What is Mathematics? An Elementary Approach to Ideas and Methods
Second Edition
Richard Courant, Herbert Robbins
0. Preface
Formal mathematics is like spelling and grammar - a matter of the correct application of local rules.
Meaningful mathematics is like journalism - it tells an interesting story. Unlike some journalism, the story has to be true.
The best mathematics is like literature - it brings a story to life before your eyes and involves you in it, intellectually and emotionally.
Before 19th Century: what the mathematical entities (point, lines, numbers, …) “actually” are?
After 19th Century: understanding the structures and relationships among “undefined objects”.
(two points determine a line, numbers combine according to certain rules to form other numbers)
1. The Natural Numbers
All mathematical statements should be reducible ultimately to statements about the natural numbers.
Numbers are an abstract concept, having no reference to the individual characteristics of the objects counted.
Begins by accepting the natural numbers as given, together with the 2 fundamental operations, addition and multiplication.
1.1. Calculation with Integers
1. Laws of Arithmetic
- Natural numbers = positive integers
- Arithmetic: mathematical theory of the natural numbers
- 5 fundamental laws of arithmetic
\(a, b, c\): natural numbers
Commutative law of addition: \(a + b = b + a\)
Commutative law of multiplication: \(ab = ba\)
Associative law of addition: \(a + (b + c) = (a + b) + c\)
Associative law of multiplication: \(a(bc) = (ab)c\)
Distributive law: \(a(b + c) = ab + ac\) - Inequality (\(<, >\))
The statements \(a < b\) (\(a\) is less than \(b\)) and \(b > a\) (\(b\) is greater than \(a\)) are equivalent.
This means that there exists a natural number \(c\) such that \(b = a + c\). - Subtraction (\(-\))
When \(b > a\) and \(b = a + c\), subtraction is defined as \(c = b - a\)
Addition and subtraction are inverse operations: \((a + d) - d = a\) - The integer zero (\(0\))
Zero is a special integer defined by the following arithmetic properties.
\(a + 0 = a\)
\(a \cdot 0 = 0\)
\(a - a = 0\)
2. The Representation of Integers
- Computation in Systems Other than the Decimal
1.2. The Infinitude of the Number System: Mathematical Induction
The Principle of Mathematical Induction
The Arithmetical Progression
The Geometrical Progression
The Sum of the First n Squares
An Important Inequality
The Binomial Theorem
Further Remarks on Mathematical Induction