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E-raamat: Mechanical Theorem Proving in Geometries: Basic Principles

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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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Springer Book Archives
Authors note to the English-language edition.- 1 Desarguesian geometry
and the Desarguesian number system.- 1.1 Hilberts axiom system of ordinary
geometry.- 1.2 The axiom of infinity and Desargues axioms.- 1.3 Rational
points in a Desarguesian plane.- 1.4 The Desarguesian number system and
rational number subsystem.- 1.5 The Desarguesian number system on a line.-
1.6 The Desarguesian number system associated with a Desarguesian plane.- 1.7
The coordinate system of Desarguesian plane geometry.- 2 Orthogonal geometry,
metric geometry and ordinary geometry.- 2.1 The Pascalian axiom and
commutative axiom of multiplication (unordered) Pascalian geometry.- 2.2
Orthogonal axioms and (unordered) orthogonal geometry.- 2.3 The orthogonal
coordinate system of (unordered) orthogonal geometry.- 2.4 (Unordered) metric
geometry.- 2.5 The axioms of order and ordered metric geometry.- 2.6 Ordinary
geometry and its subordinate geometries.- 3 Mechanization of theorem proving
in geometry and Hilberts mechanization theorem.- 3.1 Comments on Euclidean
proof method.- 3.2 The standardization of coordinate representation of
geometric concepts.- 3.3 The mechanization of theorem proving and Hilberts
mechanization theorem about pure point of intersection theorems in Pascalian
geometry.- 3.4 Examples for Hilberts mechanical method.- 3.5 Proof of
Hilberts mechanization theorem.- 4 The mechanization theorem of (ordinary)
unordered geometry.- 4.1 Introduction.- 4.2 Factorization of polynomials.-
4.3 Well-ordering of polynomial sets.- 4.4 A constructive theory of algebraic
varieties irreducible ascending sets and irreducible algebraic varieties.-
4.5 A constructive theory of algebraic varieties irreducible decomposition
of algebraic varieties.- 4.6 A constructive theoryof algebraic varieties
the notion of dimension and the dimension theorem.- 4.7 Proof of the
mechanization theorem of unordered geometry.- 4.8 Examples for the mechanical
method of unordered geometry.- 5 Mechanization theorems of (ordinary) ordered
geometries.- 5.1 Introduction.- 5.2 Tarskis theorem and Seidenbergs
method.- 5.3 Examples for the mechanical method of ordered geometries.- 6
Mechanization theorems of various geometries.- 6.1 Introduction.- 6.2 The
mechanization of theorem proving in projective geometry.- 6.3 The
mechanization of theorem proving in Bolyai-Lobachevskys hyperbolic
non-Euclidean geometry.- 6.4 The mechanization of theorem proving in
Riemanns elliptic non-Euclidean geometry.- 6.5 The mechanization of theorem
proving in two circle geometries.- 6.6 The mechanization of formula proving
with transcendental functions.- References.
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