A world-renowned mathematician takes a lighthearted look at the philosophy, form, and recreations of mathematics in this fascinating book. In lucid and witty language, he discusses and illustrates the charm of mathematics as well as the science's vast practical utility and its vital significance to our cultural history. 1958 edition.
Author(s): Nathan Altshiller-Court
Year: 2006
Language: English
Pages: 191
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
