Mathematics in Fun and in Earnest

Cover......Page 1
Copyright......Page 3
Preface......Page 4
Contents......Page 7
A. The nature of Mathematics......Page 14
B. The unity of form and number......Page 15
C. The dimensions of space......Page 16
D. Postulational Mathematics......Page 18
F. The empirical origin of the axioms......Page 22
G. The worth of deductive reasoning......Page 24
H. Imagination and imitation......Page 25
A. Origins of geometrical knowledge......Page 27
B. The sense of touch......Page 28
C. The sense of vision......Page 29
D. Metrical Geometry and Projective Geometry......Page 30
E. Conflicting testimony of the senses......Page 32
A. Reasoning and psychology......Page 34
B. The role of the body in the reasoning process......Page 35
C. A definition of reasoning......Page 37
D. Applications of the definition......Page 39
E. Pitfalls and merits of reasoning......Page 41
F. More checks on the definition......Page 43
G. Reasoning, memory, and invention......Page 45
A. The impact of non-Euclidean Geometry and of Projective Geometry......Page 46
B. The formalist approach to Geometry......Page 48
C. Role of the knower......Page 49
D. Axiomatic method......Page 51
E. Meaning of intuition......Page 52
F. Foundations of logic......Page 54
G. Symbolic representation, or miniature realization......Page 56
H. Rational theory of objective existence......Page 57
I. Logic......Page 58
A. The early beginnings of counting and reckoning......Page 62
B. Measuring. Beginnings of Geometry and Chronology......Page 66
C. The Renaissance period. The great voyages. The invention of Analytic Geometry and of the Calculus......Page 70
D. Mathematics for the modern age......Page 72
E. Conclusion......Page 73
A. The \"heroic\" and the \"objective\" interpretations of history......Page 74
B. Are inventions inevitable?......Page 76
C. Genius and environment......Page 78
D. Genius and the \"instinct of workmanship\"......Page 80
E. Mathematics - the patrimony of the race......Page 81
A. No largest number......Page 83
B. A part as big as the whole......Page 84
C. Arithmetical operations performed on the infinite......Page 87
D. No escape from the infinite......Page 89
A. Parallelism in Euclid\'s Elements......Page 90
B. The difference between Metric and Projective geometry......Page 91
C. The point at infinity of a line......Page 92
D. The line at infinity of a plane and the plane at infinity of space......Page 93
E. Advantages and limitations of the elements at infinity......Page 95
F. Could Euclid find room in his Elements for points at infinity......Page 96
G. Do the elements at infinity \"enrich\" Projective Geometry......Page 97
B. The arguments of Zeno and those of his imitators......Page 98
C. Aristotle\'s arguments about the infinite divisibility of both time and space......Page 100
D. The potential infinity and the actual infinity......Page 102
E. Motion and dynamics......Page 103
F. Motion - an \"undefined\" term......Page 105
G. Theory and observation......Page 106
H. Instantaneous velocity......Page 107
I. A modern answer to Zeno\'s paradoxes......Page 109
A. Beauty in Mathematics......Page 110
B. Mathematics in beauty......Page 111
A. Mathematics, logic, music......Page 112
B. Opinions of mathematicians about Mathematics......Page 114
C. The opinion of a poet......Page 115
D. Mathematics - a creation of the imagination......Page 116
E. Further analogies between Mathematics and the arts: symbolism, condensation, care in execution, etc.......Page 117
F. \"Movements\" in Mathematics......Page 120
G. Conclusion......Page 121
b. Is the mathematician \"objective\"?......Page 122
c. Priority disputes......Page 123
f. Disputes over results obtained......Page 124
g. The quest for rigor......Page 125
i. Formalism......Page 126
i. Logicalism......Page 128
k. Intuitionism......Page 129
m. Conclusion......Page 130
A. Winning a prize......Page 131
B. Gambling and statistics......Page 132
C. Tit-tat-toe ancient and modern......Page 133
D. New checker games for old......Page 134
E. Potato-pushing a la mode......Page 136
F. Conclusion......Page 138
A. A point fixation......Page 139
B. A square deal......Page 142
B. Four examples......Page 146
C. An explanation......Page 148
D. Analogy as a useful guide to discovery......Page 150
E. Limitations of that method......Page 151
F. The altitudes of a triangle and of a tetrahedron......Page 153
A. Historical data......Page 155
B. The theorem of Pythagoras in India......Page 156
C. The story of Lilavati......Page 157
3. Running Around In Circles......Page 158
4. Too Many?......Page 163
A. \"River crossing\" problems......Page 167
B. Multiplication performed on the fingers......Page 170
C. \"Pouring\" problems. The \"Robot\" method......Page 172
D. The \"false coin\" problem......Page 177
A. Morley\'s problem......Page 181
C. \"Fermat\'s last theorem\"......Page 182
D. Goldbach\'s conjecture......Page 183
F. The recurrence of \"famous\" problems......Page 184
G. Conclusion......Page 185
A. Mathematics and computation......Page 186
B. Problems......Page 187
C. Solutions......Page 188