Friday, October 6. 2006Book review  A quintuple of math books
In the last few years I spent my summer breaks writing all the code I couldn't write during the semester. This year it was all different. You might have guessed it already because I wrote just a single update since July. Except for the Khallenge I don't think I actually wrote any serious code. So what have I done in the mean time? I read a lot. Mainly math and economics, some history and anthropology and even one book that is vaguely related to programming (How would you move Mount Fuji; it has some awesome riddles).
This post merely serves to prove that I'm not dead yet. I merely didn't actually do anything I could present on this website during the last months. The math books might fit in here best, so let's review them. It's not quite "Programming Stuff" but I always liked writing book reviews and I haven't reviewed anything in a while. Over the summer I read seven math books. Well, technically I read only five because I'm not yet done with two but I'm making good progress. The titles of these books are "Foundations and Fundamental Concepts of Mathematics", "Concepts of Modern Mathematics", "The Equation that couldn't be solved", "The Nothing that is", "Introductory Graph Theory", "Categories and Computer Science", and "Basic Category Theory for Computer Scientists". The first four books put various aspects of math in its proper historical context. That's something I always missed in school. Ergo I need to spend my free time to learn about these things. The last three books are about actual math. For the sake of the structure of this post I'm not going to review the books in the order I read them. Instead I'm going to review them in the order that makes most sense. It's probably the order you should check them out if you feel like reading them. Let's begin with "Foundations and Fundamental Concepts of Mathematics". Originally I wasn't aware of this book. I wanted to read "Concepts of Modern Mathematics". I shied away from reading "Concepts of Modern Mathematics" first though. I felt I wasn't prepared to read that book without knowing how the earlier math developed. Amazon suggested to buy "Concepts of Modern Mathematics" together with "Foundations and Fundamental Concepts of Mathematics". I liked that idea and bought both. I read the "Foundations" book before the "Modern" book to get an idea of the math before the 20th century. "Foundations and Fundamental Concepts of Mathematics" is a nice book. It's only like $11 on Amazon. You can't go wrong with a 4.5 stars book for $11. Especially when the 1star review  a complaint that the author didn't mention India or China  is not really relevant. At least not for me. On a bit more than 300 pages the book gives a whirlwind introduction about the development of math between Euclid and Russell/Whitehead/Gödel. Between Euclidian geometry and modern set theory the author touches on subjects like nonEuclidian geometry, analytic geometry, algebraic structure (group theory), the introduction of axioms to various branches of mathematics, the development of various number systems, and the development of set theory. The book closes with a chapter about logic and philosophies of mathematics. I liked the book. It was written on a level I could comprehend. Most of the time at least. Sometimes it was over my head. There are just some aspects of math I care less about. I think most people who finished high school with decent grades in math could understand the book. The book has exercises too. Lots of exercises in fact. Maybe an overwhelming number of exercises. And thankfully the book also contains hints on how to tackle the exercises. I hate books that give you no way to check if you came to the correct solution. I didn't actually do many exercises. My goal was an idea of what came before the topics covered in "Concepts of Modern Mathematics". The book did this well and I moved on to the next book. The next book is "Concepts of Modern Mathematics". I wanted to read this book because I felt that I didn't actually have a good grasp of what mathematicians do today (except for being weird hermits that solve Millenium Prize Problems, that is). One day I noticed that all my math classes end at the beginning of the 20th century. That's just how it is if you study computer science. Well, that's actually not really true. Out of the blue I can think of complexity theory, theory of computability, aspects of cryptography, and aspects of AI as examples of 20th century math that's covered in CS classes. But that's not what I'm talking about. I wanted to know what mathematicians do besides stuff I know from my CS math classes. And I wanted it on a level I could comprehend. That means I need context and I need someone to explain it to me. Clicking from math topic to math topic on Mathworld or Wikipedia doesn't cut it. My math background is too weak. I can read the Wikipedia entry of the Taniyama–Shimura theorem all I want but I'm not going to understand it. "The Taniyama–Shimura theorem (also called the modularity theorem) establishes an important connection between arithmetic of elliptic curves over rational numbers and modular forms, analytic objects of the 19th century mathematics, which are certain periodic holomorphic functions investigated in number theory." What does that even mean? And I'm not even talking about understanding the actual math of the theorem yet. That's just not possible without educating me on all kinds of things beforehand. And that's exactly what I wanted to do with the book. OK, about the book now. The book is based on a lecture given in 1971 at Warwick University. It was no ordinary lecture for college students. It was a lecture anyone could attend to get an understanding of modern math. It was an experiment on how to teach math to regular people. Basically that's exactly what I wanted. The book doesn't really require much familiarity with math. Every necessary basic concept is properly introduced in the first few chapters. Afterwards the book takes off where "Foundations and Fundamental Concepts of Mathematics" ended. Starting from set theory it goes through abstract algebra (group theory), axiomatics, and a discussion of countability and noncountability. The next major topic is Topology. Several chapters are dedicated to this topic. The last few chapters are dedicated to linear algebra, real analysis, probability, and computers. All of this is once again touched upon in a chapter dedicated to applications of the math that was introduced in the earlier chapters. I liked what I read in the book. Sometimes I was already familiar with the material, sometimes I wasn't. Topology was completely new to me. It turned out to be more interesting than I anticipated. It's pretty interesting actually. I think I've even seen aspects of topology I might actually use in my code some day. "Concepts of Modern Mathematics" is a 5stars book for $10 on Amazon. How can you go wrong? Unless you already know about all the topics of the book it's probably worth a read. And just in case someone wonders why I mentioned the Taniyama–Shimura theorem of all theorems out there. It was mentioned in the BBC documentary about Fermat's last theorem and I liked the name. Let's move on to the third book: The Equation that couldn't be solved. I bought this book because I read the Slashdot review and I like group theory. "Group theory", I always say, "group theory is the math of gentlemen". Well, I haven't actually ever said that out loud. It would be too weird. Regardless of context and company. Especially since I picture myself wearing a top hat and a monocle when saying that. And then I need to giggle. The book describes how group theory was developed. Starting from quintic equations that can't be solved using a formula that contains only basic mathematical operations (hence the title of the book) it explores how the search for such a formula led to group theory. It tells the story of early attempts to solve polynomial functions of various degrees, the failure to find one for polynomials of degree five, and how Evariste Galois and Niels Henrik Abel developed group theory independently. Furthermore the book gives lots of practical applications of group theory. It's a fun read. Amazon reviewers gave it four stars. I think that's fair. One reviewer wrote "As I was reading this book, my interest level ebbed and flowed". That covers it well. Some parts of the book are pretty good, others not so much. Which parts are interesting and which are not probably depends on what you want from the book. Be warned though, it's less a book about group theory than about the history of group theory. The next book I read was "The Nothing That Is: A Natural History of Zero". That's a pretty weird book. Someone mentioned it somewhere on the internet and I wanted to see what a book looks like that deals with the history of a single number. The reviewers on Amazon are split about this book. Some fivestar reviews, some onestar reviews. Three and a half stars on average. I remember little about this book. This is probably a bad sign. It's been a while since I read it, I have to admit. Nevertheless I read it after the Group theory book and I remember a lot from the Group theory book. What I remember from the ZeroBook was that it introduced the various number systems that were in use in ancient times. The number system of the Babylonians, the Indians, and the Mayans. That was pretty interesting. The rest apparently less so. Otherwise I could remember it. It was only like $12, no big loss. One Amazon reviewer mentions that there's another book about the number zero which is supposedly much better. Maybe read that one instead. The fifth book I read was "Introductory Graph Theory". Actually I just read that book last weekend. I bought it because I noticed something. I didn't yet have a book about graph theory. This is pretty weird because I actually use a lot of graph theory. It's just that I didn't yet have a reference for some reason. I always looked up everything in compiler theory books or on the Internet. I wanted to have it all in one place though. That's why I bought the book. The book itself is very nice to read. I read it in one sitting. It has all the important theorems with most of the proofs. It has exercises too and it has hints on how to solve them. The exercises were so interesting that I actually did many of them. The book covers everything I expected. In fact it covers more. This book is not a dry book about graph theory. It has lots of cool examples. Examples from many different fields. Not only are there traditional examples of graph theory like the Königsberg bridges problem. There are many modern examples too. My favourite examples are those that describe how to solve games, puzzles, and riddles that were popular in recent years. Another thing I liked was that many NPcomplete problems I know from computer science were mentioned. I guess that might be the case with most introductory graph theory books. I don't know though, I've never read any other books about that topic. Reading the book is really fun. I can recommend it to everyone that wants to know about basic graph theory. The only problem I have is that it doesn't cover advanced topics. Of course that's not a real problem as the book doesn't claim to cover that. It's just that I feel I need to buy another more advanced book to satisfy my desire for graph theory. Let's see what I can dig up on Amazon. I can't review the last two books, "Categories and Computer Science" and "Basic Category Theory for Computer Scientists", yet. I haven't finished them. So far I like what I read. The books compliment each other well. I read the books parallely. That means I read a chapter dealing with a certain topic in both books before I move on to the next chapter. That works well as no book completely includes the topics covered in the other book. Trackbacks
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Thanks for the reviews, it was pretty interesting.
I've heard "Concrete Mathematics" is a great book too, though it's a bit expensive.
I've heard good things about that book too. I checked it out a while ago and put it on my massive toread list. I don't expect to read it any time soon though.

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