Introductory material
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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 subjective1
2.A mental state is subjective, for Psychology, when its immediate cause does not lie in the outer world2
3.A piece of knowledge is à priori, for Epistemology, when without it knowledge would be impossible2
4.The subjective and the à priori belong respectively to Psychology and to Epistemology. The latter alone will be investigated in this essay3
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 necessity4
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 ground5
9.Forecast of the work5
CHAPTER I.
A SHORT HISTORY OF METAGEOMETRY.
10.Metageometry began by rejecting the axiom of parallels7
11.Its history may be divided into three periods: the synthetic, the metrical and the projective7
12.The first period was inaugurated by Gauss,10
13.Whose suggestions were developed independently by Lobatchewsky10
14.And Bolyai11
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 contradictions12
16.The second period had a more philosophical aim, and was inspired chiefly by Gauss and Herbart13
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 magnitudes15
20.The second, the measure of curvature of a manifold, grew out of curvature in curves and surfaces16
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 dimension20
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 constant21
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 assumed24
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 curvature27
30.The third period abandons the metrical methods of the second, and extrudes the notion of spatial quantity27
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 convention30
34.But this view is due to a confusion as to the nature of the coordinates employed30
35.Projective coordinates have been regarded as dependent on distance, and thus really metrical31
36.But this is not the case, since anharmonic ratio can be projectively defined32
37.Projective coordinates, being purely descriptive, can give no information as to metrical properties, and the reduction of metrical to projective properties is purely technical33
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 Metageometry38
40.Klein's elliptic Geometry has not been proved to have a corresponding variety of space39
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 figures45
44.We have now seen that projective Geometry is logically prior to metrical Geometry, but cannot supersede it46
45.Sophus Lie has applied projective methods to Helmholtz's formulation of the axioms, and has shown the axiom of Monodromy to be superfluous46
46.Metageometry has gradually grown independent of philosophy, but has grown continually more interesting to philosophy50
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 Geometry52
50.Both sets of axioms are necessitated, not by facts, but by logic52
CHAPTER II.
CRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL THEORIES OF GEOMETRY.
51.A criticism of representative modern theories need not begin before Kant54
52.Kant's doctrine must be taken, in an argument about Geometry, on its purely logical side55
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 apodeictic55
54.Metageometry has upset the first line of argument, not the second56
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 space57
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 experience59
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 experience60
59.Among the successors of Kant, Herbart alone advanced the theory of Geometry, by influencing Riemann62
60.Riemann regarded space as a particular kind of manifold, i.e. wholly quantitatively63
61.He therefore unduly neglected the qualitative adjectives of space64
62.His philosophy rests on a vicious disjunction65
63.His definition of a manifold is obscure,66
64.And his definition of measurement applies only to space67
65.Though mathematically invaluable, his view of space as a manifold is philosophically misleading69
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 bodies78
73.Geometry deals with an abstract matter, whose physical properties are disregarded; and Physics must presuppose Geometry80
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 manifold82
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 application86
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 apriority89
84.Invalidate his final conclusions on the theory of Geometry90
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 conform97
90.(2) He attacks the mathematical procedure of Metageometry98
91.The attack begins with a question-begging definition of parallels99
92.Lotze maintains that all apparent departures from Euclid could be physically explained, a view which really makes Euclid empirical99
93.His criticism of Helmholtz's analogies rests wholly on mathematical mistakes101
94.His proof that space must have three dimensions rests on neglect of different orders of infinity104
95.He attacks non-Euclidean spaces on the mistaken ground that they are not homogeneous107
96.Lotze's objections fall under four heads108
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 objection111
100.Recent French speculation on the foundations of Geometry has suggested few new views112
101.All homogeneous spaces are à priori possible, and the decision between them is empirical114
CHAPTER III.
Section A. the axioms of projective geometry.
102.Projective Geometry does not deal with magnitude, and applies to all spaces alike117
103.It will be found wholly à priori117
104.Its axioms have not yet been formulated philosophically118
105.Coordinates, in projective Geometry, are not spatial magnitudes, but convenient names for points118
106.The possibility of distinguishing various points is an axiom119
107.The qualitative relations between points, dealt with by projective Geometry, are presupposed by the quantitative treatment119
108.The only qualitative relation between two points is the straight line, and all straight lines are qualitatively similar120
109.Hence follows, by extension, the principle of projective transformation121
110.By which figures qualitatively indistinguishable from a given figure are obtained122
111.Anharmonic ratio may and must be descriptively defined122
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 transformation126
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 contradictory128
117.We define the point as that which is spatial, but contains no space, whence other definitions follow128
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 equivalent129
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 given130
121.Any two projectively related figures are qualitatively equivalent, i.e. differ in no non-quantitative conceptual property131
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 space133
125.The conception of a form of externality,134
126.Being a creature of the intellect, can be dealt with by pure mathematics134
127.The resulting doctrine of extension will be, for the moment, hypothetical135
128.But is rendered assertorical by the necessity, for experience, of some form of externality136
129.Any such form must be relational136
130.And homogeneous137
131.And the relations constituting it must appear infinitely divisible137
132.It must have a finite integral number of dimensions,139
133.Owing to its passivity and homogeneity140
134.And to the systematic unity of the world140
135.A one-dimensional form alone would not suffice for experience141
136.Since its elements would be immovably fixed in a series142
137.Two positions have a relation independent of other positions,143
138.Since positions are wholly defined by mutually independent relations143
139.Hence projective Geometry is wholly à priori,146
140.Though metrical Geometry contains an empirical element146
Section B. the axioms of metrical geometry.
141.Metrical Geometry is distinct from projective, but has the same fundamental postulate147
142.It introduces the new idea of motion, and has three à priori axioms148
I. The Axiom of Free Mobility.
143.Measurement requires a criterion of spatial equality149
144.Which is given by superposition, and involves the axiom of Free Mobility150
145.The denial of this axiom involves an action of empty space on things151
146.There is a mathematically possible alternative to the axiom,152
147.Which, however, is logically and philosophically untenable153
148.Though Free Mobility is à priori, actual measurement is empirical154
149.Some objections remain to be answered, concerning—154
150.(1) The comparison of volumes and of Kant's symmetrical objects154
151.(2) The measurement of time, where congruence is impossible156
152.(3) The immediate perception of spatial magnitude; and157
154.Free Mobility includes Helmholtz's Monodromy159
155.Free Mobility involves the relativity of space159
156.From which, reciprocally, it can be deduced160
157.Our axiom is therefore à priori in a double sense160
II. The Axiom of Dimensions.
158.Space must have a finite integral number of dimensions161
159.But the restriction to three is empirical162
160.The general axiom follows from the relativity of position162
161.The limitation to three dimensions, unlike most empirical knowledge, is accurate and certain163
III. The Axiom of Distance.
162.The axiom of distance corresponds, here, to that of the straight line in projective Geometry164
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 reference165
165.There can be only one such relation166
166.This must be measured by a curve joining the two points,166
167.And the curve must be uniquely determined by the two points167
168.Spherical Geometry contains an exception to this axiom,168
169.Which, however, is not quite equivalent to Euclid's168
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 magnitude170
172.A relation between two points must be a line joining them170
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 it172
175.Hence the axiom of distance, also, is à priori in a double sense172
176.No metrical coordinate system can be set up without the straight line174
177.No axioms besides the above three are necessary to metrical Geometry175
178.But these three are necessary to the direct measurement of any continuum176
179.Two philosophical questions remain for a final chapter177
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 experience178
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-perception181
185.Externality should mean, not externality to the Self, but the mutual externality of presented things181
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-subsistent183
189.To prove that experience requires a form of externality, I assume that all knowledge requires the recognition of identity in difference184
190.Such recognition involves time184
191.And some other form giving simultaneous diversity185
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 latter186
193.How to account for the realization of this element, is a question for metaphysics187
194.What are we to do with the contradictions in space?188
195.Three contradictions will be discussed in what follows188
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 space191
199.The causal properties of matter are irrelevant to Geometry, which must regard it as composed of unextended atoms, by which points are replaced191
200.(2) The circle in defining straight lines and planes is overcome by the same reference to matter192
202.Seems to depend on the confusion of empty space with spatial order193
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 Geometry196
206.The apparent thinghood of space is a psychological illusion, due to the fact that spatial relations are immediately given196
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 relations197
208.Externality is not a relation, but an aspect of relations. Spatial order, owing to its reference to matter, is a real relation198
209.Conclusion199