Mechanical Theorem Proving in Geometries

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.

Contenu

Author's note to the English-language edition.- 1 Desarguesian geometry and the Desarguesian number system.- 1.1 Hilbert's 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 Hilbert's 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 Hilbert's mechanization theorem about pure point of intersection theorems in Pascalian geometry.- 3.4 Examples for Hilbert's mechanical method.- 3.5 Proof of Hilbert's 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 theory of 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 Tarski's theorem and Seidenberg's 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-Lobachevsky's hyperbolic non-Euclidean geometry.- 6.4 The mechanization of theorem proving in Riemann's 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.