Friday Science: Complex Number Calculus

This is my post on the seventh chapter.

This isn't going to be long. I generally read the chapter. I didn't really understand the chapter. I think what he was calling contour integrals is probably called a surface integral on this side of the pond.

Here's a quotable: "Complex smoothness throughout some region is equivalent to the existence of a power series expansion about any point in the region" (139). Here's another: "Perhaps the most important unsolved mathematical problem today is the Riemann hypothesis, which is concerned with the zeros of this analytically extended zeta function, that is, with the solutions of ζ(z)=0."

Well, those are things I don't really understand, but they seemed significant. :-) Throughout this chapter I kept thinking about something he said in his preface about being discouraged by his publisher from going so technical. I don't mind him going deep. I don't mind not understanding. I'm just pretty sure that if I understood this chapter, I still could have explained it a lot better.

Friday Science: Real Number Calculus

This is my post on the sixth chapter.

1. I'm sure there is a method to Penrose's madness. He's giving the building blocks and later he'll build. This chapter was about real number calculus. He gave a brief intro to differentiation and integration.

Basically, differentiation is the process of finding a function that describes the slope of another function at every point. For example, the function u=2x describes the slope of the function y=x² at every point. He gives the most basic formulas for figuring these slope formulas out without explanation.

2. Integration goes the other way. If you are told that u=2x is the derivative of some other function, integration figures out that the function in view is y = x² + some constant. In the case above that constant was 0, but 2x could be the derivative of an infinite number of other instances, like y=x² + 523 . Don't worry about it if you don't know calculus. Not important. The integral of a function turns out basically to be the area under it.

absolute value function, Wikipedia

3. He spends a lot of time talking about functions that Leonhard Euler would have liked (pronounced OIL-er, 1707-83). Interestingly, he doesn't really give the definition I learned. He looks at functions that are continuous, functions that are differentiable. He tells us Euler probably wouldn't have liked the absolute value function, step functions, or piecewise functions.

But Euler would have liked the hyperbolic function y=1/x, even though it doesn't have a value for x=0.

4. Apparently Euler liked functions that could be expressed as power series.

from mathwords.com

Again, I'm sure there's a reason Penrose is covering this particular territory. I look forward to finding out what it is. :-)

5. The theta function or Heaviside step function is a function that is 0 for any number less than 0 and 1 for any number greater than zero. He introduces the derivative of the theta function as the Dirac delta function. It looks like it will be significant for quantum mechanics later. The Dirac delta function is zero everywhere except for zero. At zero, the function is like an infinite line going up. The strange thing is that the "area" of that infinite spike is one.

I'm just going to leave it at that.

Friday Science: Geometry of Logarithms

This is my post on the fifth chapter. Well, Roger Penrose is a smart dude. I followed some of the material in this chapter and knew of some of it, but a lot of it is a bit beyond me. Still more fundamentally, I'm not exactly sure why we are talking about the material in this chapter. It's relevant to quantum physics I know, but I can't quite catch the context.

1. So I know the chapter dances with graphs of the addition and multiplication of complex numbers. Penrose makes a connection between the way logarithms behave and the way the graphs of complex numbers work. Complex number graphs 1) have an x axis that is used for the "real" part of a complex number (see my last post for a + bi as the standard form of a complex number, with a as the real part and bi as the "imaginary" part). Then 2) instead of a y axis they have an "i" axis.

2. Then he shifts to treat these graphs in terms of polar coordinates. Polar coordinates treat a point in terms of an angle that is at zero and then moves in a circle in a counterclockwise direction. Then there is a distance "r" from the origin (0,0) to that point. So the point is described in terms of an (r, ϑ) instead of an (x, y). I believe he is doing this because these ways of thinking about imaginary numbers help us get some little grasp of why the equations work the way they do.

3. Logarithms are a way of conceptualizing how exponential functions work, one that was developed in the 1600s. For some reason, I've always had trouble conceptualizing them. I get exponents. So the following relationship makes sense to me: bⁿ=x. Thinking about this relationship logarithmically rearranges the relationship to say that the log_bx=n.

4. Now I understand all these ingredients. I just don't quite get what these things all cook. The next ingredient is the number e (2.718281828...). It is a curious number that is somehow basic to the universe. It can be derived from the following formula:

The logarithm base e is called the natural logarithm (often abbreviated "ln"). Apparently, the logarithms of imaginary numbers can be expressed and graphed in terms of polar coordinates: z=log_er+iϑ.

5. I'm afraid I'm still struggling to have some sense of perspective on a number of other relations that are in this home stretch. e^2πi=1. And imaginary numbers can be put into polar format using the form w=re^iϑ.

6. Another relation he mentions is iⁱ. I can't entirely follow why but this is equal to e^ilogi =0.207879576...

7. The final section implies that there is a direct application of this math to the quantum world. Although I'm still not clear how. :-)

Friday Science: Do parallel lines meet?

This is my second post on Roger Penrose's The Road to Reality.

1. The second chapter is titled, "An ancient theorem and a modern question." The ancient theorem is the Pythagorean theorem (a² + b²=c²). As it turns out, the Pythagorean theorem only works if the old idea that "two parallel lines never meet" is true.

The "parallel postulate" goes back to Euclid, the old Greek (c. 300BC). Euclid was not able to prove it. In the 1700s, a man named Girolamo Saccheri devoted much of his mathematical life trying to prove the parallel postulate by pursuing its opposite. That is, he tried to show that if you assumed that parallel lines did meet, it ended with a contradiction. Perhaps he died feeling a failure, for he did not show any contradiction.

After him, another mathematician named Heinrich Lambert (1728-77) continued to develop a geometry in which Euclid's parallel lines do meet. But Carl Friedrich Gauss (1777-1855) would get the credit. Gaussian geometry, also called Lobachevskian geometry after a Russian who also explored it, is a geometry in which parallel lines do meet. It is thus a "non-Euclidean" geometry.

2. Another name for the type of geometry these individuals were developing is called "hyperbolic" geometry, and the key is that they are not exploring lines on a flat plane but on a curved one. In particular, hyperbolic geometry is the kind of geometry you end up with if you are drawing your lines on a saddle type figure. Penrose and the artist Escher (see picture) had some fruitful interchanges that resulted in several of his drawings that have "tesselations" like the one above.

From a Euclidean perspective, it looks like the fish get smaller and smaller as you approach the edges, but in fact this is an allusion caused by flattening out the saddle. In reality each fish perceives itself to be the same size as all the other fish. Similarly, the boundary is artificial because of the flattening. In reality, the fish go on forever.

3. There are two ways, if I am understanding correctly, of representing this hyperbolic saddle geometry in a flattened, Euclidean plane. The one way is called conformal and the other projective. The conformal representation has all the lines hitting the boundary circle at right angles and the lines are curved. Some sense of proportion is retained.

The other model, the projective, flattens the figure even more. The lines are straight rather than the curved lines of the conformal. The angles are distorted.

So if there is Euclidean geometry (flat), hyperbolic geometry (like a saddle), there is also spherical or elliptical geometry. Parallel lines meet on a sphere and thus defy Euclid's fifth postulate.

4. I won't pretend to understand everything in this chapter. Penrose is trying to be clear but he needs a translator. :-) The reason for the chapter is the fact that physical space exemplifies some non-Euclidean features when we get into Einstein's general relativity. He is laying some ground work for things to come.

There is debate as to whether the universe as a whole is Euclidean, hyperbolic, or elliptical. The majority I believe are currently leaning Euclidean. But I'm guessing that Penrose is a hyperbolic guy.

Friday Science: The Road to Reality 1

I haven't stopped reading Brian Greene's, The Fabric of Reality. I've read almost two more chapters since I last posted on it, the latest of which is on string theory.

But I came across a 1000 page book by Roger Penrose that has distracted me: The Road to Reality. Penrose is a 80+ year old Oxford physicist who worked in the area of cosmology, not least as a mentor to Stephen Hawking. What attracts me about the book is the way that he blends the expansion of mathematical understanding with the expansion of our knowledge of physics.

1. The first chapter is called, "The roots of science." I won't go into much detail but he basically is making an argument that mathematics is objectively true. Math is not a matter of opinion. Fermat's Last Theorem is not true because Andrew Wiles came up with a proof that was pleasing before someone came up with a disproof that was pleasing.  Fermat's Last Theorem was true before Fermat came across it, and it would be true even if no one had yet produced a mathematical proof.

The way Penrose expresses this idea is by affirming Plato. Plato believed that there was an independent reality to ideas apart from the concrete instantiations of them in the world. Penrose isn't wanting to make a big deal of this. He wants us basically to see this as a way of saying that math is objectively true. In the real world, we have approximations, but mathematical ideas are real in a way.

I would personally rather go Aristotle on him. Math is an abstraction of the real world. Math is objectively true but as an abstraction of the concrete world. One corresponds to one thing. Two to two things. Multiplication is multiple addition. Division is multiple subtraction. Exponents are a particular kind of multiplication.

We use a base ten coincidentally because we happen to have ten fingers, but this is just a way of talking about reality. "Natural math" is probably more based in e or pi. These last two paragraphs are my thoughts rather than those of Penrose.

2. The last part of this first chapter presents his sense that the whole of our mental world comes from the physical world, and the whole of math comes from our mental world, and the whole of the physical world comes from the mathematical world. He leaves room for the possibility that there may be left overs. Some of our ideas may be distinct from the physical world. Some of the physical world may be apart from math. Some of math may be independent of our mental world.

Interesting, although not why I bought the book yet.