Mechanical Theorem Proving in Geometries: Basic Principles

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Springer Science & Business Media, Apr 14, 1994 - Computers - 288 pages
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There seems to be no doubt that geometry originates from such practical activ ities as weather observation and terrain survey. But there are different manners, methods, and ways to raise the various experiences to the level of theory so that they finally constitute a science. F. Engels said, "The objective of mathematics is the study of space forms and quantitative relations of the real world. " Dur ing the time of the ancient Greeks, there were two different methods dealing with geometry: one, represented by the Euclid's "Elements," purely pursued the logical relations among geometric entities, excluding completely the quantita tive relations, as to establish the axiom system of geometry. This method has become a model of deduction methods in mathematics. The other, represented by the relevant work of Archimedes, focused on the study of quantitative re lations of geometric objects as well as their measures such as the ratio of the circumference of a circle to its diameter and the area of a spherical surface and of a parabolic sector. Though these approaches vary in style, have their own features, and reflect different viewpoints in the development of geometry, both have made great contributions to the development of mathematics. The development of geometry in China was all along concerned with quanti tative relations.
  

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Contents

Desarguesian geometry and the Desarguesian number system
13
12 The axiom of infinity and Desargues axioms
18
13 Rational points in a Desarguesian plane
25
14 The Desarguesian number system and rational number subsystem
30
15 The Desarguesian number system on a line
37
16 The Desarguesian number system associated with a Desarguesian plane
42
17 The coordinate system of Desarguesian plane geometry
55
Orthogonal geometry metric geometry and ordinary geometry
63
42 Factorization of polynomials
152
43 Wellordering of polynomial sets
159
44 A constructive theory of algebraic varieties irreducible ascending sets and irreducible algebraic varieties
169
45 A constructive theory of algebraic varieties irreducible decomposition of algebraic varieties
178
46 A constructive theory of algebraic varieties the notion of dimension and the dimension theorem
183
47 Proof of the mechanization theorem of unordered geometry
187
48 Examples for the mechanical method of unordered geometry
195
Mechanization theorems of ordinary ordered geometries
213

22 Orthogonal axioms and unordered orthogonal geometry
70
23 The orthogonal coordinate system of unordered orthogonal geometry
80
24 Unordered metric geometry
91
25 The axioms of order and ordered metric geometry
102
26 Ordinary geometry and its subordinate geometries
109
Mechanization of theorem proving in geometry and Hilberts mechanization theorem
115
32 The standardization of coordinate representation of geometric concepts
118
33 The mechanization of theorem proving and Hilberts mechanization theorem about pure point of intersection theorems in Pascalian geometry
124
34 Examples for Hilberts mechanical method
128
35 Proof of Hilberts mechanization theorem
139
The mechanization theorem of ordinary unordered geometry
149
52 Tarskis theorem and Seidenbergs method
220
53 Examples for the mechanical method of ordered geometries
228
Mechanization theorems of various geometries
235
62 The mechanization of theorem proving in projective geometry
236
63 The mechanization of theorem proving in BolyaiLobachevskys hyperbolic nonEuclidean geometry
246
64 The mechanization of theorem proving in Riemanns elliptic nonEuclidean geometry
258
65 The mechanization of theorem proving in two circle geometries
264
66 The mechanization of formula proving with transcendental functions
267
References
281
Subject index
285
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Elimination Methods
D. Wang
Limited preview - 2001
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