CHAPTER II.

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