An essay on the foundations of geometry

TABLE OF CONTENTS.

	INTRODUCTION.
	OUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC, PSYCHOLOGY AND MATHEMATICS.
		PAGE
1.	The problem first received a modern form through Kant, who connected the À priori with the subjective	1
2.	A mental state is subjective, for Psychology, when its immediate cause does not lie in the outer world	2
3.	A piece of knowledge is À priori, for Epistemology, when without it knowledge would be impossible	2
4.	The subjective and the À priori belong respectively to Psychology and to Epistemology. The latter alone will be investigated in this essay	3
5.	My test of the À priori will be purely logical: what knowledge is necessary for experience?	3
6.	But since the necessary is hypothetical, we must include, in the À priori, the ground of necessity	4
7.	This may be the essential postulate of our science, or the element, in the subject-matter, which is necessary to experience;	4
8.	Which, however, are both at bottom the same ground	5
9.	Forecast of the work	5

	CHAPTER I.
	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
114.	By the general principle of projective transformation	126
115.	The principle of duality is the mathematical form of a philosophical circle,	127
116.	Which is an inevitable consequence of the relativity of space, and makes any definition of the point contradictory	128
117.	We define the point as that which is spatial, but contains no space, whence other definitions follow	128
118.	What is meant by qualitative equivalence in Geometry?	129
119.	Two pairs of points on one straight line, or two pairs of straight lines through one point, are qualitatively equivalent	129
120.	This explains why four collinear points are needed, to give an intrinsic relation by which the fourth can be descriptively defined when the first three are given	130
121.	Any two projectively related figures are qualitatively equivalent, i.e. differ in no non-quantitative conceptual property	131
122.	Three axioms are used by projective Geometry,	132
123.	And are required for qualitative spatial comparison,	132
124.	Which involves the homogeneity, relativity and passivity of space	133
125.	The conception of a form of externality,	134
126.	Being a creature of the intellect, can be dealt with by pure mathematics	134
127.	The resulting doctrine of extension will be, for the moment, hypothetical	135
128.	But is rendered assertorical by the necessity, for experience, of some form of externality	136
129.	Any such form must be relational	136
130.	And homogeneous	137
131.	And the relations constituting it must appear infinitely divisible	137
132.	It must have a finite integral number of dimensions,	139
133.	Owing to its passivity and homogeneity	140
134.	And to the systematic unity of the world	140
135.	A one-dimensional form alone would not suffice for experience	141
136.	Since its elements would be immovably fixed in a series	142
137.	Two positions have a relation independent of other positions,	143
138.	Since positions are wholly defined by mutually independent relations	143
139.	Hence projective Geometry is wholly À priori,	146
140.	Though metrical Geometry contains an empirical element	146
	Section B. the axioms of metrical geometry.

141.	Metrical Geometry is distinct from projective, but has the same fundamental postulate	147
142.	It introduces the new idea of motion, and has three À priori axioms	148
	I. The Axiom of Free Mobility.

143.	Measurement requires a criterion of spatial equality	149
144.	Which is given by superposition, and involves the axiom of Free Mobility	150
145.	The denial of this axiom involves an action of empty space on things	151
146.	There is a mathematically possible alternative to the axiom,	152
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