Mathematics in Fun and in Earnest

Cover......Page __sk_0000.djvu
Copyright......Page __sk_0002.djvu
Preface......Page __sk_0003.djvu
Contents......Page __sk_0006.djvu
A. The nature of Mathematics......Page __sk_0013.djvu
B. The unity of form and number......Page __sk_0014.djvu
C. The dimensions of space......Page __sk_0015.djvu
D. Postulational Mathematics......Page __sk_0017.djvu
F. The empirical origin of the axioms......Page __sk_0021.djvu
G. The worth of deductive reasoning......Page __sk_0023.djvu
H. Imagination and imitation......Page __sk_0024.djvu
A. Origins of geometrical knowledge......Page __sk_0026.djvu
B. The sense of touch......Page __sk_0027.djvu
C. The sense of vision......Page __sk_0028.djvu
D. Metrical Geometry and Projective Geometry......Page __sk_0029.djvu
E. Conflicting testimony of the senses......Page __sk_0031.djvu
A. Reasoning and psychology......Page __sk_0033.djvu
B. The role of the body in the reasoning process......Page __sk_0034.djvu
C. A definition of reasoning......Page __sk_0036.djvu
D. Applications of the definition......Page __sk_0038.djvu
E. Pitfalls and merits of reasoning......Page __sk_0040.djvu
F. More checks on the definition......Page __sk_0042.djvu
G. Reasoning, memory, and invention......Page __sk_0044.djvu
A. The impact of non-Euclidean Geometry and of Projective Geometry......Page __sk_0045.djvu
B. The formalist approach to Geometry......Page __sk_0047.djvu
C. Role of the knower......Page __sk_0048.djvu
D. Axiomatic method......Page __sk_0050.djvu
E. Meaning of intuition......Page __sk_0051.djvu
F. Foundations of logic......Page __sk_0053.djvu
G. Symbolic representation, or miniature realization......Page __sk_0055.djvu
H. Rational theory of objective existence......Page __sk_0056.djvu
I. Logic......Page __sk_0057.djvu
A. The early beginnings of counting and reckoning......Page __sk_0061.djvu
B. Measuring. Beginnings of Geometry and Chronology......Page __sk_0065.djvu
C. The Renaissance period. The great voyages. The invention of Analytic Geometry and of the Calculus......Page __sk_0069.djvu
D. Mathematics for the modern age......Page __sk_0071.djvu
E. Conclusion......Page __sk_0072.djvu
A. The \"heroic\" and the \"objective\" interpretations of history......Page __sk_0073.djvu
B. Are inventions inevitable?......Page __sk_0075.djvu
C. Genius and environment......Page __sk_0077.djvu
D. Genius and the \"instinct of workmanship\"......Page __sk_0079.djvu
E. Mathematics - the patrimony of the race......Page __sk_0080.djvu
A. No largest number......Page __sk_0082.djvu
B. A part as big as the whole......Page __sk_0083.djvu
C. Arithmetical operations performed on the infinite......Page __sk_0086.djvu
D. No escape from the infinite......Page __sk_0088.djvu
A. Parallelism in Euclid\'s Elements......Page __sk_0089.djvu
B. The difference between Metric and Projective geometry......Page __sk_0090.djvu
C. The point at infinity of a line......Page __sk_0091.djvu
D. The line at infinity of a plane and the plane at infinity of space......Page __sk_0092.djvu
E. Advantages and limitations of the elements at infinity......Page __sk_0094.djvu
F. Could Euclid find room in his Elements for points at infinity......Page __sk_0095.djvu
G. Do the elements at infinity \"enrich\" Projective Geometry......Page __sk_0096.djvu
B. The arguments of Zeno and those of his imitators......Page __sk_0097.djvu
C. Aristotle\'s arguments about the infinite divisibility of both time and space......Page __sk_0099.djvu
D. The potential infinity and the actual infinity......Page __sk_0101.djvu
E. Motion and dynamics......Page __sk_0102.djvu
F. Motion - an \"undefined\" term......Page __sk_0104.djvu
G. Theory and observation......Page __sk_0105.djvu
H. Instantaneous velocity......Page __sk_0106.djvu
I. A modern answer to Zeno\'s paradoxes......Page __sk_0108.djvu
A. Beauty in Mathematics......Page __sk_0109.djvu
B. Mathematics in beauty......Page __sk_0110.djvu
A. Mathematics, logic, music......Page __sk_0111.djvu
B. Opinions of mathematicians about Mathematics......Page __sk_0113.djvu
C. The opinion of a poet......Page __sk_0114.djvu
D. Mathematics - a creation of the imagination......Page __sk_0115.djvu
E. Further analogies between Mathematics and the arts: symbolism, condensation, care in execution, etc.......Page __sk_0116.djvu
F. \"Movements\" in Mathematics......Page __sk_0119.djvu
G. Conclusion......Page __sk_0120.djvu
b. Is the mathematician \"objective\"?......Page __sk_0121.djvu
c. Priority disputes......Page __sk_0122.djvu
f. Disputes over results obtained......Page __sk_0123.djvu
g. The quest for rigor......Page __sk_0124.djvu
i. Formalism......Page __sk_0125.djvu
i. Logicalism......Page __sk_0127.djvu
k. Intuitionism......Page __sk_0128.djvu
m. Conclusion......Page __sk_0129.djvu
A. Winning a prize......Page __sk_0130.djvu
B. Gambling and statistics......Page __sk_0131.djvu
C. Tit-tat-toe ancient and modern......Page __sk_0132.djvu
D. New checker games for old......Page __sk_0133.djvu
E. Potato-pushing a la mode......Page __sk_0135.djvu
F. Conclusion......Page __sk_0137.djvu
A. A point fixation......Page __sk_0138.djvu
B. A square deal......Page __sk_0141.djvu
B. Four examples......Page __sk_0145.djvu
C. An explanation......Page __sk_0147.djvu
D. Analogy as a useful guide to discovery......Page __sk_0149.djvu
E. Limitations of that method......Page __sk_0150.djvu
F. The altitudes of a triangle and of a tetrahedron......Page __sk_0152.djvu
A. Historical data......Page __sk_0154.djvu
B. The theorem of Pythagoras in India......Page __sk_0155.djvu
C. The story of Lilavati......Page __sk_0156.djvu
3. Running Around In Circles......Page __sk_0157.djvu
4. Too Many?......Page __sk_0162.djvu
A. \"River crossing\" problems......Page __sk_0166.djvu
B. Multiplication performed on the fingers......Page __sk_0169.djvu
C. \"Pouring\" problems. The \"Robot\" method......Page __sk_0171.djvu
D. The \"false coin\" problem......Page __sk_0176.djvu
A. Morley\'s problem......Page __sk_0180.djvu
C. \"Fermat\'s last theorem\"......Page __sk_0181.djvu
D. Goldbach\'s conjecture......Page __sk_0182.djvu
F. The recurrence of \"famous\" problems......Page __sk_0183.djvu
G. Conclusion......Page __sk_0184.djvu
A. Mathematics and computation......Page __sk_0185.djvu
B. Problems......Page __sk_0186.djvu
C. Solutions......Page __sk_0187.djvu