BIBLIOGRAPHICAL NOTE
THE PROBLEM OF CHINA
By Bertrand Russell
CONTENTS
QUESTIONS
CHINA BEFORE THE NINETEENTH CENTURY
CHINA AND THE WESTERN POWERS
MODERN CHINA
JAPAN BEFORE THE RESTORATION
MODERN JAPAN
JAPAN AND CHINA BEFORE 1914
JAPAN AND CHINA DURING THE WAR
THE WASHINGTON CONFERENCE
PRESENT FORCES AND TENDENCIES IN THE FAR EAST
CHINESE AND WESTERN CIVILIZATION CONTRASTED
THE CHINESE CHARACTER
HIGHER EDUCATION IN CHINA
INDUSTRIALISM IN CHINA
THE OUTLOOK FOR CHINA
APPENDIX
INDEX
THE PRACTICE AND THEORY OF BOLSHEVISM
Bertrand Russell
CONTENTS
PAGE
PART I
THE PRESENT CONDITION OF RUSSIA
I. WHAT IS HOPED FROM BOLSHEVISM 15
II. GENERAL CHARACTERISTICS 24
III. LENIN, TROTSKY AND GORKY 36
IV. ART AND EDUCATION 45
V. COMMUNISM AND THE SOVIET CONSTITUTION 72
VI. THE FAILURE OF RUSSIAN INDUSTRY 81
VII. DAILY LIFE IN MOSCOW 92
VIII. TOWN AND COUNTRY 99
IX. INTERNATIONAL POLICY 106
PART II
BOLSHEVIK THEORY
I. THE MATERIALISTIC THEORY OF HISTORY 119
II. DECIDING FORCES IN POLITICS 128
III. BOLSHEVIK CRITICISM OF DEMOCRACY 134
IV. REVOLUTION AND DICTATORSHIP 146
V. MECHANISM AND THE INDIVIDUAL 157
VI. WHY RUSSIAN COMMUNISM HAS FAILED 165
VII. CONDITIONS FOR THE SUCCESS OF COMMUNISM 178
MYSTICISM AND LOGIC
AND OTHER ESSAYS
Bertrand Russell
CONTENTS
Chapter Page
I. Mysticism and Logic 1
II. The Place of Science in a Liberal Education 33
III. A Free Man's Worship 46
IV. The Study of Mathematics 58
V. Mathematics and the Metaphysicians 74
VI. On Scientific Method in Philosophy 97
VII. The Ultimate Constituents of Matter 125
VIII. The Relation of Sense-data to Physics 145
IX. On the Notion of Cause 180
X. Knowledge by Acquaintance and Knowledge by Description 209
Index 233
OUR KNOWLEDGE OF THE EXTERNAL WORLD
AS A FIELD FOR SCIENTIFIC METHOD IN PHILOSOPHY,
By Bertrand Russell
CONTENTS
LECTURE PAGE
I. Current Tendencies 3
II. Logic as the Essence of Philosophy 33
III. On our Knowledge of the External World 63
IV. The World of Physics and the World of Sense 101
V. The Theory of Continuity 129
VI. The Problem of Infinity considered Historically 155
VII. The Positive Theory of Infinity 185
VIII. On the Notion of Cause, with Applications to the Free-will Problem 211
Index 243
INDEX
Absolute, 6, 39.
Abstraction, principle of, 42, 124 ff.
Achilles, Zeno's argument of, 173.
Acquaintance, 25, 144.
Activity, 224 ff.
Allman, 161 n.
Analysis, 185, 204, 211, 241.
legitimacy of, 150.
Anaximander, 3.
Antinomies, Kant's, 155 ff.
Aquinas, 10.
Aristotle, 40, 160 n., 161 ff., 240.
Arrow, Zeno's argument of, 173.
Assertion, 52.
Atomism, logical, 4.
Atomists, 160.
Belief, 58.
primitive and derivative, 69 ff.
Bergson, 4, 11, 13, 20 ff., 137, 138, 150, 158, 165, 174, 178, 229 ff.
Berkeley, 63, 64, 102.
Bolzano, 165.
Boole, 40.
Bradley, 6, 39, 165.
Broad, 172 n.
Brochard, 169 n.
Burnet, 19 n., 160 n., 161 n., 170 n., 171 ff.
Calderon, 95.
Cantor, vi, vii, 155, 165, 190, 194, 199.
Categories, 38.
Causal laws, 109, 212 ff.
evidence for, 216 ff.
in psychology, 219.
Causation, 34 ff., 79, 212 ff.
law of, 221.
not a priori, 223, 232.
Cause, 220, 223.
Certainty, degrees of, 67, 68, 212.
Change,
demands analysis, 151.
Cinematograph, 148, 174.
Classes, 202.
non-existence of, 205 ff.
Classical tradition, 3 ff., 58.
Complexity, 145, 157 ff.
Compulsion, 229, 233 ff.
Congruence, 195.
Consecutiveness, 134.
Conservation, 105.
Constituents of facts, 51, 145.
Construction v. inference, iv.
Contemporaries, initial, 119, 120 n.
Continuity, 64, 129 ff., 141 ff., 155 ff.
of change, 106, 108, 130 ff.
Correlation of mental and physical, 233.
Counting, 164, 181, 187 ff., 203.
Couturat, 40 n.
Dante, 10.
Darwin, 4, 11, 23, 30.
Data, 65 ff., 211.
�hard� and �soft,� 70 ff.
Dates, 117.
Definition, 204.
Descartes, 5, 73, 238.
Descriptions, 201, 214.
Desire, 227, 235.
Determinism, 233.
Doubt, 237.
Dreams, 85, 93.
Duration, 146, 149.
Earlier and later, 116.
Effect, 220.
Eleatics, 19.
Empiricism, 37, 222.
Enclosure, 114 ff., 120.
Enumeration, 202.
Euclid, 160, 164.
Evellin, 169.
Evolutionism, 4, 11 ff.
Extension, 146, 149.
External world, knowledge of, 63 ff.
Fact, 51.
atomic, 52.
Finalism, 13.
Form, logical, 42 ff., 185, 208.
Fractions, 132, 179.
Free will, 213, 227 ff.
Frege, 5, 40, 199 ff.
Galileo, 4, 59, 192, 194, 239, 240.
Gaye, 169 n., 175, 177.
Geometry, 5.
Giles, 206 n.
Greater and less, 195.
Harvard, 4.
Hegel, 3, 37 ff., 46, 166.
�Here,� 73, 92.
Hereditary properties, 195.
Hippasos, 163, 237.
Hui Tzu, 206.
Hume, 217, 221.
Hypotheses in philosophy, 239.
Illusions, 85.
Incommensurables, 162 ff., 237.
Independence, 73, 74.
causal and logical, 74, 75.
Indiscernibility, 141, 148.
Indivisibles, 160.
Induction, 34, 222.
mathematical, 195 ff.
Inductiveness, 190, 195 ff.
Inference, 44, 54.
Infinite, vi, 64, 133, 149.
historically considered, 155 ff.
�true,� 179, 180.
positive theory of, 185 ff.
Infinitesimals, 135.
Instants, 116 ff., 129, 151, 216.
defined, 118.
Instinct v. Reason, 20 ff.
Intellect, 22 ff.
Intelligence, how displayed by friends, 93.
inadequacy of display, 96.
Interpretation, 144.
James, 4, 10, 13.
Jourdain, 165 n.
Jowett, 167.
Judgment, 58.
Kant, 3, 112, 116, 155 ff., 200.
Knowledge about, 144.
Language, bad, 82, 135.
Laplace, 12.
Laws of nature, 218 ff.
Leibniz, 13, 40, 87, 186, 191.
Logic, 201.
analytic not constructive, 8.
Aristotelian, 5.
and fact, 53.
inductive, 34, 222.
mathematical, vi, 40 ff.
mystical, 46.
and philosophy, 8, 33 ff., 239.
Logical constants, 208, 213.
Mach, 123, 224.
Macran, 39 n.
Mathematics, 40, 57.
Matter, 75, 101 ff.
permanence of, 102 ff.
Measurement, 164.
Memory, 230, 234, 236.
Method, deductive, 5.
logical-analytic, v, 65, 211, 236 ff.
Milhaud, 168 n., 169 n.
Mill, 34, 200.
Montaigne, 28.
Motion, 130, 216.
continuous, 133, 136.
mathematical theory of, 133.
perception of, 137 ff.
Zeno's arguments on, 168 ff.
Mysticism, 19, 46, 63, 95.
Newton, 30, 146.
Nietzsche, 10, 11.
No�l, 169.
Number, cardinal, 131, 186 ff.
defined, 199 ff.
finite, 160, 190 ff.
inductive, 197.
infinite, 178, 180, 188 ff., 197.
reflexive, 190 ff.
Occam, 107, 146.
One and many, 167, 170.
Order, 131.
Parmenides, 63, 165 ff., 178.
Past and future, 224, 234 ff.
Peano, 40.
Perspectives, 88 ff., 111.
Philoponus, 171 n.
Philosophy and ethics, 26 ff.
and mathematics, 185 ff.
province of, 17, 26, 185, 236.
scientific, 11, 16, 18, 29, 236 ff.
Physics, 101 ff., 147, 239, 242.
descriptive, 224.
verifiability of, 81, 110.
Place, 86, 90.
at and from, 92.
Plato, 4, 19, 27, 46, 63, 165 n., 166, 167.
Poincar�, 123, 141.
Points, 113 ff., 129, 158.
definition of, vi, 115.
Pragmatism, 11.
Prantl, 174.
Predictability, 229 ff.
Premisses, 211.
Probability, 36.
Propositions, 52.
atomic, 52.
general, 55.
molecular, 54.
Pythagoras, 19, 160 ff., 237.
Race-course, Zeno's argument of, 171 ff.
Realism, new, 6.
Reflexiveness, 190 ff.
Relations, 45.
asymmetrical, 47.
Bradley's reasons against, 6.
external, 150.
intransitive, 48.
multiple, 50.
one-one, 203.
reality of, 49.
symmetrical, 47, 124.
transitive, 48, 124.
Relativity, 103, 242.
Repetitions, 230 ff.
Rest, 136.
Ritter and Preller, 161 n.
Robertson, D. S., 160 n.
Rousseau, 20.
Royce, 50.
Santayana, 46.
Scepticism, 66, 67.
Seeing double, 86.
Self, 73.
Sensation, 25, 75, 123.
and stimulus, 139.
Sense-data, 56, 63, 67, 75, 110, 141, 143, 213.
and physics, v, 64, 81, 97, 101 ff., 140.
infinitely numerous? 149, 159.
Sense-perception, 53.
Series, 49.
compact, 132, 142, 178.
continuous, 131, 132.
Sigwart, 187.
Simplicius, 170 n.
Simultaneity, 116.
Space, 73, 88, 103, 112 ff., 130.
absolute and relative, 146, 159.
antinomies of, 155 ff.
perception of, 68.
of perspectives, 88 ff.
private, 89, 90.
of touch and sight, 78, 113.
Spencer, 4, 12, 236.
Spinoza, 46, 166.
Stadium, Zeno's argument of, 134 n., 175 ff.
Subject-predicate, 45.
Synthesis, 157, 185.
Tannery, Paul, 169 n.
Teleology, 223.
Testimony, 67, 72, 82, 87, 96, 212.
Thales, 3.
Thing-in-itself, 75, 84.
Things, 89 ff., 104 ff., 213.
Time, 103, 116 ff., 130, 155 ff., 166, 215.
absolute or relative, 146.
local, 103.
private, 121.
Uniformities, 217.
Unity, organic, 9.
Universal and particular, 39 n.
Volition, 223 ff.
Whitehead, vi, 207.
Wittgenstein, vii, 208 n.
Worlds, actual and ideal, 111.
possible, 186.
private, 88.
Zeller, 173.
Zeno, 129, 134, 136, 165 ff.
[1] Delivered as Lowell Lectures in Boston, in March and April 1914.
[2] London and New York, 1912 (�Home University Library�).
[3] The first volume was published at Cambridge in 1910, the second in 1912, and the third in 1913.
[4] Appearance and Reality, pp. 32�33.
[5] Creative Evolution, English translation, p. 41.
[6] Cf. Burnet, Early Greek Philosophy, pp. 85 ff.
[7] Introduction to Metaphysics, p. 1.
[8] Logic, book iii., chapter iii., � 2.
[9] Book iii., chapter xxi., � 3.
[10] Or rather a propositional function.
[11] The subject of causality and induction will be discussed again in Lecture VIII.
[12] See the translation by H. S. Macran, Hegel's Doctrine of Formal Logic, Oxford, 1912. Hegel's argument in this portion of his �Logic� depends throughout upon confusing the �is� of predication, as in �Socrates is mortal,� with the �is� of identity, as in �Socrates is the philosopher who drank the hemlock.� Owing to this confusion, he thinks that �Socrates� and �mortal� must be identical. Seeing that they are different, he does not infer, as others would, that there is a mistake somewhere, but that they exhibit �identity in difference.� Again, Socrates is particular, �mortal� is universal. Therefore, he says, since Socrates is mortal, it follows that the particular is the universal�taking the �is� to be throughout expressive of identity. But to say �the particular is the universal� is self-contradictory. Again Hegel does not suspect a mistake but proceeds to synthesise particular and universal in the individual, or concrete universal. This is an example of how, for want of care at the start, vast and imposing systems of philosophy are built upon stupid and trivial confusions, which, but for the almost incredible fact that they are unintentional, one would be tempted to characterise as puns.
[13] Cf. Couturat, La Logique de Leibniz, pp. 361, 386.
[14] It was often recognised that there was some difference between them, but it was not recognised that the difference is fundamental, and of very great importance.
[15] Encyclop�dia of the Philosophical Sciences, vol. i. p. 97.
[16] This perhaps requires modification in order to include such facts as beliefs and wishes, since such facts apparently contain propositions as components. Such facts, though not strictly atomic, must be supposed included if the statement in the text is to be true.
[17] The assumptions made concerning time-relations in the above are as follows:�
[18] The above paradox is essentially the same as Zeno's argument of the stadium which will be considered in our next lecture.
[19] See next lecture.
[20] Monist, July 1912, pp. 337�341.
[21] �Le continu math�matique,� Revue de M�taphysique et de Morale, vol. i. p. 29.
[22] In what concerns the early Greek philosophers, my knowledge is largely derived from Burnet's valuable work, Early Greek Philosophy (2nd ed., London, 1908). I have also been greatly assisted by Mr D. S. Robertson of Trinity College, who has supplied the deficiencies of my knowledge of Greek, and brought important references to my notice.
[23] Cf. Aristotle, Metaphysics, M. 6, 1080b, 18 sqq., and 1083b, 8 sqq.
[24] There is some reason to think that the Pythagoreans distinguished between discrete and continuous quantity. G. J. Allman, in his Greek Geometry from Thales to Euclid, says (p. 23): �The Pythagoreans made a fourfold division of mathematical science, attributing one of its parts to the how many, t? p?s??, and the other to the how much, t? p??????; and they assigned to each of these parts a twofold division. For they said that discrete quantity, or the how many, either subsists by itself or must be considered with relation to some other; but that continued quantity, or the how much, is either stable or in motion. Hence they affirmed that arithmetic contemplates that discrete quantity which subsists by itself, but music that which is related to another; and that geometry considers continued quantity so far as it is immovable; but astronomy (t?? sfa??????) contemplates continued quantity so far as it is of a self-motive nature. (Proclus, ed. Friedlein, p. 35. As to the distinction between t? p??????, continuous, and t? p?s??, discrete quantity, see Iambl., in Nicomachi Geraseni Arithmeticam introductionem, ed. Tennulius, p. 148.)� Cf. p. 48.
[25] Referred to by Burnet, op. cit., p. 120.
[26] iv., 6. 213b, 22; H. Ritter and L. Preller, Historia Philosophi� Gr�c�, 8th ed., Gotha, 1898, p. 75 (this work will be referred to in future as �R. P.�).
[27] The Pythagorean proof is roughly as follows. If possible, let the ratio of the diagonal to the side of a square be m/n, where m and n are whole numbers having no common factor. Then we must have m2 = 2n2. Now the square of an odd number is odd, but m2, being equal to 2n2, is even. Hence m must be even. But the square of an even number divides by 4, therefore n2, which is half of m2, must be even. Therefore n must be even. But, since m is even, and m and n have no common factor, n must be odd. Thus n must be both odd and even, which is impossible; and therefore the diagonal and the side cannot have a rational ratio.
[28] In regard to Zeno and the Pythagoreans, I have derived much valuable information and criticism from Mr P. E. B. Jourdain.
[29] So Plato makes Zeno say in the Parmenides, apropos of his philosophy as a whole; and all internal and external evidence supports this view.
[30] �With Parmenides,� Hegel says, �philosophising proper began.� Werke (edition of 1840), vol. xiii. p. 274.
[31] Parmenides, 128 A�D.
[32] This interpretation is combated by Milhaud, Les philosophes-g�om�tres de la Gr�ce, p. 140 n., but his reasons do not seem to me convincing. All the interpretations in what follows are open to question, but all have the support of reputable authorities.
[33] Physics, vi. 9. 2396 (R.P. 136�139).
[34] Cf. Gaston Milhaud, Les philosophes-g�om�tres de la Gr�ce, p. 140 n.; Paul Tannery, Pour l'histoire de la science hell�ne, p. 249; Burnet, op. cit., p. 362.
[35] Cf. R. K. Gaye, �On Aristotle, Physics, Z ix.� Journal of Philology, vol. xxxi., esp. p. 111. Also Moritz Cantor, Vorlesungen �ber Geschichte der Mathematik, 1st ed., vol. i., 1880, p. 168, who, however, subsequently adopted Paul Tannery's opinion, Vorlesungen, 3rd ed. (vol. i. p. 200).
[36] �Le mouvement et les partisans des indivisibles,� Revue de M�taphysique et de Morale, vol. i. pp. 382�395.
[37] �Le mouvement et les arguments de Z�non d'�l�e,� Revue de M�taphysique et de Morale, vol. i. pp. 107�125.
[38] Cf. M. Brochard, �Les pr�tendus sophismes de Z�non d'�l�e,� Revue de M�taphysique et de Morale, vol. i. pp. 209�215.
[39] Simplicius, Phys., 140, 28 D (R.P. 133); Burnet, op. cit., pp. 364�365.
[40] Op. cit., p. 367.
[41] Aristotle's words are: �The first is the one on the non-existence of motion on the ground that what is moved must always attain the middle point sooner than the end-point, on which we gave our opinion in the earlier part of our discourse.� Phys., vi. 9. 939B (R.P. 136). Aristotle seems to refer to Phys., vi. 2. 223AB [R.P. 136A]: �All space is continuous, for time and space are divided into the same and equal divisions�. Wherefore also Zeno's argument is fallacious, that it is impossible to go through an infinite collection or to touch an infinite collection one by one in a finite time. For there are two senses in which the term �infinite� is applied both to length and to time, and in fact to all continuous things, either in regard to divisibility, or in regard to the ends. Now it is not possible to touch things infinite in regard to number in a finite time, but it is possible to touch things infinite in regard to divisibility: for time itself also is infinite in this sense. So that in fact we go through an infinite, [space] in an infinite [time] and not in a finite [time], and we touch infinite things with infinite things, not with finite things.� Philoponus, a sixth-century commentator (R.P. 136A, Exc. Paris Philop. in Arist. Phys., 803, 2. Vit.), gives the following illustration: �For if a thing were moved the space of a cubit in one hour, since in every space there are an infinite number of points, the thing moved must needs touch all the points of the space: it will then go through an infinite collection in a finite time, which is impossible.�
[42] Cf. Mr C. D. Broad, �Note on Achilles and the Tortoise,� Mind, N.S., vol. xxii. pp. 318�9.
[43] Op. cit.
[44] Aristotle's words are: �The second is the so-called Achilles. It consists in this, that the slower will never be overtaken in its course by the quickest, for the pursuer must always come first to the point from which the pursued has just departed, so that the slower must necessarily be always still more or less in advance.� Phys., vi. 9. 239B (R.P. 137).
[45] Phys., vi. 9. 239B (R.P. 138).
[46] Phys., vi. 9. 239B (R.P. 139).
[47] Loc. cit.
[48] Loc. cit., p. 105.
[49] Phil. Werke, Gerhardt's edition, vol. i. p. 338.
[50] Mathematical Discourses concerning two new sciences relating to mechanics and local motion, in four dialogues. By Galileo Galilei, Chief Philosopher and Mathematician to the Grand Duke of Tuscany. Done into English from the Italian, by Tho. Weston, late Master, and now published by John Weston, present Master, of the Academy at Greenwich. See pp. 46 ff.
[51] In his Grundlagen einer allgemeinen Mannichfaltigkeitslehre and in articles in Acta Mathematica, vol. ii.
[52] The definition of number contained in this book, and elaborated in the Grundgesetze der Arithmetik (vol. i., 1893; vol. ii., 1903), was rediscovered by me in ignorance of Frege's work. I wish to state as emphatically as possible�what seems still often ignored�that his discovery antedated mine by eighteen years.
[53] Giles, The Civilisation of China (Home University Library), p. 147.
[54] Cf. Principia Mathematica, � 20, and Introduction, chapter iii.
[55] In the above remarks I am making use of unpublished work by my friend Ludwig Wittgenstein.
[56] Thus we are not using �thing� here in the sense of a class of correlated �aspects,� as we did in Lecture III. Each �aspect� will count separately in stating causal laws.
[57] The above remarks, for purposes of illustration, adopt one of several possible opinions on each of several disputed points.
AN ESSAY ON THE FOUNDATIONS OF GEOMETRY
By Bertrand A. W. Russell
Fellow Of Trinity College, Cambridge
1897
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 � 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 � priori, for Epistemology, when without it knowledge would be impossible 2
4. The subjective and the � priori belong respectively to Psychology and to Epistemology. The latter alone will be investigated in this essay 3
5. My test of the � priori will be purely logical: what knowledge is necessary for experience? 3
6. But since the necessary is hypothetical, we must include, in the � 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
[viii] 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
[ix] 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
[x] 53. Kant contends that since Geometry is apodeictic, space must be � priori and subjective, while since space is � 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 � 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
[xi] 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 Delbouf, 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 � priori possible, and the decision between them is empirical 114
[xii]
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 � 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
[xiii] 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 � 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 � 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 � priori, actual measurement is empirical 154
[xiv] 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 � 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
[xv] 175. Hence the axiom of distance, also, is � 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
[xvi] 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
WHY MEN FIGHT
A METHOD OF ABOLISHING THE INTERNATIONAL DUEL
By Bertrand Russell
CONTENTS
CHAPTER PAGE
I The Principle of Growth 3
II The State 42
III War as an Institution 79
IV Property 117
V Education 153
VI Marriage and the Population Question 182
VII Religion and the Churches 215
VIII What We Can Do 245