May, 1897.

TO

JOHN McTAGGART ELLIS McTAGGART

TO WHOSE DISCOURSE AND FRIENDSHIP IS OWING

THE EXISTENCE OF THIS BOOK.

[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 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
[153.](4) The Geometry of non-congruent surfaces158
[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
[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 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
[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 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