Number and the Nature of Mathematics


About the Author


1 Mathematics: Misconceptions

2 Early Counting – Speculation

First Steps

Then, The Babylonians, Egyptians, and The Greeks

And Elsewhere

3 Natural Numbers and Axiom Systems


Definitions and Properties


Properties of Addition – Proof Based on Axioms

The Associative Law

The Law of Substitution

The Commutative Law

What is Mathematics: A First Look

The Law of Cancellation

Summary So Far

4 An Introduction To Sets And A Bit Of Logic

Sets – Definitions and Properties

Simple Logic

A Note About Infinity

5 Zero. A First Extension – The Whole Numbers

Definitions and Properties

A new relationship: Less–Equals

Some Terminology


What is Easy, What is Hard, and Mathematical Insight

Algebra, Variables, Unknowns, and How Mathematics is Not Computer Science

Some Interesting Sums

Representing Whole Numbers

Summary So Far

6 The Integers

Equivalence Relations and Equivalence Classes

Definitions and Properties

Generalization and the Nature of Mathematics – Foreshadowing Abstract Algebra

Applied Mathematics: A Simple Example


A Note on Proof

Substitution and Cancellation Laws in Z

Integer Division and Remainders

A Note on Mathematical Insight

Summary So Far

7 The Rational Numbers

Definitions and Properties

Denseness and Distance in the Rational Numbers

Cardinality of the Rational Numbers

Decimals and the Rational Numbers

Summary So Far

8 Infinite Sequences

Definition and Properties

Infinite Series

9 The Real Numbers

Cauchy Sequences

Definition and Properties

Cardinality of the Real Numbers

Foreshadowing: Topology

Foreshadowing – Calculus and Analysis

Summary So Far

10 Polynomials

Definition and Properties

11 The Complex Numbers

Definition and Properties

Foreshadowing – Trigonometry

Foreshadowing – Matrix Theory

Summary So Far

12 Mathematics

What Do Mathematicians Do?

A Simple Example

13 Ideas for Studying Mathematics

14 Further Reading