An Essay on the Foundations of Geometry

CHAPTER I.. A Short History of Metageometry.

A SHORT HISTORY OF METAGEOMETRY.

10. Metageometry began by rejecting the axiom of parallels 7

11. Its history may be divided into three periods: the synthetic, the metrical and the projective 7

12. The first period was inaugurated by Gauss, 10

13. Whose suggestions were developed independently by Lobatchewsky 10

14. And Bolyai 11

15. The purpose of all three was to show that the axiom of parallels could not be deduced from the others, since its denial did not lead to contradictions 12

16. The second period had a more philosophical aim, and was inspired chiefly by Gauss and Herbart 13

17. The first work of this period, that of Riemann, invented two new conceptions: 14

18. The first, that of a manifold, is a class-conception, containing space as a species, 14

19. And defined as such that its determinations form a collection of magnitudes 15

20. The second, the measure of curvature of a manifold, grew out of curvature in curves and surfaces 16

21. By means of Gauss's analytical formula for the curvature of surfaces, 19

22. Which enables us to define a constant measure of curvature of a three-dimensional space without reference to a fourth dimension 20

23. The main result of Riemann's mathematical work was to show that, if magnitudes are independent of place, the measure of curvature of space must be constant 21

24. Helmholtz, who was more of a philosopher than a mathematician, 22

25. Gave a new but incorrect formulation of the essential axioms, 23

26. And deduced the quadratic formula for the infinitesimal arc, which Riemann had assumed 24

27. Beltrami gave Lobatchewsky's planimetry a Euclidean interpretation, 25

28. Which is analogous to Cayley's theory of distance; 26

29. And dealt with n-dimensional spaces of constant negative curvature 27

30. The third period abandons the metrical methods of the second, and extrudes the notion of spatial quantity 27

31. Cayley reduced metrical properties to projective properties, relative to a certain conic or quadric, the Absolute; 28

32. And Klein showed that the Euclidean or non-Euclidean systems result, according to the nature of the Absolute; 29

33. Hence Euclidean space appeared to give rise to all the kinds of Geometry, and the question, which is true, appeared reduced to one of convention 30

34. But this view is due to a confusion as to the nature of the coordinates employed 30

35. Projective coordinates have been regarded as dependent on distance, and thus really metrical 31

36. But this is not the case, since anharmonic ratio can be projectively defined 32

37. Projective coordinates, being purely descriptive, can give no information as to metrical properties, and the reduction of metrical to projective properties is purely technical 33

38. The true connection of Cayley's measure of distance with non-Euclidean Geometry is that suggested by Beltrami's Saggio, and worked out by Sir R. Ball, 36

39. Which provides a Euclidean equivalent for every non-Euclidean proposition, and so removes the possibility of contradictions in Metageometry 38

40. Klein's elliptic Geometry has not been proved to have a corresponding variety of space 39

41. The geometrical use of imaginaries, of which Cayley demanded a philosophical discussion, 41

42. Has a merely technical validity, 42

43. And is capable of giving geometrical results only when it begins and ends with real points and figures 45

44. We have now seen that projective Geometry is logically prior to metrical Geometry, but cannot supersede it 46

45. Sophus Lie has applied projective methods to Helmholtz's formulation of the axioms, and has shown the axiom of Monodromy to be superfluous 46

46. Metageometry has gradually grown independent of philosophy, but has grown continually more interesting to philosophy 50

47. Metrical Geometry has three indispensable axioms, 50

48. Which we shall find to be not results, but conditions, of measurement, 51

49. And which are nearly equivalent to the three axioms of projective Geometry 52

50. Both sets of axioms are necessitated, not by facts, but by logic 52

CHAPTER II.. Critical Account of Some Previous Philosophical

CRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL THEORIES OF GEOMETRY.

51. A criticism of representative modern theories need not begin before Kant 54

52. Kant's doctrine must be taken, in an argument about Geometry, on its purely logical side 55

53. Kant contends that since Geometry is apodeictic, space must be à priori and subjective, while since space is à priori and subjective, Geometry must be apodeictic 55

54. Metageometry has upset the first line of argument, not the second 56

55. The second may be attacked by criticizing either the distinction of synthetic and analytic judgments, or the first two arguments of the metaphysical deduction of space 57

56. Modern Logic regards every judgment as both synthetic and analytic, 57

57. But leaves the à priori, as that which is presupposed in the possibility of experience 59

58. Kant's first two arguments as to space suffice to prove some form of externality, but not necessarily Euclidean space, a necessary condition of experience 60

59. Among the successors of Kant, Herbart alone advanced the theory of Geometry, by influencing Riemann 62

60. Riemann regarded space as a particular kind of manifold, i.e. wholly quantitatively 63

61. He therefore unduly neglected the qualitative adjectives of space 64

62. His philosophy rests on a vicious disjunction 65

63. His definition of a manifold is obscure, 66

64. And his definition of measurement applies only to space 67

65. Though mathematically invaluable, his view of space as a manifold is philosophically misleading 69

66. Helmholtz attacked Kant both on the mathematical and on the psychological side; 70

67. But his criterion of apriority is changeable and often invalid; 71

68. His proof that non-Euclidean spaces are imaginable is inconclusive; 72

69. And his assertion of the dependence of measurement on rigid bodies, which may be taken in three senses, 74

70. Is wholly false if it means that the axiom of Congruence actually asserts the existence of rigid bodies, 75

71. Is untrue if it means that the necessary reference of geometrical propositions to matter renders pure Geometry empirical, 76

72. And is inadequate to his conclusion if it means, what is true, that actual measurement involves approximately rigid bodies 78

73. Geometry deals with an abstract matter, whose physical properties are disregarded; and Physics must presuppose Geometry 80

74. Erdmann accepted the conclusions of Riemann and Helmholtz, 81

75. And regarded the axioms as necessarily successive steps in classifying space as a species of manifold 82

76. His deduction involves four fallacious assumptions, namely: 82

77. That conceptions must be abstracted from a series of instances; 83

78. That all definition is classification; 83

79. That conceptions of magnitude can be applied to space as a whole; 84

80. And that if conceptions of magnitude could be so applied, all the adjectives of space would result from their application 86

81. Erdmann regards Geometry alone as incapable of deciding on the truth of the axiom of Congruence, 86

82. Which he affirms to be empirically proved by Mechanics. 88

83. The variety and inadequacy of Erdmann's tests of apriority 89

84. Invalidate his final conclusions on the theory of Geometry 90

85. Lotze has discussed two questions in the theory of Geometry: 93

86. (1) He regards the possibility of non-Euclidean spaces as suggested by the subjectivity of space, 93

87. And rejects it owing to a mathematical misunderstanding, 96

88. Having missed the most important sense of their possibility, 96

89. Which is that they fulfil the logical conditions to which any form of externality must conform 97

90. (2) He attacks the mathematical procedure of Metageometry 98

91. The attack begins with a question-begging definition of parallels 99

92. Lotze maintains that all apparent departures from Euclid could be physically explained, a view which really makes Euclid empirical 99

93. His criticism of Helmholtz's analogies rests wholly on mathematical mistakes 101

94. His proof that space must have three dimensions rests on neglect of different orders of infinity 104

95. He attacks non-Euclidean spaces on the mistaken ground that they are not homogeneous 107

96. Lotze's objections fall under four heads 108

97. Two other semi-philosophical objections may be urged, 109

98. One of which, the absence of similarity, has been made the basis of attack by Delbœuf, 110

99. But does not form a valid ground of objection 111

100. Recent French speculation on the foundations of Geometry has suggested few new views 112

101. All homogeneous spaces are à priori possible, and the decision between them is empirical 114

CHAPTER III.. Section a. the Axioms of Projective Geometry.

SECTION A. THE AXIOMS OF PROJECTIVE GEOMETRY.

102. Projective Geometry does not deal with magnitude, and applies to all spaces alike 117

103. It will be found wholly à priori 117

104. Its axioms have not yet been formulated philosophically 118

105. Coordinates, in projective Geometry, are not spatial magnitudes, but convenient names for points 118

106. The possibility of distinguishing various points is an axiom 119

107. The qualitative relations between points, dealt with by projective Geometry, are presupposed by the quantitative treatment 119

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