frege - the foundatins of arithmetic

8/4/2019 Frege - The Foundatins of Arithmetic

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The Foundations ofArithmetica logico-mathematicalinvestigation into theconcept of numberl

[Die Gru dlase der Aithnetih was published in 1884. \]Vlar folowshere is the Introduction, SSl-a (which funher explain Frege's lask),S545-69 (which establish th philosophical fowdations of Frege's logicistproiect), and SS87-91 and 104 9 (from the Conclusion). Summ"des ofrhe rcmaining sections are provided at the relevant points.l

IntroductionIfwe ask what rhe number one is, or what rhe symbol I means,: we aremore often than not given the answer a rhing. And if we then point outrlat rhe prcposirion

"I he number one is , thing'is not a dfinition, since ir has the definite adicle on one side and theindennire on rne orher, and that it only says that th number oncbelongs to the class of things, but nor which thins it is, then we maywell be invired ro choose wharever we like to call the number one. Butif everyone was allowed .o understand by rhis name whalever he likd'&en r}te same prcposition about the number one would mean differentthings to different people; such proposirions would have no commoncontent. Some may reject the question, noting that rhe meaning of drc' 'ltanshtcd by Michacl tsc.ncy. I'.se nunbels in dre malgin ue ftom rhe..isin!lI -I hroughout thn trandntirm ol OI-, !nle$ oiheNise indiclted. Bedeurung rnd its roA-nJtesbav.b.en trn\hicd rs mcrnns rnd nscognirri ()nrh txnnxlnn ol llcrlcultnA',{r thc I rodu.lnr\ i1 rb.vr.

The Fou/dations ol Aithnet;c 85

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leller a in aridrmeric cannor be given eirher; and ifit were said: a meansi Dumber, tlen rhe same misrake would be made as in rhe defir tion:,r'e is a_ rhing. Now the rejection of the question in the case ofa is quitet(stilied: it means no panicular, specmabte number, but serves insieadtr cxpress the generality of propositions. Ii in a + a d = a, we sub_'trtute for a any number we like, bur rhe same rhroughout, rben a &uerrturrion is alwa]s obtahed.r ft is in tlis sense that the letrer d is used.lhrt in rhe case ofoner the maner is essentially difierenr. Can we. in the,,I'r.rLion I + I = 2, subsLirurc for I boLh rimes rhe same objecr. sa) dreAl()on? ft seems rarher rhar we must subsriture something ditrerent forrlr. iirst I as for rhe second_ \j{tly is ir rhar we must do here precisely\ hxr would be a mistake in the other case? Arirhmetic does noimanagetrrrh the letru a alone, but musr also use other lett$, D, ., et"., L,',(lcr to express in general form relarions berween differcnr numbersr,r) it might be supposed that fie symbot I cannot be sufficienr either.,r ,r r(ned in a similar wav to confer generstiry on p,opo.irjons. Bul,l(r's lhe number one rot appear as a parricular oblecl with specifiableItr,)r)crries, e.g. that of remainins unchanged when multiplid by irself?lr rh;s sense, there are no propenies of d rhar can be specifiei; sincerllrircvcr is assened of d is a common prcperty of numlen, whereasI I assens nothinC ofthe Moon, nor ofthe Sun, nor ofrhe Sahara. no."l rl,L l'eJk ofTenenfe; Ior whar coutd dre sense ofsuch an as.enion be?l,) such quesrions not even a mathematician is likely to have a satis_li( r("t' answr rcady to give. yer is it nor shameful that a science shouldl* ..(' unclear about irs most prominent obiect, which is apparently so,r"t)icl Small wonder rhan no one can say what number is. Ifa conceptrlri r is fundamenral to a geat science poses dimculties, then it is surjv.", ,,Dlrcrative rask ro inve:ligare il in more deiarl and ovcrcome Lhese,illr(ulries, especialty since complere ctariry wilt hardly be achieved con_,, r,nrs negative, fractional and complex numbers, so long as insight

,,,r,, rhc foundation of rhe whole strucrure of arithmetic is deficient..\ u,,irrudly. mrny wiU nor rhrnk rhis worrh the Foubte. This concept.rl,, v suppose, is quire adequatety Feated in rhe etementary textUo&s, ,rL tr|ung' h.s, throughout rhn rohme, becn tnnsiar.d !s .cquatin,, which n what, 'L ' ,h!,uoudv rndans. Ho{,de., .s norld abov. (p. 64, an. 24), it is nevertnetess ctcar' ' r'.r. undcntoo.t Olcic|heit (,cqu!rity') in thc scnsc of.idcn ty,, and.esard.d| ' r ,, n\ i\ idcnritics. (Ct aS, \B ( pD. 64 j above) ) wherc lis symbol tu .InlMhsAeich_" i n.A iDroduccdi and .Sa, In. A, p. t5t bclow.) rr w.s rhis rhd led Au$in to ren-I..r,lL,rlrns as identiry,in what n $i rhe only.onplcte tnnslaiion oi GZ G.e FZ,r rr i' ). llut ir is retrainly oore naru.al to cal I + t = 2, say, an "araao4 mtrur rhan' , , ,r^ . iDd Lhn h.s bcon .esp.cte.t hcrc. Since Fruge,s primrry concdm in Ct ni, rlvrnh.rnllmetic,.cleicbhcir. lnd.gtcich, toohavc nom.llvbeen ii,nsl,rd rjrn,,'.J' .',.1 iqu.,!.Jlh.-jl'.n,-ury'rrd rdcnrnrt r,"" o. ,ror,.r, .r.. r,.,.-q . J .rr. t!r\ r,. L sn,A rhrm rh.1. rh!\ -rr . !.,,tv


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86 'lhe Fomdanons of Arnhnetitand tlereby setded once and for all. \vho can then believe that he stillhas something to leam about so simple a maftr? So free from anydimculty is fie concept of positive whole number taken to be, that itis assumed tlat it can be erTlained scientmcaly and definitiveiy ro chil-dren, and that everyone, without funher reflectron or acquaintance withwhat othe.s have thought, knows all about it. The fust precondition forlearning is thus frequently lacking: the knowledge that we do rot know,The result is rhat we remain content with a cnde conception, evenrhough He$anA has already pmvided a better one. It is depressing anddiscoumeine thar aeain and ag3in an insight once achieved threatens tobe lost in rlis way, and that so much wo* appears to be done in vainJbecause in our irflated conceit we do not rhink it necessary to appropli-ate its iuits. My work tooJ I am well awarer is exposd ro such a danger.This crudiw of conception surfaces when calculation is described agaggreAativeJ mechanical thought.B I doubt that &ere is any such thought.Aggregative imagrnation there may well be; but that has no signifrcanccIBedzutunel for calc.rlation. Thought is essentially lhe same everlwherc:it is not the case tlat drere are different kinds of laws of drought depend-ing on the objecr lof fioughr] . The differences lin thoughil merely con"sisr in fie grater or lesser purity and independence from psychologicalinfluencs and external aids such as ordinary language, numerals andsuchlike, and also in rhe dgree of refinement in the structure of con-cepts; but ir is precisely in this respect that mathematics aims not to bcsurpassed by any ofier science, not even philosophy.It will be seen from the present work that even an inference like thatfrom n ro , + l, which is apparenrly peculiar ro mathematics, is basedon general loeical laws, and rlat there is no need of special laws foraggregative thought. Admittedly, ir is possible ro manipulate numeral!mechanically, iust as it is possible to speak like a panot; but thar canscarcely be called thinking. It only bcoms possible after mathematicalsymbolism has been so developed, through genuine thinking, that ildoes the thinking for us, so to speak. This does not show that numbe$are formed in a panicularly mechanical wayr as sand, say, is fomdfrom erains of quart. It is in the irterest of ma&ematicians, I think,to counter such a view, which is characterized by a disparagemenr ofthe principal object of their science and thereby that science itselt Ycteven mathemaricians are pron to say such things. Sooner or later, how-everr the concpt ofnumber must be recognized as having a finer structure than most of the concepts of other sciences, even though it is stillone of the simplest in arithmetic.\ Cdl/..rd ltlor&j, ed. Hlncnstein, Vol. x, Irdrt I, Um.iss t:idasogis.hr V.d$unscni\2j2) t;. 2i Two ddes not m.in {r&Je ril',1 Mo things. but doubling ctc.! K. r:i\.hcr, lrv,r dr l.,fr| untl LIodl'Itik rn. Watnl .ldl^r.r'r, 2nd cdn., $9,r.

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The Foundations of Arithnetic 8j. In order, rhen, ro dispel rhis ilusion that no ditficufties ar ail are posdI v r1r-e,p-o.iuve whole oumber\. bur rhar g.n"rut ugre.nle.rt pr.ui,l",-it(('ned ro me a good idea ro discuss some otrhe vrew" ot ph,lo,ophe,s:rnd mathematicians on the quesrions mised here. h will ie seen;;;I :lh fol9 i. r: be found. even ourishr conradidions cmersins. some,\. ror example. .uni$ a. idenrcal lsieirl wirh one ano*rer.l oUe."r!'rd thrt rhey are differenLr and borh (ides have rea.onr for rtreir ctaimrlrrr.cannot be rejected our ofhand. Here I shalt try to motivut" *"l;.,1j"-1^1i:T :*:, I'vesrisarion Ar rhe same rime. Lh,s preriminary, ir.rcdrron or Lhe vje$s expre\sed by orher, will clear L_he ground for, r!,.wn con(eprion, b] convincing peopte be,orehand Lhar rhe\e othef'l '11. !: ':, Iedd ro rhe sor. and drar my opinion i. ,., r,.; ";;';i,.nrv equalty rLrsLfied opinion\i and so I hope ro \eLrle rhe quesrion,lt.linitively, at leasr ir essemiais.Admir.edly, rhis has led me ro take a more philosophical approach tlan,,'.,r,! rnarhemarhians mr) deem 3ppropriarei bur u f,"arrn.nruf ;,,u.",_,).rr.\n ol tne concepr of number wilt inevirabtv rum our ro be somerlnr phiiosophical. The task is shared by mathem"ti"" ""d p;l;*;;.rr tt'e co-operation berween rhese *L""*, a."p.. #;;;,J;i;r,,i.r bodr_ sides, is nor as productive * _ight b.-rlr"h"; ;;;'";;;;t",,\rble, rhen tlis seems to me ro be due to rh" p."rrl""". .f p";;i;:l,,u,crl modes of investigation, which have even penetrated logic. Wirh'1., . 'r(nd. mrlhemarics ha. s6 po;n," or conraci at "r,, ""j;t. .;";i;' ,r,t,,,n, dre aversion or manv msrhcmari\ianr . ph,b";;h; tr;;:,i:.r(rns. \Vhen, for example, SDickerc calls rte iaeu" ot"u*t.. _ot*t,h,,,,m(na. dependent on mu\le senrarions. no m"themarr.ian can,,,,,qnize his numbers in this or knows where to b"gi"\virh ,;;;;r' ",rx)n. An arirhmedc founded on mnscle ,""""ti;". *."[ ;;":;l"y, rr sltionat, bur it would also rum our to be iusr as *C"" ;"-,-h;,, ,ll...r;,r1:J't No. a:,!hnl:Lic ha. nofiins a, al ro d" *r,r, ,*,i,;.rl1.1*,

Lo do $ir} mcnrrl jmage.. compound


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88 The Foundations ol Aithuetiofdle mafter as it concems us here, just as incidental as blackboard andchalk, and that they do not desewe to be called ideas of the numberone hundred at all? The essence ofthe matter should not be seen to liein such ideas! The description of ihe origin of an idea should not betaken for a dfinition, nor should the accounr of the mental and ph)s-icat conditions for becoming aware of a proposition be taken for a proof,and nor should tlre discovery [Ge,l.afitwettun] of a proposition be con-fused with its nuth! We must be remindedi ir seems, that a propositionjusr as little ceases to be tlue when I am no longer thinking of it as theSun is extinguishd when I close my eyes. Othenvise we would end upfinding it necessary to take account of the phosphorous contnt of ourbrain in proving Plthagoras' theorem, and astronomers would shy awayfrom extnding their conclusions to the distant past' for fear of theobjection: 'You reckon that 2 x 2 = 4 held then; but the idea of numberhas a development, a history! One can doubt whether it had reachedtiat stage by then. How do you know that this proposition akeadyexisted at that point in the past? Might not the creatures living at thattime have held the proposition 2 x 2 = 5, ftom which the proposition2 x 2 = 4 only evolved through narural selction in the struggle forexisrence; and might not this in tum, perhaps, be destined in the sameway to develop turther into 2 x 2 = 3?' Est modus in rebu:' sunt certide iq e finesla T}Je historical mode of investigation, which seeks to tracethe development of rhings fiom which to understand their nature, iscenainly legitimate; but it also has its limitations. If eve4'thing were incortinual flux and nothins remained fixed and etemal, then knowledgof the wo{d would cease to be possibl and everything would be drcwninro confusion. We imagine, it sems, that conceprs originate in thcindividual mind like leaves on a tree, and we suppose that thir natuccan be undersrood by investigating their orisin and seeking to explainthem psychologicaly through the working of the human mind. But thisconception makes vert'thing subjective, and taken to its logical conclu-sion, abolishes truth. \ihat is called the history of concepts is really ahistory eidter of our knowledge ofconcepts or of t}le meanings ofwords.Often it is only through enomous intellectual work' which can last forhundreds of years, that knowledge ofa concept in its purity is achieved,by peeling off the alien clothing that conceals it ftom lhe mind's eye.'lfhat are we then to say when someone, instead of canying on thiswork where it still seems incomplete, ignorcs it entirely, and enters thcnursery or takes himself back to the earliest conceivable stage of humandevelopment, ir order there ro discover, like John Stuan Mill, some gin-ge$read or pebble adthmetic! It remains only ro ascribe to the flavour' lbdc is DodcmioD nr .l rhingsi thero !rc, id shorl. lilcd [nns'; a qu.trtioD fro ltldrcc, Sdhl. Il(nk I. I. lnrc l(J6.

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The Foundariots oJ Airhmeti. 89of the cake a speciat meaning for the concepr ofnumber. This is suretyrhe cxact oplosite of a rational procedure and ir, *y "nr" u" ,r_rr",fr"_cmatical as it could possibly be. No wonder that mathematicians want[othing to do widr ir! Instead of finding conceprs in parricular pudty,Krr ro rheir imagined source, , every'rhine is sien btuned and undii_irrrnLiarect as rhrough a log. I is as rhough someone who wanied rolcrm about America ried ro take himsetf back to the position of -Jwxbu. as he calghr hjs firsr dubiour glimpse of his supposed tndia./\rm,rredty. such a comprrison proves nodringi bur ir does. I hope,!r.,Rc my poinr. Ir may well be drai rhe h,srory of di


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90 lhe Foundatbns oJ Aithnettcif coinections aie revealed between apparentlv distant drings' and if thisyields $eater order and regularity, then rhe definition is usuallv regadedas sumcienrly stablished and few questions are asked about its logicaliusrification. This procedure has in any case the advantage tlat it isunlikely entirely to fail in its purpose. I too rhink dtat definitions mustshow theb wonh by their ftuitfulness, by their usefulness in constuct_ing proofs. But it is well to observe that &e rigour of a proof remainsan illusionJ however complete r}le chains of inference mav be, if thedefinitions are only justified retrospectively, by the non-appearance ofany contradiction. Fundamentally, then, only an empirical cenaintv i3ever achieved, and it must tcally be accepted that in the end a contra-diction might still be encountered rlat brings the whole edfice downin ruins. That is why I have flt obliged to so back somewhat turtherinto the general logical foundations than mosr mathemaricians, perhaps,would regsrd as necessary. LX In this invesrigation I have adhered to the following fundamenralprincipls:

There must be a sharp separltion of the psychological from the logical, thesubjecrivc from the objcctiveiThe meanins of a word must be asked for in the contqt of ! proposition,not in iolation;The disdnction between concepr snd object musl bc kept in mind.

To comply with dre first, I have used the word'idea' I'Vo$te ung')always in fie psychological sense, and have disringuished ideas ftomboth concprs and obiects. If the second principle is not obsen'ed, thnone is almosr forced to take as t}re meaning of words mefltal images oracts of an individual mind, and thereby to offend against the first aswell. As concerns the third point, it is a mere illusion to suppose thata concepr can be made into an obiect without altering it From rlis itfollows thar a widcly held formalist theory of ftactional, negative num-bels, etc., is unlenable. How I intend to improve on it can be onlv indic-ated in the prcsent work. In all these cases, as with &e positive wholcnumbers, it will come down m fl\iDe the sense of an equation.5My results will, I think, at least in essentials, win the apprcval ofthose mathematicians who take th trouble to consider my argumentsThey seem to me to be in rhe air, and separately tiev have, perhaps,already been stated, at least in rough form; though thev may well bc newin their connections with one anorher. I have sometimes been suryrisedthar accounts thar come so close to my conception on one point deviatcso sharply on anothr.The reception by philosophers will be varied, depending on thcir' Sc! \\6211 hp. l0i)tr b.ldv)

The Fo datinns oJ Aithnetic 9l,rindpoinq it will cenainly be worst by I those empnicists who would

r rcognize only induction as the original mode of inference, and even thatr!,r really as a mode of inference, but as habituation. One or another,t( draps, will take this opportunitv to examine afresh rhe foundarions ofl,r\ fieor.J of knowledge. To those who might want ro declare my de6ni-rr s unnatural, I would suggest rhat the quesrion here is nor whedrerrl,ey are natural, but whether they go to the han of rhe maner and arel,'ricaily unobiectionable.I circrish the hope that even philosophers will find something usetulrrr the present work, if thy examine it without preiudice. I

ril . After depafing for a long time from Euclidean rigour, marhemar-i I i\ now retuming to it, and even striving to rake ir funher. In arirh-l,,.ric, simply as a result of the origin in India of many of its merhodsd,rl concepts, reasoning has raditionally ben less srricr 1han in geo-,rrrry, ivhich had mainly been developd by the Greeks. This was onlyrquli,rced by the discovery of higher analysisj since considerable, almost|r,upcrable dimcddes srood in the way of a dgorous trearment of this',rl)icct, whilst ar rhe same time rhere seemed lirrle profir in tlte expend-rr|t c of effort in overcoming them. I-rter developmentsr however, havelxrvn more and more clearly that in mathematics a mere moml convic-r!r), based on many successful applications, is insufficient. A proof is,r,trv demanded of many things that previously counted as self-evidenr.lr rs only in this way ihat the limits to their validity have in many casesl-dr derermined. The concepts oftunction, continuity, limit and infinityl,,,vc bcen shown to require sharper definition. Negative and inarional,i,,,trbers, which have long been accepted in science, have had ro sub-,',Lr ro a more exacting test of their laitimacy.

lhus eve4'ivhere efforts are being made ro provide rigorous proofs,t,,..isc determinations ofthe limits of validiry and, as a means ro rhis,ll Ip dcnnitions of concepts. Irl. 'lhis path must eventually lead to the concept of Numbel' andr1,,. simplest propositions holding ofthe positive whole numben, which

I l.lltrv Aunin lErc (ct ?ir) t. 2, tu.) in t.anshtins 'Anzahl' by 'Nunbcr' (wirh a{, r Ll N ). lcivins nun6er' f.. the more gene.al tcm 'Zahl'. Thc distincrion playsI ' i '.le in 6I- (ct l:reqc s own ln. G bclow), bui n d.es a.qui.e rignificance in GG,rr rirr,|rtcd at 6G, I, $$41 2), shcn Frcsc disrinsuishcs ihe rdal nuobeE ( reelen', il liom the nrtural or cardinal nunrbms ('Anzrhl.n'), irhi.h fc now ro be udet' ,,I rs dillorent liom thc positne wholc nunrhcrs ('rositivcn grnzen z,hlen ). lhct'r .' L n ntbuN rnswer thc que{bn 'rlow many obiccts oa . .ctuin kind fe lhcre? ',. ,li rl:( ' l nnmhers crn b. rclr3Rlr.l a\ meisurcnrcnr Dur$us [,ttu$j,r]Lrl, *lri.nr ir ll{v lrnrc mrlr ilutle ,s ro rtJrrc(l \nh x unn nrr{nnu(le (a;C, t. \157).


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tl? 't ltt I\ oxiutio s oJ Aithnen.1"r,, r1,,. l,,lr(li !rr ('t thc lvhole of arirhmctic. Admirredty, numedcalI'nor,li( \r(t,,,\ t r7.- t2andlaws such as that ofrhe associativiry',1 i,lllr,r' i,r. s(' lrcqucntly confirmed by the counrless applicarion;rl,ir ,,,, ,|ir,lc ol rhcm every day, thar ir can seem almost tudicrous tot,,ll rl,.rr rDro qucsrion by denanding a prcot Bur it lis deep in thc",,r ( r. ot mdrhcmatics alivays to prefer proof, wherever it is pos;ibte, toukluerivc conlirmation. Euclid proved many things that would have beenAnDlcd him anlavay. And ir was the dissaiisfacrion even with Euclideanrigour rhat led to rhe invesrigation of the A\iom of paralcls.?Thus ttris movement towards ever sreater rigour has aheady in manylvays left trehind rhe originally felt need, and the need has itsetf grownn1ore and more in strcngrh and extcnr.'fhe aim of prool h not only ro ptace rhc truth of a proposirion bey"ond ali doubt, bur atso ro afford insight into rhe aepenaence of truftron one anoder. After one has been convinced of rhe immovabilitv ofa boulder b) vain aLremprs ro ,hrE ir. dre qu..rion rhcn arircr as rowhat secures ir so firmty. The further rhese invcsrigations arc pursucd,Ihe fewer become the primirive rrurhs to which evirything is reduceajand ihis simplification is in itself a wonhwhitc goal. perhaps the lopi

is even raised thar, by bringins ro lighr rhe gcneral principles involv;din what people have instincrivcty done in the simptcst cases, generalmcthods of concepclolmarion and justitication may be discovered tharwill also b usctul in more complicated cases. I

The Foundarion, of Anthnetic 93) ,,,f:::l :: rhus jusr a. much an ubsurdrry a.. say. a brue con_., 11. ,, a propo\ruon js Lrlied J ,osi?ab" or dnaly,j. i" .nt ..nr.. ;;;;". F r iL,dsemenL nor abour rhe p\v.hotosicat. pr,r"r.i."ri"r ",ra'"ill,l.,r..ondjrron. rhdr havc made ir porsible ro form rhe conrenr of th;;;"_t{srton ln ourmind, nor abour how sdm,,,,., ome Lo bord ir ro bc.^., u u, ,., n,1n.,.,'.. l*rap. cr.on eoL,jy.,,,

I,lt:t ," j,.; o:;;";;,;"i.,t;i"ij l?"1,, jTil -"" ".*.,, tr,'s wry Lhe quesrjon is rcmc,vcd fr,| ," , .r 'isned '. *;' .i .",i,.*","r.1' "om rhe domain or psv(hologv, ,r rr no_ aepcna, o;.;;;:;;i;;l'",.ii],.;ff ff::._T:,,:,1r,, r)ruve ruths. If, on rhe way, only eeneraj logi.rl bd;d G;;;:, drcounrercd, rhcn tle Irllth is rnalwi," {l,,ch $e ,d,n;r,io,u', "1""" j'""i',j,'c r"umins $at propositions,,,.,, nr. ,i, i. no, pos.ili. ,; ;;;;-J';;f.;,:ilffl,:,:,i:J:f,,,T::':.1... r'1arc no: oiJ c.n.,,r Lgi.or,u,,,.. u,,,J";;;;.,.;;,;'il:h{,,.,'n or a prrricutrr scrence. rh

frege - the foundatins of arithmetic

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