From Mathematics to Generic Programming” review. A book by Alexander Stepanov and Daniel Rose. Published by Addison-Wesley.

From Mathematics to Generic Programming

Book review

Let the title of the book, From Mathematics to Generic Programming, doesn’t confuse you. My first impression after seeing it was that the book will teach me how to write generic code including constructs like iterators and the mathematical theory being behind them. After reading it, it turned out that the book is in fact not about what I had excepted from the title. From Mathematics to Generic Programming is really a book about mathematics in programming in general, with focus on mathematics behind the generic programming. There is not much about generic programming itself. A much better title would be “The Mathematical Theory behind Generic Programming”.

If you are looking for a book about generic programming, you may be now wondering if you should get this book or not. The answer to this question is, in my opinion, definitely yes. From Mathematics to Generic Programming may not contain much about generic programming itself, but it contains a lot of well explained concepts from the mathematical background of generic programming. All of this theory is a fundamental that allows to truly understand the concept behind the generic programming.

From Mathematics to Generic Programming will teach you how to practically use various mathematical axioms, theorems, structures, algorithms and techniques that are very useful not only in generic programming, but in general programming as well. Starting with as simple concepts as ancient multiplication algorithms, through monoids, groups, rings or fields, the book will get you to the point where you will be able to very easily understand how RSA algorithm works and even to implement it yourself.

The book is written in light non-academic style, rather than in style of academic technical books. It is filled with short biographies of important mathematicians along with stories about how they developed the knowledge that strongly influenced generic programming. All of that make this book a very interesting read, making the entire experience of reading it similar to reading a history book. You simply don’t want to put the book away, because you are very curious of what is next in that great history of mathematics. After finishing the book I placed orders for next books, like Euclid’s Elements, because the author really made me interested in pursuing more knowledge about the topics he wrote about.

Since this is a book about mathematics, you may be wondering what prerequisites are needed to understand it. The good news is that anybody who understand high-school mathematics should have no problem with reading it. If you think that your knowledge is a little rusty, read about high-school-level algebra and geometry to recall most of the information you need. Beside mathematics, you will need some basic knowledge of C++ to understand the code examples (there are not much of them, however). The examples are simple enough to make them understood by non-C++ programmers. If you program in languages with a similar syntax, like Java, then you should have not much trouble with them. In case you would have any troubles with the examples, on the end of the book is an appendix “C++ for Non-C++ Programmers” which should dispel any doubts.

In a summary, the book is a great and unique position for people wanting to learn the mathematical theory behind generic programming, or learn about mathematics in computer programming in general. While this book will not teach you how to write generic code, it will teach you fundamental knowledge that will allow you to understand the concept behind generic programming. As mathematics is the fundamental of computer programming, learning mathematics is a certain way toward becoming a better programmer.

I have really enjoyed the book and I recommend it for anybody who wants to become a better programmer or wants to better understand generic programming.

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Table of Contents

Chapter 1: What This Book Is About

1.1 Programming and Mathematics

1.2 A Historical Perspective

1.3 Prerequisites

1.4 Roadmap

Chapter 2: The First Algorithm

2.1 Egyptian Multiplication

2.2 Improving the Algorithm

2.3 Thoughts on the Chapter

Chapter 3: Ancient Greek Number Theory

3.1 Geometric Properties of Integers

3.2 Sifting Primes

3.3 Implementing and Optimizing the Code

3.4 Perfect Numbers

3.5 The Pythagorean Program

3.6 A Fatal Flaw in the Program

3.7 Thoughts on the Chapter

Chapter 4: Euclid’s Algorithm

4.1 Athens and Alexandria

4.2 Euclid’s Greatest Common Measure Algorithm

4.3 A Millennium without Mathematics

4.4 The Strange History of Zero

4.5 Remainder and Quotient Algorithms

4.6 Sharing the Code

4.7 Validating the Algorithm

4.8 Thoughts on the Chapter

Chapter 5: The Emergence of Modern Number Theory

5.1 Mersenne Primes and Fermat Primes

5.2 Fermat’s Little Theorem

5.3 Cancellation

5.4 Proving Fermat’s Little Theorem

5.5 Euler’s Theorem

5.6 Applying Modular Arithmetic

5.7 Thoughts on the Chapter

Chapter 6: Abstraction in Mathematics

6.1 Groups

6.2 Monoids and Semigroups

6.3 Some Theorems about Groups

6.4 Subgroups and Cyclic Groups

6.5 Lagrange’s Theorem

6.6 Theories and Models

6.7 Examples of Categorical and Non-categorical Theories

6.8 Thoughts on the Chapter

Chapter 7: Deriving a Generic Algorithm

7.1 Untangling Algorithm Requirements

7.2 Requirements on A

7.3 Requirements on N

7.4 New Requirements

7.5 Turning Multiply into Power

7.6 Generalizing the Operation

7.7 Computing Fibonacci Numbers

7.8 Thoughts on the Chapter

Chapter 8: More Algebraic Structures

8.1 Stevin, Polynomials, and GCD

8.2 Göttingen and German Mathematics

8.3 Noether and the Birth of Abstract Algebra

8.4 Rings

8.5 Matrix Multiplication and Semirings

8.6 Application: Social Networks and Shortest Paths

8.7 Euclidean Domains

8.8 Fields and Other Algebraic Structures

8.9 Thoughts on the Chapter

Chapter 9: Organizing Mathematical Knowledge

9.1 Proofs

9.2 The First Theorem

9.3 Euclid and the Axiomatic Method

9.4 Alternatives to Euclidean Geometry

9.5 Hilbert’s Formalist Approach

9.6 Peano and His Axioms

9.7 Building Arithmetic

9.8 Thoughts on the Chapter

Chapter 10: Fundamental Programming Concepts

10.1 Aristotle and Abstraction

10.2 Values and Types

10.3 Concepts

10.4 Iterators

10.5 Iterator Categories, Operations, and Traits

10.6 Ranges

10.7 Linear Search

10.8 Binary Search

10.9 Thoughts on the Chapter

Chapter 11: Permutation Algorithms

11.1 Permutations and Transpositions

11.2 Swapping Ranges

11.3 Rotation

11.4 Using Cycles

11.5 Reverse

11.6 Space Complexity

11.7 Memory-Adaptive Algorithms

11.8 Thoughts on the Chapter

Chapter 12: Extensions of GCD

12.1 Hardware Constraints and a More Efficient Algorithm

12.2 Generalizing Stein’s Algorithm

12.3 Bézout’s Identity

12.4 Extended GCD

12.5 Applications of GCD

12.6 Thoughts on the Chapter

Chapter 13: A Real-World Application

13.1 Cryptology

13.2 Primality Testing

13.3 The Miller-Rabin Test

13.4 The RSA Algorithm: How and Why It Works

13.5 Thoughts on the Chapter

Chapter 14: Conclusions

Further Reading

Appendix A: Notation

Appendix B: Common Proof Techniques

B.1 Proof by Contradiction

B.2 Proof by Induction

B.3 The Pigeonhole Principle

Appendix C: C++ for Non-C++ Programmers

C.1 Template Functions

C.2 Concepts

C.3 Declaration Syntax and Typed Constants

C.4 Function Objects

C.5 Preconditions, Postconditions, and Assertions

C.6 STL Algorithms and Data Structures

C.7 Iterators and Ranges

C.8 Type Aliases and Type Functions with using in C++11

C.9 Initializer Lists in C++11

C.10 Lambda Functions in C++11

C.11 A Note about inline

About the authors

Alexander Stepanov is a computer programmer, mathematician and pioneer of generic programming. He is most known by creation and major contribution to the C++ STL (Standard Template Library), for which he received Dr. Dobb’s Journal Excellence in Programming Award. Currently he is working for A9, a subsidiary of Amazon that works on developing product search and advertising technology. His previous employers include GE, Polytechnic University, Bell Labs, HP, SGI and Adobe.

Daniel Rose is a computer programmer and research scientist specializing in search technologies, ranging from low-level algorithms for index compression to human-computer interaction issues in web search. Currently he works as Chief Scientist at A9, together with Alexander Stepanov. His previous employers include Yahoo!, AltaVista, Xigo and Apple. He holds a PhD in cognitive and computer science from University of California and a B.A. in philosophy from Harvard.