logicomix glossary

8/2/2019 Logicomix Glossary

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ThE!following notes a V" e . b'{ no I"ne.cmsnece..ssaV"'{ f oV" the en;o' {l "ne.nt of L o g ico miJ


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to the left ..:" and so on. Pvobabl,/ the eCll'"l iestsophisticated ~\Iestel'"nalgOl'"ithISthe one 91venin Euclid'5 Elemer'S fol'" compLri-ing the gr-eate.st common divis;of two non-negative number-so Algor-i thms gained pl' "ominence in the lVest i +h15th centl.lry with the intr-odl.lction of the decimal s'fstem, which, in staY"kGOn;as~Withthe Roman nl. lmer-icals),stem, was amenable to fast calwlafons sl. lch .u .0. 'bed b ' CIS meone e.scn a ove.Numer -ical a lgol'"ith~ pla,/ed a centl'"Cllpan in the scientificand technologlcalr-evolLri-lons. Toda,!, algov-ithm5 eweuSl.lall),coded in advancednotations called programming languages. The), ar-e often tl'"ansmiHed ovev-theIntev-net, and const itute the soffware that ISthe woV"khor-se plc:ltfoV"m 0 .backbone of compLri-el'"sand the Inter-net. " em

Ari~+otle BOl'"nin 38


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"ov" and "not ", ~an be tv-ac.edd ivedl '{ back to hi s i deas. yet , desp ite thegveCl t volve of h is wovk in mathemClti' :ing 10gicCiIavgumen+S, Boole did notoffe:- an' { gveCl : ins ights into the stud' { of logic i tsel f, hav ing wovked whollywithin Aristotle 5 classical model. In Boole's system, v m b o l such as X and y(essentlCllly they ave vav iables that ccm take onl '{ the two volves a and 1) oreJOinedvia the thvee connectives mentioned obovc, 05 wellos the "implies"connect ive envisaged b'{ Anstotle. ( Intevest ingl '{ , the Stoic Chvys ippus hadalV'ead'{ Identif ied these connectives in the ~vd centurv BeE) Th I' t,~ 'J e app ICOIonof algebvaic ident it ies, such as the thV'eebelow, allow a logician to simplif'{logical expvesslons and deduce useful conclusions fV'om them:

e x OV' Y) = e y ov X)not (not X) = Xnot (X and Y) = (not X) OV' (not Y)

What this 10gicCl Ifovmalism is lacking is the abi li ty to cxPV"csssemanticconnections between pmposit ions. So, fov example, theV'e is no WCl'{to deno te In the above tha t X Clnd Y ma'{ stand fov the +WopV'opos't,"PI ro i Id th I ionsa 0 IS 0 evan Socmtes" and "Socvates is oldeV' t han Ploto."Thi.s weClkne;,sI;' vemedied in the predicofe calcv.vs. Boole died in 186~.

Ca rd 'o r- , O e ol "' g BoV'nin 18~5,CantoV' studiedundeV' soroe o f the ! gvcote; ,t mo themot ici an; , o f h istime, includil'1g RichoV'd Dedekind (md KaV'1WeicV'stvClss,f.lespent the gV'eotest PClvt of hiscaveev teaching at the Univevs it '{ of f .lal le , wheve hewV'otehis seminal papevs demonstn::l ting the gveatpowel ' of the ideas of set the,, .. ,y . f .l is most fClmoustheoV'em is thot the set of 50-coiled rea! nvmbei-s(ClIIthe numbeV'son the number line, i.e. the natvralnl,lmbevs 1, 2,3. .. e+c., togethev with the decimals,including a and the nega ti ves ) is uncounrable in otheV"wovds wnl' 1ot be pu t i nto 0one-to-ol'1e 'covV'espondence with the whole numbevs 1, 2, 3,. .. e tc . On the contV'aV'y, as

CClntOV'hod olV'eady pvoved, the set of ~II vafional numbers, i.e, oll fv~ct,on;,of nat~val numbevs, such o.s 2/3 01" 11/~76, is countable and can be put insuch ,a Covvespondence, As both countoble and non-countable sets have anInflnlt-r of elevncn+s, cantoV";, vesults eS.5entiall '( pvoved thot theve av evaV'lOU.5,mutually exc lusive k inds of inf in it '{ . As his thcoV'ems were extvemel' (countev-intL.Jitive ~nd thus totall' { unexpected, the'{ cveoted mL.Jchskepticismabout set theovy Inthe mathemotlcal communi fJ '.

One of Cal'1tov-'stoochevs, the gV'eot mothematician Leopold KmneckeV', as wellas the mathematicalgiont ~enri Poineal"eweVe5tl'"Ongl'(cvitical of seh, thoughthe othev-mCithematicCllgiont of that t ime, David ~ilberl -, was one of Cantov'sgV'ootest .suppodevs, The.identificotion of two distinct 'sizes' of infil '1ity in+beserof veal numbevs, a smolleV'cmd olavgev one, usheved inthe question of whethevtheve could exist 0 thi rd k ind: GOuldheve be a sub;,et of the V'eo.lnumbevs that isneithev GOuntClble1'101"an be put in one-to-one cov-vespondence with the veals?CcmtoV'conjectuV'edthat none exists, 0. guess ever slnce called "the ContinuumH'fPothes is" - the Coniirwom being Clnothev l'1amefov the numbev line. ContoV'wov-kedtowav-ds0proof of the Continuum f.I '(pothesis fov many ,{eavs, butnever' achieved it. In19~O,Kuri COdel pv-ovedthat the Continuum ~'(pothesis isconsisrent with the pV'evoiling ClxiomClticsystem of set theoV'{ (which does nofamount +00 pvccf of it). In 1963, the young Americon mothemoticiCln PaulCohenpv-ovedthot i t is independent of it, i.e, that 1'10 V'ealpvoof of the f.lypothesis canbeestoblished fV'om it, OV', olteV'not lvely, that the axiom;, of set theovy aV'econsistent with the H'{pothes is being eithev tv-ue or Folse. This discovev'{eavned Cohel'1a Fields Medal, CI di; ,t inct ion often wiled "the NobelPv ize ofmClthematics". CClntOV'suffeved fvom severe emotionol pV'oblemsand wasV'epeatedly hospitolizeo with 0diagnosis of meloncholia, which cerl-ain histovion;,of mathematics have oscV'ibed to the hosti le veactions of some mothematicians+0set theoV'{, Clndo+hevsto the constont anxiefJ' vesl,ll ting fvom his fV\Ji+lessattempt to pv-ovethe Conti l'1uum H'{pothes is . Inthe last decades of his l ifeCal'1toV'did no mCithematical woi-k, but wV'oteextensively trying to sl,Jbstal'1tiotetwo stvonge theovies; a) that the pla '{s of Shakespeave wer e il'1fact wvittel'1 bythe Elizabethan phil050phev- Siv FVClncisBacol'1,Clndb) that ChV'ist wasthenotl,Jvo.lson of Jo;,eph of AvimGthea, The secol'1dof these is a bosic componentof mal'1'( vC1V'iotionsof the Hol'{ Gmil legend, ond a standov-d porl- of esoteric10V'e.CC1ntordied in CI mental a;,yll,lm, wheV'ehe had been intev-nedC1gC1instiswill, in 1918.

Euclid Bov-1'1vound 32":BeE, Euclid is the eavliestGy-eekmathematiciC1n whose wovk isextant in thefoV'm Inwhich heC1ctual lygove i t - theovem;, ofeavliev mothe!maticiCll'1s sUl'"Viveonl'{ CiStV'onscV'ibedb'{ othevs. f.lelived o.l'1dwoV'kedin Alexandv-iC1,wheV'ehewas as.sociated with the GV'eatLibvary. His opusmagnum, thc Elements, has been 0 best-selieI' fov

twent-r-thV'ee centuvies, and is the bookwi th the most edi tions inthe WestevnwoV'ld,ofteV"the Bible. Thol,Jghmany of the theovems oppeaving in it aV'epV'obobl" not Euclid's own discovevies, the wovk of compilotion, ciassificotion


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. f"" T . __ _ " . , ,' ,' " ' . ' W O . .. I.:>~ H , , ! : : ? , 1' . 1' ''V. , 1' " 1 , _ . , ,,,_ . " 1 , - ' ' . .. . . . . . . . 11 'f r I " , . ,- """" __Elements is a majestic conceptuClI edi fice which, inspiv-ed b'{ Ar-istotle (melhiswov-kon logic, stm-+s fv-om defil'1;+ions al'1dfir-st pv-inciples, the ClxiomsCademafa - Iitev-all 'f "yeC1uests"- in Euclid's ov-iginal Gveek) C1ndhen pr-oceedsto av-rive at Cli lthe theon:ms thmugh v-igomus pr-oof. Though ICl ter students ofl og ic , espec ia ll 'f a t the t ime of the Cjues t fo v the four1dCltions of mCithemCiticsClndaftcv, have cv-iticizcd Euclidfor v-el'fing too muc.h on geometv-ic.insight Or"tC1kingman'f mov-ethings fov-gmnted thCln his oxicms, the inf luence of theElemerri has been colossal, Clndi t is v-ightl 'f consideved to bethe fOlJntainheaelo f the mathemat iw l method. Eucl id d ied Clmund 265 BeE.

Foundations of Mathematics Since the t imeof P'{thagovas, mathemClt ic ians have wonde..-edobou+ the nCiture of mathemCit icCl I tyuth, theontolog 'f o f mathemCit ical ent it ies and the reasonsfo v the vCl li di t' ! o f pr"oof and, more generall 'f ,mClthematical knowledge, Fv-omthe Enlightenmentuntil the middle of the 19th centur,/, the prevailingscienti fi c ideol09 '{ saw mathemati cs as the onl 'fWCl'fof f'eachin9 a t f' uth that i s f ina l, C lbso lU+eandtotall y indcpcndeV1t of the human mind 's cCipaci~

to unde,stand i t. The bas ic not ions of mathemat ics wer ethought to v-ef lectessential pvopedie .s of the cosmos ond the theoV"em5 to bethe tV"u+h5of C lh ighev veoli+y . Th is Clbsolute foi th in mClthemCit ics i s vef lected in thecvowning of the di5cipl ine ClSthe "Queen of the Sciences" Cltitle whose previousholdev, signi ficClntl ,{ , WClSheology. This view is usuc&"j teV"med moihernaiica!OICi foni ::m , hClv ing i ts roots in the v iew5of PICI to- Clnd, o + l eo s + padl,/,P'{thagoV"Cls befoV"e him - on the +V"Cln5cendent IdeCls (eide). Yet, in the 19fucentu"f this t f'adit ional bel ief WClSundeV"mined in the minds of some peopleClnd eventuCllly led +0a sevious fmmdClfiol1al C,~i::'5 in mCithemCltic~, The firstof the discover i es which CClusedthis loss of faith, dat ing fV"om the t ime ofthe RenaissCince, WClShat of the imaginary rwmbers ( i.c, those involving thesCjuaV"eroot o f minus one). But in the 19 th ccntuV",/ the oppeClV"onc.eof non-Euclidean geamefne:: stV"engthened the avguments C1gClinstthe "self-eviden+"tr-uth of the cxierns. The rr-os+ tV"ouble:;ome of 0.11mClthema+iwl cOl'1cep+~,though, wa5 that of iV1finl f' j. PI"oblem5 concel"V1ingthe mothematiwl handl ingof the In fin it e had fir -st been alluded to by Zeno, 1V 1 his pCli"C1doxes,'e5uvfClcedwith the inventioV1of the calculv5 In the 18,hcentuv"y and the countcvintvit iveand ill-defined concep t of Cln inhnrtesimal, Clnd peaked in the ICist +wo decode~of the 19-" cen+lJvy, most e~peciall 'f with :;et tl-leoV"'(Clnd Oeol'9 Cantor':;

. - _ chief among them Russell ': ; Par-ac lox - culm lnClTe" In",,.,,,,~~--__+heol"~, e lf -eviden+" h-uths and thus, ind ivec tl '( , about the value of a llo'oout sCltiwl knowledge. It WClSprinc.ipClII' / the wish to ove..-co,:nethe5~ doubtsn1o+::mll d the C1uestfov :;ecure foundCltions. The "pvogl"om pr"oclalmedHV;1 u~ci~il 'oed in the eal"l '{ 1920's 'oeClV"inghis nClme,expv"e5Se5 .b'{ 00" + t'mi5+ic veV"siono f the foundationClI dn~Clm; the CYeClt'On .the m05 o~ I + foV"a ll mathemo. ti c! '> , olso contClining a proof +hCl+thiSof 0 foV"mCi : ; ' 1 5 em . f ~f ( '1 ~ cCin lead +0 no contV"adict"lons), camp/de. +Izotlon I:; con::/5 e" .~. . ~ .OXlomo . '01 t ths) and decidable (one isob le to dec lQe In ever' f( Ieave~ no ~ntPhvovCl:r~ lJICl fol lows fvom the Clxioms or not , thvough theoccaSion wne er Clappliwtion of a set of Cligorithm:;.)

Fl'"ege CoHlob BoV"nin 1848, Fvege spent thet 't ~..1. of hi5 ma+hemCltiwl life ClSC1 pV"ofe50vgreo es p~n . dat the UniveV"si~ of JcnCl.~e 'ISgeneV"ClII' fconsiderc

+0be the fCl theV"of modeV"n logic, who:;e notot lon andmethod he expounded fiV"~t in hi~ Begnfh:chnff ~hlChlitev"ClII'{tron:;lotes f,,-om the CeV"man ClS concep5cV"ipt"), publ15hed in 1879. Init, F,,-egedepClrl-~d fV "omthe covlicv 10giciClnswovking Inthe wakeof Anstotle,by cxp li cit ly ln t, ,-oduc in9 the not ion of variaole Inlogiwl stotements. In the pbce of ~~e oldeV" t'{ pe ofstotements l ike "50cl '"otes is a mCin, he Introduced

t Ike "x ISClmon" pvoposi tions t 'nCl t wnpVOpOSI Ions I ,. t th lue iven +0 x - thi~ podiculoV" one,be true OV"alse accor"dlng 0 c v~ g.. t f I if it's "Mongo.". Fl"egcFo v iOXo.m Ie is kue if x i5 equell to Alecos bu Clse . . ha lso inve: ted the not ion ~ ,f fqIiCl i1fi6er~: :~~ ~~:e:~:lf~~;i:lt~:V"~~e~h~~mClkesCl5+otement tvue ov eveY"~x '. k + + mcnt +rve.t "th . t on x which ma e.s0s Cl ewhich SCl ,/S+hCl . eve cxis Sit t the ,-,uest f o Y " the fovnda+ion:;I t I d h s new l oglco. 5,/S em 0 VI~e 0 eV"opple I .. + 1 f ' < C"heBa


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a sevies of Y"abid+reo+ises attacking paY"liamentaY"'{ democY"ac,{, bbouY"unions, foY"eigneY"sand, especiall,{, the Jews, evensuggesting "final soluti to the "Jewish pY"oblem". ~e died in 1925. ens

Codel , Kurt ~e was boY"nin1906 in the townof BY"unn," ' \ oY"avi a, then a part of the AustY"o-~ungaY"ianEmpiY"e(the city now called BY"noin the Czech Republic). Codel studied mathe~aticsin Vienna, wheY"ehe became fascinated withmathematical logic and the Cjuestionof thefoundations of mathematics. Inhis dodoY"aldisser+o+ion, he advanced ~ilbert's Programb'{ pY"Ovinghis compleieness Theorema result establishing that all valid st~temen+Sin Frege's [irst-order logic can be proved froma set of simple axioms. In1931,however; he provedthe Incompleteness Theorem for second-order

logic, i.e. foY"a logicpoweY",fulenough to support aY"ithmetic and eCjuall'{ormove complex mathematical theories. Codel became one of the '{oungestmembev-sof the Vienna Circle, though his deepl'{-ingmined, idealist belief inthe independent, Platonic existence of mathematical Y"ealit'{ eventuall'{alienated him f Y " o m the otheY"membeY"5,who embraced a mateY"iali5t-empiriccl wov-Idview.DUY"inghe late thidies, Codel was hospitalized twicefov- severe melancholia. In 19~0, afteY"the Anschluss, i.e.the annexationof Austv-ia to Nazi Cev-man'{, he managed to escape the countY"'{with hisWifeand took the tmns-Sibev-ian roo+e to the United states. l-lebecameone of the fiY"st member's of the Institute foY" Advanced Stud'{at Pnnceton, wher'e he spent the Y"estof his life, ~is most importantmathematical Y"esultFrom this pev-iodis the pv-oof that Cantor's Continuum~'{pothesis is consisrenr with the axioms of set theor'{ ( i.e. that it would notbein contmdiction with them, if tv-ue). At Princeton, Codel developed a closefV"iendship with Albed Einstein and worked fo v a while on the theoY"'{of velativit'{, establishing the mathematical p05sibilit '{ of a non-expanding,v-otating vnivev.sc,in which time tv-avelcan bea ph{5iwl veality. In latev life,Codel became incveasingl'{ pamnoid. ~e died in Januav'{ 1978, at thePvinceton hospital, wheve he hCldbeen admitted fov- the tv-eatmentof a non-life-thY"eatening uvinav-'{ tY"ad pvcblcm. The cause of hi5 deathWclSmalnutv-ition: he vefused to eat fo v feClV-hat the hospital5taffWClSttempting to poison him.

l4i1bed, D~vid ~ilbed was boY"nin 186ein Konigsbevg, PY"ussio(now KoliningY"od, Russio)ond spent the gveatest port of his life Clt theUnivevsit'{ of Cottingen, the wovld's mostvenowned mathemotical cen+re ot that time.~e is one of the gveotest mathemoticions inhistoY"'{ ond, with ~enri Poincare, the gY"eatestof his evCl.~e mode impodont contY"ibutionsto mCln,{bY"onchesof mathematics including

. 'nt theov'{ algebvoic numbev theoY"'{,functional anol,{sis, the calculusInvanCl , ' 'f nviations the theo"" of diffevential eCjuotions ond move, 01050ploneenngo v~ , ' J f 'e+hods of "roof. In 1899he published Crundlagen der aeome rienewm ,.. f' b' 'th(Povndafions of ceomeirv), a book which gove geo~etY"'{ 0 IV'm 005105,I

newaxioms, theY"ein impvoving on the work of Euclid. InhiS famous 1900 talko+the InteY"nationClI Cong!'"ess of Mathemoticians, in Pavis, he attemptedto give0bivd's-e'{e viewof the mothematics of the twentieth cenrvrv.b'{ wa'{ of twenty-thvee gV"eCltopen Cjuestions. Of these, now V"enowned"L


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Incomple+ene:;:; Theor-em In 1931, the 2.5 ' {em"-old Kur+ Oo~el provedtwo theorems that Cl resometimes rcfer red to as " the" Incomplete l" lessTheorem - though occas io l" lCl II ,{ thi s form isused to del "lo te thC! fi rst o fthC!se. The c~mplefene.5.5 of a logical s'{stem is the pmpe.t-y that evewell-folr~Tled ( i.e ..grammatica ll ,/ con-ed by the rvles of the s ,/stem) I'"'{proposition 1 1 " \ It con be proved or disv>roved from the svs+em' ,C"d I' . ,.. ,s aXiomso e 5.earlie.r Compldenes< Theorem show~ thn+ th . . I .- ~ ~ ere ISa simp esoch aXIOmGltlCs,/stem for fir.5f-orde/- fogic. ~owevey the h I '1Hnb + ' ' o'{ g..-alo er 5 Program WGISo create a complete cmd ccnsisren axiomatics ,/stem that can support arithmetic, i .e , the mathemati cs of whole numberSuch a s,/stcm would rcC]uire .5econd-order l ogi c , i.e, a s'{stem thGlt is also s,ab le to accep t seh as values of variables. Codel shocked the mClthemat Iwo~ld b '{ proving, in h iS fClmous paper "On UndecidClb le Propos it ions in~~Pr,ncp.a ,ltafhemaf:t!a and Related S, /s tems " , that an, / consistent axiomaticsystem f~r ari thmetic, 1 1 ' 1 the form developed in the P ri.ncipia, mustof neceS5lt, / be irccmpteie. More pl'"ec isely, the f i . . -s to f the +WoIncompleteness heorems C!5tabl ishes that in a logica l axiomatic s ,/stemnch enough to descnbe propert ies of the whole numbers and ord inal '" '{onthmetlc opeV"atIOI'15,there wil l alwGlYsbe proposi tions that Glregmmmdiwll,/ con-ed b,/ the rules of the s,/stem, ond mOl'"eover rrue,but cannot beproven within the s ,/stem. The second IncompletenessTheorem states t hat if such a s,/stem were to prove i ts own cOl"Isistenc,/I t would be Inconsl5tent. ThiS WGIS new,devastat ing blow to Hilbert 'sProg..-am, with i ts goal thClt CIstrong axiomatic s,/stem should be equippedWith CIproof of I ts own consi .stency.

Infui+ioni.sm This isthe phi losoph,!of mathematics created by the greClt DutchmathemClt ic iCln Lui tzen Egbedus Jan Brouwe..-(1881-1966), though some conSider' ~ent' i POinecwewith his strong belief in the role of Intuition in 'mClthemCltics, CIcleGlr precursor. Intvitioni.sm isbased on the belief thot intuition and time ClrefundamentGl I to mathemati cs , which cannot bemade a-tempol'al 0....0n11C1/ in t he sense of~ Ibed. Contml '" '{ to what 10gici.5f.5 like F....geGlndRussell thought, Bl'"ouwer was convincedthClt logic is fovnded upon mathemClt ics ... .ther

. than the othel'" WCl,/ V"ound.Also, he wos totC\ll,/Glga lnst the theorems of Oeorg Ccmto. .. .i n the theory of 5e+s, considering

them i ll -formed. Time-hCll lowed logical . lows, s~ch CISthClt of the excluded((lIddle, Clndmathemati ca l techni' 1ves Inuse s ince the t ime of the anc ientcreeks, 5uch CISthe reductio ad absvrdorn, were put on t l' "l al and thew

O,~demned 1 1 1 fact Br'ouweV"bel ieved that Cli l the theoV"ems makingu5e c ,. . . I of these in the ir I'....ofs - wnel' "e i l1 fin ite sets of mathemClt lca l obJechuse th twere conceY'ned - shovld be excised from the bod,! of ma e.mCl ICS,ewthot made the bl' "i ll iant Br' it ish logician and matl1emCltlclana VI " Itn h hFrank RClmse'l ca ll in tu it ionism "mathemClt icC lI Boishev i.sm . A oug IS

logic al1d mCl+hemGltics were formalized b,/ his studel1t Arend I-Ieyh"g,Brouwer I '"emained .skept iw l towCll '"ds an, ! such attempt to the endof h is l ife.

Lelbniz , OoHfl"ied This g..-eCltGermanphi l050pher, mClthematician, sciEmtist andstvdent of logic was born il11b~6. \-Iesel'"vedin the couds of severo l Cerman rvler sCISdiplomGlt, politicol odvisor Cll '\dhistorian,GIl lthe whi le pursuing his theoret ical studies.Heinvented the inflnifesimal cakulos concuV"rentl,!wit I- ., but independentl, ! f rom, IsOoGlcNewton,

Gli soproposing the notClt ion for i ts opemt ions that i ss+i !l i n use toda, !,\-Iewas a s trong pr'oponent of philosophicai optimism, With hiS theol'"'{+hd OUI'"world isthe "best of 011possible worlds", hov'lng been creCltedb,/ GICod who is both loving ond Gllmight' /' ~e is c ;ons idered the mostimpodClnt logician Clf ter Ar;stot le Glndbefore Eloole,haVing envlslo~edtne calculus raf/odna/or. This was GIkind of compu+atlonClI PI'"OposltlonClIlogic thClt wovld el'1ablecompletel,! V"igorous GlndrCltional decision-makil"lgwhich could elimil1C1te011disagl'"cemcnt Clmol'1gV"GltionClICISLelbr'llZ thoughtthem) humal '\ beings. SCldl ,/ , Leibniz did not mClI1C1geo V"eCl iizehis mostcoveted of I-. ismCln,{projects. Hedied in 1716.

Logic The term cover's CIbl' "oGldspectrum of discipl ines - not ur' lexpededl' !,as i t del- ives f rom Oneof the semClnticol l, / r iches t Creek words , l ogos,d -h fh ' !}hr ' ~eason raf io r -af ionali fy ,some of whose meol1ings a..-ewor ,speec, O Li , ' , 'Clnd/01'" concept - but can perhaps be best described 05 the stud,!of methodical thinking, deduction Glnddemonst. .-Cl tion. The books of~i"'sto+le's Org(ll10n present Clnextensive stud,! of the dedudlve pClttel '"nscalled syllogi.5ms, which for over two mi llenn ia we. .-econsidered pradlca ll, / .s,!non,/mous with logical thi r' lk il '1g. Ul'1t ll the mddle of the 19th centuY"/, logiC

f h h B t th th Cldvent of Boo Ie Cll'1dhiSwas cOl'\sideV"edCIbvanch 0 p 110501''/. u WI e


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aI9~bv.~ of pvoposi ti ons and, move impodantly , FV"egeand his "conceptSCV"lpt which led to a t:wedicatp. calculus, i t incV"easingl' { came withinthc Pf'OVW1Cef mathematics. The new logic vevealed both the undevlyinmathemati ca l natuvc of the sLJbjed and i ts potentia l v-ole in the cv-eat io~of solid foundations of mathemati cs . The bas ic c la im of the schoolIn the philosophy of mathematics known as logic ism - the school fOLJndedb,! Fvege, of which SertV"and Russell was one of the pf' imaV'f exponents_was that all of mathematiCs cem be veduced to logic ov; in othef ' wov-dsthat mothemahcs isessentioll ,! 0 branch of logic. Aftev- the ,!eav-s 'of the foundational quest, howevev, ond especiall ,! Clf tev- Oodel 's f 'esults,logiC became a wel l-developed, divev-sif ied f ield in the intev-face betweenphilosophy and mathematics. In the second half of the 201" centLJV'fI t also found unexpected appl. iwtions in computev- science, wheve i t pv-ovide.sso lid foundat ions fov the deSign and vev if iwt ion of softwave and hClv-dwoveas wel l as fov- databases and adificial intelligence. '

Ore.sfe!a Wv-it ten b,! Aeschylus and fivstpev-fovmed in the theatve of D ionysus, in A thens,two yeavs befove the poet's death, in ,"58 SeE,i t i s the onl ,! extant tvi logy of Gv-eekdv-amas- although the satif'ical pia,! Proteus, intendedto be pevfov-med Clftev the tv-ilogy, is missing.In the tv-ilogy's fivst pial', the Agamemnon,the epon, !mous hem and leadev- of the Cv-eekfovces in Tvo,! v-etuvns a victov to his hometown

of Avgos, w ith the capti ve pvophetes5, Cassandm. Though his w ife,CI' !temnestva, at f lvst ClppeaV'5to vejoice Clt his vetuvn, she hClsothev plan5.She and hev lover; Agamemnon's cousin AegisthLJs, muV'dev AgClmemnonClndbecome the new soveV'eigns of Av-gos. In the Libation Bearers, the secondplay, the chorvs of women Clccompanies Agamemnon's daughtev- Eledv-ato heV 'fathev' s tomb. The fov lov-n E lectva ishoping f oV ' V'evenge, whichshe CClnonly cavV'f out with the help of hev-bvothev; Or es+es who is in exi le.When Orestes c landes tine l'{ vetuvns to Avgos, he ClndE lectva p lan andexecu.tethe muv-dev-of Aegisthus Clndthen, in a highl ,! dvamatic sceneIn which CI'{femnestv-a baV'es hev-bV'easts befov-e his nClkedswoV'd Ov-estesa lso ,~i ll s hev , h is own mothev . heth iv -d p lay, the Ecrnenidas, ov "beneficentones, IS one of the most unusual in the h is tov, ! o f dV'ama: a ll it s speak ingPClv-tS,apav-t fV 'om thClt of OV'estes him~elf, aV 'e bken up b'{ gods ov - o+hersLJpevnatuV'al er+i+ies, The choVLJ.5consists of the E rin ve s o v- Fuvies, avchaicgoddesses of v-evenge, who chase Ovestes fvom the temple at Delph i, wheve

he has been V' ituCl II '{pur if ied b,! the god Apollo, to A thens.Ina to+al l, ! unprecedented move - fov a god Cln'fWa,!- AthenCl, the patY'Onod of Athens, decides to let the citizens of Athens judge Ovestes' cose,~V1USiving a mythological ov-lgin-.5toV'f fov- the democV'atic innovCl+ionof a couv- t o f law, w ith c it izen juV' f. The tvial and i ts aftevma+h developa~ sV10wnn ouv book's f inale, though OLJVext is onl '{ an appv-oximatetvan5lation, slightl ,! Cldap+ed, of Aesch,! lus' ov-iginal woV'ds.

PeC lno , C i v s. eppe Bom in 1858, this gv-eatI taliCln mathematiciCln and 10giciCln spent thegveatest pav-t of his creo+ive life ClSClpvofessovClt the UnivcV's it '! o f Tor in , Though his ideas wcvcnot as influential as Frege's in the seavch foV' t hefOIlYldCltions of mothemCltics, Peono, like Fvegc,cv-eClted a notation foV' firsr-order logiC andCls,!stem of oxiom.; f oV ' Clv-ithmetic, +hat i~stil li n use - in fact , ow Clv -i thmeti c is fov-mall '{ cal ledPeano arirhmetic. ~e inf luenced Bedl"onci Russell

gvcatl,!, especiali '{ with his 10giwlnotCltion, whichwas much move us~V'-fviendl'lthan Fv-ege's. Peano bel ieved thClt all mathematiCs could be fov-mait; :edClndexpvessed in a common, minimClI language that oviginates fvom justa few Clx ioms. But when he t f' ied to pvesent h is own version of thi s univevsalmCl themClt ics in tex tbook fovm and usc i t fov- teaching, his student;;V'evolted, eventuall,! wusing the book's withdV-Clwal. InspiV'ed b'{ his attemptsTO unif ,! a ll mClthematics b 'l use of a common 10giwllClnguage, Peano ICltevcV'cated an intev-national auxiliaV''{ nCltLJvallanguage, f ov - v se among peopleof d if fevent li nguist ic backgV'Ounds, based on a s impli fi ed fov-m of Lat inwhich he called l.aiino sine Flexione. ~owevev-,like 0,0 mCln,! othev adificialntevnationClI languClges, such ClSEsperanio, Volapuk,Ido - Cli l of them theoffspV' ing of C lnovevopt imist ic C lge- Peano's bra inch ild pv-oved to beClmev-epipe dveam. PeClnodied in193~.

PO inCC l I" 6 , l 4 en l "i BoV'nin 185'1 irt Nanc,!,Fvance. Although he studied en9ine~V' ingo+the Ecole Pol 'l techniC1ue artd the Ecole deI\\ine:> PoinCClveWClSo become, with DClvid~ 'Ibe;t , the gvea+est mathemClt ician of h ist ime. ~e has been cal led the " last univevsa lmathemati cian", i .e . the IClst one to havepV'ofound knowledge of all the mathematiCS


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of his time. ~e made important conh"ibvt ions to man)' divev'se f ieldsof mathematiC5, among them ~iffeY'ential eGjuation5, automoY'phic funct ionsthe theoror of s(!veml complex variables, pmbabili~ and statistics. '\~i th his Analy.5ls s/fIJ.5he essentiall)' cr eated the majo r 20th field of algebraictopology, and hiSwork on the 3-bod)' problem laid the g . ..oundwork for whatISnow called chaos theorv. Despite his man'{ gY'eat innovCltion5, Poincan~WCl5an cx+r'ernelv p.-o.ctical man, involved to the end of his life - alongsideof hiS mClthemCltlcal ...eseo...ch - wit h the most down-to-eClrth of affairsCl5fo ...example the inspect ion of mines Clndan enginee ...ng p...olcc+ to mClkethe Elffel towe...function a5 a huge antenna bV'oCldca5ting time signalsto ncwigatoV's. ~c was pmbabl) ' the last of the gV'eat mathematiciansto adheV'c to emoldeV'conception of mathematics, which championedCI V'omantic fai th in intui tion ever: V'igouV'ClndfoV'malism. This stancewas made famous b '{ h is V' caC+ionto the set theoV ') ' o f Ceorg CantoI'"a5 CI "diseuse, from which mathematic5 wil l eventuClI I) ' be cun::d." ~ is viewson mathemC1ticClI cY'eation, encapsulClted in hi,;,sCl)' ing thClt "logic is bClv'V'enunles.s feY"til ized b'{ intuition," c1I"'eseen by many as the p...ecvV'soY'of Lui+zenBmuweV"s school of intui tionism, CItheo ...y CItthe Clntipodes of ~ ilbed'sstV'id formalism. PoincaV'e died in 191~.Pl"'eclicate C",ICIJIiJ5 Often used s) ,nonymously w ith predicate log(cClnd fil...t-order loq, the pV'ediwte calcvlus is FV'ege'sextension of thepmposit ionClI logic developed by Boole. In the pn?dicClte calculus. elementarypl"opositions (0V' predicates Cln~composi te objects of the foV'm pea, b, c,oo.),wheV'eP is CIsvmbo; i n the bnguage, C lnda, b,c , e tc. a l"e ccnsianis0... variables. Fov example, i f . .olde v-" is a pV'opositionClI symbol, "Plato"is a constant and Y' is a vaV'iClble,then "oldeV'(Plato, x)" is a well-foV'medpmposit ion, descV'ibin9 that PICltois oldeV' thCln x. PV'oposit ions of thistype cClnthen be combined b '{ Boole 's connect ives "and" , "0V''', "not" and"implies" Clnd pV'efixed by F'r-ege's quantifieV's, such C1s"For all x" (wv-i+ten ' r : / )ClndutheV'eexists ' /" (wV'iHen 3). Thus, "there exists x cldcr C x , Plato)"meClns thot theV'e is (at least ) one ind iv idua l who iso lde . .. than Plato.Ev identl y, thi s i s 0 much moV'c ambit ious oHempt at cV'eClt ingLeibniz 'scakolo raiiocinaior than Boole's simpleV' foV'mClI logic. B,/ employingsymbols fV'om var-ious fields of mathemCltics (.sl,Jchas "e ll' s ParC ldox . The t it le PrincipiaII:;dhemafica ( i.e, "Principles of MathemClt ics") in i tsel f pV'ovokedcontrovcl"s, {, as i t is the exClct same CIS thClt of Newton's .gV'eatest wOl"k;man'{ in the BY' it ish mathemCl ti ca l communi ty thought thiS chOiceto be in bCld t aste, if not Clctvolly blasphcmous. The thV'ee volumesof the PrinCipia, published in 1910, 1912and 1915, wer'e based on a developedver sicn of Russell 's fheory o f ivpes, the so-called "Y'amif iedoo , whichmposcd CIhieY"aV'chlwl stV"uctuV'e on the objects of . $ e t theo..-,{. This ~~uldnot be mClde to ,/ield the l"eGjviY'ed l"I?sulh, howeveV',wlthovt the Clddl tlOnof what Russel l cCl ll ed an ax 'om of / " 'edvcibil i fy , which eventuall'{ becClmeone of the mClin V 'easons foV' negat ivc cV' it ici sm of the whole wor'k .Logicians found this axiom extremely counteV'- intvit ive, a fClI"-fetchedand bas ical ly adi fi cial mC+hod to .sweep the vel" '/ pV'oblem i t was tV' yin9_to solve undeV' the V'ug. Despite t he fact thClt the Principia fell s~od of IhauthoV's' immense ambition it hod a hvge inf luence on the shapingof modcV'n logic , i ts gV'eate~t effec t poss ib ly being the in: ;p iV 'a tionand context i t pY'ovided Kud Codel Fov his gV'oundb ...eClklng dlsCOVE!V'Y,the Incomple+ene$$ Theorem.


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Proof The pv-oce:;:; of clV"v-ivingo+ the 10gicClIver ifico+ion of 0 mothemoticol ov-logicol :;totement:;tav-ting fv-om 0 set of ogv-eed-upon fiv-:;t pv-incipl~(the:;e could be ei+he--Clxioms ov-olv-eod'{ pv-oven:;totemenh, dev-iving fv-om the:;e oxioms) ondpv-oceeding b'{ totoll'{ unombiguous ond u'nobV-id dlogical steps ov rules of inference The demo~~t gte. . ,,_ v-o Ion:;of geometnc pv-opositions in Euclid's ElemenT-.5weconsidev-ed fov- ovev-two millennio to set the :;tan;:V-dof excellence to which mothemoticol pv-oof :;houldo:;piv-e. Yet, towov-d:; the end of the 19th centuv-his method come undev-Iogico I ond philo:;ophic~scv-utln,{ ond wo:; found to lack, pv-incipoll'{, in twodwedlon:;: 0) in i+ssense of the logicol "obviou:;ne:;s"of the oxioms, ond b) in its logicol gop:;, whev-eintuition.- which, in Euclid's case wa:; mostl'{ visvol-geometnc - took ovev-fv-om :;tv-id opplicotion

of 0 fov-mol :;' {stem of roles. In0 sense, Fv-ege'sond Rus:;ell ond Wht h d'I .. f . tie eCl s.oglc/". ~v-o)ec wos developed 0:; 0 v-eodion to the impev-fedions foundInEuclid s proofs, as well os 011those developed in his woke. The logicistsos wel los the formalisfs wov-kingon the founaClt;ons of mathematic:; ,olmed ot a full'{ developed theov-'{ ond pv-odice of v-igov-ouspv-oof, ,b'{ which onthmetlc (os the bo.sisof 011mothematics) would beginfv-om a small numbev- of consistent axiom:;, cmd eventuall '{lead, via pv-oof,to the ful l mnge of tv-uth. ~Ibed's seminol quest ion, which he calledthe [nf"chelaung"problem ("decision pv-oblem"l, posed in 1928ond answev-edseven '{eav-s lotcv- b'{ AICln Tuv-;ng, is equivolent to the demond fov-o totoll'{ powev-ful oppav-atus of pv-oof, which can pv-ovide a provableov-unprovable response to on'{ mothcmatical stotement b'{ vidueof 0 V"lgov-ousCllgov-ithm.

Ru.s .se l l, Berf rClnd Bov-nin \voles, in 1872, Bedv-ondAdhuv- \Vill iam, the Thiv-d Eav-IRussell - this ishis ful l nome, b'{ vir+oe of his noble descent - wosthe gv-ondson of the impodcmt politicion Lov-dJohnRussell, whose title he eventuall'{ inhev-ited. An ov-phano+the oge of fouv-,Russell wos mised b'{ his potev-nalgmndpov-ents, ond oftev- his gv-cmdfothev-'s deathtwo '{cav-s lotev-, exclvsivelv b'{ his gv-ondmothev-,Lod'{ Russell. He gv-ewup at +bc fomil'{ horne of

pembv-okeLodge, in Richmond Pav-k,to the wesr OT Lonoon. r


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but the set of all "umbev-s is "at a numbeV" and thus not contoinedin i tsel f. ~y vir+ve of thi5 pV"opev"1"' (,we c on define the "set of 01 1 5etswhlchdonf conto in themselves' : and ask , wi th the young Russell , thegu." !s tl on: Does this set conta in i tsel f 01" not?" See what happehs:If It d~es contain itself, it follows that it is one of the sets which don'tcontain themselves (as this is the pv-opedy that chav-adeV"iz.e5 elementsof thl5 set) and thus conno+ contain ihelf. But if it doesnr contain itselfthen It does not have the pv-opedy of not containing itself, and thus 'does conta in i tself . Th is si tuat ion, in which aS5uming someth ing impliesIts negafion, and vice vC!v-sa,is cal led a paradox, When a pcwadoxsuch as Russel l's , av- ises in a theoV"y, i t i s a sign that one of its b~sicpY"emises, definitions ov - Clxioms is fault'{. Though histov-icoll)' developedwithin the context o f the theov-) ' o f sets, Russel l h imsel f l ateV"v iewedhis paV-CldoxCl5eS5entiall'{ having to do with :;elf-Y"efev-ence, i.e, withstatements Y'efevY"ing to themselves, such as Euboulides' "I am nowlying to ) ,ou."

Self-reference LiteY"all,{, the qualit)'of a statement of v-efeV-Y'ingto itself.Howevev; i t is also used rncr'e geneY"ClI I,{in log 'c to char-ader-i ze s ta tements whichinell/de themselves within theiY"scopeof Y"efen::nce, CIS in the "barbeY'' ' stoY"'{ usedto explain Russell 's PClr-Cldox.The bClY"beY'l ives in a town whel'ein a law decY"eesthat

"all residents of the town mU5t ci t heY"shave themselves oY " be shavedby the bav-bcY"."This low i5 self-referenfial CISthe barber- , apad f l'"om being" the bar-ber -" Y 'e fe l'" r-edto, i s a lso one of the "V"e5 idenb of the town".Self-Y"efev-ence has pla)'ed a seminal Y'ole in logic and mathematics,alY"eady fv-om the time of the Gv-eeks. FY"omEuboulides' self-Y"efeY"ential5 ta tements, to Cantov- , whose pV"oof of the ron-derwmerobilirv of the realnumbers r elics heavi l) ' on a numeY"ical vaY"iant of scl f-refeY"ence to RussellClndhis pClradox, and to Oodel . In fact, Godel pV"oved his l r1com~letenessTheorem by creating, in the context of modeY"n logic, 0 statement thoti s C ju ites imi lav- in spi ri t to that o f Euboulides, wi th one c l' "ucia l d if fcv -ence:whi le Euboulides states this statement is faI5e': Godel 's ingenious val '" iantessentiol l, / 50'15, in the longuoge of ari thmeti c, " th is s tC ltementis unprovabl e." An'{ cons is tent axiomatic them' ), i n which one can fov-mulc1tesuch 0statement must be necessClY"i l, !lncomplete: fol '" elthfY this5tatement i5 fal se, in which case it is both false an d pv-ovable,

contY"adiding the consi s iencv of the axiomat ic s ),5tem, o Y " f r u e , in wniChCClseit is both +rve an d unpV"ovable, establishing its incompleteness.Se t t l- 1 eo r 'l The stud,! of collections of objects united by 0commonpvopedy - in some cases this proped)' can be nothing mor e t han thefad thClt the,! a Y " e defined to be member5 of the same set, as f o Y "example in the oV'bi tV 'aY' ily def ined se+ whose elements a Y " e the numbeY"sc - 8 1311579. Sets weY"eiY"st 5tudied bv the Czech mathematician BeY"nav-dr;..,;), I , 1Bolzano (1781-18~8), who also introduced the teV'm Menge Cset") ond defined thenotion of a 5et's cardinaliiv, i .e ,of i ts "size" in0.wo.ynot div-ectl'! involvingmeasuV"ement. Thus, one wn speak of two sets hoving the same CClY"dinali-k{ftheiv-elemenh can beput in a one-to-one coY"v-espondence- without ever-needingto knowviathe pV"ecisenvmbeY"of these elements. This ha5 the gr-eat oclvanto.gethat it abo woV'k.sov - infinite sets, wheY"ehe notion of numbeY"does not appl,/:mathematician5 don't think of "infini-k{" as a numbeV".~oweveV',some seemingparadoxes, such as the fact that the whole cmd the e v e n numbeY"swn be putinto Clone-to-one coY"Y"e5pondence(jvst b,! multipl,/ing each whole numbeY"b)'0, oY "dividing each even number b,/ 2), thu5 showing a subset + 0 have the.same ccwdinali t' l as the containing set, pV"evented Bolzano fV 'om developingthe theol '")' fudher-. The advanced mathematical discipl ine of 5et theor'l WCSo.rguabl), bern on DecembeV"7, 1873, when Ceorg Gcm+or WY"O+Cl+OiS teocheY",~chard Dedekind descY"ibing his I 'r 'oof of the non-denum erabl i dy of the rea!numbev-s (the set of the whole numbeV-5,decimals, zer'o and the negativenumbClrs), as opp05ed to the denl .Jm ef'obi l i fy of the rationals (ClII fnlctions),which CantoV' al50 pl' "oved - denvmerabil it '{ is def ined as a one-to-onecoY"v-espondencewith the natura! numbeY"5 1,2, 3.. .etc. The concept of o.set0, alm05t too primi tive to meri+ a mathematical definition, Clnd15pY"actlcall,!mpossible to def ine informal I ,! w ithout the use of some s, (non 'lm (he.. -ewe .,-,sedthe woY"d"collection"). It is pY"cci5elythi5 "natuY"Cllness" of the concept InBolzano's and Cantov-'s woV"khot led to Ru.;,sell 's ParCldox. Toovercome it,ond to Y"u leov+ the f lClwed concept of . .the set of 01 1 se+s" i t allowed fo..-, onehos to come up wi th boHom-up con5tv -uct ion5 and axioms foY" sets, 05 In t heI'rnc 'p'a ltarltemar:ca and, lateY", the s'(stem cal led "ZFC", fv-om the namesof its two cY"eatoY"s EY"nstZeY"melo ond Ab"-ClhClmFY"oenkel,ond the Axiom ofChoice a necessav- ,{ add it ional axiom that a ll ows the theo"\ { to deal w ithinf init ; sets. set theoY"' { is considev-ed b'( some the most bClsicbY"onch ofmathemotics a5 01 1 otheY"s can bedefined in tcY"ms of I t. Th iS was the gls+of an oveY'-a~bitioi.Js pl'"ojed vndedoken, fv-om the 1930s onWO..-d5,by thegV"ovpof bl' "i ll io.nt Fv-ench mathematicians wri ting undeV"thCl pen name of"Nicolas Bouv-boki ".


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Tracfafvs Logico-Phi/o, : ;ophicv, : ; I.udwig Wi++gens+ein wmtehi" "emlnal phil,050phical work Ju..-ing 11 '11 '1 , building on his p..-e-wav-~otebook.5 and ideas on logic, It con+oiris hi" "olution of (in his own wov-dsall the pv-oblems of philo"oph(", dealing with the wov-Id, v-epv-esentat')cmd la 0 ,. II .. Ion" ,nguage. V'lglna '( called Logische-Phi losophische Abhandiung ,

( Loglcal-Phil050ph,col Tr eo+ise"), it wa" v-enamed fov- its Englishpubkat,on undev- the influence of C. E. ''''OOv-e,with his p..-edilection fov-Latin t l+l:S. In the Tractaiv, Wi++genstein uses man'( techniqves and ideasfv-om logiC, especlall'( those of Fv-egeClnd Russell, as well as insights fl"omtotall'( dlffev-ent philosophical positions, mostl'( that of Av-thuv-S~hopenhauev-. Though publication b'( the then totall'( unknownv ~I++gensteln was 01'11'(made possible when Russell accepted + 0 wri+e anIntV-O~,uctIOn.;.allin,g the book" a)1 impodant event in the philosophicalwov-Id,the Traciaiv: was the cause of the two men's fClll ing-out.Wi++genstein conslden:!d Russell's - not Clltogethev- appv-eciative -i)'1tv-oductio)'1o his work to be fV"ought with misunde..-stClndings andp~'losophICCl~eVTOV-S,while Russell saw in the Tractatus the fi..-st signs ofv ~I++gensteln 5decline - as he SClWt - into m,(sticism, The tight s+....ctuv-eof the book _proceeds with seven main pv-opositions, etlch developed in 0chapte v, which av-e furthe ....developed in pmpositions CI. . . . . . . .anged b'( 0.v-athe....pedcmt,c, and often somewhat confusing, s'(stem of numbev-ing.The f,..-st two p..-opositions (1 cmd 2) expClnd the positions thClt "the wov-Id15 all thClt 15 the cose",and +hc+ "whot isthe case" av-e Fact, ondcombination ~f fClds, This is Cldepadu ....f ....m clClssiwl philosoph( and.the metClph,(slcs of A~'stotle in pmticulav; tlccovding + 0 which th~ wo....dconsists of objects. Inthe logical language of the Traciaivs, objects do figuv-eWithinstates of tlffai ...., but in complex combinations and ....lCltionships witheochothe....and not as elementa..-,(units. The next two pl"opositions (3 andy) develop ~ost l' ( what hels been wiled the pictvre theo ....( of language,wheV'eb,(tl thought 15 a pv-oposit,on with sense." Passing he.... tov-epv-esentation and bnguoge, \'Yittgenstein delimits thoughts to logicalpv-opositions, but within tl context Clndin v-efel"enceto the wodd. This ispe....haps the most subtle pad of the book, and. ClI50the one which I"elatesto IYit+genstein's idea of mathemCltics cmd logic as machines fo....pv-od.ucing iavtotcqics. Propositions 5 Clnd6 develop the idea thClt..p....positions av-e tv-uth functions of clcmcn+o ....( pv-opositions", in whichmathemat,cal-s,(mbolic notation is used + 0 explain pv-ecisel,(whClt tl tv-uthfund,on 15. ~e..-eWI+tgenstein uses logic to define p..-opositions (ClndthvsIcmgutlgc. Clndthought) tlS the combinCltions of atomic, ov- elementClV-,(PV-Opo5Itlons,combined thv-ough BoDie's laws of composition. This pad of

the book adual l,! contains the fiv-s+ mention of what isnowknown as the"tr-vth table method" Fcv dealing with BooleClnfunctions. The book's finolclause, p....position 7,is: "What wecannot speClkof, wemust PQSSovev-insilence," (This and othev- '1uotes a..-efrom the D.F. PeClV-sClnd B.F .]llcCuinness trClnslQtion,) This IQst pv-oposition wClSgiven two high'(dive..-gent inter-p..-e+ations, the extv-eme posiiivist one of the ViennCl Ci....le,b,! which whClt one "wnnot speak of" (logicQII'() is, '1vite litevClII'(, non-sense, and the one that V,Iittgenstein and o+ne.... himself late ....gave , whichRussell te..-med "m'(stical", accov-ding to which whot "one connot speak of"is the tvu l,! important. The TradcdLJs isone of the most inf luential andclosel,!-stvdied books in IVeste.... phil050ph(. Its influences an~ legion Clndit mCl '(hClvealso 'Influenced - cmd certainl '( WClSindicated b'( - the wa'!in which compute..-s tlnd databClses model the wOl"ldtodCl,('

TUl"ir' lg, AI~r'I Bom in London in 1911:, this gn~atB..-itish mClthemCltician is gene..-all'( conslde ....dto bethe fathe..- of compute r science. Tv... .ngcontributed to man'( cweClsof mathematics,but is mostl,! ....membel"ed fov- one of his eo... .iest....5ults in logic. vvhile a student at Cambv-idge,he became fascinClted b'( the founelatior'l~of matne.mCl+ic~Clnde5peciClII,(the lr'lcompletene.s5Tneo~em of Kvd Ooelel, which inspi ....d him tostud,! ~;Ibeds Enfscheldvn9sproblem ("deciSionpv-oblem"), a '1uestion that had 5uv-vived Codel'sanal,(sis. The Ei1f.5cheid(J/19.5pi'oblemaskswhethev-, give)'10. 10gicClI5'(stem, +her e is Cln

I. algorithm fo....deciding whe+hev-a p....position ispvovable within the s'(stem 0v not. Tu..-ing's Clnswe...WClSa devastCltin.9"no". Toveach this, he fiv-st had to define I"igov-ousl'{ the notion ofCllgov-ithm. ~is ingenious definitiol'l in ter ms of a theov-etiwl "mochine"with tl central COi1r,roiand a tape fo.. -memory , inpv] Clnd outpu",ant icipated in important WCl '(She digital compute v t lnd hos had, sincethen Qnenov-movSinfluence on computCltionCiI pl"actice and thought.TI.i!'il~9machines - as the'( ave now cCilled - shal'e with todCl,!'5 computev-sthe b~'( pV'oped,! of univev.5aiify, in thtlt 0 machine con w..- . . . ..! out Clnycomputational tosk, pl"ovided it is supplied with ClnClppvopn~te ,pl"o9rarnfo....t. Two otnev- mathemClticicms, Alonz.o Chu....h [btev- Tunng s theSISadvisov tlt Pv-inceton) and Emil Post, CClmeup independentl,!, al'ld at aboutthe same time, with al9o ... .thm foV'malism5 that were ultimatel,! shown


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eCjuivalent to TuY"ing's, Yet his foY"mal izat ion had the gY"eatest impact,malnl'{ becouse of the extY"eme simplicit'{ o f its basic constY"uction h' ht ,w ICcem, neveY"heless, ach ieve ext . .. .mel' / complex Y"esults The wo. .. .o f T .. Unng- as well as that of the othel's mentioned - on algol'ithms and methodsfol' the geneY"a lsolvablht '/ o f pY"ob lems, i s an obv ious outg . .. .wth of thefoundat iona l C lues t and thus, in a sense, i ts culmination. DUl' ing Wo. .. .dWal ' I I, Tu... .ng p. .. .s ided ovev the des ign and constY"uc tion of two sel 'i es ofelectY"onlc ccrnpv+cr s the" Bombe" cmd the "Colossus". These weY"eusedsuccessfull ,/ - and cvucial l' { foY" the w a Y " eHod - fov b....oking seveY"alCel 'man cv'{ptog . .. .phic codes, inc luding the no+oviouslv haY"d "Enigma"of the Cevman nav,/. AfteY" t he waY", TUY"ing wOY"kedin the fledgling BY"itishcompute ... .ndust! '' {, did impor+on+ woY"kin b io log, / and founded the f ie ldof ar tiiicia! mieiiigeV1ce b,/ pY"oposing whC1tbecame known as the Tvringtest, a method fo!' dcteY"minlng whethcY" cm adifad "can think", Alwa'{sInteV"ested Inspods and games - hewas an accompli shed long-distanceY"unneY" TUY"ingwas thc fil '"st to develop Ideas foY" a chess-pia,/ing pl'"ogY"ammakll~g mC1steY",{n the game one of the goal; ; towcwds which the designeV"sofIntel ligent machines should str ive. In 1952 he was pl' "osecuted on occovn+ ofhis hom05exuC1l it ,/ , then a punishable offense in Bvi tain. A5 C1nalteY"nat iveto a Jai l sen+ence, he agreed to undeY"goC1nexpeY"imental "tl '"eatmcnt" withest l' "ogens, which pY"obabl' { caused the seveY"edepl' "es5ion which led him totake his own life, if'"195~,

V ie n na C iv -c le A gl'"oup of philosophel'"sand philosophicall,/-minded scief'"1tish,who met in V ienna between 1n14 and 1936.Theiv mC1ina im Wa5two-fold: to bui lda s trong empiV" ic is t philosoph'{ using theinsights into scief'"1tific me+hodolog,/garneY"ed fY"om n~cent advaf'"1ces In logic,mothematics and ph'{ 5ics, and to appl,/

the methodolog,/ o f the ph'{si ca l sciences to the soc ia l. The scient if icall ,/ -tv-ained phi losopheY" of science j \\ovih Schlick is gef '"1evoll ,/ n~cognizedto be the group 's leadeY" .Some of the most p l' "ominef' "1 tmember s were:the mathematic ians Han5 Hohn, Olga Hahn- euV"oth, Cu.: ;tav BcY"gmaf' "1n,Ka.. ..I~\ '?I'1geY",nd Kvr+ Code I foY"a shod peY"iod of t ime; the ph' {sic istPh il ip I' Fraf '"1k ; the soc ic l scienti st o tto NeuY"ath and the phi losopheY"sVik+ov K. .-of t and Rudol f Cal' "nap. The grour met I l'1 fovmall ,! Thu. . ..da' {evening5 at Vicnnos "Cafe Cent .... l . . but was lo+cv constituted asa societ,/ wit h public meetings, DC5pite the g roup's info ....mall'1atuY"e,

the membe . .. . had 0 common coY"eof phi losophical bel iefs, expre5sedi f' "1Clsod of manifesto, t it led "The Scienti fi c Concept ion of the \~oY" ld",The member5 of t he Civcie declared thClt the wor k of Fl'"ege, Ru.;..;.ellGlnd Eif'"1steinpY"ovided theil'" fiY"st if'"1spiY"atiof'"1,hile the Tracfafl/~I.og.'co-Phiiosophici/s of Ludwig Wi++genstein funct ioned a:: . their divedmodel, The phi losophies of loqica! posiiivisr and logical emplI'"ICISm,expY"e5sit1g t he world view of t he membe .... of the Civcle, sta te thot .knowledge comes fY"om expeY"ience - and +hu5, ba5icall,/, fY"om SCientifiCobseY"vat ion af' "1dexperimet1t - developed into theoY"' { thvough logicalCif'"1al,/5isand s,!nthesi::'. Still, following the Tractaius, membevs of the\ cnna Civcie he ld that logic ond mathematiC':; ot1I'{deallt1 rauioloqies,and thus do no t pl'"ovide knowledge as such , but onl'{ one of t he toolsfoV"the elobovation of empivical kf' "1owledge.Accovding to the worldviewof the CiV"cle, stotemenh that wnt10 t be reduced to expe . ...ence (such astheologicClI or ethical PVonouf'"1cemef'"1ts)cannot be V"ight OY" wl'"ong, as the,!Cif""e_ (juite liteV"all,! - nOV1-SeY1.Se, having no meanif'"1g.T\-"Iemost ex+remevCI '5 ion of this tenet , due to CaV"nap, ociua ll ,! reC ju ived thot fov a5tatement to be meaningful, its tV"uth or fal si t' / mu. :; t be vel '" if iable b '{on Cl lgoY"i thm reduc ing i t to obseY"vable tY"uths - a new incarnation ofe'bn'z'5 "catcolemos". CaY"nap latel '" tY"ied to reconci le thiS view Withthe Incompleteness Theorem, Though the Vienna CiV"Cle, in i ts o~igi f' "1alToY"m,was dissolved in 1936, ClfteV" Schlick's muvdeY" b'{ C1pavaf'"1old3x-student (md Nozi 5'!mpo+hizeY", i ts spiY"it cOf '"1t inuedto l ive on.'\05t of its membe l's managed to flee AustY'ia and emigyate to Ef'"1glandond the Un ited st ates, when~ the,! had a ma jov inflvef'"1ceonthe development of pos t-war phi losoph '{ .

V on N eum eln n , J ol- ln Born iY\Budope,."t in 1903("John" is the C1ngl ic iz.edform of the I-\vngal' " ion"Janos"), von Neumann showed vev'{ eaY"l,/ :;ignsof unusual intel ledual pvowe:;s, bei f' "1gable +0 domental division of 8-digit f'"1umbeV":;f'"1dcov;veY"seif'"1af'"1cientCY"eekb,/ the age of six. He 5tudiedmothematics in Budapest, obtaining CI PhD a t ~2,mcanw\-"lile 0150 wOl'"kingtowaY'ds a degV"eeinchemical cf'"1gif'"1eeY'if'"1gt the Y"enowf'"1edechnicalUnive.. .. it '/ of ZVl '" ich, to plwse his. fat\-"le.....

He. . -orid l' { become the s tar mathemati cian of h is geneV"otlon, legendaY" ,!foY" his penetY"atif'"1g af'"1drapid-fiye mathematical genius. Upon at+endlt1gthe lecture wheY"eCladel Clnnouf'"1ceclhe fiV"5t lt1comple+ene5$ Theorem,


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von Neumcmn was +hc fi v-st to v -eQl ize the v-esul t's impov- f Qnd did ' ..JI' "i+' II .. ' In",eedpv-oc aim I :: Q ovev- aftev- fl' e tQlk. Buf he made crvciol svgge t-to Codel v-ight aftev- it, Qnd wenf on + o pv-ove the second Incompl~tlonsTh h' '" veness. eov-crn - w Ich howevev- Codel himself had olso pv-oven independentlIn the meClntlme. Von NeumQnn n(!vc;v-wov-ked on fhe foundClt icns of ymClfhemofics ClgQin. Possessing Clwide-I'"Clnging mClfhemQticQlgeni hd t 'b f' t us, emCl e con V" I u Ions 0 mQny diffev-ent bV-Qnches, he hClSbeen cQlled "thIClst of the gl'"eQt mQthemQticiClns", hQvlng mClde gl'"CCltcontl'"ibutions ~omany dIf fev-ent bV-Qnches .of mClthematics, among them set theol '"Y,opemtol '" Qlgebv-as, ev-godlcfheov-y ClndstCl tist ics. ~e Qiso did importQntwOl'"kIV1 ' 1uantum theon/ , f lUid mechanics and mQthematica l econom'lb' fh t: t cs,e lng e co- , oundel'" (wi h economist Oscal '" MOl' "genstel '"n) o f the f ie ldof game theor v. DUl'"ing \1 '\\111,he was one of the bv-ains behind the Cltomicbomb, and aHel'" it headed the u.s. govel '"nment committee in chal' "geof the constnxtion of the hydv-ogen bomb. Pev-haps most impodantof all his wOI'k, however; was his contv- ibut ion to the cr eo+ion of compvt\ rh'l h . el'"s.\ lee WQSwOl'"king QSQ consultant in the design of one of the fil'"stelectl'"onic computel'"s, in 19"6, and inf luenced by AICln Tu.-ing's ideCls,von Neumann developed an ClV-l '"ayof fundamenbl design pY' inciples,poStvlQtlng, Qmong o+her s, a central prccessinq unit Clnd sepclI'"atememory deviccs whel'"c both data and proqrams aV'eboth stov-ed.Pv-act iw ll y al l subsequent computeI '" des igns have been bQsed on this bas icmodel , now known QSthe von Neumann architecrore. Von Neumann went onto become one of the f il '"st gl' "f :!af computeI' " scientists, especially excel lingII what now would be cal led ::cienfific compurlng, i .e. the use of computel' "sfOI~SCient if iC veseCll' "ch. ! -Ied ied of conccr - possib ly the vesu lt o f h isattendance of fhel'"monucleal" tests - In 1957.

Whitehead, AlfY'eci Nol"'th EnglishmClthematicicm and philosophel'". BOl'"nin 1861,h~ studied mathemCltics o+ CQmbl '" idge, whel' "ehe also tClught fov many decades. In 1891he mal'"l'"ied Evelyn Wade, an lv-ish woman muchyoungcl '" thCln himsclf . Bcfon; his intense,deCClde-long collabol'"ation with Bedn::H,d Ru!>sellon the Pr.ric 'pa nathemar.ca, Whiteheadpubli shed his book Universal Algebra, an oHempt

. . +o sfud'{ the types of symbolic v-easoningW 1 vanous Cllgebn:: lic sY5tems fvom a vev-y modevn - fov- i+s time - fov-malv iewpoin t, A ftel '" Russe ll 's abandonment of the Prircipia, in 191.3, Whitehead

tl'"ied +o wl '"i te a foudh volume, on geometv- '{ , but nevel' "completed it ,The two men had vel'" '1 li t+le intev-ad ion aftel' " the publicCl ti on of theF,-inc/pia, and Whitehead did not confvibute + o the 1925, second editionof the book, hav ing moved on to mafhemCl ticCl\ ph'{si cs Clnd ICl te l' "phi losophy. ! -Ied ied in 19"7.

WiHgenstein, Ludwig Wittgens te in isconsidel' "ed by mClYlYto be the gl"eatestphi losophel '" of fhe (;0+'0 centuv-y. ~e WCl5one of fheeight childven of industl '"iCll ist K CI I'"I Wit+genstein,one of Aust l' "i a's weal th ies t and most powel' "fulmen, and C\ gveQt PCltl'"Onof the ods. Of hisfou l'" bvo+her s . +nr eecommiHed suicide in eavlymanhood, while the foudh, Paul, went on

, to become a venowned concev+ pianist, Aftel '": : : : : ": : . . . 1 . J~, ' - : : : : : : : ; ; : : : : = : 1 two yeal'"s of eY'gineeV"ing studies, IViHgenstein

developed CI stl "ong intcl '"est in logic. and the foundCltions of mClthem",ti.!-Iewent to see Fr-ege, who suggested that he go to Cambvidge to studywi th Rus5el l, Clp iece of advice Wi ttgenstein fol lowed. The assoc ia tion deeplyin fluenced both men, but p l' "obabl '{ the teacheV" mOl'"e than the s tudent .DUl"ing hi: ; service with the Austl '"o-!-Iungal '" iCln avmy i~ IvIVI, Wit,~genstein .won s(!vel'al medals fol '" hi5 valoul' ", his citat ions undel '" lInlng hiS sang-froidundel '" f in :" . He was eventuall'{ captwed b'{ the enem'{ Clnd completed hismagnum opus the Tracratu l .og/co-Philosophiw5, in Cln Italian pl'"isoYlel"s'camp, Aftel '" the wal'" he donClted the huge fodune left to him b'{ hisfathe l" to h is thl '"ee s lstel '"s and, hav ing, as he believed and declal'"ed,"so lved al l the p l'"ob lems of phi losoph,! " w ifh the Tracraivs, he workedas a gal"denel", aV"Chited, and eventually as a teachev in a small vil lage'n Lowev Austvia. In 1929, possibly inspived by intel '"Clcf lons With membev-sof the V'eM' " C. il '"cleas well as Clftending a kct'.lv-e on the philosophyof mClthemotics b'{ LUitzen Bl' "ouwel '" ,on intui t'onism, WiHgenstein I '"ehw~edto Cambi' idge and phi I050ph '{ . ~e I '"etvacted h iseavliev work ~s dogmaticand went on to cv-eCl teanew, ext remel ,! i nf luential philosophica l staYlceoften I"efevl'"ed to as "the late WiHgenstein", Unlike the ideas in the"racioivs, Wit+genstein d id not Cl t+empt to put h is la te l'" phi losophy. . t t d th . sevies of mOl'"e 01" \e.5sin a s,{stemotlc tv-eah"e, bu prcsen e em In a .i ndependent n~mal' "ks. Man' { o f these he saw as fov-ming Clbook, . whichwas posthumousl '{ publ ished as . C lh i lo s op h i, :; o l ln v e sh g ah o I1 5 - thiS,ClSwell as a few books based on his notebooks, 01' "tYClnscl' "ipt .s of lectul' "es01' discussion5, al'"e all that we have of his latev thought. This is C\


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philo:;ophiG,,1 po:;ition of "n ex+vcme "nti-dogm"tio n"tuV"e, foou:;ing on/ongvoge "nd p.5ychology (wh,,+ wenow coll coqnitive psychology), in:;te"dof logio ond objective tV"uth, "nd on fuzz,/ concepts :wch "s "f

logicomix glossary

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