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