Бертран Рассел

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УКАЗАТЕЛЬ ПРОИЗВЕДЕНИЙ БЕРТРАНА РАССЕЛА В ПРОЕКТЕ «ГУТЕНБЕРГ»

Составитель: Дэвид Уидджер

CONTENTS

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## PROPOSED ROADS TO FREEDOM

## THE ANALYSIS OF MIND

## POLITICAL IDEALS

## THE PROBLEMS OF PHILOSOPHY

## THE PROBLEM OF CHINA

##

BOLSHEVISM

## MYSTICISM AND LOGIC

## OUR KNOWLEDGE OF EXTERNAL WORLD

FREE THOUGHT AND OFFICIAL PROPAGANDA

## ON FOUNDATIONS OF GEOMETRY

## WHY MEN FIGHT

ОГЛАВЛЕНИЯ ТОМОВ

«ПРЕДЛАГАЕМЫЕ ПУТИ К СВОБОДЕ»

Бертран Рассел

CONTENTS

INTRODUCTION

PART I

HISTORICAL

CHAPTER I

MARX AND SOCIALIST DOCTRINE

CHAPTER II

BAKUNIN AND ANARCHISM

CHAPTER III

THE SYNDICALIST REVOLT

PART II

PROBLEMS OF THE FUTURE

CHAPTER IV

WORK AND PAY

CHAPTER V

GOVERNMENT AND LAW

CHAPTER VI

INTERNATIONAL RELATIONS

CHAPTER VII

SCIENCE AND ART UNDER SOCIALISM

CHAPTER VIII

THE WORLD AS IT COULD BE MADE

«АНАЛИЗ СОЗНАНИЯ»

Бертран Рассел

1921

CONTENTS

ФИЛОСОФСКАЯ БИБЛИОТЕКА МЬЮРХЕДА ПРЕДИСЛОВИЕ «АНАЛИЗ СОЗНАНИЯ»

LECTURE I.

RECENT CRITICISMS OF "CONSCIOUSNESS"

LECTURE II.

INSTINCT AND HABIT

LECTURE III.

DESIRE AND FEELING

LECTURE IV.

INFLUENCE OF PAST HISTORY ON PRESENT OCCURRENCES IN LIVING

LECTURE V.

PSYCHOLOGICAL AND PHYSICAL CAUSAL LAWS

LECTURE VI.

INTROSPECTION

LECTURE VII.

THE DEFINITION OF PERCEPTION

LECTURE VIII.

SENSATIONS AND IMAGES

LECTURE IX.

MEMORY

LECTURE X.

WORDS AND MEANING

LECTURE XI.

GENERAL IDEAS AND THOUGHT

LECTURE XII.

BELIEF

LECTURE XIII.

TRUTH AND FALSEHOOD

LECTURE XIV.

EMOTIONS AND WILL

LECTURE XV.

CHARACTERISTICS OF MENTAL PHENOMENA

«ПОЛИТИЧЕСКИЕ ИДЕАЛЫ»

Бертран Рассел

CONTENTS

I:

Political Ideals

II:

Capitalism and the Wage System

III:

Pitfalls in Socialism

IV:

Individual Liberty and Public Control

V:

National Independence and Internationalism

«ПРОБЛЕМЫ ФИЛОСОФИИ»

Бертран Рассел

CONTENTS

PREFACE

CHAPTER I.

APPEARANCE AND REALITY

CHAPTER II.

THE EXISTENCE OF MATTER

CHAPTER III.

THE NATURE OF MATTER

CHAPTER IV.

IDEALISM

CHAPTER V.

KNOWLEDGE BY ACQUAINTANCE AND KNOWLEDGE BY DESCRIPTION

CHAPTER VI.

ON INDUCTION

CHAPTER VII.

ON OUR KNOWLEDGE OF GENERAL PRINCIPLES

CHAPTER VIII.

HOW A PRIORI KNOWLEDGE IS POSSIBLE

CHAPTER IX.

THE WORLD OF UNIVERSALS

CHAPTER X.

ON OUR KNOWLEDGE OF UNIVERSALS

CHAPTER XI.

ON INTUITIVE KNOWLEDGE

CHAPTER XII.

TRUTH AND FALSEHOOD

CHAPTER XIII.

KNOWLEDGE, ERROR, AND PROBABLE OPINION

CHAPTER XIV.

THE LIMITS OF PHILOSOPHICAL KNOWLEDGE

CHAPTER XV.

THE VALUE OF PHILOSOPHY

BIBLIOGRAPHICAL NOTE

«ПРОБЛЕМА КИТАЯ»

Бертран Рассел

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

«ПРАКТИКА И ТЕОРИЯ БОЛЬШЕВИЗМА»

Бертран Рассел

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

«МИСТИЦИЗМ И ЛОГИКА»

И ДРУГИЕ ЭССЕ

Бертран Рассел

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

«НАШЕ ЗНАНИЕ ВНЕШНЕГО МИРА» КАК ОБЛАСТЬ ПРИМЕНЕНИЯ НАУЧНОГО МЕТОДА В ФИЛОСОФИИ

Бертран Рассел

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

УКАЗАТЕЛЬ

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.

Noël, 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.

Poincaré, 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] Encyclopædia 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 mathématique,” Revue de Métaphysique 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 Philosophiæ Græcæ, 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-géomètres de la Grèce, 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-géomètres de la Grèce, p. 140 n.; Paul Tannery, Pour l'histoire de la science hellène, 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 über 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 Métaphysique et de Morale, vol. i. pp. 382–395.

[37] “Le mouvement et les arguments de Zénon d'Élée,” Revue de Métaphysique et de Morale, vol. i. pp. 107–125.

[38] Cf. M. Brochard, “Les prétendus sophismes de Zénon d'Élée,” Revue de Métaphysique 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.

«ОПЫТ ОБ ОСНОВАНИЯХ ГЕОМЕТРИИ»

Бертран А. У. Рассел

Член Тринити-колледжа, Кембридж

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 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 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 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

«ПОЧЕМУ ЛЮДИ ВОЮЮТ»

МЕТОД УПРАЗДНЕНИЯ МЕЖДУНАРОДНОЙ ДУЭЛИ

Бертран Рассел

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

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