Ryerson Polytechnic University Foundations of Mathematical Thought MTH 599 Course Notes by P. Danziger c P. Danziger 1997, 1998, 2004 Contents 1 What is Mathematics? 8 1.1 A Definition of Mathematics? . 8 1.2 Mathematical Systems . 10 2 Early History 15 2.1 Pre-Hellenic Civilisations . 15 2.1.1 The Early Development of Mathematics . 16 2.1.2 Ancient Number Systems . 17 2.2 The Early Greeks . 22 2.3 Thales of Miletus . 24 2.3.1 Thales 1st Theorem . 25 2.3.2 Thales 2nd Theorem . 26 2.3.3 Thales 3rd Theorem . 28 2.3.4 Thales 4th Theorem . 30 2.4 Pythagoras and the Pythagoreans . 30 2.4.1 Pythagoras' Theorem . 30 2.4.2 The Pythagorean School . 33 2.4.3 The Golden Ratio . 34 1 2.4.4 Pythagoreans and Number Theory . 36 2.4.5 Figurate Numbers . 41 2.4.6 The Later Pythagoreans . 42 2.4.7 Irrational Numbers . 43 2.5 Zeno of Elea and his Paradoxes . 47 2.5.1 Zeno's Paradoxes . 47 2.5.2 Zeno's Paradoxes Explained . 49 3 The Athenians 52 3.1 Athens . 52 3.2 Hippocrates of Chios and Quadrature . 53 3.2.1 Quadrature . 54 3.2.2 The Method of Exhaustion . 57 3.2.3 The Three Great Problems of Antiquity . 59 3.3 The Athenian Philosophers . 60 3.3.1 Socrates . 60 3.3.2 The Sophists . 60 3.4 Plato . 61 3.4.1 The Platonic Ideal . 62 3.4.2 Plato's Academy . 64 3.5 Eudoxus of Cnidus . 65 3.5.1 Eudoxus' Definition of Ratio . 65 3.6 Aristotle . 68 3.6.1 Aristotelian Logic . 69 3.7 Euclid and Alexandria . 70 2 4 Euclid 71 4.1 Summary of the 13 Books of Euclid . 72 4.2 Euclid { Book I . 74 4.2.1 Definitions . 74 4.2.2 Postulates . 76 4.2.3 Common Notions . 76 4.3 Analysis of Euclid's Book I . 77 4.3.1 The Definitions . 77 4.3.2 The Postulates . 81 4.3.3 The Common Notions . 84 4.3.4 Propositions 1 - 4 . 85 4.3.5 The Other Propositions . 88 5 Euclid to the Renaissance 91 5.1 The Later Greek Period . 91 5.1.1 Apollonius of Perga . 91 5.1.2 Archimedes of Syracuse . 92 5.1.3 The Alexandrians . 94 5.1.4 Mathematics in the Roman World . 95 5.2 The Post Roman Period . 96 5.2.1 The Hindus . 96 5.2.2 The Arabic World . 97 5.2.3 Fibonacci . 98 5.3 The Renaissance . 100 5.4 Early Rationalism . 102 3 5.4.1 Descartes . 102 5.4.2 Fermat . 102 6 The Infinitesimal in Mathematics 104 6.1 The Development of Calculus . 104 6.1.1 Newton . 105 6.1.2 Leibniz . 105 6.1.3 English Mathematics . 106 6.1.4 The Bernoullis . 106 6.2 Limits and the Infinitesimal . 107 6.2.1 Integration . 107 6.2.2 Differentiation . 109 6.2.3 Limits . 109 7 The Modern Age 117 7.1 The Age of Rationalism . 117 7.1.1 Euler . 117 7.1.2 The French School . 119 7.1.3 Gauss . 120 7.2 Non Euclidean Geometry . 121 7.2.1 Riemann . 123 7.3 Cantor and the Infinite . 124 7.3.1 Cardinal vs. Ordinal Numbers . 124 7.3.2 Cantor . 125 8 Formal Logic 130 4 8.1 Statements . 131 8.2 Logical Operations . 133 8.2.1 NOT, AND, OR . 133 8.2.2 Implication . 137 8.2.3 Necessary and Sufficient . 140 8.3 Valid Forms . 140 8.4 Set Theory . 142 8.4.1 Basic Definitions . 142 8.4.2 Notation . 143 8.4.3 The Universal Set . 144 8.4.4 Some Useful Sets . 144 8.4.5 Set Comparisons . 145 8.4.6 Operations on Sets . 146 8.4.7 Set Identities . 147 8.5 Quantifiers . 149 8.5.1 Domain Change . 151 8.5.2 Negations of Quantified Statements . 152 8.5.3 Vacuosly True Universal Statements . 153 8.5.4 Multiply Quantified Statements . 153 8.5.5 Scope . 155 8.5.6 Variants of Universal Conditional Statements . 156 8.5.7 Necessary and Sufficient . 157 8.6 Methods of Proof . 158 8.6.1 Direct Methods . 159 8.6.2 Indirect Methods . 160 5 8.6.3 Induction . ..