{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_59391/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_59391","id":"gutenberg_59391","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/59391","summary":null,"tags":["Category: Essays, Letters & Speeches","Category: Philosophy & Ethics","Indexes","Nobel Prizes in Literature"],"text":"Produced by David Widger\n\n\n\n\n\n\n\n\nINDEX OF THE PROJECT GUTENBERG\n\nWORKS OF\n\nBERTRAND RUSSELL\n\n\n\nCompiled by David Widger\n\n\n\n\n\nCONTENTS\n\n##  PROPOSED ROADS TO FREEDOM\n\n##  THE ANALYSIS OF MIND\n\n##  POLITICAL IDEALS\n\n##  THE PROBLEMS OF PHILOSOPHY\n\n##  THE PROBLEM OF CHINA\n\n##  BOLSHEVISM\n\n##  MYSTICISM AND LOGIC\n\n##  OUR KNOWLEDGE OF EXTERNAL WORLD\n\nFREE THOUGHT AND OFFICIAL PROPAGANDA\n\n##  ON FOUNDATIONS OF GEOMETRY\n\n##  WHY MEN FIGHT\n\n\n\n\n\n\n\nTABLES OF CONTENTS OF VOLUMES\n\n\n\n\n\nPROPOSED ROADS TO FREEDOM\nBy Bertrand Russell\n\n\n\nCONTENTS\nINTRODUCTION\nPART I \tHISTORICAL\nCHAPTER I \tMARX AND SOCIALIST DOCTRINE\nCHAPTER II \tBAKUNIN AND ANARCHISM\nCHAPTER III \tTHE SYNDICALIST REVOLT\nPART II \tPROBLEMS OF THE FUTURE\nCHAPTER IV \tWORK AND PAY\nCHAPTER V \tGOVERNMENT AND LAW\nCHAPTER VI \tINTERNATIONAL RELATIONS\nCHAPTER VII \tSCIENCE AND ART UNDER SOCIALISM\nCHAPTER VIII \tTHE WORLD AS IT COULD BE MADE\n\n\n\n\n\nTHE ANALYSIS OF MIND\nBy Bertrand Russell\n1921\nCONTENTS\nMUIRHEAD LIBRARY OF PHILOSOPHY\n\nPREFACE\n\nTHE ANALYSIS OF MIND\nLECTURE I. \tRECENT CRITICISMS OF \"CONSCIOUSNESS\"\nLECTURE II. \tINSTINCT AND HABIT\nLECTURE III. \tDESIRE AND FEELING\nLECTURE IV. \tINFLUENCE OF PAST HISTORY ON PRESENT OCCURRENCES IN LIVING\nLECTURE V. \tPSYCHOLOGICAL AND PHYSICAL CAUSAL LAWS\nLECTURE VI. \tINTROSPECTION\nLECTURE VII. \tTHE DEFINITION OF PERCEPTION\nLECTURE VIII. \tSENSATIONS AND IMAGES\nLECTURE IX. \tMEMORY\nLECTURE X. \tWORDS AND MEANING\nLECTURE XI. \tGENERAL IDEAS AND THOUGHT\nLECTURE XII. \tBELIEF\nLECTURE XIII.     \tTRUTH AND FALSEHOOD\nLECTURE XIV. \tEMOTIONS AND WILL\nLECTURE XV. \tCHARACTERISTICS OF MENTAL PHENOMENA\n\n\n\n\n\nPOLITICAL IDEALS\nBy Bertrand Russell\n\n\n\n\nCONTENTS\nI:   \tPolitical Ideals\nII:   \tCapitalism and the Wage System\nIII:   \tPitfalls in Socialism\nIV:   \tIndividual Liberty and Public Control\nV:   \tNational Independence and Internationalism\n\n\n\n\n\nTHE PROBLEMS OF PHILOSOPHY\nBy Bertrand Russell\n\n\n\nCONTENTS\nPREFACE\nCHAPTER I. \tAPPEARANCE AND REALITY\nCHAPTER II. \tTHE EXISTENCE OF MATTER\nCHAPTER III. \tTHE NATURE OF MATTER\nCHAPTER IV. \tIDEALISM\nCHAPTER V. \tKNOWLEDGE BY ACQUAINTANCE AND KNOWLEDGE BY DESCRIPTION\nCHAPTER VI. \tON INDUCTION\nCHAPTER VII. \tON OUR KNOWLEDGE OF GENERAL PRINCIPLES\nCHAPTER VIII. \tHOW A PRIORI KNOWLEDGE IS POSSIBLE\nCHAPTER IX. \tTHE WORLD OF UNIVERSALS\nCHAPTER X. \tON OUR KNOWLEDGE OF UNIVERSALS\nCHAPTER XI. \tON INTUITIVE KNOWLEDGE\nCHAPTER XII. \tTRUTH AND FALSEHOOD\nCHAPTER XIII.     \tKNOWLEDGE, ERROR, AND PROBABLE OPINION\nCHAPTER XIV. \tTHE LIMITS OF PHILOSOPHICAL KNOWLEDGE\nCHAPTER XV. \tTHE VALUE OF PHILOSOPHY\nBIBLIOGRAPHICAL NOTE\n\n\n\n\n\nTHE PROBLEM OF CHINA\nBy Bertrand Russell\nCONTENTS\nQUESTIONS\nCHINA BEFORE THE NINETEENTH CENTURY\nCHINA AND THE WESTERN POWERS\nMODERN CHINA\nJAPAN BEFORE THE RESTORATION\nMODERN JAPAN\nJAPAN AND CHINA BEFORE 1914\nJAPAN AND CHINA DURING THE WAR\nTHE WASHINGTON CONFERENCE\nPRESENT FORCES AND TENDENCIES IN THE FAR EAST\nCHINESE AND WESTERN CIVILIZATION CONTRASTED\nTHE CHINESE CHARACTER\nHIGHER EDUCATION IN CHINA\nINDUSTRIALISM IN CHINA\nTHE OUTLOOK FOR CHINA\nAPPENDIX\nINDEX\n\n\n\n\n\nTHE PRACTICE AND THEORY OF BOLSHEVISM\nBertrand Russell\nCONTENTS\n  \t  \tPAGE\nPART I\nTHE PRESENT CONDITION OF RUSSIA\nI. \tWHAT IS HOPED FROM BOLSHEVISM \t15\nII. \tGENERAL CHARACTERISTICS \t24\nIII. \tLENIN, TROTSKY AND GORKY \t36\nIV. \tART AND EDUCATION \t45\nV. \tCOMMUNISM AND THE SOVIET CONSTITUTION \t72\nVI. \tTHE FAILURE OF RUSSIAN INDUSTRY \t81\nVII. \tDAILY LIFE IN MOSCOW \t92\nVIII. \tTOWN AND COUNTRY \t99\nIX. \tINTERNATIONAL POLICY \t106\nPART II\nBOLSHEVIK THEORY\nI. \tTHE MATERIALISTIC THEORY OF HISTORY \t119\nII. \tDECIDING FORCES IN POLITICS \t128\nIII. \tBOLSHEVIK CRITICISM OF DEMOCRACY \t134\nIV. \tREVOLUTION AND DICTATORSHIP \t146\nV. \tMECHANISM AND THE INDIVIDUAL \t157\nVI. \tWHY RUSSIAN COMMUNISM HAS FAILED \t165\nVII. \tCONDITIONS FOR THE SUCCESS OF COMMUNISM \t178\n\n\n\n\n\nMYSTICISM AND LOGIC\nAND OTHER ESSAYS\nBertrand Russell\nCONTENTS\nChapter \t  \tPage\nI. \tMysticism and Logic \t1\nII. \tThe Place of Science in a Liberal Education \t33\nIII. \tA Free Man's Worship \t46\nIV. \tThe Study of Mathematics \t58\nV. \tMathematics and the Metaphysicians \t74\nVI. \tOn Scientific Method in Philosophy \t97\nVII. \tThe Ultimate Constituents of Matter \t125\nVIII. \tThe Relation of Sense-data to Physics \t145\nIX. \tOn the Notion of Cause \t180\nX. \tKnowledge by Acquaintance and Knowledge by Description \t209\n  \tIndex \t233\n\n\n\n\n\nOUR KNOWLEDGE OF THE EXTERNAL WORLD\nAS A FIELD FOR SCIENTIFIC METHOD IN PHILOSOPHY,\nBy Bertrand Russell\nCONTENTS\nLECTURE \tPAGE\nI. \tCurrent Tendencies \t3\nII. \tLogic as the Essence of Philosophy \t33\nIII. \tOn our Knowledge of the External World \t63\nIV. \tThe World of Physics and the World of Sense \t101\nV. \tThe Theory of Continuity \t129\nVI. \tThe Problem of Infinity considered Historically \t155\nVII. \tThe Positive Theory of Infinity \t185\nVIII. \tOn the Notion of Cause, with Applications to the Free-will Problem \t211\n  \tIndex \t243\nINDEX\nAbsolute, 6, 39.\nAbstraction, principle of, 42, 124 ff.\nAchilles, Zeno's argument of, 173.\nAcquaintance, 25, 144.\nActivity, 224 ff.\nAllman, 161 n.\nAnalysis, 185, 204, 211, 241.\nlegitimacy of, 150.\nAnaximander, 3.\nAntinomies, Kant's, 155 ff.\nAquinas, 10.\nAristotle, 40, 160 n., 161 ff., 240.\nArrow, Zeno's argument of, 173.\nAssertion, 52.\nAtomism, logical, 4.\nAtomists, 160.\nBelief, 58.\nprimitive and derivative, 69 ff.\nBergson, 4, 11, 13, 20 ff., 137, 138, 150, 158, 165, 174, 178, 229 ff.\nBerkeley, 63, 64, 102.\nBolzano, 165.\nBoole, 40.\nBradley, 6, 39, 165.\nBroad, 172 n.\nBrochard, 169 n.\nBurnet, 19 n., 160 n., 161 n., 170 n., 171 ff.\nCalderon, 95.\nCantor, vi, vii, 155, 165, 190, 194, 199.\nCategories, 38.\nCausal laws, 109, 212 ff.\nevidence for, 216 ff.\nin psychology, 219.\nCausation, 34 ff., 79, 212 ff.\nlaw of, 221.\nnot a priori, 223, 232.\nCause, 220, 223.\nCertainty, degrees of, 67, 68, 212.\nChange,\ndemands analysis, 151.\nCinematograph, 148, 174.\nClasses, 202.\nnon-existence of, 205 ff.\nClassical tradition, 3 ff., 58.\nComplexity, 145, 157 ff.\nCompulsion, 229, 233 ff.\nCongruence, 195.\nConsecutiveness, 134.\nConservation, 105.\nConstituents of facts, 51, 145.\nConstruction v. inference, iv.\nContemporaries, initial, 119, 120 n.\nContinuity, 64, 129 ff., 141 ff., 155 ff.\nof change, 106, 108, 130 ff.\nCorrelation of mental and physical, 233.\nCounting, 164, 181, 187 ff., 203.\nCouturat, 40 n.\nDante, 10.\nDarwin, 4, 11, 23, 30.\nData, 65 ff., 211.\n\ufffdhard\ufffd and \ufffdsoft,\ufffd 70 ff.\nDates, 117.\nDefinition, 204.\nDescartes, 5, 73, 238.\nDescriptions, 201, 214.\nDesire, 227, 235.\nDeterminism, 233.\nDoubt, 237.\nDreams, 85, 93.\nDuration, 146, 149.\nEarlier and later, 116.\nEffect, 220.\nEleatics, 19.\nEmpiricism, 37, 222.\nEnclosure, 114 ff., 120.\nEnumeration, 202.\nEuclid, 160, 164.\nEvellin, 169.\nEvolutionism, 4, 11 ff.\nExtension, 146, 149.\nExternal world, knowledge of, 63 ff.\nFact, 51.\natomic, 52.\nFinalism, 13.\nForm, logical, 42 ff., 185, 208.\nFractions, 132, 179.\nFree will, 213, 227 ff.\nFrege, 5, 40, 199 ff.\nGalileo, 4, 59, 192, 194, 239, 240.\nGaye, 169 n., 175, 177.\nGeometry, 5.\nGiles, 206 n.\nGreater and less, 195.\nHarvard, 4.\nHegel, 3, 37 ff., 46, 166.\n\ufffdHere,\ufffd 73, 92.\nHereditary properties, 195.\nHippasos, 163, 237.\nHui Tzu, 206.\nHume, 217, 221.\nHypotheses in philosophy, 239.\nIllusions, 85.\nIncommensurables, 162 ff., 237.\nIndependence, 73, 74.\ncausal and logical, 74, 75.\nIndiscernibility, 141, 148.\nIndivisibles, 160.\nInduction, 34, 222.\nmathematical, 195 ff.\nInductiveness, 190, 195 ff.\nInference, 44, 54.\nInfinite, vi, 64, 133, 149.\nhistorically considered, 155 ff.\n\ufffdtrue,\ufffd 179, 180.\npositive theory of, 185 ff.\nInfinitesimals, 135.\nInstants, 116 ff., 129, 151, 216.\ndefined, 118.\nInstinct v. Reason, 20 ff.\nIntellect, 22 ff.\nIntelligence, how displayed by friends, 93.\ninadequacy of display, 96.\nInterpretation, 144.\nJames, 4, 10, 13.\nJourdain, 165 n.\nJowett, 167.\nJudgment, 58.\nKant, 3, 112, 116, 155 ff., 200.\nKnowledge about, 144.\nLanguage, bad, 82, 135.\nLaplace, 12.\nLaws of nature, 218 ff.\nLeibniz, 13, 40, 87, 186, 191.\nLogic, 201.\nanalytic not constructive, 8.\nAristotelian, 5.\nand fact, 53.\ninductive, 34, 222.\nmathematical, vi, 40 ff.\nmystical, 46.\nand philosophy, 8, 33 ff., 239.\nLogical constants, 208, 213.\nMach, 123, 224.\nMacran, 39 n.\nMathematics, 40, 57.\nMatter, 75, 101 ff.\npermanence of, 102 ff.\nMeasurement, 164.\nMemory, 230, 234, 236.\nMethod, deductive, 5.\nlogical-analytic, v, 65, 211, 236 ff.\nMilhaud, 168 n., 169 n.\nMill, 34, 200.\nMontaigne, 28.\nMotion, 130, 216.\ncontinuous, 133, 136.\nmathematical theory of, 133.\nperception of, 137 ff.\nZeno's arguments on, 168 ff.\nMysticism, 19, 46, 63, 95.\nNewton, 30, 146.\nNietzsche, 10, 11.\nNo\ufffdl, 169.\nNumber, cardinal, 131, 186 ff.\ndefined, 199 ff.\nfinite, 160, 190 ff.\ninductive, 197.\ninfinite, 178, 180, 188 ff., 197.\nreflexive, 190 ff.\nOccam, 107, 146.\nOne and many, 167, 170.\nOrder, 131.\nParmenides, 63, 165 ff., 178.\nPast and future, 224, 234 ff.\nPeano, 40.\nPerspectives, 88 ff., 111.\nPhiloponus, 171 n.\nPhilosophy and ethics, 26 ff.\nand mathematics, 185 ff.\nprovince of, 17, 26, 185, 236.\nscientific, 11, 16, 18, 29, 236 ff.\nPhysics, 101 ff., 147, 239, 242.\ndescriptive, 224.\nverifiability of, 81, 110.\nPlace, 86, 90.\nat and from, 92.\nPlato, 4, 19, 27, 46, 63, 165 n., 166, 167.\nPoincar\ufffd, 123, 141.\nPoints, 113 ff., 129, 158.\ndefinition of, vi, 115.\nPragmatism, 11.\nPrantl, 174.\nPredictability, 229 ff.\nPremisses, 211.\nProbability, 36.\nPropositions, 52.\natomic, 52.\ngeneral, 55.\nmolecular, 54.\nPythagoras, 19, 160 ff., 237.\nRace-course, Zeno's argument of, 171 ff.\nRealism, new, 6.\nReflexiveness, 190 ff.\nRelations, 45.\nasymmetrical, 47.\nBradley's reasons against, 6.\nexternal, 150.\nintransitive, 48.\nmultiple, 50.\none-one, 203.\nreality of, 49.\nsymmetrical, 47, 124.\ntransitive, 48, 124.\nRelativity, 103, 242.\nRepetitions, 230 ff.\nRest, 136.\nRitter and Preller, 161 n.\nRobertson, D. S., 160 n.\nRousseau, 20.\nRoyce, 50.\nSantayana, 46.\nScepticism, 66, 67.\nSeeing double, 86.\nSelf, 73.\nSensation, 25, 75, 123.\nand stimulus, 139.\nSense-data, 56, 63, 67, 75, 110, 141, 143, 213.\nand physics, v, 64, 81, 97, 101 ff., 140.\ninfinitely numerous? 149, 159.\nSense-perception, 53.\nSeries, 49.\ncompact, 132, 142, 178.\ncontinuous, 131, 132.\nSigwart, 187.\nSimplicius, 170 n.\nSimultaneity, 116.\nSpace, 73, 88, 103, 112 ff., 130.\nabsolute and relative, 146, 159.\nantinomies of, 155 ff.\nperception of, 68.\nof perspectives, 88 ff.\nprivate, 89, 90.\nof touch and sight, 78, 113.\nSpencer, 4, 12, 236.\nSpinoza, 46, 166.\nStadium, Zeno's argument of, 134 n., 175 ff.\nSubject-predicate, 45.\nSynthesis, 157, 185.\nTannery, Paul, 169 n.\nTeleology, 223.\nTestimony, 67, 72, 82, 87, 96, 212.\nThales, 3.\nThing-in-itself, 75, 84.\nThings, 89 ff., 104 ff., 213.\nTime, 103, 116 ff., 130, 155 ff., 166, 215.\nabsolute or relative, 146.\nlocal, 103.\nprivate, 121.\nUniformities, 217.\nUnity, organic, 9.\nUniversal and particular, 39 n.\nVolition, 223 ff.\nWhitehead, vi, 207.\nWittgenstein, vii, 208 n.\nWorlds, actual and ideal, 111.\npossible, 186.\nprivate, 88.\nZeller, 173.\nZeno, 129, 134, 136, 165 ff.\n[1] Delivered as Lowell Lectures in Boston, in March and April 1914.\n[2] London and New York, 1912 (\ufffdHome University Library\ufffd).\n[3] The first volume was published at Cambridge in 1910, the second in 1912, and the third in 1913.\n[4] Appearance and Reality, pp. 32\ufffd33.\n[5] Creative Evolution, English translation, p. 41.\n[6] Cf. Burnet, Early Greek Philosophy, pp. 85 ff.\n[7] Introduction to Metaphysics, p. 1.\n[8] Logic, book iii., chapter iii., \ufffd 2.\n[9] Book iii., chapter xxi., \ufffd 3.\n[10] Or rather a propositional function.\n[11] The subject of causality and induction will be discussed again in Lecture VIII.\n[12] See the translation by H. S. Macran, Hegel's Doctrine of Formal Logic, Oxford, 1912. Hegel's argument in this portion of his \ufffdLogic\ufffd depends throughout upon confusing the \ufffdis\ufffd of predication, as in \ufffdSocrates is mortal,\ufffd with the \ufffdis\ufffd of identity, as in \ufffdSocrates is the philosopher who drank the hemlock.\ufffd Owing to this confusion, he thinks that \ufffdSocrates\ufffd and \ufffdmortal\ufffd 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 \ufffdidentity in difference.\ufffd Again, Socrates is particular, \ufffdmortal\ufffd is universal. Therefore, he says, since Socrates is mortal, it follows that the particular is the universal\ufffdtaking the \ufffdis\ufffd to be throughout expressive of identity. But to say \ufffdthe particular is the universal\ufffd 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.\n[13] Cf. Couturat, La Logique de Leibniz, pp. 361, 386.\n[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.\n[15] Encyclop\ufffddia of the Philosophical Sciences, vol. i. p. 97.\n[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.\n[17] The assumptions made concerning time-relations in the above are as follows:\ufffd\n[18] The above paradox is essentially the same as Zeno's argument of the stadium which will be considered in our next lecture.\n[19] See next lecture.\n[20] Monist, July 1912, pp. 337\ufffd341.\n[21] \ufffdLe continu math\ufffdmatique,\ufffd Revue de M\ufffdtaphysique et de Morale, vol. i. p. 29.\n[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.\n[23] Cf. Aristotle, Metaphysics, M. 6, 1080b, 18 sqq., and 1083b, 8 sqq.\n[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): \ufffdThe 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.)\ufffd Cf. p. 48.\n[25] Referred to by Burnet, op. cit., p. 120.\n[26] iv., 6. 213b, 22; H. Ritter and L. Preller, Historia Philosophi\ufffd Gr\ufffdc\ufffd, 8th ed., Gotha, 1898, p. 75 (this work will be referred to in future as \ufffdR. P.\ufffd).\n[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.\n[28] In regard to Zeno and the Pythagoreans, I have derived much valuable information and criticism from Mr P. E. B. Jourdain.\n[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.\n[30] \ufffdWith Parmenides,\ufffd Hegel says, \ufffdphilosophising proper began.\ufffd Werke (edition of 1840), vol. xiii. p. 274.\n[31] Parmenides, 128 A\ufffdD.\n[32] This interpretation is combated by Milhaud, Les philosophes-g\ufffdom\ufffdtres de la Gr\ufffdce, 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.\n[33] Physics, vi. 9. 2396 (R.P. 136\ufffd139).\n[34] Cf. Gaston Milhaud, Les philosophes-g\ufffdom\ufffdtres de la Gr\ufffdce, p. 140 n.; Paul Tannery, Pour l'histoire de la science hell\ufffdne, p. 249; Burnet, op. cit., p. 362.\n[35] Cf. R. K. Gaye, \ufffdOn Aristotle, Physics, Z ix.\ufffd Journal of Philology, vol. xxxi., esp. p. 111. Also Moritz Cantor, Vorlesungen \ufffdber 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).\n[36] \ufffdLe mouvement et les partisans des indivisibles,\ufffd Revue de M\ufffdtaphysique et de Morale, vol. i. pp. 382\ufffd395.\n[37] \ufffdLe mouvement et les arguments de Z\ufffdnon d'\ufffdl\ufffde,\ufffd Revue de M\ufffdtaphysique et de Morale, vol. i. pp. 107\ufffd125.\n[38] Cf. M. Brochard, \ufffdLes pr\ufffdtendus sophismes de Z\ufffdnon d'\ufffdl\ufffde,\ufffd Revue de M\ufffdtaphysique et de Morale, vol. i. pp. 209\ufffd215.\n[39] Simplicius, Phys., 140, 28 D (R.P. 133); Burnet, op. cit., pp. 364\ufffd365.\n[40] Op. cit., p. 367.\n[41] Aristotle's words are: \ufffdThe 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.\ufffd Phys., vi. 9. 939B (R.P. 136). Aristotle seems to refer to Phys., vi. 2. 223AB [R.P. 136A]: \ufffdAll space is continuous, for time and space are divided into the same and equal divisions\ufffd. 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 \ufffdinfinite\ufffd 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.\ufffd Philoponus, a sixth-century commentator (R.P. 136A, Exc. Paris Philop. in Arist. Phys., 803, 2. Vit.), gives the following illustration: \ufffdFor 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.\ufffd\n[42] Cf. Mr C. D. Broad, \ufffdNote on Achilles and the Tortoise,\ufffd Mind, N.S., vol. xxii. pp. 318\ufffd9.\n[43] Op. cit.\n[44] Aristotle's words are: \ufffdThe 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.\ufffd Phys., vi. 9. 239B (R.P. 137).\n[45] Phys., vi. 9. 239B (R.P. 138).\n[46] Phys., vi. 9. 239B (R.P. 139).\n[47] Loc. cit.\n[48] Loc. cit., p. 105.\n[49] Phil. Werke, Gerhardt's edition, vol. i. p. 338.\n[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.\n[51] In his Grundlagen einer allgemeinen Mannichfaltigkeitslehre and in articles in Acta Mathematica, vol. ii.\n[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\ufffdwhat seems still often ignored\ufffdthat his discovery antedated mine by eighteen years.\n[53] Giles, The Civilisation of China (Home University Library), p. 147.\n[54] Cf. Principia Mathematica, \ufffd 20, and Introduction, chapter iii.\n[55] In the above remarks I am making use of unpublished work by my friend Ludwig Wittgenstein.\n[56] Thus we are not using \ufffdthing\ufffd here in the sense of a class of correlated \ufffdaspects,\ufffd as we did in Lecture III. Each \ufffdaspect\ufffd will count separately in stating causal laws.\n[57] The above remarks, for purposes of illustration, adopt one of several possible opinions on each of several disputed points.\n\n\n\n\n\nAN ESSAY ON THE FOUNDATIONS OF GEOMETRY\nBy Bertrand A. W. Russell\nFellow Of Trinity College, Cambridge\n1897\nCONTENTS\n\tINTRODUCTION.\n\tOUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC, PSYCHOLOGY AND MATHEMATICS.\n\t\tPAGE\n1. \tThe problem first received a modern form through Kant, who connected the \ufffd priori with the subjective \t1\n2. \tA mental state is subjective, for Psychology, when its immediate cause does not lie in the outer world \t2\n3. \tA piece of knowledge is \ufffd priori, for Epistemology, when without it knowledge would be impossible \t2\n4. \tThe subjective and the \ufffd priori belong respectively to Psychology and to Epistemology. The latter alone will be investigated in this essay \t3\n5. \tMy test of the \ufffd priori will be purely logical: what knowledge is necessary for experience? \t3\n6. \tBut since the necessary is hypothetical, we must include, in the \ufffd priori, the ground of necessity \t4\n7. \tThis may be the essential postulate of our science, or the element, in the subject-matter, which is necessary to experience; \t4\n8. \tWhich, however, are both at bottom the same ground \t5\n9. \tForecast of the work \t5\n\n\tCHAPTER I.\n\tA SHORT HISTORY OF METAGEOMETRY.\n10. \tMetageometry began by rejecting the axiom of parallels \t7\n11. \tIts history may be divided into three periods: the synthetic, the metrical and the projective \t7\n12. \tThe first period was inaugurated by Gauss, \t10\n[viii] 13. \tWhose suggestions were developed independently by Lobatchewsky \t10\n14. \tAnd Bolyai \t11\n15. \tThe 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 \t12\n16. \tThe second period had a more philosophical aim, and was inspired chiefly by Gauss and Herbart \t13\n17. \tThe first work of this period, that of Riemann, invented two new conceptions: \t14\n18. \tThe first, that of a manifold, is a class-conception, containing space as a species, \t14\n19. \tAnd defined as such that its determinations form a collection of magnitudes \t15\n20. \tThe second, the measure of curvature of a manifold, grew out of curvature in curves and surfaces \t16\n21. \tBy means of Gauss's analytical formula for the curvature of surfaces, \t19\n22. \tWhich enables us to define a constant measure of curvature of a three-dimensional space without reference to a fourth dimension \t20\n23. \tThe 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 \t21\n24. \tHelmholtz, who was more of a philosopher than a mathematician, \t22\n25. \tGave a new but incorrect formulation of the essential axioms, \t23\n26. \tAnd deduced the quadratic formula for the infinitesimal arc, which Riemann had assumed \t24\n27. \tBeltrami gave Lobatchewsky's planimetry a Euclidean interpretation, \t25\n28. \tWhich is analogous to Cayley's theory of distance; \t26\n29. \tAnd dealt with n-dimensional spaces of constant negative curvature \t27\n30. \tThe third period abandons the metrical methods of the second, and extrudes the notion of spatial quantity \t27\n31. \tCayley reduced metrical properties to projective properties, relative to a certain conic or quadric, the Absolute; \t28\n32. \tAnd Klein showed that the Euclidean or non-Euclidean systems result, according to the nature of the Absolute; \t29\n33. \tHence Euclidean space appeared to give rise to all the kinds of Geometry, and the question, which is true, appeared reduced to one of convention \t30\n34. \tBut this view is due to a confusion as to the nature of the coordinates employed \t30\n[ix] 35. \tProjective coordinates have been regarded as dependent on distance, and thus really metrical \t31\n36. \tBut this is not the case, since anharmonic ratio can be projectively defined \t32\n37. \tProjective coordinates, being purely descriptive, can give no information as to metrical properties, and the reduction of metrical to projective properties is purely technical \t33\n38. \tThe 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, \t36\n39. \tWhich provides a Euclidean equivalent for every non-Euclidean proposition, and so removes the possibility of contradictions in Metageometry \t38\n40. \tKlein's elliptic Geometry has not been proved to have a corresponding variety of space \t39\n41. \tThe geometrical use of imaginaries, of which Cayley demanded a philosophical discussion, \t41\n42. \tHas a merely technical validity, \t42\n43. \tAnd is capable of giving geometrical results only when it begins and ends with real points and figures \t45\n44. \tWe have now seen that projective Geometry is logically prior to metrical Geometry, but cannot supersede it \t46\n45. \tSophus Lie has applied projective methods to Helmholtz's formulation of the axioms, and has shown the axiom of Monodromy to be superfluous \t46\n46. \tMetageometry has gradually grown independent of philosophy, but has grown continually more interesting to philosophy \t50\n47. \tMetrical Geometry has three indispensable axioms, \t50\n48. \tWhich we shall find to be not results, but conditions, of measurement, \t51\n49. \tAnd which are nearly equivalent to the three axioms of projective Geometry \t52\n50. \tBoth sets of axioms are necessitated, not by facts, but by logic \t52\n\n\tCHAPTER II.\n\tCRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL THEORIES OF GEOMETRY.\n51. \tA criticism of representative modern theories need not begin before Kant \t54\n52. \tKant's doctrine must be taken, in an argument about Geometry, on its purely logical side \t55\n[x] 53. \tKant contends that since Geometry is apodeictic, space must be \ufffd priori and subjective, while since space is \ufffd priori and subjective, Geometry must be apodeictic \t55\n54. \tMetageometry has upset the first line of argument, not the second \t56\n55. \tThe 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 \t57\n56. \tModern Logic regards every judgment as both synthetic and analytic, \t57\n57. \tBut leaves the \ufffd priori, as that which is presupposed in the possibility of experience \t59\n58. \tKant's first two arguments as to space suffice to prove some form of externality, but not necessarily Euclidean space, a necessary condition of experience \t60\n59. \tAmong the successors of Kant, Herbart alone advanced the theory of Geometry, by influencing Riemann \t62\n60. \tRiemann regarded space as a particular kind of manifold, i.e. wholly quantitatively \t63\n61. \tHe therefore unduly neglected the qualitative adjectives of space \t64\n62. \tHis philosophy rests on a vicious disjunction \t65\n63. \tHis definition of a manifold is obscure, \t66\n64. \tAnd his definition of measurement applies only to space \t67\n65. \tThough mathematically invaluable, his view of space as a manifold is philosophically misleading \t69\n66. \tHelmholtz attacked Kant both on the mathematical and on the psychological side; \t70\n67. \tBut his criterion of apriority is changeable and often invalid; \t71\n68. \tHis proof that non-Euclidean spaces are imaginable is inconclusive; \t72\n69. \tAnd his assertion of the dependence of measurement on rigid bodies, which may be taken in three senses, \t74\n70. \tIs wholly false if it means that the axiom of Congruence actually asserts the existence of rigid bodies, \t75\n71. \tIs untrue if it means that the necessary reference of geometrical propositions to matter renders pure Geometry empirical, \t76\n72. \tAnd is inadequate to his conclusion if it means, what is true, that actual measurement involves approximately rigid bodies \t78\n73. \tGeometry deals with an abstract matter, whose physical properties are disregarded; and Physics must presuppose Geometry \t80\n74. \tErdmann accepted the conclusions of Riemann and Helmholtz, \t81\n[xi] 75. \tAnd regarded the axioms as necessarily successive steps in classifying space as a species of manifold \t82\n76. \tHis deduction involves four fallacious assumptions, namely: \t82\n77. \tThat conceptions must be abstracted from a series of instances; \t83\n78. \tThat all definition is classification; \t83\n79. \tThat conceptions of magnitude can be applied to space as a whole; \t84\n80. \tAnd that if conceptions of magnitude could be so applied, all the adjectives of space would result from their application \t86\n81. \tErdmann regards Geometry alone as incapable of deciding on the truth of the axiom of Congruence, \t86\n82. \tWhich he affirms to be empirically proved by Mechanics. \t88\n83. \tThe variety and inadequacy of Erdmann's tests of apriority \t89\n84. \tInvalidate his final conclusions on the theory of Geometry \t90\n85. \tLotze has discussed two questions in the theory of Geometry: \t93\n86. \t(1) He regards the possibility of non-Euclidean spaces as suggested by the subjectivity of space, \t93\n87. \tAnd rejects it owing to a mathematical misunderstanding, \t96\n88. \tHaving missed the most important sense of their possibility, \t96\n89. \tWhich is that they fulfil the logical conditions to which any form of externality must conform \t97\n90. \t(2) He attacks the mathematical procedure of Metageometry \t98\n91. \tThe attack begins with a question-begging definition of parallels \t99\n92. \tLotze maintains that all apparent departures from Euclid could be physically explained, a view which really makes Euclid empirical \t99\n93. \tHis criticism of Helmholtz's analogies rests wholly on mathematical mistakes \t101\n94. \tHis proof that space must have three dimensions rests on neglect of different orders of infinity \t104\n95. \tHe attacks non-Euclidean spaces on the mistaken ground that they are not homogeneous \t107\n96. \tLotze's objections fall under four heads \t108\n97. \tTwo other semi-philosophical objections may be urged, \t109\n98. \tOne of which, the absence of similarity, has been made the basis of attack by Delbouf, \t110\n99. \tBut does not form a valid ground of objection \t111\n100. \tRecent French speculation on the foundations of Geometry has suggested few new views \t112\n101. \tAll homogeneous spaces are \ufffd priori possible, and the decision between them is empirical \t114\n [xii]\n\tCHAPTER III.\n\tSection A. the axioms of projective geometry.\n102. \tProjective Geometry does not deal with magnitude, and applies to all spaces alike \t117\n103. \tIt will be found wholly \ufffd priori \t117\n104. \tIts axioms have not yet been formulated philosophically \t118\n105. \tCoordinates, in projective Geometry, are not spatial magnitudes, but convenient names for points \t118\n106. \tThe possibility of distinguishing various points is an axiom \t119\n107. \tThe qualitative relations between points, dealt with by projective Geometry, are presupposed by the quantitative treatment \t119\n108. \tThe only qualitative relation between two points is the straight line, and all straight lines are qualitatively similar \t120\n109. \tHence follows, by extension, the principle of projective transformation \t121\n110. \tBy which figures qualitatively indistinguishable from a given figure are obtained \t122\n111. \tAnharmonic ratio may and must be descriptively defined \t122\n112. \tThe quadrilateral construction is essential to the projective definition of points, \t123\n113. \tAnd can be projectively defined, \t124\n114. \tBy the general principle of projective transformation \t126\n115. \tThe principle of duality is the mathematical form of a philosophical circle, \t127\n116. \tWhich is an inevitable consequence of the relativity of space, and makes any definition of the point contradictory \t128\n117. \tWe define the point as that which is spatial, but contains no space, whence other definitions follow \t128\n118. \tWhat is meant by qualitative equivalence in Geometry? \t129\n119. \tTwo pairs of points on one straight line, or two pairs of straight lines through one point, are qualitatively equivalent \t129\n120. \tThis 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 \t130\n121. \tAny two projectively related figures are qualitatively equivalent, i.e. differ in no non-quantitative conceptual property \t131\n122. \tThree axioms are used by projective Geometry, \t132\n[xiii] 123. \tAnd are required for qualitative spatial comparison, \t132\n124. \tWhich involves the homogeneity, relativity and passivity of space \t133\n125. \tThe conception of a form of externality, \t134\n126. \tBeing a creature of the intellect, can be dealt with by pure mathematics \t134\n127. \tThe resulting doctrine of extension will be, for the moment, hypothetical \t135\n128. \tBut is rendered assertorical by the necessity, for experience, of some form of externality \t136\n129. \tAny such form must be relational \t136\n130. \tAnd homogeneous \t137\n131. \tAnd the relations constituting it must appear infinitely divisible \t137\n132. \tIt must have a finite integral number of dimensions, \t139\n133. \tOwing to its passivity and homogeneity \t140\n134. \tAnd to the systematic unity of the world \t140\n135. \tA one-dimensional form alone would not suffice for experience \t141\n136. \tSince its elements would be immovably fixed in a series \t142\n137. \tTwo positions have a relation independent of other positions, \t143\n138. \tSince positions are wholly defined by mutually independent relations \t143\n139. \tHence projective Geometry is wholly \ufffd priori, \t146\n140. \tThough metrical Geometry contains an empirical element \t146\n\tSection B. the axioms of metrical geometry.\n141. \tMetrical Geometry is distinct from projective, but has the same fundamental postulate \t147\n142. \tIt introduces the new idea of motion, and has three \ufffd priori axioms \t148\n\tI. The Axiom of Free Mobility.\n143. \tMeasurement requires a criterion of spatial equality \t149\n144. \tWhich is given by superposition, and involves the axiom of Free Mobility \t150\n145. \tThe denial of this axiom involves an action of empty space on things \t151\n146. \tThere is a mathematically possible alternative to the axiom, \t152\n147. \tWhich, however, is logically and philosophically untenable \t153\n148. \tThough Free Mobility is \ufffd priori, actual measurement is empirical \t154\n[xiv] 149. \tSome objections remain to be answered, concerning\ufffd \t154\n150. \t(1) The comparison of volumes and of Kant's symmetrical objects \t154\n151. \t(2) The measurement of time, where congruence is impossible \t156\n152. \t(3) The immediate perception of spatial magnitude; and \t157\n153. \t(4) The Geometry of non-congruent surfaces \t158\n154. \tFree Mobility includes Helmholtz's Monodromy \t159\n155. \tFree Mobility involves the relativity of space \t159\n156. \tFrom which, reciprocally, it can be deduced \t160\n157. \tOur axiom is therefore \ufffd priori in a double sense \t160\n\tII. The Axiom of Dimensions.\n158. \tSpace must have a finite integral number of dimensions \t161\n159. \tBut the restriction to three is empirical \t162\n160. \tThe general axiom follows from the relativity of position \t162\n161. \tThe limitation to three dimensions, unlike most empirical knowledge, is accurate and certain \t163\n\tIII. The Axiom of Distance.\n162. \tThe axiom of distance corresponds, here, to that of the straight line in projective Geometry \t164\n163. \tThe possibility of spatial measurement involves a magnitude uniquely determined by two points, \t164\n164. \tSince two points must have some relation, and the passivity of space proves this to be independent of external reference \t165\n165. \tThere can be only one such relation \t166\n166. \tThis must be measured by a curve joining the two points, \t166\n167. \tAnd the curve must be uniquely determined by the two points \t167\n168. \tSpherical Geometry contains an exception to this axiom, \t168\n169. \tWhich, however, is not quite equivalent to Euclid's \t168\n170. \tThe exception is due to the fact that two points, in spherical space, may have an external relation unaltered by motion, \t169\n171. \tWhich, however, being a relation of linear magnitude, presupposes the possibility of linear magnitude \t170\n172. \tA relation between two points must be a line joining them \t170\n173. \tConversely, the existence of a unique line between two points can be deduced from the nature of a form of externality, \t171\n174. \tAnd necessarily leads to distance, when quantity is applied to it \t172\n[xv] 175. \tHence the axiom of distance, also, is \ufffd priori in a double sense \t172\n176. \tNo metrical coordinate system can be set up without the straight line \t174\n177. \tNo axioms besides the above three are necessary to metrical Geometry \t175\n178. \tBut these three are necessary to the direct measurement of any continuum \t176\n179. \tTwo philosophical questions remain for a final chapter \t177\n\n\tCHAPTER IV.\n\tPHILOSOPHICAL CONSEQUENCES.\n180. \tWhat is the relation to experience of a form of externality in general? \t178\n181. \tThis form is the class-conception, containing every possible intuition of externality; and some such intuition is necessary to experience \t178\n182. \tWhat relation does this view bear to Kant's? \t179\n183. \tIt is less psychological, since it does not discuss whether space is given in sensation, \t180\n184. \tAnd maintains that not only space, but any form of externality which renders experience possible, must be given in sense-perception \t181\n185. \tExternality should mean, not externality to the Self, but the mutual externality of presented things \t181\n186. \tWould this be unknowable without a given form of externality? \t182\n187. \tBradley has proved that space and time preclude the existence of mere particulars, \t182\n188. \tAnd that knowledge requires the This to be neither simple nor self-subsistent \t183\n189. \tTo prove that experience requires a form of externality, I assume that all knowledge requires the recognition of identity in difference \t184\n190. \tSuch recognition involves time \t184\n191. \tAnd some other form giving simultaneous diversity \t185\n192. \tThe 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 \t186\n193. \tHow to account for the realization of this element, is a question for metaphysics \t187\n[xvi] 194. \tWhat are we to do with the contradictions in space? \t188\n195. \tThree contradictions will be discussed in what follows \t188\n196. \t(1) The antinomy of the Point proves the relativity of space, \t189\n197. \tAnd shows that Geometry must have some reference to matter, \t190\n198. \tBy which means it is made to refer to spatial order, not to empty space \t191\n199. \tThe causal properties of matter are irrelevant to Geometry, which must regard it as composed of unextended atoms, by which points are replaced \t191\n200. \t(2) The circle in defining straight lines and planes is overcome by the same reference to matter \t192\n201. \t(3) The antinomy that space is relational and yet more than relational, \t193\n202. \tSeems to depend on the confusion of empty space with spatial order \t193\n203. \tKant regarded empty space as the subject-matter of Geometry, \t194\n204. \tBut the arguments of the Aesthetic are inconclusive on this point, \t195\n205. \tAnd are upset by the mathematical antinomies, which prove that spatial order should be the subject-matter of Geometry \t196\n206. \tThe apparent thinghood of space is a psychological illusion, due to the fact that spatial relations are immediately given \t196\n207. \tThe 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 \t197\n208. \tExternality is not a relation, but an aspect of relations. Spatial order, owing to its reference to matter, is a real relation \t198\n209. \tConclusion \t199\n\n\n\n\n\nWHY MEN FIGHT\nA METHOD OF ABOLISHING THE INTERNATIONAL DUEL\nBy Bertrand Russell\nCONTENTS\nCHAPTER \tPAGE\nI \tThe Principle of Growth \t3\nII \tThe State \t42\nIII \tWar as an Institution \t79\nIV \tProperty \t117\nV \tEducation \t153\nVI \tMarriage and the Population Question \t182\nVII \tReligion and the Churches \t215\nVIII \tWhat We Can Do \t245","title":"Index of the Project Gutenberg Works of Bertrand Russell","tradition":"philosophy-ethics","translated_text":null,"word_count":6648}