be explained, things happen not just because, but for particular reasons. It's mechanistic in that the world is seen as being made up of stuff, material objects. Even forces that act apparently at a distance, like gravity, were viewed as being transmitted by a tangible physical medium. So it's a world of things, of stuff, in which things happen for a reason. And lastly the world is seen as being mathematical. The laws of nature, and maybe nature itself, is seen as being mathematical. And Galileo has a quote that summarizes this idea really nicely. So Galileo says: "Philosophy is written in this grand book of the Universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and to read the alphabet in which it is composed. It is written in the language of mathematics. So although that quotation is hundreds of years old, I could imagine a physics professor saying that today. So in the Newtonian framework the world is mechanistic, material and mathematical. The picture that emerges is of the Universe as a giant clock, or a giant machine. Gears and levers pushing on each other, making things happen, making other things happen, and then making other things happen, but always following the laws of motion. The Newtonian world is one of cause and effect. There are laws and a certain orderliness out there. And even if we don't understand all of those laws, the Newtonian world holds out the promise that the world is fundamentally an understandable and logical place.