기하학의 철학에 기여하는 수학의 기초 책.The Book of The Foundations of Mathematics,A Contribution to

기하학의 철학에 기여하는 수학의 기초 책.The Book of The Foundations of Mathematics,A Contribution to the Philosophy of Geometry. by Paul Carus
TABLE OF CONTENTS.
the search for the foundations of geometry:
historical sketch.
PAGE
Axioms and the Axiom of Parallels . . . . . . . . . . . . . 1
Metageometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Precursors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Lobatchevsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Bolyai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Later Geometricians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Grassmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Euclid Still Unimpaired. . . . . . . . . . . . . . . . . . . . . . . . . 32
the philosophical basis of mathematics.
PAGE
The Philosophical Problem . . . . . . . . . . . . . . . . . . . . . 36
Transcendentalism and Empiricism . . . . . . . . . . . . . 39
The A Priori and the Purely Formal . . . . . . . . . . . . 41
Anyness and its Universality . . . . . . . . . . . . . . . . . . . . 46
Apriority of Different Degrees. . . . . . . . . . . . . . . . . . . 49
Space as a Spread of Motion . . . . . . . . . . . . . . . . . . . . 56
Uniqueness of Pure Space. . . . . . . . . . . . . . . . . . . . . . . 61
Mathematical Space and Physiological Space. . . . 63
Homogeneity of Space Due to Abstraction . . . . . . 66
Even Boundaries as Standards of Measurement . 69
The Straight Line Indispensable . . . . . . . . . . . . . . . . 72
The Superreal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Discrete Units and the Continuum . . . . . . . . . . . . . . 77
mathematics and metageometry.
Different Geometrical Systems . . . . . . . . . . . . . . . . . . 81
Tridimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Three a Concept of Boundary. . . . . . . . . . . . . . . . . . . 86
Space of Four Dimensions. . . . . . . . . . . . . . . . . . . . . . . 88
The Apparent Arbitrariness of the A Priori . . . . . 94
Definiteness of Construction . . . . . . . . . . . . . . . . . . . . 97
PAGE
One Space, but Various Systems of Space Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Fictitious Spaces and the Apriority of all Systems
of Space-Measurement . . . . . . . . . . . . . . . . . 107
Infinitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Geometry Remains A Priori . . . . . . . . . . . . . . . . . . . . 116
Sense-Experience and Space . . . . . . . . . . . . . . . . . . . . 120
The Teaching of Mathematics. . . . . . . . . . . . . . . . . . . 125
EPILOGUE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135