Good math: a Geek's Guide to the beauty of numbers, logic, and computation

Table of Content:

Pt. I Numbers
1.Natural Numbers
1.1.The Naturals, Axiomatically Speaking
1.2.Using Peano Induction
2.Integers
2.1.What's an Integer?
2.2.Constructing the Integers-Naturally
3.Real Numbers
3.1.The Reals, Informally
3.2.The Reals, Axiomatically
3.3.The Reals, Constructively
4.Irrational and Transcendental Numbers
4.1.What Are Irrational Numbers?
4.2.The Argh! Moments of Irrational Numbers
4.3.What Does It Mean, and Why Does It Matter?

Pt. II Funny Numbers
5.Zero
5.1.The History of Zero
5.2.An Annoyingly Difficult Number
6.e: The Unnatural Natural Number
6.1.The Number That's Everywhere
6.2.History
6.3.Does e Have a Meaning?
7.φ: The Golden Ratio
7.1.What Is the Golden Ratio?
7.2.Legendary Nonsense
7.3.Where It Really Lives
8.i: The Imaginary Number
8.1.The Origin of i
8.2.What i Does
8.3.What i Means

Pt. III Writing Numbers
9.Roman Numerals
Contents note continued: 9.1.A Positional System
9.2.Where Did This Mess Come From?
9.3.Arithmetic Is Easy (But an Abacus Is Easier)
9.4.Blame Tradition
10.Egyptian Fractions
10.1.A 4000-Year-Old Math Exam
10.2.Fibonacci's Greedy Algorithm
10.3.Sometimes Aesthetics Trumps Practicality
11.Continued Fractions
11.1.Continued Fractions
11.2.Cleaner, Clearer, and Just Plain Fun
11.3.Doing Arithmetic

Pt. IV Logic
12.Mr. Spock Is Not Logical
12.1.What Is Logic, Really?
12.2.FOPL, Logically
12.3.Show Me Something New!
13.Proofs, Truth, and Trees: Oh My!
13.1.Building a Simple Proof with a Tree
13.2.A Proof from Nothing
13.3.All in the Family
13.4.Branching Proofs
14.Programming with Logic
14.1.Computing Family Relationships
14.2.Computation with Logic
15.Temporal Reasoning
15.1.Statements That Change with Time
15.2.What's CTL Good For?

Pt. V Sets
Contents note continued: 16.Cantor's Diagonalization: Infinity Isn't Just Infinity
16.1.Sets, Naively
16.2.Cantor's Diagonalization
16.3.Don't Keep It Simple, Stupid
17.Axiomatic Set Theory: Keep the Good, Dump the Bad
17.1.The Axioms of ZFC Set Theory
17.2.The Insanity of Choice
17.3.Why?
18.Models: Using Sets as the LEGOs of the Math World
18.1.Building Natural Numbers
18.2.Models from Models: From Naturals to Integers and Beyond!
19.Transfinite Numbers: Counting and Ordering Infinite Sets
19.1.Introducing the Transfinite Cardinals
19.2.The Continuum Hypothesis
19.3.Where in Infinity?
20.Group Theory: Finding Symmetries with Sets
20.1.Puzzling Symmetry
20.2.Different Kinds of Symmetry
20.3.Stepping into History
20.4.The Roots of Symmetry

Pt. VI Mechanical Math
21.Finite State Machines: Simplicity Goes Far
21.1.The Simplest Machine
21.2.Finite State Machines Get Real
Contents note continued: 21.3.Bridging the Gap: From Regular Expressions to Machines
22.The Turing Machine
22.1.Adding a Tape Makes All the Difference
22.2.Going Meta: The Machine That Imitates Machines
23.Pathology and the Heart of Computing
23.1.Introducing BF: The Great, the Glorious, and the Completely Silly
23.2.Turing Complete, or Completely Pointless?
23.3.From the Sublime to the Ridiculous
24.Calculus: No, Not That Calculus-A Calculus
24.1.Writing A Calculus: It's Almost Programming!
24.2.Evaluation: Run It!
24.3.Programming Languages and Lambda Strategies
25.Numbers, Booleans, and Recursion
25.1.But Is It Turing Complete?
25.2.Numbers That Compute Themselves
25.3.Decisions? Back to Church
25.4.Recursion: Y Oh Y Oh Y?
26.Types, Types, Types: Modeling λ Calculus
26.1.Playing to Type
26.2.Prove It!
26.3.What's It Good For?
27.The Halting Problem
27.1.A Brilliant Failure
27.2.To Halt or Not To Halt?

Mathematics is beautiful—and it can be fun and exciting as well as practical. Good Math is your guide to some of the most intriguing topics from two thousand years of mathematics: from Egyptian fractions to Turing machines; from the real meaning of numbers to proof trees, group symmetry, and mechanical computation. If you’ve ever wondered what lay beyond the proofs you struggled to complete in high school geometry, or what limits the capabilities of the computer on your desk, this is the book for you.

Why do Roman numerals persist? How do we know that some infinities are larger than others? And how can we know for certain a program will ever finish? In this fast-paced tour of modern and not-so-modern math, computer scientist Mark Chu-Carroll explores some of the greatest breakthroughs and disappointments of more than two thousand years of mathematical thought. There is joy and beauty in mathematics, and in more than two dozen essays drawn from his popular “Good Math” blog, you’ll find concepts, proofs, and examples that are often surprising, counterintuitive, or just plain weird.

Mark begins his journey with the basics of numbers, with an entertaining trip through the integers and the natural, rational, irrational, and transcendental numbers. The voyage continues with a look at some of the oddest numbers in mathematics, including zero, the golden ratio, imaginary numbers, Roman numerals, and Egyptian and continuing fractions. After a deep dive into modern logic, including an introduction to linear logic and the logic-savvy Prolog language, the trip concludes with a tour of modern set theory and the advances and paradoxes of modern mechanical computing.

If your high school or college math courses left you grasping for the inner meaning behind the numbers, Mark’s book will both entertain and enlighten you.

https://pragprog.com/titles/mcmath/good-math/