Contents Author’s note to the English-language edition 1 1 Desarguesian geometry and the Desarguesian number system 13 1.1 Hilbert’s axiom system of ordinary geometry 13 1.2 The axiom of infinity and Desargues’ axioms 18 1.3 Rational points in a Desarguesian plane 25 1.4 The Desarguesian number system and rational number subsystem 30 1.5 The Desarguesian number system on a line 37 1.6 The Desarguesian number system associated with a Desarguesian plane 42 1.7 The coordinate system of Desarguesian plane geometry 55 2 Orthogonal geometry， metric geometry and ordinary geometry 63 2.1 The Pascalian axiom and commutative axiom of multiplication-(unordered) Pascalian geometry 63 2.2 Orthogonal axioms and (unordered) orthogonal geometry 70 2.3 The orthogonal coordinate system of (unordered) orthogonal geometry 80 2.4 (Unordered) metric geometry 91 2.5 The axioms of order and ordered metric geometry 102 2.6 Ordinary geometry and its subordinate geometries 109 3 Mechanization of theorem proving in geometry and Hilbert’s mechanization theorem 115 3.1 Comments on Euclidean proof method 115 3.2 The standardization of coordinate representation of geometric concepts 118 3.3 The mechanization of theorem proving and Hilbert’s mechanization theorem about pure point of intersection theorems in Pascalian geometry 124 3.4 Examples for Hilbert’s mechanical method 128 3.5 Proof of Hilbert’s mechanization theorem 139 4 The mechanization theorem of (ordinary) unordered geometry 149 4.1 Introduction 149 4.2 Factorization of polynomials 152 4.3 Well-ordering of polynomial sets 159 4.4 A constructive theory of algebraic varieties-irreducible ascending sets and irreducible algebraic varieties 169 4.5 A constructive theory of algebraic varieties-irreducible decomposition of algebraic varieties 178 4.6 A constructive theory of algebraic varieties-the notion of dimension and the dimension theorem 183 4.7 Proof of the mechanization theorem of unordered geometry 187 4.8 Examples for the mechanical method of unordered geometry 195 5 Mechanization theorems of (ordinary) ordered geometries 213 5.1 Introduction 213 5.2 Tarski’s theorem and Seidenberg’s method 220 5.3 Examples for the mechanical method of ordered geometries 228 6 Mechanization theorems of various geometries 235 6.1 Introduction 235 6.2 The mechanization of theorem proving in projective geometry 236 6.3 The mechanization of theorem proving in Bolyai-Lobachevsky’s hyperbolic non-Euclidean geometry 246 6.4 The mechanization of theorem proving in Riemann’s elliptic non-Euclidean geometry 258 6.5 The mechanization of theorem proving in two circle geometries 264 6.6 The mechanization of formula proving with transcendental functions 267 References 281 Subject index 285
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