加载中…希尔伯特《几何基捶的写作大纲
(2019-03-13 10:07:27)希尔伯特《几何基础》的写作大纲
袁萌在初中阶段就迷恋于组装“矿石”收音机，经常去旧货市场找电子“零件”。有一次在一个旧书摊上发现一本旧书希尔伯特《几何基础》。当时，袁萌并不懂得这本的价值，只是觉得这本书的名也许有用”。当时，袁萌没有钱买这本书（5分钱），就用一个小电子元件换回了这本书。
本文附件是希尔伯特《几何基础》的完备写作大纲，其其中第四章“面积论”是现代积分学的最根本原理，是我们的“最爱”。数学公理化就从这里开始！什么叫“面积”（这就是积分定义的根据）？……
袁萌
附件：
CONTENTS
PAGE Introduction ..............1
CHAPTER I. THE FIVE GROUPS OF AXIOMS.
§ 1. The elements of geometry and the ve groups of axioms ............. 2 § 2. Group I: Axioms of connection ......................... 2
§ 3. Group II: Axioms of Order ................ 3
§ 4. Consequences of the axioms of connection and order ...... 5
§ 5. Group III: Axiom of Parallels (Euclid’s axiom) .......... 7
§ 6. Group IV: Axioms of congruence ........ 8
§ 7. Consequences of the axioms of congruence ............. 10
§ 8. Group V: Axiom of Continuity (Archimedes’s axiom) ...... 15
CHAPTER II. THE COMPATIBILITY AND MUTUAL INDEPENDENCE OF THE AXIOMS.
§ 9. Compatibility of the axioms.. 17
§10. Independence of the axioms of parallels. Non-euclidean geometry ... 19
§11. Independence of the axioms of congruence .............. 20
§12. Independence of the axiom of continuity. Non-archimedean geometry 21
CHAPTER III. THE THEORY OF PROPORTION.
§13. Complex number-systems ....23
§14. Demonstration of Pascal’s theorem ............... 25
§15. An algebra of segments, based upon Pascal’s theorem ........... 30
§16. Proportion and the theorems of similitude ............... 33
§17. Equations of straight lines and of planes .................... 35
CHAPTER IV. THE THEORY OF PLANE AREAS.
§18. Equal area and equal content of polygons ................ 38
§19. Parallelograms and triangles having equal bases and equal altitudes . 40 §20. The measure of area of triangles and polygons ............... 41
§21. Equality of content and the measure of area ............ 44
CHAPTER V. DESARGUES’S THEOREM.
§22. Desargues’s theorem and its demonstration for plane geometry by aid of the axioms of congruence ..........
...
48
§23. The impossibility of demonstrating Desargues’s theorem for the plane without the help of the axioms of congruence ........... 50
§24. Introduction of an algebra of segments based upon Desargues’s theorem and independent of the axioms of congruence .......... 53
§25. The commutative and the associative law of addition for our new algebra of segments .......... 55
§26. The associative law of multiplication and the two distributive laws for the new algebra of segments .............................
.... 56
§27. Equation of the straight line, based upon the new algebra of segments ... 61
§28. The totality of segments, regarded as a complex number system ..... 64
§29. Construction of a geometry of space by aid of a desarguesian number system .................... 65
§30. Signicance of Desargues’s theorem ....... 67
CHAPTER VI. PASCAL’S THEOREM.
§31. Two theorems concerning the possibility of proving Pascal’s theorem .... 68
§32. The commutative law of multiplication for an archimedean number system ................. 68
§33. The commutative law of multiplication for a non-archimedean number system ....................... 70
§34. Proof of the two propositions concerning Pascal’s theorem. Non-pascalian geometry. ..................72
§35. The demonstration, by means of the theorems of Pascal and Desargues, of any theorem relating to points of intersection ............. 73
CHAPTER VII. GEOMETRICAL CONSTRUCTIONS BASED UPON THE AXIOMS I–V.
§36. Geometrical constructions by means of a straight-edge and a transferer of segments .............. 74
§37. Analytical representation of the co-ordinates of points which can be so constructed ...................... 76
§38. The representation of algebraic numbers and of integral rational functions as sums of squares .......................... 78
§39. Criterion for the possibility of a geometrical construction by means of a straight-edge and a transferer of segments ....80
Conclusion ..... 83-92
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