-r18

sU 5

10212122 Thus what appaar as member. to one p8nCll1 IIB¥ not appear 88 ••••• to uotheroIn this emmple, the OIl8odD.ch leDCtbBwould be•••••• t.raa QIIo8

po1Dt of view, 'but not frca the other",

lo212l23 Tbu the 0ba8r1tJr dotem1Des the nature of the f1014 of attet1GD. aa4 thereia DO auch thing _ a f1eld of attent10D 1Ddep8DdeDt of t.bD0.•••••••

102122 In order to test for 'the preBentati~ of a set. the 8811I8GpIIl ~caD be used as in 10120

..• 102l22l A turtber test is the open smt4mce "this of 111••• , of _~~~,JI ••

1Cbere·the first bl.e.nk conta1Da the pr88entat1~ arMl • ..ca. aII4 W1"4b1pJrs cOntain tbG content-aameo

. 1021222 Thws the particular pre88lltaUQl';MWle used will be a opec1al ••• of tbepresentation-llWDe "sat" 0

A b9& of oranees is B set of oranaesJ a group of •• 18 a aet ofmen, a pile of stones is a set of stoneso

1021' W. B8k Dext of a countable field, of attention the question of a8J*Jt. tba 1I'qin vhich tho presentation of the giV8ll content- La cona1derec1o

1021301 It 1s poaaible to OCft.9ider a countable f1e.L<Lof attent1cm trca _ WlCOUlltableaspect" .

It 1e pa,,~.Lbl.eto tlWlk about tho we1&ht of a l;Ia£ ot oraD&Mo

1021302 But such pointe of viow are best COD81dered as a. aub-aectiCID of 1.220

loml The aatlmll aspect UDder which to cCDSider a given set of _~ COUDUDs .18mZllibe:ro

1021'U JJumber can ~ thus at this pre11m1MJ7 ata&o be def'1Da4 as tbD coaDtableaspect of a set of memberso

1021312 V. ld.ll find later that number 1s also relcmmt to ncDoCOUIltable aets. s1Doeit appea:roat the stage of ftlUfWVUM or DlIIlber-llamao

lo213121 'lh1s will require a mod1t1cat1on aDdelaboration of tho prel:lm"V def1iU.t1G1l. of mabero

1•.•21'122 HowYer. this new def1nit1C1111f1ll be ~ onq aD u;t;anmc of .• 014W'in14don to take S:t into account a :2._ at tvatiOllo

To be .f.ecJ"fJ~ and thUs to anticipate wbat 1s to CQII8 latei'. INChDiab8r8 aa ~a l'4t1anal ad real ntDIlbers must be 1Jmmtecl to __ careot tbo a1Wtt+'i&'i1fh1ch arisee 1f1th-DOD-CoUntable f1elda·or ~U&lJ

102lJ2 The opeD. IIeIltanoe 'l.\aed in 101' 1f1ll demo:D8trate that maher 1a a pro,..aa~ for a set 01: III8IIlberse.

102J32l To 4aDautn.te tb1s turthcr we can use the 0lJ8n &aDteDce "I &Illtbin)1aa of. toe •• bar of this .t CJt It" ~ tbe blaDk 81ft. 1:beogGt •.••••• o

1021'22 There are. ho1Amtl','nc otber aapect.Guaea m1ch are special __ ot tbt •• ~••• "zNalber" 0

102m Nuaber ia the Datural aapeot UDdel"1dd.ch to ccaa1der tile iJIolaw. dt~1D41V14uala 1n a aato

1<>2l3:51 li\IIlber is the property whioh a eet baa when it 18 COD81dctre48OCOl'41Ds to 1.diacxreteaauo

102l.D2 Other propert:Les at a presentation cOil81dor it apart trOll •••. poea:t.l»le d18an ••..•Deas it IIIB1' d1apq e

102l,,21 Tbeae other properties are thuD to be oOJ1B14ered as UIlOOUIlta'b1e •••••• 0

Weighte as a.ppl1~ to a eet or 01"8Di88 does DOt taU into accountthe cUbCretenos8 of tho OrarJp8. and could equalq appq: to the __oranpS IJIBShedinto a pulp"

10213322 'rbi3 opllt between nUl!lb9l.~nnd all otllez' SSpe(ltl3 is the most bu1c feature a£the oplit betwsOll ~vnt-t)J8.e and uncountable f1elde of atteDt1cDo

10214 Wenak next of a cali ft:fr~PI(! field of attention the Q.U8oticm of moawro, the WJq 111lAtch tho a:1ven as,ecl- c1F the preoentllt1011l of thu given oaa~ 1.-' Yallald.o

1021401 It'is poss1bl.e to consider a cOlmtable t1e14 ot attentiCil tr. aspeota,aDd thll8 to iIlpoee uncountable measures upcIl1to

It we aro th:lnldng ot a •• of C"aDi88. _ mustmeasure that weicht 1n a un1t such aa poUDdeo

1021402 aat 8UCb pointo of view are best oODSi4ereci 88 a sub-aeotiOll of 10220

102141 Tban U'8 eneral po88ib1e 1D888Ure8 which can be 1mpoeId ell tlIe CCUl1a~ aapeotwb1C1b18 •• bero

Wecan. tor instance. OG:USiderthe number of a group or people 111tcmDa ot 1Dd1Y1duala ar pa1l"8 or t01Jl"llOlll88o . ,

lo214U 1JoIIeftr, each of these 18 a IIIUlUple of the baa10 •• auze .1n .•••• of iDtividllti180

102l.4l2 '.rima _ will take aLl the b&e1o JDINUIUJ'8 at number the 1D41v1dlBl. 1fb1ch cum alaobe called the iteac. ' ' , , '

10214121 The 1tem, 1s one 6inela memberof a d1aarete set. oonsidere a)l8.ri frail :l.tsCCIIlteato

ThUll. considered tl'om this a~traot point ot v:I.ev. the -He•••tartn. set of ar aJ1(pI8 is the 88118 as tho 1.• tor the oet of peoJleo

~214122 The COI'ltententers at an 811Z'l1c'a'tap. and 1ibuBDeed 110lcJVr -. 0GDa:I.d-.re4at tb1a' ; po1Dto

.$0

10214123 'I'ha item is the measure of the unit set, and allows WI to compare the meaat.Il'eof the 8iven presentation,. or set, with the measure of the unit sotQ

lo21412! 'I'hs compuison is done in terms of number, which is the asp8c'to

10214125 Thus a countable field of attention can be defined as one ~ch can bertw&d 88 a camb~nt:t.tion 01' d1screte tm1t fields 01' attenU.o

~i "'8 cOYlsideredfrom this point of view is cO\DltaQle, 8'YeIla ruler whJch -is looked GIl as series of incl),..lengthe placed ~to-oendc>

10214126 The normal way of' looking at certain fields of attention 18 as COUIltableD

tel'llS ot itelil8~ as is shown b,y the plural in the contem.t...naMo

Thus the D8IJl8 "oranps" with its plural 1'01'111 ' . ehows that a setof oranges is looked at as consisting of a collect1an ot e1ngl.e 1teEo

1,)21413 Other measures can be constructed ,by putting together items into iargermaasureso

1",2142 Th& cpon aontenoe used in 1014 will demonstrate that item io a proper ~t~ (\Nt of membereo

1021422 NOI16 of the mee.svre=namesdl3rived fran '.:ll,"ouutable aspects Will be specialCl'USG8 of this measur&~c;>

It -lie th-1••21k (\f tha number of inches in a piece at atri.ngp we oreth:J.Dk:1njof .J.he s1Tl'.illg as uncotmtnble, in terms of langtho

1.:;2143 ItGSil is the Mtl~l'lt.l 1'lJe,~.sUl:ewhich should be imposed en the isolated!) d1.econtiPr.R.'.OUS individuals in a seta

10214;1 Item i.8 the lll£U:l.SUre imposed on a oet which it is cor~sider6d accordin8 to itsd.iBo.retmesso

1021432 Otbar mea~~e. 01' a pl'esantation consider it under some aspect ubichio notrelated to its dJ.Bcretenesso

10214321 Thwl theoe other D!.eas~1l'eSare considered to be uncountable measureao

The pounds ~ which a set of orange£!are measured do nottake into account the discreteness of tho oraJl88s, and could equalq&PPlJ' to the same oranges maahed into a pulpo

10214322 ThiD split between item and ~.ts multiples aild all other measures in basicto the apU t between countable and uncountable fields of attantiODo

10215 We ask last of a coun-taltlElfield. of attention tba questicm ot 'Ialua., the ~in lfh1cb the given measure of the aspect of the presentation of the content isactualq D\IIIlberecl,the range of po8sibilities of numbero

- lII""1. 2~~ j ~l'hol' ;'ro Govaral posll1h1ft nUllbcr systems which can 'be ue~~ tn :n£Ike spoc.1.f.~~ the

cOI.ll\'tal>la.lfIIIIQS\Jl"e." '..:lidl 4,8 ,·ou.:ed f:OJ: item ana its Ul'.lltlpleao

1. 2:,5U rfhe~c number ec&les 8l'C those mentioned in 1015. which are of oor.::plc'.:o;~eral-lty and apply equally to all fields of attention.

le2151,2 In partloul."..l', it 1s possible to uae the range none-somc-a11, or the r:~ dordinnry oounting numbers.

10215J'1 The ord.1n.£ryoo,.,nt1nenuabera give. the most natural, and yet procL.:;. vnlue.tt> the :;lf~ure by i tema. •

1,,2: c) .22 Thc:,Goow:tinC numbers enable us to giV'l~ ~ .•..•Neiss comparaeon between the givenpreot'.nta.tion and the standard uni f prc.aent«.-tlono

1,,;?:.:~~3 TIme CY<.lry tifft., ",,'IiGoM i~ a presentation of a countoo.le field of attontion, haaBe a propertJ YlIJMbe •••.I ':hioh iB me8nurcdby i ternef and valued in oounting numbers.

Thus h· car- opeak of a set of seven orangea,

J .215.3 The oount'foe number3apply equally to the moasure by pail"'3 or fouraor.;ce or anyothor ::!Ul!;iple of un1tHeme •

. •2152 The open son+ence used in 1.15 will show the names of these numbers to be value-naaea for the oct of :tcrubers oonsidered from the aepect of number and mea'urad inite 13 or eor;e m:.ltiple of items.

1 • .?lS:2l Th.::: rnn.:.;oe~l-&:>~orle will fit th.is open aerrtonce ,

1021522 The ron.'.J"eof ordinary counting numbers will fit this open sentence.

1.215> ather nunbert: Will be found which fit the countalle fields of attention only in-diroctly.

For inat&.,c~;e we can use special fractions, D8l!lely. those which areequivalent to "tIhc. J.e numbus f as numbers to f1t countable :t'ields of at-tOIltion, Bu"tt: if, is clearly art1:t'icial. The real use of fractions willbe eoen unde;: 1022 to be in oonnection with unc.luntable fields of atten-tiono

1.,21531 This Dplit bOWdencounting numb.::rsand other extended number ayste&l8 will befound to be the split between countable and uncountable fields of attention.

L215)~ Th(,:~cothJ1' nunber- ayet(Jl!lB Will be cons1dered uncmmtable value soaleoo

1,..>2154 The countin& numbers thl.Wnumber discrete sets of pointe.

1,215(1 A Line consisting of a finite number of discrete points will have 11 nuaber , butnot a length. since 1ts meacure. in the uncountable aenae will be 0, and its num-bar :L'1 a countnble sonae will be fin1 tee

1~21~..;.2 A line conaisti!lB of all points in a neighborhood will have a length, but not anuabcr , Gince i to measure in an \mcountable acnae will be fin! te, and i to nUI:lbarin II countcbl.e SOlleewill be 1nt1n1tee

-r6~-• 113 ¢o

1022 We CODSidernext uncountable f1elds of attent10n0

10221 W. aak first of an uncountable field the question of content.'

102211 The 'basic tact about an uncountable field of a.ttention 18 that theCODtent-warki8 1nd1tterent eitber to s1ngularity or plural1tyo

For instanCt~" rice and water are neither s1ngular ••• plural, butare s1mPq V1t<OLJfl~ble•. wheroas o~s and people 81'8 countablop sincethe 8inguJll-V'i-tyor the plurality are ind1cated in the canten"'!- aamBo

lQ22111 Thus we do nO~GlelU'al an.ythiIlg about the valua of the meesure ot the uncountablefiaM of attent10D fl~omthe 1fS'¥ in which 1ts contenklame appeara in thedescr:Lpt1ano

102212 A tel'll wbieh will cover all content-names for uncountable fields of attenti=18 "material tI 0)

It I attend to an uncolmtable field of attention, I can considerwhatever I find in it., whether rice or red/Vitero as mater1alo .

lo221Zl Material is not iDberently individual or 1aolableQ

1022]22l This continuity need not be ~icalt 8patial contimait7. but can be SODI8other tRe of cMtr)'l~tyo

lo221?22 the CClIltinuitj is it. ~stion of the wfq in which the fio14 of attention isCODS1clered,sinev .LD. one way the f1eld of attention 'I1BJ' be CGDt1nou8•• andin· another way disoontinuouao

As in 10211222p a group ot peoJl,e 1s countable l11th respect tozwmbor, and thus is discountinUDUBo However. vith r8speot to velshtit i8 caat1nuous, since the· p}\ysical separation s-. ...between the pe1'8QD8 does not .: . I.US8Da neceS88l1' p\ys1cal aopazationaD the level of weisbtQ From this point of View, tho pJva1cal separao-ti.GIl is 81mpq not' attended too Likeu1Bep it 1s poss1ble to vi.., aruler either au 8. continuous l.eDgtb~.or as·8. discrete colleotiOD incb=~ placed end to endo .

1022122:5 We1"8IIIIIlber from lo21~ that the field of attention 16 tho content as it ispresented. considered from an aspect, measured by a meaGUr8,aDd numbered by valueo

10221224 The COIltinuity' of the field of attention is wbat makes it imposs1ble for the:field to be counted, since there are DO items to be counted. UI1leos orbitrarp.m-d1'1i1d0D8 are made in the f18ld. accOrding to the UDitao

1022123 Thus a test tor material 1s an open 8CmUnco of the type •• is tbe_ter1al in the field of attention." . 4rbere the blank gives the cOIltcM-orko

·-ltS3 -sa

For instance,. ve can fI83 that "rice is the material in the fieldof attention" or' "sound is the ma:tel'ial in the fl8ld or atteGtiGD"0

This test Will .svcQm for uncountable fields of attantica.

lo~2 '1'h18test wil.l fa: I ntr countable fields of attentiODo

10222 W~ask next of an uucQ!Jnt;ablefield of attention the queatiarl of pl"e80ntatiOD6the wq tho content is presentedo

L,222Ol It is difticult to find a geaeral _y ~ describing Uncountable fields of'attenti<mo

Such fielde of attention vary wide~, from.a pool of wator. toa note 01' musio to a IDOIDentof thOU8ht. and are not 80 88s11l' character!.".* as are sets 01' oountable 11I8.lIlbers<) .

1-l222Ct2 Tlwa arq deacr:i.ption which is obtained JID13t Il8cesear1q be sOIDI"nataad al.ieD to Ol.'dinary speacho

. abstracto

102221. Cant~ is a ( genenl presentation-word tor the way the oontent of anuocountable field of attention is formedo

We Onn speak of a continuum of ric~3, or a cont1D\UI or IIl1l.ke ora continuum of YE,j.l.OW, but not of a continuum of 0raDBell. or a continuumof people" or ~ GOI"l;1nuum ot ideaso Or, it we do oODS1deruncountableth1 HiS ill teI1'''' ,oj: continuum. we are consider1De thom in GJl 'UDCO\mtable ~o

10222ll 'rhus.we can define Cj, contimDII 88 presentation of mate~1alo

lo222lU''l'hio is a . detinition in tems of prev1ousJ.r def1ned 'tel'mB, aDd proceeds frauthe III01'e genenl to the more specifiOo

lo222lll The term cCllltinUUlllt:. . is the ~ preaentat~D8II8 tor a presentation of an lmccnmtable field of attention, and is appl1catieto ~ pr8sr.ntation of an UDCouutable .. field of attentiC11lo .

lo222JX; The term OCl'ltinUum is not applicable to 8ZI3 countable fl8ld or' attention!) \Dllessit is cCllUlidered tram someuncountable point of vi8W1I in vh1ch coae it is in.tact an l1I1countable field of attentioDo

Wemust l'8IIIU"k tbnt we can think of CIl'8Jl&Qoin terms of thelrve1cbt. in which case we are cODSidcr1Dg the oranges 1'roIl a CGIlt1D.uouBothus uncountable nepeoto

1022212 WeGaD also see that tl1e cOl1tinuual i8 8. presentation of 118.ter1Bl cona1deredwithout 8D¥ i/JOlat£.'Claeemonts or discont1iNoua 1Ddividualao

loW.l2l 'rhis is a 1pI8t1on of the vq the tield of attention is viftD40

lomA22 Thus wbat appears 88 mater1al. to one person rlJIX3" appear otberVioe to anotheroTo quote the example under If)2l2l2l and 102l2l22. w see that a

-IS'+'-

•.mruler can be considered as an uncountable maoa. aa voll GIJ B •• ~ otons-:lnch lengths set end to endl,\

10222123 Tbu.'m.CIlC8 Il&'Bin we note that the observer detel'lll1Des the zmture or thG fieldof attention. and that there is DO such th1Dg as a t1e14 of a~ODt1cmindependent of the obaer'lero

102222 In o;rder to teat tor the proaentation-oame of a continuum, the __ OpeD sentencecan 'be used as in 10120

lo~.l A turther test is the 01- sentence "this at 18 a cont1Duumof • $I" whe..\"e the f:l.rat blank contains the preaenta~, and the

8OCOD!1 aDd third blmllm c:ontaiDthQ content-oameQ

1022222 Thus the particular :9l'8f,ents.t1QDooonaJDe will be a apec:f.aJ.case of tbe Feaentati~nama -cont1D.uura""

A plOl of lmter 1s a continuum of vater; a breath l)t tzoeeh a1ris a continuum of fresh air; 0. bag of' rice 18 a contiuum or 2."1080

10223 Weaaknext of an uncmm'b:,:ble :f1eld of attention the question ot •• peat. thD W8¥in which tho preoont&Uon of the given content 1s consideredo

lo~l It is possible to consi(ier an uncomtable field of attent1cabcm a ccnmtableaapecto

It is possible. tor 1nfItanceo to think about tbe mlDber ot ~in a be.gof rice, or the number of molecules in a bottle ot _ter.

lo~2 :&It in this W8¥ the field of AttEmt10D115 actuaJ.4r than cour1tableo ODd thecOllteil.t and presentatiC»l names should be altered.

lo~ It i8 clear that a cOlltUl'DmI,from the point of view at ~; 18 what _attend to and describe b.' our ~. but not nece~ tba~tbe thiDa 18in itself, wlU.ch 1s in fact unimportant and irrelevant tmd .,. ,mdiecuaableo

1022'31 There Us DO s1.ncle natural aspoct under \d11ch to consider a g:s.ftD CODtimNmof JlBter1al1l other than quality" ~

Obv1ousl,y lie can think of a pool of water in te1wt or -1&h~.'901\1118" temperature. color. uaetulnesa, 8D4 so tortbo

. lo~U Tbomio a cm~ uniformity amongall the•• aapeata •. in tbat tM;y all8Z'Obaaac1 on thu continuity ~ by the .lDBter1al. aDdcan be called.qWLl1t1eso

loWW. Thus all those aepecte 81"0 mass p:roperties or propert1ae ot tho ator1a11D tb8Jar., DOt propert1ea which depend on atG' essential diSOClDt:1m11V 1D theoem_to

lo223U2 It 1s true 'that ~ 18 also a property depending on tbo 1tbole or the SiY8lllfCUIiitati.on of canten.t. but 1t also depmds on ~ d1aoc:atilDllV or theCGD'MDto and thus 18 test aalled qusnUt.Y :l.DBtead of qual.11;ro .: .

-185-

•lo223m T'.beee aspect-\ which 'ore call qml.1t1ea, can be .' marate4. ao •• u:Ul

•• 1,JDder 10224and lc225, bu.t this numeration baa a spec18l chDacwo

lo~l14 Tbe d18t1llctian madehere. tberetcre, between quaJ.1ty aDd qUELlltit7. 18 at'Ql'l4amentald18t1ncti(:nlJ aDd oorrespcmds to that between COWltable audUllCOUIltableo

Thill is the traditior? 1 distinction of the loeiCal cateeoriea ofAristotle and Otl:lors. It is hare Been in a clearer l.:14hto u.4 UDderatcD:lin terms of the total picture of the descriptica. of tlel4a or attenticmoThe categoriea can onq be basic divisions in the wa;r at 4eecriblAi thecontent of' a field of attention. and thus this analiJs1a a:1vea the beat~.soo.ri:ptionof tl;,o oatee;or1es, one which is guaranteed to be CGIIp1ewo

1t.2232 The cp8Il eentenca usod 1.['. 10131fill delll(t~'~":l'&t.sthat quality 10 p:ropor aapect=rlaI!I8 tar a cantimnlU1 O~;-11l8.teria1o

1022521 To demOlllStml;~ #1; s fUI-'i:her W8 can use the open sentencep "I em tb'Dk1Dg of aq1&~Jj:t,y of. i;hi s coVl+i.l.l.um ot _ p" where tho blaDk gives tb&. content-oameo

1022!:22 TMre are tIlB..1lY ~"'Ji.,.e"';=nam8S which are sPl'cial cases of tba aspeciHwm "qual1tT'.as \Ie have seen a.lxWt!o

lo22'!:} QL1a.1.ity uAtbe natural. a.e:poct under which to consider the coDt1mxN8o _.i:d,ded2ter1&1 in a continUUi.D" .

lo~;l QualJ.ty is t·J.8 propel"t~·which a. Clmt:bmm has wlumit is CODBidared aooordiDgto its ccmtiJl\wuanoa~<l .

loZ-"!-J2 T.ba CIIll,y otmr proPOl~Y of a presentation apart from UlJ,COUIltable aapectap orquallt1e8o .is numbero C'1' quantityo

10223321 'J.lb:1e~\"tliaH· :tn"Operty tz thus tile oountable aspect of a preee1:aUGDo

The numoor c·f a set of O1'8ngeS, just as the nlllibor at 1nch$a ina rulero takes tn.to account d1scretaneE8p but not azw qual1Q'o

~2m22 '1'hu8 cmce ~ \m o.')~ that thie split betweennumberand all otbeJ.-aspects18 tbe moat ~3ic foakro of the split between countable 8DIlUDOOmltable t1eldaor attent~

10224 We8sk DeZt at an Ul'WOtmtable f18ld of attention the quast1cD of a.sura, the 118.7in 1Ib1oh the &1van aspect of the presentaUOD of the gi'9U. COIlteDt :la ftl.-'o

1022401 It 18 lJQasible to c0ll81.der an uncountable fiold of attenticm t1'CIIl a countableaopeotcs, thus ccms1deril161to in ettect, ao a countable fiel4 of attent1cmo

We can think at a bottle of water in terms ot the JWIIlbel"ofmolecules ~ it, thus canvert1Dg it into a Bet of IDOlecu1ee at _taro

1022401 M iNCh points of view are best cOllDidered as a 8\ll).aectiGll of 10210

·-ltJ6 -41 ••

10224.1. There (J,l'e 2levt):ral possible 1II88oures which can be 1mpo88ClOIl .,.-h qU8l1V, aruncountable aspect, byl,tdch we may consider a continuum at zraWrialo

Wecan mea5\:\l'e the length ot n piece of 8~ in te.z. c4 inchesor feet or centin'flterso Wecan msasur::, the beauV of a pa:lnt1rl6 interms of l.>:l"l'SowY omoticm or cr~:tical. l1'.celaiIIQ

.10224111 Theunit meaaUl"~1.San erb1~ determinoo field of attontiOD, 1fh1.ch18~j with tz'1.t) s;lven field of attention, end uhich lu:\S tho fJ8me contentp

b Bru:10prGs~'t3tian" the aameaBpact, the same measure, wt a cl1tfcront"md.lIIO I)

For inBte.n¢S, the inch of strin8 is the same as the p1eoe of stringin ever,)Tt!line axef.,pt valuao

l.C'22U:t2 There ean be muny S'Cal'i.c'.ard lDliu o~ m3aBUre& since each of them ill! ebol38ll pgre~l'.rbitl'ar:Uy fr{~ the; ~()ntinUUlll'ilihich is T.1ewedunder the giVQlrl qunl1i'.7o

~ ,Tlr~a the 1nd: ~e cent1matar are incOl1llIl8illS1n'blo ~8, since

t)s;y- .ar'3 chcsen .sl1t.lt:rar~o This could not happen. of cout'8e~ Withc;)wJ;able lI'..GaSUr8S, s:i.nce they: msut all • related to tho 1temo

j,(j2;~',:? The 188i.c ~.»:a is tlwn \Wed as a aW1'd41('i\ ')f compsrison, l:u . idlich the81.m present.'l.t:L01l C8Il 110 gi\'8n .A-BVJIII,-per 1n '<~ of tho staMardo

102242 The opan I!!eJlt0.''''l~H 'USe~ ,r" 1014 v:Ul domonstr&te tha.t unit 18 a proper ~J:Zllme tor a giv .en ~1Ia.1.:-+y of a cCIIlt1nUUlllof me.terislo

1022421 L1kniae each of 't~oe. ,1Iec1al measure-names ldl1 fit this open eentenceo

lo224Z~ Nonoof thr) m2S3Ul'&-&"la1r..fi'S related to the countable aspect of q\8ltit)r 1f111 bespec1al cases of this mea~o

Wecannot thlnk of a piece of Btr:lDg as being moawred1n termsowhereas it can be:maasuredin un1tso

1;;224' t1n1':is the JlB:tural meM~i'N which should be imposed. on the coniotmaouBtI DCJDoo~ted material in a coot1nUUlllo

102?-4Jl Unit i~ mooaure im:f<;sod on a continuum when it is considered accorcl1JJi toita ccntinuityl'l rathol' than accorcl1.De to ar.y type of d1aoretcmeaao

1022432 Thus it is d1stiIJeu;iahsd tram the basic comtable measure, 1Ihich 18 in 'tel'IIIS of items. or 1!IIlltiples thereoto

10224" Thus ve 889 once agal.."'l. that th1e split bob'een unit, vh1ch 18 ohociaD trcaa cODtimP!!D a:rb1trBl~y~ and item, ~ch is a naturalll8lUJU1'e ot diaa:l'etenesso

, 1e fundamental to the: - 3plit between countable and 1mCoun1ab1e t1elds atA~tiClllo

-/f7-•••••~-,- .

10225 Weuk last of Eon uncovrl'l.;.aole field of attention the quesUca tit "Ialueo the 'Wl\Yin which the g1:~'e2l Iooasm'} of the aDpoct at tho presentation of tile content 18s.otu.~Uy muubarOO.. t..he :ren,,~ of posaiblitiee of nUlIlber9

1,,2Z'5Cii 'l':"l8 most noulble fact ~If way of ~.ntroduction is that the UD1t of measure andt~ given prtlfW.ntatioll do not ha"e to be CClIIIIDeDS\IX'te in termc of count1rgnWJlbarsll

For i.nste.nCI~., there is no reason ~ a piece of S'tr111B baa to'b!t an 5.utegral niz"lber of inches in le;neth" There is no reaDon wb,)' anemission of sO\r'.I\l.l baa to bn a\n intogral n~t cycleD ~r secondin. fxsquen<'7o

1()2t:5(2 T"bioit:! in sllai:ll (!0Tt:' ~iJtwith tile c:oun:j;a.lll,~~field of attentlcm 1fhet.'8 th£.,civen field ~mpi be.ell ~ntegral. number of items in valuoo

102251. 'rlr<J.6 the numbHI' ~cal ~~ d.ppl1ed to l.Dl1ts of measure must be more CCDplez tl:an·~O~ of tho (!O"~1.:Qt.bJ.(:;;~ields ot attentiono

102(511 It :ioBo of courBo", J,.'Oooiblothat tho simple valUft<aS"Jutems. auch o.a all-s<mJoan~, and tho countin,~ nunibers. can apply to an uncountable f10ld of attl}ntiono

It io lXlsnit1j 0 to have a piece of string throe liD.chGa in le11«t.11oor a" :xnintin.£: emctly aa ooautifui a:jtlL stazJd.ard po1ntineo

lo225l:L' In tb.ia caw, tJ1r.) ol'(lilJ,ary count:! :Jg numbers, which arc tho countaDJ.o valUt~sccl.ao apply t;o the unc:e'Wltable 11e1<1of attention, which io ';ms conside;:'efiam 1.1' it were in 01'1'001; countable"

'.rO lOOatJW:'6 tba piece of strine as exactlJ three wJws in leDg'mi:}) to c,cm.n:tdm: ;i.t as a set ·Q·f t111'oo inchmoleDJth8o

102251.:12 In thia caee , t;P.oanalJlsis under 1<;1::0;3.,~..~Il'~,and. 10215 app~ to tho UI1!~ount-abl.5field of attm:\~io.l. c<XiW1de:t"~l .'~3 ;U'~~Jro countableo

lc225D.' Thu.o tilD inDtr,\C~.+;on !;. "Ol.1ut ~it J~\.U:~ .All te ••~ of o;;d1i1iB,';.y~t1n'1m.wber~i13~ocq,nsl.:(der. tho,) fiold oi' attention !LOcompoood of theee unitmeasures ccac.,; ,,:ti'f\:t.her to :makcthe field of attcnt1ano

10225114 1t~ tho ~lfJ.~ .;;.nthis ~ .or the lJII..oountablecase is a repot:Uin'lot thebehavior praviouol.V C.EIBCri'bed for tile c()urrtablo case, .

. 1022512 It ~ ba., of course. t.hat ·tho countins num~ra do not aYJ?q in wbich cas.~ otb8rval~-"aystems are nece:,aary, in order to make the 6ivon presentation and

the ntan.dB.rd presentati(iIl commene.uratso .

102251:21 O.no8UOhvaltl6 aystOOl :i.s that 01 o:..-11nary f:reet10JlOD in which CIl.OOthe c1.venprosantat1on, is , :in c:j'tect. compared with atanda.rdport1QD8 of tho etaDjardunit moaaureo

For :1nstanC(l ve m1ght say that a piece of a~!Dg 18 1 5/6 1nchas ink....,JGtho by whioh \10 mean that we measure it in terms of l/&-hlch 'Dl1t8~and. we find tba.t i;here are l' ot" thoDk.

-ise-28 -

lo225l2U Th10 ftJ.ue system if) just an extension of tho preYioua ftl\I8 ~, o1Doe ituaea a smaller unit meaoure, a mvltiple of which is tho Olictul lID1tlI8G8U1'eo

lo22~l2l2 Thus onco ap.1n the ~ siTeD i"or oountablo tielda or attant1cm.CODoic1er1Dg the aap9Ct as number. the measure as itea 8M 'the 'I&l.uesyataa 88 aouz;.~::.~:;~umberS,is applicableo

11.'~ A more complex case iSlIbere DO taction at the standard ua1t JDGaIN1'e isCOCDmsurate with the given preaentat10n0

For inBtfmce, the c1reumference of a circle 18 pJ. tilDes the dia-meter of the ni:rc1e~ and pi is not a fractiOllo IJ.kInr1ae the d1aw,cmalof a uni:ii Bq'Ual"0 is the square root of two~which 10 not a traoUono

10225f22l ID this eaae tbe i;.:TJl. t10nal numbora are introdUCGClas a IIlOl.'e oc:a:Plex wlue-oystam. includ.:l:lg more numbers than the countiDg numboro or eve the

fraotiQtlD.,

10225122_ II'bj,s value-systelll -oo:ae.veaas the lim1ting ease of the f:ract1cmao ODd is them.MD8 by uhich every point on an ordered continuum receiveD a number !ran the.'tal.W system"

10225123 Another complex oa,le is whore the ,ie1~, 0' :ttcntian oos1at:s not of pos1tivecan'tBt, but 01' t::_e:ll.ek of contento

1. CaY! ~ tM,ild to a debt which I owe to aJ:lOthsrmoDo ThUl debtmight ~ COJ.lI'C~ble, as a debt of 10 orOJlg'9S. or miGht ~ DQDIIoCountablo~as a debt OJ;" :ricec.

lo'225~ In the ~~&O~d3:tic1ency is countUle, I can refer to it vit.b the eo-calleduso.tive ~"""ilacti0D8 or the nesativo ir.rat1onal mHDberso

1Q225l232 In case the de:f1c:lenc;y is UlWountab1e. I can refer to it nth tho IlOPtiwbact10ns or the ll&gativo 1rrat1onal numberso

102252 Tfte open aontenco used in 1.15 v1l1 show that aU these d1ff'orent value aystemaprovide Ynlue-namos;t'or the oOllUmn. of material couiderod f'rcD the aspector qualiV and. measu:red 1n UD1tB.

1022522 'l'bG r&IJ68 of orcl:1.narycount1DB 1NDlbez'8 vUl tit tb1s open ecmteDce.

lo225~ '1'hc Z'8D88 of ~t1:ma ~ll tit 'this open aont8D.C8.

1022524 '!'be ftD88 of :1rr8.t1mal mllDboi-e v1ll tit th1a opeD aenteDce.

1022526 Tlaere 8l"O other val~. ~.:':;:'" _ D88Cl not OQlU'1cJ •• boro, 1Ib1ch'tdll tit tb1s open sentence.r/

·~'19··J..,~;~5:~ 01'10:' 'J.;"; n , tr·~,~ is ;:i f'.'.I:.t:.rr:.o{!tal r;p~1.t hare hQ+'w;ei~OOUlltinc;;...t\'1'~1)11"]'),,-"; L r,··

ciudi.(; counUlg UUWDel'U c j.Ai:egcrs, and frc.ctions) and uncountable-'t!p8 numbers(inoludinc pricar1.ly the irrational mab$l"s).

1022531 We900 t1'>..:ltthe one type is countable and i8 ba81cally related to oounto.ble :fieldsof attC!ltion.

1.22532 We see tll4t the other tJ'pe is uncountable, and i8 basically ralaW to uncountablefields or attention.

lo22~;4 Thus we 880 that the \Dloountable nuabere, as well &e the oounto.ble nUllbors, oons1.-dared as special or lim1t1ng case. or the 1aOOuntablenuabera, •• uure oontinUO\J9sets af points.

1~22541 J.. line sognont; consisting of a finite numberof diacrete points will not l:tlve aleng~;~~,and th11Swill have measure _1'0 in thia aen88, even thougl: its oC\Dltablenumbornll be .fin! to and non-eero.

1.22542 A line scgmoo.t: corrtain:1ngall points in a Deishborhood11111tmaa have a length,which oa.. be l;iven a fin! te uncountable n\IDbor, but it will not have 0. oountablemznbez ir~ the, proper and original aenee, lI1noei to measure will be fin1 te, butits nunb·:.!r01?pointe will be 1nf1ni tee

1022543 Thus. even thcU8h the counting l1Wlborecan be -.beddK 1L tM \IDOOaDtoblonumbersae linitincr I)r epoo1al CaDOS, there 1. baa1oall.7a fund-eta! IPlit between thetwo ld.ndo of numbers, j1,;,St as there 18 betwoen the two f1el4a of attention.

103 Wenowconsider the f1e1d of attent101L re8trJ.0te4 eo that our cU_·",~ is at thethird level of cunarali ty 0

10301 Wehave t1lUD far considered f1elde of attenUOIL of abeolute ~V, and thenconBidcr'fJifieldo of att811.t1011vh10b •••• IIS.tber ooaDtable or lInOOaIltGble,andhave ir: ~VCTJ cae.. given a tbo1'O~ ~. of each of the fi'Y8 oontont-centcredfeatures or desor.1.binathese fi.l.cla tit atteDti.OIL.

10302 We~t ihu3 DOli' l\onalder tho bu1c 1P1lte at tJpee of oountable fte14 end typesof uncou!-t;ablefiEld, but •• v111not ••• a oa.plet. anal..J'a. fill laneurce behavior.

1031 W'3 will Q-matderfirst countable t1el4a of attalt1on, md "O"d t Ir t:bo typeo of suchfields of attS4tion,

10311 The f1ro; typ!) of countable field of attctioo oomd..te of ~ ob~oot8, en-cOW'lto:rol 1.'1the world of space. 1Ib1ohII8\r be .mate 01' ' •••••- •••

103111 PhysicaL ob:ecte occur 88 membersof oollecUOIl., aDd th~ir _"OS" upoot is mea-sured 1" 1'tr.ta.

103112 This t:'}pe"t field ot attelition oorre~ to ext8Dd.ecl•• , •• the stuff out ofwhioh j.t iJ D3de

Wecan see tl'Iio in ID8nY eDIIIPlea. Cona:ldv trMe, 1I11oh oro r.I..'ldeout01 'WOod, as the stuff or mahrial. !he treee are OCMltuble,pb;rGioal ob-;;'ots, mn.deout of '.;heWlCountable_terial. which i. WOOd.

1031?- l'ho accori type ot countablo fi.ld af attention oonll1l1taot oornpletod C1"<lIlta,..rhicllc:: enoounte:r..d in the world of time.

_Iqc;~p-

We can speak 1D w.o ocmaeot1.aD at tho .re and ~ 1IIb1abtook ploce in tho paat. 1Ib1.ab•• =__lire olee.rl: ooun.tD.bl.a.

103121 Tbe8e ne'lte can alao 1» ooam •••••• 110be MMmI or -aaND •••• IlD4 GDOe ap1ntb81r ~ 1a pee••••. i--.

1••'122 Th1a tne of field ~ atteDUGD oar:NapDSa to aot1.n.tiJ, •• the .taft •• ofldWJb 1.t 18 1IIIIde.

'1'buaa war:La..a oat of t.t.IbUDs. 'Ib1ah U l'\aelt WI ~,

al tbouBb the 1I(.l1'II are OOUDtable.

1031.3 'l'U third tne at OOl~tab1e t1el4 at &t'tellu. oaaalata of pTOPJ81u.& or1I1ou. 1Ib1ah ant 8DCOUn1:e1"ll4 SA~ wor14 of Id.Ddo

Ve 06IL thiDk of tbG J4'OP081t1aD "a ~ 2 4tq\I&la 4-, 1Ih1oh 18& a:lDgle. 00tl,Il'.;ab!e JII'O'.PM1t1al1a ~ nala ell ••••

l.o:;JJl 1'bMe propoa1t1Cll.~. can be ocaa1clered to ~ of relaUAma, and tbe7 are1IIiBIUNZ"8Cl, 8CCoI""....ilJ(: ';0 thu1r n~peCt, b;11tOll8. aDd aro joined toaotberbJ 10Bioal prooedUl 'OAo

1.,3132 'l'h1o t;ypa of field of attention cOITOspc:o:la to qual1 ty or aapoet, ac the stuff'aut ~ which abe pl'Opoait1C1ls 6DD'pwc)M.W;, are prediatecL.

'l'"rw6:J:: 1~ tl qw.llty or Attribute, an.et so is fouruaa, aDd the propooi tia"2 plu.u 2 c4,ua1a 411 1'1 O0l21lO000. out of uttr1butell of th1IJ trPeeoo

1,32 i'e will cCll18i4CJr D6xt UIlOOuntable fields of attention, aDd. cona1dor ths types otsuch fields or attenUono

1:.321 The t1rat VPe of uncountabl ..•t'1eldak atttmt1an co~t$'t. " f!'Il~"'" "'SC,~tored :U1the 1'orld of' ~~08

1032ll E:xt<mded II1sa occurs as a oont:1.mDa of matter, aDd has ~ upt'c~. l4U.ah oa.nbe meaam-.u &COCJl"d.iJ:Ig to unitBo

.1.,3212 Th1a ~ of field. 01' Htt~t1aP QOl"l"eap<mWi to ..,a108l O~. ~ca &I'b 1n-"(Il'au.. 111\0a.u.uJU •••• aDd 0CXDp1et04 aole or \.bo ••• 4 •.••t.rial.

We thu!l eee that the 1ntecrat1J:m at a JI&8D at woo4 18 • U .••or tho 1nte{;rntiOZl of a mass of lumber is a hou8eo

1,,32121 Wo OIW above under 1,,225 tilnt unoountable number appUoa to & oetiDI1U. 01._tcJrial" maasur1n{; it in campar18011 to a ataDdard, ODd :l4at1tJ1,Da 1iJMtIIOOS\U"e of the 1nfj.nite point seto

1..3~l.22 The integrated mu~rial then becClli88 a discrete Nt aDd 18 matced GEl.& tet0( acmbera e.ccordillB to 1tomoo

1. :1)2 'llh.e aeccmd type of Ul'lcountable field of attont101l ~ at .at1T1t,y,. onccmnter.i.n tho ¥OX'ldof time (

..:.. .\c;.-t1v1~r 0CC\.U"5 ae s. (!ont1nUllll of snarQ. and has ~ &spB~O, uhich c:an be~x.a..·'NX'od /:;(:cort!1ng to tmi ta"

-t~/-•

L,222 Th1atJpe of t1e14 or .tt.UGD UClCX"bta \0 0CIIIplne4...ta. 1ddaIa •••~'1C1D8 into a siJIulteeoue end ~ whole of .t6 _.,.

We tmn nee tbat tM ~tiAa of • __ of ~ 18 •wa:r~ or the 1ntoecrat:Lc:mof a __ fit e1ecrb1oftl-.,. ia a·:' ':. IIIboak.

1.'2221 (Ace ap1D we 888 that ~ _baa ~ to •••• ". g fill ft.,.."~ it in ~ to a IItaDdar4eaDd.1.drt1f)i,11c" .11_""iJItiB1te point set!ll

10'2222 'l'hc intecrated matorial becQ'Sn • as.-••)ICd.IRen fit aNllfD. •• )U •• bared••• at of •• bfmJ accOl"4iD8 1:0 1--.;J

10,2'.5 The th1ri type at UDCOUIltablll&14 of a~ oaaat-.., .-u•. ~in the world or 1IiDd••

103231 Qualit.J occurs as a c<mtjmUl at a'b.traoticm. aDd baa .aI\1 upeota, IIa10Ia CODbe ~ e.occmii.na to \JD1b.

Far 1nstance. color CD be OGDS1der8d. aooaI"d..1Dc to tat.aal_or m..

10'232 'l'Id..DQpe of field of attoot1oD oarre8p(DSa to nowplete4 pro~u.a. 1Ib.1chare 1lrtegrat1oas into pred.1oUcma of 1:be__ of a quali1¥.

It is an 1ntegat1cm \Jt J'8llo1r to ~ -rtda n...r 1a ,.ua.-.a1nce it appllH the q\Bl1V to • 00ItplArte4 ••• 1IIdnAI4 1IbDJa.

III th1a cue alao _ ••• that lIDCOUIltaMe •••••• a~ to _ ..u-of 'tbo pl'OJ*Ct.t en-qual1t7. ~ :1\ ill a.Jen.ca to • ~ ..sstaztinc the _sure at the iDnD1te pcdat -' 1d:d.OIl Sa •• ..u.tr.

-. --- .•.TM iDtesraW quality beoaIIee a diaarete )ropoa1Uaa. aD418 _ •••••ao a oe\ or rDIIIIbera b11'--

, .