,,works ,,of ,,bertrand ,,russell ,compiled by ,david ,widger ,,contents ,,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 ,,tables ,,of ,,contents ,,of ,,volumes ,,proposed ,,roads ,,to ,,freedom ,by ,bertrand ,russell ,,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 ,,the ,,analysis ,,of ,,mind ,by ,bertrand ,russell #aiba ,,contents ,,muirhead ,,library ,,of ,,philosophy ,,preface ,,the ,,analysis ,,of ,,mind ,,lecture ,i4 ,,recent ,,criticisms ,,of ,,consciousness ,,lecture ,,ii4 ,,instinct ,,and ,,habit ,,lecture ,,iii4 ,,desire ,,and ,,feeling ,,lecture ,,iv4 ,,influence ,,of ,,past ,,history ,,on ,,present ,,occurrences ,,in ,,living ,,lecture ,v4 ,,psychological ,,and ,,physical ,,causal ,,laws ,,lecture ,,vi4 ,,introspection ,,lecture ,,vii4 ,,the ,,definition ,,of ,,perception ,,lecture ,,viii4 ,,sensations ,,and ,,images ,,lecture ,,ix4 ,,memory ,,lecture ,x4 ,,words ,,and ,,meaning ,,lecture ,,xi4 ,,general ,,ideas ,,and ,,thought ,,lecture ,,xii4 ,,belief ,,lecture ,,xiii4 ,,truth ,,and ,,falsehood ,,lecture ,,xiv4 ,,emotions ,,and ,,will ,,lecture ,,xv4 ,,characteristics ,,of ,,mental ,,phenomena ,,political ,,ideals ,by ,bertrand ,russell ,,contents ,i3 ,political ,ideals ,,ii3 ,capitalism and the ,wage ,system ,,iii3 ,pitfalls in ,socialism ,,iv3 ,individual ,liberty and ,public ,control ,v3 ,national ,independence and ,internationalism ,,the ,,problems ,,of ,,philosophy ,by ,bertrand ,russell ,,contents ,,preface ,,chapter ,i4 ,,appearance ,,and ,,reality ,,chapter ,,ii4 ,,the ,,existence ,,of ,,matter ,,chapter ,,iii4 ,,the ,,nature ,,of ,,matter ,,chapter ,,iv4 ,,idealism ,,chapter ,v4 ,,knowledge ,,by ,,acquaintance ,,and ,,knowledge ,,by ,,description ,,chapter ,,vi4 ,,on ,,induction ,,chapter ,,vii4 ,,on ,,our ,,knowledge ,,of ,,general ,,principles ,,chapter ,,viii4 ,,how ,a ,,priori ,,knowledge ,,is ,,possible ,,chapter ,,ix4 ,,the ,,world ,,of ,,universals ,,chapter ,x4 ,,on ,,our ,,knowledge ,,of ,,universals ,,chapter ,,xi4 ,,on ,,intuitive ,,knowledge ,,chapter ,,xii4 ,,truth ,,and ,,falsehood ,,chapter ,,xiii4 ,,knowledge1 ,,error1 ,,and ,,probable ,,opinion ,,chapter ,,xiv4 ,,the ,,limits ,,of ,,philosophical ,,knowledge ,,chapter ,,xv4 ,,the ,,value ,,of ,,philosophy ,,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 #aiad ,,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 ,i4 ,,what ,,is ,,hoped ,,from ,,bolshevism #ae ,,ii4 ,,general ,,characteristics #bd ,,iii4 ,,lenin1 ,,trotsky ,,and ,,gorky #cf ,,iv4 ,,art ,,and ,,education #de ,v4 ,,communism ,,and ,,the ,,soviet ,,constitution #gb ,,vi4 ,,the ,,failure ,,of ,,russian ,,industry #ha ,,vii4 ,,daily ,,life ,,in ,,moscow #ib ,,viii4 ,,town ,,and ,,country #ii ,,ix4 ,,international ,,policy #ajf ,,part ,,ii ,,bolshevik ,,theory ,i4 ,,the ,,materialistic ,,theory ,,of ,,history #aai ,,ii4 ,,deciding ,,forces ,,in ,,politics #abh ,,iii4 ,,bolshevik ,,criticism ,,of ,,democracy #acd ,,iv4 ,,revolution ,,and ,,dictatorship #adf ,v4 ,,mechanism ,,and ,,the ,,individual #aeg ,,vi4 ,,why ,,russian ,,communism ,,has ,,failed #afe ,,vii4 ,,conditions ,,for ,,the ,,success ,,of ,,communism #agh ,,mysticism ,,and ,,logic ,,and ,,other ,,essays ,bertrand ,russell ,,contents ,chapter ,page ,i4 ,mysticism and ,logic #a ,,ii4 ,the ,place of ,science in a ,liberal ,education #cc ,,iii4 ,a ,free ,man's ,worship #df ,,iv4 ,the ,study of ,mathematics #eh ,v4 ,mathematics and the ,metaphysicians #gd ,,vi4 ,on ,scientific ,method in ,philosophy #ig ,,vii4 ,the ,ultimate ,constituents of ,matter #abe ,,viii4 ,the ,relation of ,sense-data to ,physics #ade ,,ix4 ,on the ,notion of ,cause #ahj ,x4 ,knowledge by ,acquaintance and ,knowledge by ,description #bji ,index #bcc ,,our ,,knowledge ,,of ,,the ,,external ,,world ,,as ,a ,,field ,,for ,,scientific ,,method ,,in ,,philosophy1 ,by ,bertrand ,russell ,,contents ,,lecture ,,page ,i4 ,current ,tendencies #c ,,ii4 ,logic as the ,essence of ,philosophy #cc ,,iii4 ,on our ,knowledge of the ,external ,world #fc ,,iv4 ,the ,world of ,physics and the ,world of ,sense #aja ,v4 ,the ,theory of ,continuity #abi ,,vi4 ,the ,problem of ,infinity considered ,historically #aee ,,vii4 ,the ,positive ,theory of ,infinity #ahe ,,viii4 ,on the ,notion of ,cause1 with ,applications to the ,free-will ,problem #baa ,index #bdc ,,index ,absolute1 #f1 #ci4 ,abstraction1 principle of1 #db1 #abd ff4 ,achilles1 ,zeno's argument of1 #agc4 ,acquaintance1 #be1 #add4 ,activity1 #bbd ff4 ,allman1 #afa n4 ,analysis1 #ahe1 #bjd1 #baa1 #bda4 legitimacy of1 #aej4 ,anaximander1 #c4 ,antinomies1 ,kant's1 #aee ff4 ,aquinas1 #aj4 ,aristotle1 #dj1 #afj n41 #afa ff41 #bdj4 ,arrow1 ,zeno's argument of1 #agc4 ,assertion1 #eb4 ,atomism1 logical1 #d4 ,atomists1 #afj4 ,belief1 #eh4 primitive and derivative1 #fi ff4 ,bergson1 #d1 #aa1 #ac1 #bj ff41 #acg1 #ach1 #aej1 #aeh1 #afe1 #agd1 #agh1 #bbi ff4 ,berkeley1 #fc1 #fd1 #ajb4 ,bolzano1 #afe4 ,boole1 #dj4 ,bradley1 #f1 #ci1 #afe4 ,broad1 #agb n4 ,brochard1 #afi n4 ,burnet1 #ai n41 #afj n41 #afa n41 #agj n41 #aga ff4 ,calderon1 #ie4 ,cantor1 vi1 vii1 #aee1 #afe1 #aij1 #aid1 #aii4 ,categories1 #ch4 ,causal laws1 #aji1 #bab ff4 evidence for1 #baf ff4 in psychology1 #bai4 ,causation1 #cd ff41 #gi1 #bab ff4 law of1 #bba4 not a priori1 #bbc1 #bcb4 ,cause1 #bbj1 #bbc4 ,certainty1 degrees of1 #fg1 #fh1 #bab4 ,change1 demands analysis1 #aea4 ,cinematograph1 #adh1 #agd4 ,classes1 #bjb4 non-existence of1 #bje ff4 ,classical tradition1 #c ff41 #eh4 ,complexity1 #ade1 #aeg ff4 ,compulsion1 #bbi1 #bcc ff4 ,congruence1 #aie4 ,consecutiveness1 #acd4 ,conservation1 #aje4 ,constituents of facts1 #ea1 #ade4 ,construction v4 inference1 iv4 ,contemporaries1 initial1 #aai1 #abj n4 ,continuity1 #fd1 #abi ff41 #ada ff41 #aee ff4 of change1 #ajf1 #ajh1 #acj ff4 ,correlation of mental and physical1 #bcc4 ,counting1 #afd1 #aha1 #ahg ff41 #bjc4 ,couturat1 #dj n4 ,dante1 #aj4 ,darwin1 #d1 #aa1 #bc1 #cj4 ,data1 #fe ff41 #baa4 hard and soft1 #gj ff4 ,dates1 #aag4 ,definition1 #bjd4 ,descartes1 #e1 #gc1 #bch4 ,descriptions1 #bja1 #bad4 ,desire1 #bbg1 #bce4 ,determinism1 #bcc4 ,doubt1 #bcg4 ,dreams1 #he1 #ic4 ,duration1 #adf1 #adi4 ,earlier and later1 #aaf4 ,effect1 #bbj4 ,eleatics1 #ai4 ,empiricism1 #cg1 #bbb4 ,enclosure1 #aad ff41 #abj4 ,enumeration1 #bjb4 ,euclid1 #afj1 #afd4 ,evellin1 #afi4 ,evolutionism1 #d1 #aa ff4 ,extension1 #adf1 #adi4 ,external world1 knowledge of1 #fc ff4 ,fact1 #ea4 atomic1 #eb4 ,finalism1 #ac4 ,form1 logical1 #db ff41 #ahe1 #bjh4 ,fractions1 #acb1 #agi4 ,free will1 #bac1 #bbg ff4 ,frege1 #e1 #dj1 #aii ff4 ,galileo1 #d1 #ei1 #aib1 #aid1 #bci1 #bdj4 ,gaye1 #afi n41 #age1 #agg4 ,geometry1 #e4 ,giles1 #bjf n4 ,greater and less1 #aie4 ,harvard1 #d4 ,hegel1 #c1 #cg ff41 #df1 #aff4 ,here1 #gc1 #ib4 ,hereditary properties1 #aie4 ,hippasos1 #afc1 #bcg4 ,hui ,tzu1 #bjf4 ,hume1 #bag1 #bba4 ,hypotheses in philosophy1 #bci4 ,illusions1 #he4 ,incommensurables1 #afb ff41 #bcg4 ,independence1 #gc1 #gd4 causal and logical1 #gd1 #ge4 ,indiscernibility1 #ada1 #adh4 ,indivisibles1 #afj4 ,induction1 #cd1 #bbb4 mathematical1 #aie ff4 ,inductiveness1 #aij1 #aie ff4 ,inference1 #dd1 #ed4 ,infinite1 vi1 #fd1 #acc1 #adi4 historically considered1 #aee ff4 true1 #agi1 #ahj4 positive theory of1 #ahe ff4 ,infinitesimals1 #ace4 ,instants1 #aaf ff41 #abi1 #aea1 #baf4 defined1 #aah4 ,instinct v4 ,reason1 #bj ff4 ,intellect1 #bb ff4 ,intelligence1 how displayed by friends1 #ic4 inadequacy of display1 #if4 ,interpretation1 #add4 ,james1 #d1 #aj1 #ac4 ,jourdain1 #afe n4 ,jowett1 #afg4 ,judgment1 #eh4 ,kant1 #c1 #aab1 #aaf1 #aee ff41 #bjj4 ,knowledge about1 #add4 ,language1 bad1 #hb1 #ace4 ,laplace1 #ab4 ,laws of nature1 #bah ff4 ,leibniz1 #ac1 #dj1 #hg1 #ahf1 #aia4 ,logic1 #bja4 analytic not constructive1 #h4 ,aristotelian1 #e4 and fact1 #ec4 inductive1 #cd1 #bbb4 mathematical1 vi1 #dj ff4 mystical1 #df4 and philosophy1 #h1 #cc ff41 #bci4 ,logical constants1 #bjh1 #bac4 ,mach1 #abc1 #bbd4 ,macran1 #ci n4 ,mathematics1 #dj1 #eg4 ,matter1 #ge1 #aja ff4 permanence of1 #ajb ff4 ,measurement1 #afd4 ,memory1 #bcj1 #bcd1 #bcf4 ,method1 deductive1 #e4 logical-analytic1 v1 #fe1 #baa1 #bcf ff4 ,milhaud1 #afh n41 #afi n4 ,mill1 #cd1 #bjj4 ,montaigne1 #bh4 ,motion1 #acj1 #baf4 continuous1 #acc1 #acf4 mathematical theory of1 #acc4 perception of1 #acg ff4 ,zeno's arguments on1 #afh ff4 ,mysticism1 #ai1 #df1 #fc1 #ie4 ,newton1 #cj1 #adf4 ,nietzsche1 #aj1 #aa4 ,nol1 #afi4 ,number1 cardinal1 #aca1 #ahf ff4 defined1 #aii ff4 finite1 #afj1 #aij ff4 inductive1 #aig4 infinite1 #agh1 #ahj1 #ahh ff41 #aig4 reflexive1 #aij ff4 ,occam1 #ajg1 #adf4 ,one and many1 #afg1 #agj4 ,order1 #aca4 ,parmenides1 #fc1 #afe ff41 #agh4 ,past and future1 #bbd1 #bcd ff4 ,peano1 #dj4 ,perspectives1 #hh ff41 #aaa4 ,philoponus1 #aga n4 ,philosophy and ethics1 #bf ff4 and mathematics1 #ahe ff4 province of1 #ag1 #bf1 #ahe1 #bcf4 scientific1 #aa1 #af1 #ah1 #bi1 #bcf ff4 ,physics1 #aja ff41 #adg1 #bci1 #bdb4 descriptive1 #bbd4 verifiability of1 #ha1 #aaj4 ,place1 #hf1 #ij4 at and from1 #ib4 ,plato1 #d1 #ai1 #bg1 #df1 #fc1 #afe n41 #aff1 #afg4 ,poincar1 #abc1 #ada4 ,points1 #aac ff41 #abi1 #aeh4 definition of1 vi1 #aae4 ,pragmatism1 #aa4 ,prantl1 #agd4 ,predictability1 #bbi ff4 ,premisses1 #baa4 ,probability1 #cf4 ,propositions1 #eb4 atomic1 #eb4 general1 #ee4 molecular1 #ed4 ,pythagoras1 #ai1 #afj ff41 #bcg4 ,race-course1 ,zeno's argument of1 #aga ff4 ,realism1 new1 #f4 ,reflexiveness1 #aij ff4 ,relations1 #de4 asymmetrical1 #dg4 ,bradley's reasons against1 #f4 external1 #aej4 intransitive1 #dh4 multiple1 #ej4 one-one1 #bjc4 reality of1 #di4 symmetrical1 #dg1 #abd4 transitive1 #dh1 #abd4 ,relativity1 #ajc1 #bdb4 ,repetitions1 #bcj ff4 ,rest1 #acf4 ,ritter and ,preller1 #afa n4 ,robertson1 ,d4 ,s41 #afj n4 ,rousseau1 #bj4 ,royce1 #ej4 ,santayana1 #df4 ,scepticism1 #ff1 #fg4 ,seeing double1 #hf4 ,self1 #gc4 ,sensation1 #be1 #ge1 #abc4 and stimulus1 #aci4 ,sense-data1 #ef1 #fc1 #fg1 #ge1 #aaj1 #ada1 #adc1 #bac4 and physics1 v1 #fd1 #ha1 #ig1 #aja ff41 #adj4 infinitely numerous8 #adi1 #aei4 ,sense-perception1 #ec4 ,series1 #di4 compact1 #acb1 #adb1 #agh4 continuous1 #aca1 #acb4 ,sigwart1 #ahg4 ,simplicius1 #agj n4 ,simultaneity1 #aaf4 ,space1 #gc1 #hh1 #ajc1 #aab ff41 #acj4 absolute and relative1 #adf1 #aei4 antinomies of1 #aee ff4 perception of1 #fh4 of perspectives1 #hh ff4 private1 #hi1 #ij4 of touch and sight1 #gh1 #aac4 ,spencer1 #d1 #ab1 #bcf4 ,spinoza1 #df1 #aff4 ,stadium1 ,zeno's argument of1 #acd n41 #age ff4 ,subject-predicate1 #de4 ,synthesis1 #aeg1 #ahe4 ,tannery1 ,paul1 #afi n4 ,teleology1 #bbc4 ,testimony1 #fg1 #gb1 #hb1 #hg1 #if1 #bab4 ,thales1 #c4 ,thing-in-itself1 #ge1 #hd4 ,things1 #hi ff41 #ajd ff41 #bac4 ,time1 #ajc1 #aaf ff41 #acj1 #aee ff41 #aff1 #bae4 absolute or relative1 #adf4 local1 #ajc4 private1 #aba4 ,uniformities1 #bag4 ,unity1 organic1 #i4 ,universal and particular1 #ci n4 ,volition1 #bbc ff4 ,whitehead1 vi1 #bjg4 ,wittgenstein1 vii1 #bjh n4 ,worlds1 actual and ideal1 #aaa4 possible1 #ahf4 private1 #hh4 ,zeller1 #agc4 ,zeno1 #abi1 #acd1 #acf1 #afe ff4 #a ,delivered as ,lowell ,lectures in ,boston1 in ,march and ,april #aiad4 #b ,london and ,new ,york1 #aiab 7,home ,university ,library74 #c ,the first volume was published at ,cambridge in #aiaj1 the second in #aiab1 and the third in #aiac4 #d ,appearance and ,reality1 pp4 #cbcc4 #e ,creative ,evolution1 ,english translation1 p4 #da4 #f ,cf4 ,burnet1 ,early ,greek ,philosophy1 pp4 #he ff4 #g ,introduction to ,metaphysics1 p4 #a4 #h ,logic1 book iii41 chapter iii41 #b4 #i ,book iii41 chapter xxi41 #c4 #aj ,or rather a propositional function4 #aa ,the subject of causality and induction will be discussed again in ,lecture ,,viii4 #ab ,see the translation by ,h4 ,s4 ,macran1 ,hegel's ,doctrine of ,formal ,logic1 ,oxford1 #aiab4 ,hegel's argument in this portion of his ,logic depends throughout upon confusing the is of predication1 as in ,socrates is mortal1 with the is of identity1 as in ,socrates is the philosopher who drank the hemlock4 ,owing to this confusion1 he thinks that ,socrates and mortal must be identical4 ,seeing that they are different1 he does not infer1 as others would1 that there is a mistake somewhere1 but that they exhibit identity in difference4 ,again1 ,socrates is particular1 mortal is universal4 ,therefore1 he says1 since ,socrates is mortal1 it follows that the particular is the universaltaking the is to be throughout expressive of identity4 ,but to say the particular is the universal is self-contradictory4 ,again ,hegel does not suspect a mistake but proceeds to synthesise particular and universal in the individual1 or concrete universal4 ,this is an example of how1 for want of care at the start1 vast and imposing systems of philosophy are built upon stupid and trivial confusions1 which1 but for the almost incredible fact that they are unintentional1 one would be tempted to characterise as puns4 #ac ,cf4 ,couturat1 ,la ,logique de ,leibniz1 pp4 #cfa1 #chf4 #ad ,it was often recognised that there was some difference between them1 but it was not recognised that the difference is fundamental1 and of very great importance4 #ae ,encyclopdia of the ,philosophical ,sciences1 vol4 i4 p4 #ig4 #af ,this perhaps requires modification in order to include such facts as beliefs and wishes1 since such facts apparently contain propositions as components4 ,such facts1 though not strictly atomic1 must be supposed included if the statement in the text is to be true4 #ag ,the assumptions made concerning time-relations in the above are as follows3 #ah ,the above paradox is essentially the same as ,zeno's argument of the stadium which will be considered in our next lecture4 #ai ,see next lecture4 #bj ,monist1 ,july #aiab1 pp4 #ccgcda4 #ba ,le continu mathmatique1 ,revue de ,mtaphysique et de ,morale1 vol4 i4 p4 #bi4 #bb ,in what concerns the early ,greek philosophers1 my knowledge is largely derived from ,burnet's valuable work1 ,early ,greek ,philosophy 7#bnd ed41 ,london1 #aijh74 ,i have also been greatly assisted by ,mr ,d4 ,s4 ,robertson of ,trinity ,college1 who has supplied the deficiencies of my knowledge of ,greek1 and brought important references to my notice4 #bc ,cf4 ,aristotle1 ,metaphysics1 ,m4 #f1 #ajhj;b1 #ah sqq41 and #ajhc;b1 #h sqq4 #bd ,there is some reason to think that the ,pythagoreans distinguished between discrete and continuous quantity4 ,g4 ,j4 ,allman1 in his ,greek ,geometry from ,thales to ,euclid1 says 7p4 #bc73 ,the ,pythagoreans made a fourfold division of mathematical science1 attributing one of its parts to the how many1 t8 p8s881 and the other to the how much1 t8 p8888882 and they assigned to each of these parts a twofold division4 ,for they said that discrete quantity1 or the how many1 either subsists by itself or must be considered with relation to some other2 but that continued quantity1 or the how much1 is either stable or in motion4 ,hence they affirmed that arithmetic contemplates that discrete quantity which subsists by itself1 but music that which is related to another2 and that geometry considers continued quantity so far as it is immovable2 but astronomy 7t88 sfa8888887 contemplates continued quantity so far as it is of a self-motive nature4 7,proclus1 ed4 ,friedlein1 p4 #ce4 ,as to the distinction between t8 p8888881 continuous1 and t8 p8s881 discrete quantity1 see ,iambl41 in ,nicomachi ,geraseni ,arithmeticam introductionem1 ed4 ,tennulius1 p4 #adh47 ,cf4 p4 #dh4 #be ,referred to by ,burnet1 op4 cit41 p4 #abj4 #bf iv41 #f4 #bac;b1 #bb2 ,h4 ,ritter and ,l4 ,preller1 ,historia ,philosophi ,grc1 #hth ed41 ,gotha1 #ahih1 p4 #ge 7this work will be referred to in future as ,r4 ,p474 #bg ,the ,pythagorean proof is roughly as follows4 ,if possible1 let the ratio of the diagonal to the side of a square be m_/n1 where m and n are whole numbers having no common factor4 ,then we must have m#b #bn#b4 ,now the square of an odd number is odd1 but m#b1 being equal to #bn#b1 is even4 ,hence m must be even4 ,but the square of an even number divides by #d1 therefore n#b1 which is half of m#b1 must be even4 ,therefore n must be even4 ,but1 since m is even1 and m and n have no common factor1 n must be odd4 ,thus n must be both odd and even1 which is impossible2 and therefore the diagonal and the side cannot have a rational ratio4 #bh ,in regard to ,zeno and the ,pythagoreans1 ,i have derived much valuable information and criticism from ,mr ,p4 ,e4 ,b4 ,jourdain4 #bi ,so ,plato makes ,zeno say in the ,parmenides1 apropos of his philosophy as a whole2 and all internal and external evidence supports this view4 #cj ,with ,parmenides1 ,hegel says1 philosophising proper began4 ,werke 7edition of #ahdj71 vol4 xiii4 p4 #bgd4 #ca ,parmenides1 #abh ,a,d4 #cb ,this interpretation is combated by ,milhaud1 ,les philosophes-gomtres de la ,grce1 p4 #adj n41 but his reasons do not seem to me convincing4 ,all the interpretations in what follows are open to question1 but all have the support of reputable authorities4 #cc ,physics1 vi4 #i4 #bcif 7,r4,p4 #acfaci74 #cd ,cf4 ,gaston ,milhaud1 ,les philosophes-gomtres de la ,grce1 p4 #adj n42 ,paul ,tannery1 ,pour l'histoire de la science hellne1 p4 #bdi2 ,burnet1 op4 cit41 p4 #cfb4 #ce ,cf4 ,r4 ,k4 ,gaye1 ,on ,aristotle1 ,physics1 ,z ix4 ,journal of ,philology1 vol4 xxxi41 esp4 p4 #aaa4 ,also ,moritz ,cantor1 ,vorlesungen ber ,geschichte der ,mathematik1 #ast ed41 vol4 i41 #ahhj1 p4 #afh1 who1 however1 subsequently adopted ,paul ,tannery's opinion1 ,vorlesungen1 #crd ed4 7vol4 i4 p4 #bjj74 #cf ,le mouvement et les partisans des indivisibles1 ,revue de ,mtaphysique et de ,morale1 vol4 i4 pp4 #chbcie4 #cg ,le mouvement et les arguments de ,znon d'le1 ,revue de ,mtaphysique et de ,morale1 vol4 i4 pp4 #ajgabe4 #ch ,cf4 ,m4 ,brochard1 ,les prtendus sophismes de ,znon d'le1 ,revue de ,mtaphysique et de ,morale1 vol4 i4 pp4 #bjibae4 #ci ,simplicius1 ,phys41 #adj1 #bh ,d 7,r4,p4 #acc72 ,burnet1 op4 cit41 pp4 #cfdcfe4 #dj ,op4 cit41 p4 #cfg4 #da ,aristotle's words are3 ,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-point1 on which we gave our opinion in the earlier part of our discourse4 ,phys41 vi4 #i4 #ici;,b 7,r4,p4 #acf74 ,aristotle seems to refer to ,phys41 vi4 #b4 #bbc;,,ab ,r4,p4 #acf;,a3 ,all space is continuous1 for time and space are divided into the same and equal divisions4 ,wherefore also ,zeno's argument is fallacious1 that it is impossible to go through an infinite collection or to touch an infinite collection one by one in a finite time4 ,for there are two senses in which the term infinite is applied both to length and to time1 and in fact to all continuous things1 either in regard to divisibility1 or in regard to the ends4 ,now it is not possible to touch things infinite in regard to number in a finite time1 but it is possible to touch things infinite in regard to divisibility3 for time itself also is infinite in this sense4 ,so that in fact we go through an infinite1 space in an infinite time and not in a finite time1 and we touch infinite things with infinite things1 not with finite things4 ,philoponus1 a sixth-century commentator 7,r4,p4 #acf;,a1 ,exc4 ,paris ,philop4 in ,arist4 ,phys41 #hjc1 #b4 ,vit471 gives the following illustration3 ,for if a thing were moved the space of a cubit in one hour1 since in every space there are an infinite number of points1 the thing moved must needs touch all the points of the space3 it will then go through an infinite collection in a finite time1 which is impossible4 #db ,cf4 ,mr ,c4 ,d4 ,broad1 ,note on ,achilles and the ,tortoise1 ,mind1 ,n4,s41 vol4 xxii4 pp4 #cahi4 #dc ,op4 cit4 #dd ,aristotle's words are3 ,the second is the so-called ,achilles4 ,it consists in this1 that the slower will never be overtaken in its course by the quickest1 for the pursuer must always come first to the point from which the pursued has just departed1 so that the slower must necessarily be always still more or less in advance4 ,phys41 vi4 #i4 #bci;,b 7,r4,p4 #acg74 #de ,phys41 vi4 #i4 #bci;,b 7,r4,p4 #ach74 #df ,phys41 vi4 #i4 #bci;,b 7,r4,p4 #aci74 #dg ,loc4 cit4 #dh ,loc4 cit41 p4 #aje4 #di ,phil4 ,werke1 ,gerhardt's edition1 vol4 i4 p4 #cch4 #ej ,mathematical ,discourses concerning two new sciences relating to mechanics and local motion1 in four dialogues4 ,by ,galileo ,galilei1 ,chief ,philosopher and ,mathematician to the ,grand ,duke of ,tuscany4 ,done into ,english from the ,italian1 by ,tho4 ,weston1 late ,master1 and now published by ,john ,weston1 present ,master1 of the ,academy at ,greenwich4 ,see pp4 #df ff4 #ea ,in his ,grundlagen einer allgemeinen ,mannichfaltigkeitslehre and in articles in ,acta ,mathematica1 vol4 ii4 #eb ,the definition of number contained in this book1 and elaborated in the ,grundgesetze der ,arithmetik 7vol4 i41 #ahic2 vol4 ii41 #aijc71 was rediscovered by me in ignorance of ,frege's work4 ,i wish to state as emphatically as possiblewhat seems still often ignoredthat his discovery antedated mine by eighteen years4 #ec ,giles1 ,the ,civilisation of ,china 7,home ,university ,library71 p4 #adg4 #ed ,cf4 ,principia ,mathematica1 #bj1 and ,introduction1 chapter iii4 #ee ,in the above remarks ,i am making use of unpublished work by my friend ,ludwig ,wittgenstein4 #ef ,thus we are not using thing here in the sense of a class of correlated aspects1 as we did in ,lecture ,,iii4 ,each aspect will count separately in stating causal laws4 #eg ,the above remarks1 for purposes of illustration1 adopt one of several possible opinions on each of several disputed points4 ,,an ,,essay ,,on ,,the ,,foundations ,,of ,,geometry ,by ,bertrand ,a4 ,w4 ,russell ,fellow ,of ,trinity ,college1 ,cambridge #ahig ,,contents ,,introduction4 ,,our ,,problem ,,defined ,,by ,,its ,,relations ,,to ,,logic1 ,,psychology ,,and ,,mathematics4 ,,page #a4 ,the problem first received a modern form through ,kant1 who connected the priori with the subjective #a #b4 ,a mental state is subjective1 for ,psychology1 when its immediate cause does not lie in the outer world #b #c4 ,a piece of knowledge is priori1 for ,epistemology1 when without it knowledge would be impossible #b #d4 ,the subjective and the priori belong respectively to ,psychology and to ,epistemology4 ,the latter alone will be investigated in this essay #c #e4 ,my test of the priori will be purely logical3 what knowledge is necessary for experience8 #c #f4 ,but since the necessary is hypothetical1 we must include1 in the priori1 the ground of necessity #d #g4 ,this may be the essential postulate of our science1 or the element1 in the subject-matter1 which is necessary to experience2 #d #h4 ,which1 however1 are both at bottom the same ground #e #i4 ,forecast of the work #e ,,chapter ,i4 ,a ,,short ,,history ,,of ,,metageometry4 #aj4 ,metageometry began by rejecting the axiom of parallels #g #aa4 ,its history may be divided into three periods3 the synthetic1 the metrical and the projective #g #ab4 ,the first period was inaugurated by ,gauss1 #aj viii #ac4 ,whose suggestions were developed independently by ,lobatchewsky #aj #ad4 ,and ,bolyai #aa #ae4 ,the purpose of all three was to show that the axiom of parallels could not be deduced from the others1 since its denial did not lead to contradictions #ab #af4 ,the second period had a more philosophical aim1 and was inspired chiefly by ,gauss and ,herbart #ac #ag4 ,the first work of this period1 that of ,riemann1 invented two new conceptions3 #ad #ah4 ,the first1 that of a manifold1 is a class-conception1 containing space as a species1 #ad #ai4 ,and defined as such that its determinations form a collection of magnitudes #ae #bj4 ,the second1 the measure of curvature of a manifold1 grew out of curvature in curves and surfaces #af #ba4 ,by means of ,gauss's analytical formula for the curvature of surfaces1 #ai #bb4 ,which enables us to define a constant measure of curvature of a three-dimensional space without reference to a fourth dimension #bj #bc4 ,the main result of ,riemann's mathematical work was to show that1 if magnitudes are independent of place1 the measure of curvature of space must be constant #ba #bd4 ,helmholtz1 who was more of a philosopher than a mathematician1 #bb #be4 ,gave a new but incorrect formulation of the essential axioms1 #bc #bf4 ,and deduced the quadratic formula for the infinitesimal arc1 which ,riemann had assumed #bd #bg4 ,beltrami gave ,lobatchewsky's planimetry a ,euclidean interpretation1 #be #bh4 ,which is analogous to ,cayley's theory of distance2 #bf #bi4 ,and dealt with n-dimensional spaces of constant negative curvature #bg #cj4 ,the third period abandons the metrical methods of the second1 and extrudes the notion of spatial quantity #bg #ca4 ,cayley reduced metrical properties to projective properties1 relative to a certain conic or quadric1 the ,absolute2 #bh #cb4 ,and ,klein showed that the ,euclidean or non-,euclidean systems result1 according to the nature of the ,absolute2 #bi #cc4 ,hence ,euclidean space appeared to give rise to all the kinds of ,geometry1 and the question1 which is true1 appeared reduced to one of convention #cj #cd4 ,but this view is due to a confusion as to the nature of the coordinates employed #cj ix #ce4 ,projective coordinates have been regarded as dependent on distance1 and thus really metrical #ca #cf4 ,but this is not the case1 since anharmonic ratio can be projectively defined #cb #cg4 ,projective coordinates1 being purely descriptive1 can give no information as to metrical properties1 and the reduction of metrical to projective properties is purely technical #cc #ch4 ,the true connection of ,cayley's measure of distance with non-,euclidean ,geometry is that suggested by ,beltrami's ,saggio1 and worked out by ,sir ,r4 ,ball1 #cf #ci4 ,which provides a ,euclidean equivalent for every non-,euclidean proposition1 and so removes the possibility of contradictions in ,metageometry #ch #dj4 ,klein's elliptic ,geometry has not been proved to have a corresponding variety of space #ci #da4 ,the geometrical use of imaginaries1 of which ,cayley demanded a philosophical discussion1 #da #db4 ,has a merely technical validity1 #db #dc4 ,and is capable of giving geometrical results only when it begins and ends with real points and figures #de #dd4 ,we have now seen that projective ,geometry is logically prior to metrical ,geometry1 but cannot supersede it #df #de4 ,sophus ,lie has applied projective methods to ,helmholtz's formulation of the axioms1 and has shown the axiom of ,monodromy to be superfluous #df #df4 ,metageometry has gradually grown independent of philosophy1 but has grown continually more interesting to philosophy #ej #dg4 ,metrical ,geometry has three indispensable axioms1 #ej #dh4 ,which we shall find to be not results1 but conditions1 of measurement1 #ea #di4 ,and which are nearly equivalent to the three axioms of projective ,geometry #eb #ej4 ,both sets of axioms are necessitated1 not by facts1 but by logic #eb ,,chapter ,,ii4 ,,critical ,,account ,,of ,,some ,,previous ,,philosophical ,,theories ,,of ,,geometry4 #ea4 ,a criticism of representative modern theories need not begin before ,kant #ed #eb4 ,kant's doctrine must be taken1 in an argument about ,geometry1 on its purely logical side #ee x #ec4 ,kant contends that since ,geometry is apodeictic1 space must be priori and subjective1 while since space is priori and subjective1 ,geometry must be apodeictic #ee #ed4 ,metageometry has upset the first line of argument1 not the second #ef #ee4 ,the second may be attacked by criticizing either the distinction of synthetic and analytic judgments1 or the first two arguments of the metaphysical deduction of space #eg #ef4 ,modern ,logic regards every judgment as both synthetic and analytic1 #eg #eg4 ,but leaves the priori1 as that which is presupposed in the possibility of experience #ei #eh4 ,kant's first two arguments as to space suffice to prove some form of externality1 but not necessarily ,euclidean space1 a necessary condition of experience #fj #ei4 ,among the successors of ,kant1 ,herbart alone advanced the theory of ,geometry1 by influencing ,riemann #fb #fj4 ,riemann regarded space as a particular kind of manifold1 i4e4 wholly quantitatively #fc #fa4 ,he therefore unduly neglected the qualitative adjectives of space #fd #fb4 ,his philosophy rests on a vicious disjunction #fe #fc4 ,his definition of a manifold is obscure1 #ff #fd4 ,and his definition of measurement applies only to space #fg #fe4 ,though mathematically invaluable1 his view of space as a manifold is philosophically misleading #fi #ff4 ,helmholtz attacked ,kant both on the mathematical and on the psychological side2 #gj #fg4 ,but his criterion of apriority is changeable and often invalid2 #ga #fh4 ,his proof that non-,euclidean spaces are imaginable is inconclusive2 #gb #fi4 ,and his assertion of the dependence of measurement on rigid bodies1 which may be taken in three senses1 #gd #gj4 ,is wholly false if it means that the axiom of ,congruence actually asserts the existence of rigid bodies1 #ge #ga4 ,is untrue if it means that the necessary reference of geometrical propositions to matter renders pure ,geometry empirical1 #gf #gb4 ,and is inadequate to his conclusion if it means1 what is true1 that actual measurement involves approximately rigid bodies #gh #gc4 ,geometry deals with an abstract matter1 whose physical properties are disregarded2 and ,physics must presuppose ,geometry #hj #gd4 ,erdmann accepted the conclusions of ,riemann and ,helmholtz1 #ha xi #ge4 ,and regarded the axioms as necessarily successive steps in classifying space as a species of manifold #hb #gf4 ,his deduction involves four fallacious assumptions1 namely3 #hb #gg4 ,that conceptions must be abstracted from a series of instances2 #hc #gh4 ,that all definition is classification2 #hc #gi4 ,that conceptions of magnitude can be applied to space as a whole2 #hd #hj4 ,and that if conceptions of magnitude could be so applied1 all the adjectives of space would result from their application #hf #ha4 ,erdmann regards ,geometry alone as incapable of deciding on the truth of the axiom of ,congruence1 #hf #hb4 ,which he affirms to be empirically proved by ,mechanics4 #hh #hc4 ,the variety and inadequacy of ,erdmann's tests of apriority #hi #hd4 ,invalidate his final conclusions on the theory of ,geometry #ij #he4 ,lotze has discussed two questions in the theory of ,geometry3 #ic #hf4 7#a7 ,he regards the possibility of non-,euclidean spaces as suggested by the subjectivity of space1 #ic #hg4 ,and rejects it owing to a mathematical misunderstanding1 #if #hh4 ,having missed the most important sense of their possibility1 #if #hi4 ,which is that they fulfil the logical conditions to which any form of externality must conform #ig #ij4 7#b7 ,he attacks the mathematical procedure of ,metageometry #ih #ia4 ,the attack begins with a question-begging definition of parallels #ii #ib4 ,lotze maintains that all apparent departures from ,euclid could be physically explained1 a view which really makes ,euclid empirical #ii #ic4 ,his criticism of ,helmholtz's analogies rests wholly on mathematical mistakes #aja #id4 ,his proof that space must have three dimensions rests on neglect of different orders of infinity #ajd #ie4 ,he attacks non-,euclidean spaces on the mistaken ground that they are not homogeneous #ajg #if4 ,lotze's objections fall under four heads #ajh #ig4 ,two other semi-philosophical objections may be urged1 #aji #ih4 ,one of which1 the absence of similarity1 has been made the basis of attack by ,delbouf1 #aaj #ii4 ,but does not form a valid ground of objection #aaa #ajj4 ,recent ,french speculation on the foundations of ,geometry has suggested few new views #aab #aja4 ,all homogeneous spaces are priori possible1 and the decision between them is empirical #aad xii ,,chapter ,,iii4 ,section ,a4 the axioms of projective geometry4 #ajb4 ,projective ,geometry does not deal with magnitude1 and applies to all spaces alike #aag #ajc4 ,it will be found wholly priori #aag #ajd4 ,its axioms have not yet been formulated philosophically #aah #aje4 ,coordinates1 in projective ,geometry1 are not spatial magnitudes1 but convenient names for points #aah #ajf4 ,the possibility of distinguishing various points is an axiom #aai #ajg4 ,the qualitative relations between points1 dealt with by projective ,geometry1 are presupposed by the quantitative treatment #aai #ajh4 ,the only qualitative relation between two points is the straight line1 and all straight lines are qualitatively similar #abj #aji4 ,hence follows1 by extension1 the principle of projective transformation #aba #aaj4 ,by which figures qualitatively indistinguishable from a given figure are obtained #abb #aaa4 ,anharmonic ratio may and must be descriptively defined #abb #aab4 ,the quadrilateral construction is essential to the projective definition of points1 #abc #aac4 ,and can be projectively defined1 #abd #aad4 ,by the general principle of projective transformation #abf #aae4 ,the principle of duality is the mathematical form of a philosophical circle1 #abg #aaf4 ,which is an inevitable consequence of the relativity of space1 and makes any definition of the point contradictory #abh #aag4 ,we define the point as that which is spatial1 but contains no space1 whence other definitions follow #abh #aah4 ,what is meant by qualitative equivalence in ,geometry8 #abi #aai4 ,two pairs of points on one straight line1 or two pairs of straight lines through one point1 are qualitatively equivalent #abi #abj4 ,this explains why four collinear points are needed1 to give an intrinsic relation by which the fourth can be descriptively defined when the first three are given #acj #aba4 ,any two projectively related figures are qualitatively equivalent1 i4e4 differ in no non-quantitative conceptual property #aca #abb4 ,three axioms are used by projective ,geometry1 #acb xiii #abc4 ,and are required for qualitative spatial comparison1 #acb #abd4 ,which involves the homogeneity1 relativity and passivity of space #acc #abe4 ,the conception of a form of externality1 #acd #abf4 ,being a creature of the intellect1 can be dealt with by pure mathematics #acd #abg4 ,the resulting doctrine of extension will be1 for the moment1 hypothetical #ace #abh4 ,but is rendered assertorical by the necessity1 for experience1 of some form of externality #acf #abi4 ,any such form must be relational #acf #acj4 ,and homogeneous #acg #aca4 ,and the relations constituting it must appear infinitely divisible #acg #acb4 ,it must have a finite integral number of dimensions1 #aci #acc4 ,owing to its passivity and homogeneity #adj #acd4 ,and to the systematic unity of the world #adj #ace4 ,a one-dimensional form alone would not suffice for experience #ada #acf4 ,since its elements would be immovably fixed in a series #adb #acg4 ,two positions have a relation independent of other positions1 #adc #ach4 ,since positions are wholly defined by mutually independent relations #adc #aci4 ,hence projective ,geometry is wholly priori1 #adf #adj4 ,though metrical ,geometry contains an empirical element #adf ,section ,b4 the axioms of metrical geometry4 #ada4 ,metrical ,geometry is distinct from projective1 but has the same fundamental postulate #adg #adb4 ,it introduces the new idea of motion1 and has three priori axioms #adh ,i4 ,the ,axiom of ,free ,mobility4 #adc4 ,measurement requires a criterion of spatial equality #adi #add4 ,which is given by superposition1 and involves the axiom of ,free ,mobility #aej #ade4 ,the denial of this axiom involves an action of empty space on things #aea #adf4 ,there is a mathematically possible alternative to the axiom1 #aeb #adg4 ,which1 however1 is logically and philosophically untenable #aec #adh4 ,though ,free ,mobility is priori1 actual measurement is empirical #aed xiv #adi4 ,some objections remain to be answered1 concerning #aed #aej4 7#a7 ,the comparison of volumes and of ,kant's symmetrical objects #aed #aea4 7#b7 ,the measurement of time1 where congruence is impossible #aef #aeb4 7#c7 ,the immediate perception of spatial magnitude2 and #aeg #aec4 7#d7 ,the ,geometry of non-congruent surfaces #aeh #aed4 ,free ,mobility includes ,helmholtz's ,monodromy #aei #aee4 ,free ,mobility involves the relativity of space #aei #aef4 ,from which1 reciprocally1 it can be deduced #afj #aeg4 ,our axiom is therefore priori in a double sense #afj ,,ii4 ,the ,axiom of ,dimensions4 #aeh4 ,space must have a finite integral number of dimensions #afa #aei4 ,but the restriction to three is empirical #afb #afj4 ,the general axiom follows from the relativity of position #afb #afa4 ,the limitation to three dimensions1 unlike most empirical knowledge1 is accurate and certain #afc ,,iii4 ,the ,axiom of ,distance4 #afb4 ,the axiom of distance corresponds1 here1 to that of the straight line in projective ,geometry #afd #afc4 ,the possibility of spatial measurement involves a magnitude uniquely determined by two points1 #afd #afd4 ,since two points must have some relation1 and the passivity of space proves this to be independent of external reference #afe #afe4 ,there can be only one such relation #aff #aff4 ,this must be measured by a curve joining the two points1 #aff #afg4 ,and the curve must be uniquely determined by the two points #afg #afh4 ,spherical ,geometry contains an exception to this axiom1 #afh #afi4 ,which1 however1 is not quite equivalent to ,euclid's #afh #agj4 ,the exception is due to the fact that two points1 in spherical space1 may have an external relation unaltered by motion1 #afi #aga4 ,which1 however1 being a relation of linear magnitude1 presupposes the possibility of linear magnitude #agj #agb4 ,a relation between two points must be a line joining them #agj #agc4 ,conversely1 the existence of a unique line between two points can be deduced from the nature of a form of externality1 #aga #agd4 ,and necessarily leads to distance1 when quantity is applied to it #agb xv #age4 ,hence the axiom of distance1 also1 is priori in a double sense #agb #agf4 ,no metrical coordinate system can be set up without the straight line #agd #agg4 ,no axioms besides the above three are necessary to metrical ,geometry #age #agh4 ,but these three are necessary to the direct measurement of any continuum #agf #agi4 ,two philosophical questions remain for a final chapter #agg ,,chapter ,,iv4 ,,philosophical ,,consequences4 #ahj4 ,what is the relation to experience of a form of externality in general8 #agh #aha4 ,this form is the class-conception1 containing every possible intuition of externality2 and some such intuition is necessary to experience #agh #ahb4 ,what relation does this view bear to ,kant's8 #agi #ahc4 ,it is less psychological1 since it does not discuss whether space is given in sensation1 #ahj #ahd4 ,and maintains that not only space1 but any form of externality which renders experience possible1 must be given in sense-perception #aha #ahe4 ,externality should mean1 not externality to the ,self1 but the mutual externality of presented things #aha #ahf4 ,would this be unknowable without a given form of externality8 #ahb #ahg4 ,bradley has proved that space and time preclude the existence of mere particulars1 #ahb #ahh4 ,and that knowledge requires the ,this to be neither simple nor self-subsistent #ahc #ahi4 ,to prove that experience requires a form of externality1 ,i assume that all knowledge requires the recognition of identity in difference #ahd #aij4 ,such recognition involves time #ahd #aia4 ,and some other form giving simultaneous diversity #ahe #aib4 ,the above argument has not deduced sense-perception from the categories1 but has shown the former1 unless it contains a certain element1 to be unintelligible to the latter #ahf #aic4 ,how to account for the realization of this element1 is a question for metaphysics #ahg xvi #aid4 ,what are we to do with the contradictions in space8 #ahh #aie4 ,three contradictions will be discussed in what follows #ahh #aif4 7#a7 ,the antinomy of the ,point proves the relativity of space1 #ahi #aig4 ,and shows that ,geometry must have some reference to matter1 #aij #aih4 ,by which means it is made to refer to spatial order1 not to empty space #aia #aii4 ,the causal properties of matter are irrelevant to ,geometry1 which must regard it as composed of unextended atoms1 by which points are replaced #aia #bjj4 7#b7 ,the circle in defining straight lines and planes is overcome by the same reference to matter #aib #bja4 7#c7 ,the antinomy that space is relational and yet more than relational1 #aic #bjb4 ,seems to depend on the confusion of empty space with spatial order #aic #bjc4 ,kant regarded empty space as the subject-matter of ,geometry1 #aid #bjd4 ,but the arguments of the ,aesthetic are inconclusive on this point1 #aie #bje4 ,and are upset by the mathematical antinomies1 which prove that spatial order should be the subject-matter of ,geometry #aif #bjf4 ,the apparent thinghood of space is a psychological illusion1 due to the fact that spatial relations are immediately given #aif #bjg4 ,the apparent divisibility of spatial relations is either an illusion1 arising out of empty space1 or the expression of the possibility of quantitatively different spatial relations #aig #bjh4 ,externality is not a relation1 but an aspect of relations4 ,spatial order1 owing to its reference to matter1 is a real relation #aih #bji4 ,conclusion #aii ,,why ,,men ,,fight ,a ,,method ,,of ,,abolishing ,,the ,,international ,,duel ,by ,bertrand ,russell ,,contents ,,chapter ,,page ,i ,the ,principle of ,growth #c ,,ii ,the ,state #db ,,iii ,war as an ,institution #gi ,,iv ,property #aag ,v ,education #aec ,,vi ,marriage and the ,population ,question #ahb ,,vii ,religion and the ,churches #bae ,,viii ,what ,we ,can ,do #bde