Table of Contents — An Essay on the Foundations of Geometry by Bertrand Russell

	INTRODUCTION.
	OUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC, PSYCHOLOGY AND MATHEMATICS.
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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
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 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.
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
108.	The only qualitative relation between two points is the straight line, and all straight lines are qualitatively similar	120
109.	Hence follows, by extension, the principle of projective transformation	121
110.	By which figures qualitatively indistinguishable from a given figure are obtained	122
111.	Anharmonic ratio may and must be descriptively defined	122
112.	The quadrilateral construction is essential to the projective definition of points,	123
113.	And can be projectively defined,	124
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
147.	Which, however, is logically and philosophically untenable	153
148.	Though Free Mobility is à priori, actual measurement is empirical	154
149.	Some objections remain to be answered, concerning—	154
150.	(1) The comparison of volumes and of Kant’s symmetrical objects	154
151.	(2) The measurement of time, where congruence is impossible	156
152.	(3) The immediate perception of spatial magnitude; and	157
153.	(4) The Geometry of non-congruent surfaces	158
154.	Free Mobility includes Helmholtz’s Monodromy	159
155.	Free Mobility involves the relativity of space	159
156.	From which, reciprocally, it can be deduced	160
157.	Our axiom is therefore à priori in a double sense	160
	II. The Axiom of Dimensions.
158.	Space must have a finite integral number of dimensions	161
159.	But the restriction to three is empirical	162
160.	The general axiom follows from the relativity of position	162
161.	The limitation to three dimensions, unlike most empirical knowledge, is accurate and certain	163
	III. The Axiom of Distance.
162.	The axiom of distance corresponds, here, to that of the straight line in projective Geometry	164
163.	The possibility of spatial measurement involves a magnitude uniquely determined by two points,	164
164.	Since two points must have some relation, and the passivity of space proves this to be independent of external reference	165
165.	There can be only one such relation	166
166.	This must be measured by a curve joining the two points,	166
167.	And the curve must be uniquely determined by the two points	167
168.	Spherical Geometry contains an exception to this axiom,	168
169.	Which, however, is not quite equivalent to Euclid’s	168
170.	The exception is due to the fact that two points, in spherical space, may have an external relation unaltered by motion,	169
171.	Which, however, being a relation of linear magnitude, presupposes the possibility of linear magnitude	170
172.	A relation between two points must be a line joining them	170
173.	Conversely, the existence of a unique line between two points can be deduced from the nature of a form of externality,	171
174.	And necessarily leads to distance, when quantity is applied to it	172
175.	Hence the axiom of distance, also, is à priori in a double sense	172
176.	No metrical coordinate system can be set up without the straight line	174
177.	No axioms besides the above three are necessary to metrical Geometry	175
178.	But these three are necessary to the direct measurement of any continuum	176
179.	Two philosophical questions remain for a final chapter	177
	CHAPTER IV.
	PHILOSOPHICAL CONSEQUENCES.
180.	What is the relation to experience of a form of externality in general?	178
181.	This form is the class-conception, containing every possible intuition of externality; and some such intuition is necessary to experience	178
182.	What relation does this view bear to Kant’s?	179
183.	It is less psychological, since it does not discuss whether space is given in sensation,	180
184.	And maintains that not only space, but any form of externality which renders experience possible, must be given in sense-perception	181
185.	Externality should mean, not externality to the Self, but the mutual externality of presented things	181
186.	Would this be unknowable without a given form of externality?	182
187.	Bradley has proved that space and time preclude the existence of mere particulars,	182
188.	And that knowledge requires the This to be neither simple nor self-subsistent	183
189.	To prove that experience requires a form of externality, I assume that all knowledge requires the recognition of identity in difference	184
190.	Such recognition involves time	184
191.	And some other form giving simultaneous diversity	185
192.	The above argument has not deduced sense-perception from the categories, but has shown the former, unless it contains a certain element, to be unintelligible to the latter	186
193.	How to account for the realization of this element, is a question for metaphysics	187
194.	What are we to do with the contradictions in space?	188
195.	Three contradictions will be discussed in what follows	188
196.	(1) The antinomy of the Point proves the relativity of space,	189
197.	And shows that Geometry must have some reference to matter,	190
198.	By which means it is made to refer to spatial order, not to empty space	191
199.	The causal properties of matter are irrelevant to Geometry, which must regard it as composed of unextended atoms, by which points are replaced	191
200.	(2) The circle in defining straight lines and planes is overcome by the same reference to matter	192
201.	(3) The antinomy that space is relational and yet more than relational,	193
202.	Seems to depend on the confusion of empty space with spatial order	193
203.	Kant regarded empty space as the subject-matter of Geometry,	194
204.	But the arguments of the Aesthetic are inconclusive on this point,	195
205.	And are upset by the mathematical antinomies, which prove that spatial order should be the subject-matter of Geometry	196
206.	The apparent thinghood of space is a psychological illusion, due to the fact that spatial relations are immediately given	196
207.	The apparent divisibility of spatial relations is either an illusion, arising out of empty space, or the expression of the possibility of quantitatively different spatial relations	197
208.	Externality is not a relation, but an aspect of relations. Spatial order, owing to its reference to matter, is a real relation	198
209.	Conclusion	199

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