Benis Sinaceur

Hourya

Introduction

wthemtil strtion is the proess of onsidering nd mnipulting opE ertionsD rulesD methods nd onepts divested from their referene to rel world phenomen nd irumstnesD nd lso deprived from the ontent onE neted to prtiulr pplitionsF o the strt onept of numer does not ome down to ny rel ggregte @of sheepD ensD penilsD etFA nor to ny oneived olletion @of geometril pointsD numeril elementsD unspei(ed elE ementsD etFA nd it inludes sets of numers with rules of lultion di'erent from the usul onesD suh s the rules for quternionsD otonionsD etF etullyD in mthemtisD one enounters from the very eginning not one ut severl strtion proessesD whih onstitute spei( nd permE nent wys of developing the mthemtil oreF sn modern timesD espeilly from the IW th entury onwrdsD strtion )ourishesD nd vrious proesses re more systematically piled upD ontentedD nd lended for produing proeduresD entitiesD struturesD nd theories t higher nd higher levels of strtionF estrting is n ongoing innovation proessingD whih expnds the mthemtil stu' nd mkes it still riher nd more nd more intrite nd lyeredF s m not iming t tkling hedEon the fundmentl questionX ht is n strt ojetc or sn whih sense strt ojets exist9 c 1 wy purpose is muh more modest nd my method is minly desriptiveF s wnt to estlish piture of di'erent nd reurring proedures of mthemtil strtionF husD s will fous on di'erent fetures of mthemtil prtie while s will disregrd @expliit or impliitA ontologil stnds out the nture of mtheE mtis nd the sttus of its strt ojetsF wy purpose is epistemologilD nd it onerns the tul wys of performing strtion in mthemtil doingF 2 yn the wy s shll inevitly disply how s see the mens nd prodE uts of mthemtil tivityF s think tht foussing on tul mthemtil strtion proesses my 'ord positive piture of wht is mthemtil strtionF s men tht we my ome ross riteri for eing strt tht re not otined y the lssil wy of negtionD n strt ojet eing 1. For discussing such questions belonging to the heroic tradition in the philosophy of mathematics [Kreisel 1985], see among others [Burgess & Rosen 1997] 2. My approach seems to be similar to Jean-Pierre Marquis' approach in [Marquis forthcoming]. J.-P. Marquis makes ne distinctions between symbolic, formal, and abstract, and also between abstraction and generalization.

Facets and Levels of Mathematical Abstraction 83 not loted in spe nd time nd not uslly tiveF 3 s m rther tking the wy of exmple despite its limitsF s will egin y rpid inursion into the philosophil orpusF es method s will fous on how the terms strt nd strtion hve een nd re usedY perfore s shll get informtion out the terms onept nd onepE tuliztionD thnks to whih s n mke preise my understnding of mtheE mtil oneptsF s will then try to prllel the outome of my inquiry with spei( mthemtil tehniquesX s result we will notie tht mthemtil strtion is not reduile to logil strtionD t lest s it ws underE stood in the eristotelin trditionF hirdlyD s ttempt to desrie the funE dmentl thinking proesses underlying the min wys to get nd inrese strtionD with speil ttention to reurrent mthemtil tions tht produe more nd more strt ojetsF es spei( illustrtionD s m giving in fourth setion signi(nt or emlemti exmplesY relying on them s wnt to stress tht in mthemtil prtie severl strting proesses work siE multneously nd intert togetherD oneptuliztion nd xiomtiztion eE ing n importnt but only one ftor in the jo of systematic nd uniform prolem solvingF 1 Philosophical background hilosophers my hve reourse to mthemtil prtie nd history of mthE emtis for mking more preise nd more sustntil the understnding of some fundmentl thought proessesD suh s abstractingF e philosophil men t hnd is to fous on the hnges in mening of the terms strtion nd strtF uh semnti nlysis provides indeed ruil sis for ontemporry linguistiD ulturlD nd oneptul understndingY it is lrgely used in oneptul historyD whih my e internl @onsidering the rtionl links etween mthemtil onepts nd methodsA or externlD onsidering the institutionlD politilD nd soil environment whih promotes or (ghts some typil wy of thinking nd tingX for instneD in mthemtis strE tion hs een viewed s royl route of invention in rilert9s nd iF x÷ther9s shool ndD t nerly the sme timeD s degenerte trend destroying the viE tlity of intuition in the ideology of the heutshe wthemtik hmpioned y vudwig fieerh nd yswld eihmüllerF por my prtD s see no unE ridgele gp etween strtion nd intuitionD sine insights my ring in strtion proessing nd follow from it s wellF es some mthemtiins @iF ertinD eF eilD nd othersA mintinD there is indeed symoli nd E strt intuitionF enywyD s m not iming t disussing here the question of xiomti or logi versus intuitionD whih ws the fous of intense detes 3. See [Burgess & Rosen 1997, 20]. The now standard expression way of negation was coined by Lewis in his book [Lewis 1986].

Hourya Benis Sinaceur during the PH th entury nd still is one of the min issues of the philosophy of mthemtisF o the question ht is meningcD s will give uine9s nswerX wening is wht essene eomes when it is divored from the ojet of referene nd wedded to the wordF vet me quote the whole pssgeX he eristotelin notion of essene ws the forerunnerD no doutD of the modern notion of intension or meningFFF hings hd essenes for eristotleD ut only linguisti forms hve meningsF wening is wht essene eomes when it is divored from the ojet of referene nd wedded to the wordF uine IWSID PPD see lso uine IWWHD VV etullyD s m tking the divore of mening from ojet of referene s methodologil devie for voiding ontologil onsidertions nd foussing on wht nd how we know rther thn on wht we believe or wht we assume or must ssume in order to give philosophil ount of some mthemtE il tions or ttitudesF s m not sying tht epistemologil views do not ommit to ontologil ssumptionsD s m just sying tht s will leve side those possile ommitments nd the in)uene tht they might hve on tul knowledge proesses nd on our theoretil explntion of those proessesF 1.1 Abstraction and concept-formation estrtion is n essentil knowledge proessD the proess @orD to someD the lleged proessA y whih we form oneptsF st onsists in reognizing one or severl ommon fetures or ttriutes @propertiesD preditesA in individE ulsD nd on tht sis stting onept subsuming those ommon fetures or ttriutesF gonept is n ideD ssoited with word expressing propE erty or olletion of properties inferred or derived from di'erent smplesF usumption is the logil tehnique to get generlity from prtiulrsF his rough desription omplies with eristotle9s ount of ἀφαίρεσιςX gonsidering di'erent things we subtractD removeD tke wy their prtiulriE ties nd retin only wht they hve in ommonF he onept of mn pplies to ll humnsD mle or femleD tll or shortD lond or rownD etFY the onept of tringle pplies to any tringleD retngleD equilterl or isoselesF eording to eristotleD onepts re immteril ideas tthed to mteril thingsY they exist within things on whih they re preditedF 4 here is disussion out the nture of eristotelin strtionF pregeD nd some eristotle9s experts suh s hvid oss nd rFqF epostle give psyhologil interprettionF fy ontrst tohn glery lims 5 tht ἀφαίρεσις 4. A little more on Aristotle's abstraction in [Szczeciniarz 1999, 45]. More in [Cleary 1985[Cleary , 1345]]. [Cleary 1985, 3536]. On Aristotle's view about abstract objects as a result of subtraction: τὰ ἐξ ἀφαιρέσεως λεγόμενα, τὰ δι΄ ἀφαιρέσεως, τὰ ἐν ἀφαιρέσει λεγόμενα see Metaphysics,µ,13. Facets and Levels of Mathematical Abstraction 85 tht he rightly trnsltes y sutrtionD deprivtion @in ontrst with πρόσθεσιςD to whih orresponds dditionAD is the logil method whih is used to identify nd isolte the primry sujet of predition for ny given ttriutes (Posterior Analytics)D nd whih onsequently legitimates the intelE letul seprtion of strt ojetsF enywyD strtion is the proess of pssing from things to idesD propE erties nd reltionsD to properties of reltions nd reltions of propertiesD to properties of reltions etween propertiesD etF feing fundmentl thinking proessD strtion hs two fesX logil fe nd evidently psyhologil spet tht is the trget of ognitive sienesF tohn voke @ITQPEIUHRA introdued particular ideas etween individuls nd general ideasF yn (rst stepD prtiulr ides gther individuls into lssY on seond stepD generl ides re created through the proess of strtingD drwing wyD or removing the unommon hrteristis from severl prtiulr idesF por exmpleD the strt generl ide or onept tht is designted y the word red is tht hrteristi whih is ommon to the prtiulr ides @prtiulr oneptsA of pplesD herriesD nd loodF husD is pointed out the ft tht the strting proess forms scale with t lest two stepsD nd generl onepts come loose from things. voke writes indeedX qenerl nd universl elong not to the rel existene of thingsY ut re snventions nd retures of the understndingD mde y it for its own useD nd onern only signsD whether words or idesF voke ITVW sn ontrst with eristotle9s ontologil nd logil point of viewD voke9s stndE point is squrely epistemologilF xote lso tht ides my ply the role of signsY lter onD ghrles nders eire @IVQWEIWIRA developed his semioti phiE losophy on very similr perspetiveF 1.2 Concepts heveloping further on voke9s pprohD let us ndon lto9s nd eristotle9s view tht onepts re universlD unhnging ideal objects grsped y the unE derstnding or mde up in onformne with preEexistent reltions in the rel worldF gonepts result indeed from the logil opertion of sutrtion ut they do not hve n eternl existene in some hevens of universl formsD sepE rte from prtiulrs s thought lto or not seprte s rgued eristotleF sn my opinionD onepts re historical products of the mind9s tivity nd their emergene depends on mny theoretilD ulturlD soilD eonomilD nd poE litil dtF xevertheless onepts re or my e ojetiveD sine they help to grspD to express in most ommunile wy @t lest in prinipleAD nd to msterD within vrile limitsD phenomen of the rel worldF o stress the objectivity of sienti( oneptsD the semnti trdition in philosophy folzno @IUVIEIVRVAD prege @IVRVEIWPSAD russerl @IVSWEIWQVAD nd their followers proposed to onsider the sphere of onepts s autonomousF he im ws to seprte semnti phenomen from their linguisti expresE sions nd from their mentl representtions VorstellungenF fut grounding the semnti sphere on itself my led to erse the historil hrter of its elements nd to give them n immutle ontologil sttusF st is well known thtD in order to ground the ojetivity of sienti( oneptsD qottlo prege proposed to lote onepts in third relmD the relm of strt oE jetsD whih re neither sensile nor mentlF prege9s strt ojets re not objectsD they re meaningsD more preisely timeless everlsting meningsF qiven linguisti expression F D prege nmed the mening of F its oneptul ontent begriicher Inhalt F e oneptul ontent is either lwys true or lwys flseF prege rgues tht we nnot rete meningsD nd tht we n only grsp themY he onsiders lso menings s if they were a priori essenes tht we hve to disoverF estrt ojets reD in prege9s perspetiveD menE ings in themselvesD just like the old things in themselvesF ht gve irth to philosophil endless nd urrently ongoing disussionsD with revivl of ltoni tendeniesF he semnti trdition ws retion ginst the promotion of the ujet y hesrtesD untD nd regel mong othersD nd n ttempt to sve the lleged eternl hrter of sienti( truthsF fut from more prgmti point of view there is no need to ground semnti ojetivity on ojets (xed nd independent from the mindD whose essiility would then e questionleD s pointed out F fenerrf fenerrf IWUQF he divore of oneptul oE jetivity from (x nd everlsting ojets is not newF qeorg ureisel hs pointed it out mny times in his ppersF qrounding on tht s wnt to onsider ojeE tivity s resulting from suessful intertion etween the rtionl tivity of the understnding nd the environmentF gonepts re lso produts nd tools of thinking nd resoningY nd they do not exist in the mind efore the strting tF sn unt9s terms they re a posterioriD iFeFD they rise out of experieneD experiene eing tken y me in n s wide s possile senseD nd not in its untin senseD whih is limited to pereptul or physil experieneF s would then sy tht onept is thought-objectD whih results from sutrtive proess tht onstruts the unity under whih severl spei( thought-objectsD rther thn severl rough physil ojetsD my e gtheredF wthemtil tivity is onerned with thoughtEojets rther thn with ojetsD even in the se of importnt impulse given y physilD iologilD eonomiD or soiologil phenomenF wthemtil entities re produts of the tivity of the understndingY they pper in prtiulr presenttion Darstellung 6 D whih might e modi(ed or repled y nother oneF sn other 6. It is necessary to make a distinction between the word presentation [Darstellung ], which means the objective mathematical way of introducing or using a concept, and the word: representation [Vorstellung ], which has here its usual meaning of a subjective mental content. Moreover, when an element a of a set E Facets and Levels of Mathematical Abstraction 87 wordsD we hve ess to mthemtil entities only through the onepts we form for expressing some of the properties we wnt to tke s sis for developing our knowledge onerning those entities nd mny others tht ome to e relted to themF estrtion involves pereiving somethingD relting it to other thingsD grsping some ommon trit of those thingsD nd oneivE ing of the ommon trit s to it n e relted not only to those things ut lso to other similr thingsF voke ITVWD ID PH e mthemtil onept is the ssoition of mening @oneptul ontentA with signF qenerllyD one dopted y mthemtil ommunityD sign does not hngeD for instneD the nottion dxD the nottion D or the eri numerlsF fut the mening ssoited with sign my evolveF xotlyD for instneD the onept of funtion nd the sign hve now mening di'erE ent from the one tht they hd (rst in the IU th nd IV th enturiesY nd they hve now di'erent menings in set theory nd in tegory theoryF etullyD mthemtil tivity is onerned with the proesses of ontinuous trnsforE mtion of given presenttion into othersX mening hngesD 'ording new onepts for the presumed sme entityY new proedures re introdued t some point of time nd revel new spets of our most fmilir toolsD new nottions re proposed for designting the innovtive oneptsF pinllyD mthemtil entity is the pir onstituted y the ide of supposed unique sustrte desE ignted y nme nd its mny tul nd potentil spets or presenttionsD inluding the opertions nd rules of lultion set up in eh seF 7 sn other wordsD mthemtil entity is the virtul referentD supposedly ommon to similr ut distint oneptsF hedekindEeno onept of positive integers is not the sme s iulid9s oneptD even though oth refer to the more or less same entityF gonepts re formed grdullyD through reson9s indeftigle strting workD orgnizing similrities nd di'erenesD dissolving hidden links nd reE ting links tht were unnotiedF hey re not ovious to whom who is not trined in this kind of workF xot everyody knows hedekindEeno de(nition or even iulid9s de(nition of numersF ixperiene rther thn pure intuition is t workF xew insights re gined thnks to growing knowledge nd experieneF 1.3 Abstract and concrete concepts yne distinguishes sometimes strt onepts from onrete oneptsF ine ny onept results from n strting proessD wht is onrete oneptc belongs to some equivalence class A ⊂ E, we say that a is a representative for A; that means that a stands for any element belonging to A, what again has nothing to do with a subjective (mental) representation.

See

7. In my view, it is hard to isolate completely the substrate entity from the operations attached to it. From an abstract point of view operations and properties are even more important than their specic substrate. sn vtinD concretus mens mixedD ompositeD ompoundD while the vtin word abstractus mens withdrwnD tken out ofD extrted @or isoltedAD estrngedF ht is ll tht is ontined in the originl etymologiE l mening of these wordsF he rest pertins to the philosophil oneption tht is expressed through themF 1.3.1. sn one senseD onrete onept is the onept of one or mny onE rete sensile thingsX so the onepts of this tleD of one ppleD of (ve penilsD s onepts of pereived thingsF gonrete pertins to the diret sensory referents understood under oneptD while strt hints to nonEsensory referentsD whih re the result of repeted opertion of extrting generl ide from more prtiulr idesF sn this sense one usully mkes rdil ut rough di'erene etween onrete nd strtD tul nd unrelD pereptile nd impereptileF rowever in siene to e given to the senses is n unstE isfying riterion for the demrtion etween onrete nd strt entitiesX elementry prtiles re nonEsensile entities nd onrete dt of physil experimentF he prolem of (nding riterion stisfying in ny se is di0E ult oneD nd s will not undertke to solve it euseD from n epistemologil point of viewD the distintion etween strt nd onrete is reltive nd unstleX onept F my e more strt thn onept GD whih my itself e strt ut less strtD iFeFD more onrete thn F F veiniz sid in Nouveaux Essais tht onreteness nd strtness re orreltedY tht mens tht onreteness nd strtness re question of more or less rther thn question of yes or noF gognitive sientists on(rm experimentlly inE deed the grdtion of the proess eginning with diret tegoriztion on pereptul ojets nd ontinuing with tegoriztions t higher nd higher levels on more nd more strt ojetsF woreoverD n interesting view omes from results of psyhologil experE imentX onreteness is mostly ssoited with pereptul fetures of some spei( situtionD whih is generlly ught in global viewD while strtE ness points to wide rnge of diverse situtions emedding di'erent @kinds ofA entitiesD onneted in some wyD nd vriety of proesses tthed to these @kinds ofA entitiesF end it is suggested tht there is greter enggeE ment of the verbal rin @left ererl hemisphereA system for proessing of strt onepts nd greter enggement of the perceptual rin system @right ererl hemisphereA for proessing of onrete oneptsF en strt onept is understood through verlEthinking working outD onrete onE ept is visulizedX s hve either diret pereption or t lest mentl imge of tle or of (ve pplesF ht my explin how mthemtil working onE sists prtly in mking esier the ess to mthemtil onepts nd their hndling through visuliztion on the lkord or on sheet of pper or in the imgintionX we use symolsD we drw (gures nd digrmsD nd we write down lultions nd formuleF e my even mintin tht resoning nd proving through mere nlysis of symoli formuleD s in turmEviouville theE ory of di'erentil equtionsD 8 or through digrmsD s in tegory theoryD 9 re onrete hndling with strt onstrutionsF e mnipulte formule nd digrms s eing themselves mthemtil ojetsD deteting properties not eing otherwise disernedF st is known tht prodigy people who re ple to mke quikly lultions with gret numers pereive sounds or pitures emotionlly ssoited with numersF hniel mmet sys tht when he is lulting the deimls of π he sees the numerls pssing efore his eyes like the pitures of movieF 10 st seems indeed tht gifted mthemtiins see the world through mthemtil (ltersF he prenh neurosientist tnisls hehene thinks tht resumlyD one n eome mthemtil genius only if one hs n outstnding pity for forming vivid mentl represenE ttions of strt mthemtil onepts"mentl imges tht soon turn into n illusionD elipsing the humn origins of mthE emtil ojets nd endowing them with the semlne of n independent existeneF hehene PHIID PPS he irresistile lening to relist view of the mthemtil universe of onE epts nd tehniques hs its roots in the tul proess of visulizing strt proeduresF

1.3.2.
sn seond senseD onrete pertins to our usge nd triningF pmilir onepts re tken to e onrete nd intuitively @visullyA grspleD eFgFD the positive integersD whih re lled nturl numers qu eing the si representtion of the t of ountingF hus onreteness is developed or developing hrterF eording to unt9s LogicX he expressions abstract and concrete refer not so muh to the onepts themselves"for ny onept is n strt onept"s to their usageF end this usge n gin hve di'erent grdesY" ording s one trets onept now moreD now less strt or onreteD tht isD tkes wy from or dds to it now moreD now fewer de(nitionsF unt IVHHD ITD enmerk ID ISR sn this seond perspetiveD the distintion strtGonrete is lerly n epistemic distintion nd it is relative in sense di'erent from tht ment y veinizX not only strt nd onrete re orrelted oneptsD ut n E strt onept or onstrution my eome onrete or more onrete nd it my e visulized through symol or imge or digrm stnding mterilly for itF ht mens in ft tht we my formD through some kind of drwingD onrete representtions of strt oneptsF Hourya Benis Sinaceur regel @IUUHEIVQIA introdued importnt re(nements in the distintion onE reteGstrtF re ssumed tht ny onept is lwys strtD ut he dded tht a genuine concept is not only strtD ut lso onreteD in the sense tht its de(nitions @wht old logi lls feturesA re omined in it in single omE plex expressing its individul unityF e onept is onrete euse it ontins ll the ontent of its genesis within itF fy ontrst immedite pereption is strt in the sense tht its determintions remin undevelopedF11 e onept is the onrete unity of di'erent determintionsF hus the onreteness of onept lies in the meaningful cohesion of its feturesD whih my e developed t di'erent moments of timeF por instneD out of ontextD verl de(nition is strt nd strt onlyF smmersed into the ontext of sienti( theE oretil disourseD ny strt de(nition eomes onrete @in n epistemi senseAF he onreteness of onept is therefore lwys expressed through unfolding ll its possile de(nitionsGfetures in their mutual connections rther thn through n isolted de(nitionD nd in immersing the onept into we of interonneted oneptsF st is s to sy tht )esh is given y the mutul onnetions etween di'erent fetures of the onept under onsidertion nd y the links with other oneptsF uh onsidertion my well e pplied to mthemtisX the imge of dense network for representing the mthemtil stu' hs eome ommonple y nowF 2 Mathematical practice 2.1.

wthemtil onepts my generlly e introdued or de(ned in difE ferent wysF he more presenttions Darstellungen onept hs nd the more it is emodied y di'erent proedures performed in di'erent resD the more onrete it is tken to eF his my hppen through two wysF A hen onept is repetedly used in di'erent ontextul theoriesD eFgFD when we add numersD vetorsD vetor spesD etFD we get meaninggenerality @semnti generlityAD whih is n extensive generalityD trnsversl generlity of useF sn the sme wiseD we use produts for veE tor spesD groupsD topologil spesD fnh spesD utomtD etF sn eh se we hve to tell whih properties mong ll the possile properE ties of the opertion + or ×D suh s ommuttivityD ssoitivityD etFD re preserved nd whih must e droppedF he fewer re the properties onE sideredD the greter is generlityF he very generl onept of ddition is illustrted y the struture of monoidD whih is instntited y so mny di'erent modelsF he onreteness omes from the repeted use under rules spei(ed in eh seF A he more onnetions onept hsD tully or potentillyD with other oneptsD the more intrite is its own meningF hrough its meaningcomplexity @semnti omplexity or rihnessA 12 onept gets some kind of onretenessD it looks like n individuted entity euse severl operE tions long with their properties re omined under itF gonreteness here is tken in regel9s senseF sn ontrst with the trditionl rtio etween extension nd intension of oneptD it is not the se in mthemtis tht inresing mening omplexity entils deresing mening generlityY for instneD tegoryD let us sy Grp is t the sme time more generl nd more omplex mthemtil ojet thn the group strutureF husD nlysis of mthemtil strtion does not give the sme results s the trditionl grmmtil or logil nlysis of onept formtionF 2.2.

gretive mnipultion of mthemtil onepts pertins their menE ingD not just their nmes or nominl de(nitionsF xmes designte thingsD while onepts ondense mening even when they pper t (rst sight very strtF por instne rel numers my seem so strt tht their mathematical existence is hllengedF hey reD indeedD rejeted y some onstrutivist mthemtiinsX eFgFD insted of speking of real roots of n lgeri equE tionD uroneker onsidered intervls ounded y rational quantitiesD rtionl quntities eing constructed y (nite numer of opertions from the inteE gersF roweverD there is lrger notion of onstrutive existeneD s it ws mde expliit y rermnn eylD who rgued tht we re entitled to lim tht there exists n α only fter hving instantiated α eyl IWPID SR!SSF sn this viewD rel numers exist sine we hve enountered instnes of them @eFgFD rtio of the side of squre to its digonlD πD the se e of the nturl logrithmAF he concept of rel numerD though strt in the doule sense tht we n neither survey all its individul instnes nor hve (nite lultion for eh instneD needs not to e elimintedY we rightly reson with the concept of real number s setD olletionD nd s domin equipped with more thn only one structureF utting struture on set is stipulting reltions nd opertions @funtionsA etween the elements of the set nd stipE ulting rules for working with themF sn ddition to lgeri strutures suh s groupsD ringsD (eldsD modulesD vetor spesD etFD we hve order struturesD metri struturesD topologiesD di'erentil struturesD tegoriesD mong othersF he structural omplexity of the rel numer system emerged grdully @nd minly in the IW th enturyA through suessive strtive opertionsD 12. I thank one of the referees who suggested to use richness rather than complexity. I am taking indeed meaning-complexity not as a kind of mathematical complexity, algorithmic or measurable in some other way, but as an expression for the polysemous character of many, if not all mathematical concepts. Number, for instance, has dierent meanings depending on whether you consider integers or rational numbers, or real numbers, or quaternions, etc. The polysemous character of mathematical concepts and symbols has been put to the fore by the rise of abstract axiomatics (comments might be found in [Benis Sinaceur 1991]).

92

Hourya Benis Sinaceur disentngling di'erent strutures tht were mixed togetherD dissoiting espeE illy topologil notions from lgeri opertionsD from the order reltionD nd from the metriF es erer of severl struturesD the rel numers pper compound nd multiEfetedD just like individuted physil ojetsF etullyD the set of rel numers rries the following stndrd struturesX ! n orderX eh numer is either less or more thn every other numerF ! n lgeri strutureX multiplition nd ddition mke it into (eldF ! mesureX intervls long the rel line hve spei( lengthD whih n e extended to the veesgue mesure on mny of its susetsF ! metriX there is notion of distne etween pointsF ! geometryX it is equipped with metri nd is )tF ! topologyX there is notion of open setsF wore signi(nt is the possiility of hyrid struturesY for instneX ! the order ndD independentlyD the metri struture indue the stndrd topologyD ! the order nd the lgeri struture mke this set into totlly ordered (eldD ! the lgeri struture nd the topology mke it into vie groupF ht mtters with strutureD tht ws lled onept y qermn mthemtiins of the qöttingen hoolD is tht it provides us with new strt oneptD ndD t the sme timeD it gives more determined mening to the underlying set of spei(ed or unspei(ed elementsF st is to sy tht strtion rings riherD not poorer meningD even for more generl onE eptsF sn other wordsD struturl omplexity rings simultneously syntti nd semnti rihnessF es FvFyF uine stressed mny timesD the retion of strt onepts is semnti sentD 13 whih goes hnd in hnd with the syntti sentF 2.3. husD we oserve in mthemtis something whih is lose to regel9s desription of strt nd onreteF ht is of onern to us in this deE sription is tht it develops further unt9s epistemological distintion etween strt onept nd onrete oneptF eording to untD very strt onepts give little informtion out mny thingsD while through onrete onepts we know muh out few thingsF 13. The semantic ascent is the shift from talk of miles to talk of mile, it is what leads from the material (inhaltlich) mode into the formal mode, to invoke an old terminology of Carnap; [...] The strategy is one of ascending to a common part of two fundamentally disparate conceptual schemes, [Quine 1960, 271272]; [Quine 1990, 33].

Facets and Levels of Mathematical Abstraction 93 o ount for the fruitfulness of mthemtisD unt rgues tht mthemtE il knowledge proeeds in concretoD iFeFD presents the onept into pure nd a priori ut singular intuitionF hus the division strtGonrete integrtes the division generlGprtiulr nd the division lssGindividulF unt tells tht the onrete usge of onept is tht whih is most lose to the individulF fy ontrstD regel onsiders not only the form of knowledgeD ut lso its ontentD nd he dethes the onreteness from its referene to relEworld existent individulX we my hve very strtD very poor knowledge of n individul or of singulr situtionD while oneptD s produt of knowlE edgeD is n evolving concrete unityD whih my get more nd more mening determintions ndD thenD eome more nd more onreteF e thus go from abstract to concrete nd not vie versF ht mtters is how muh nd via how mny wys or viewpoints we know out something t some point of timeY wht mtters is the knowledge-contentD the inresing rihness nd the progressive diversi(tion of knowledgeF Knowledge-content is semantic content in its historical dimensionF estrtness nd onreteness re not (xed forms of the sujetiveD empiril or trnsendentlD t of knowingD they re hrteristis of knowledge as suchD of knowledge s historical and objective produt of olletive tivityF en importnt gin of tht view is tht it is now ler tht the division strtGonrete oinides neither with the division generlGprtiulr nor with the division lssGindividulF 2.4. vet us return to mthemtil prtieF sf n individul thing @pheE nomenonD ftD entityD oneptD proedureD theoryA is not understood through the onrete interonnetion within whih it tully emergedD existsD nd developsD tht mens tht only abstract knowledge hs een otinedD eFgFD when one hs lerned wht is group y lerning the group xioms without knowing the ontext of their emergene @historyA nd t lest some of the difE ferent situtions where they n e pplied fruitfully for reveling the struture of domin or suggesting solution for some prolem @tul prtie nd prolemEsolvingAF sullyD lgeri onepts produe knowledge when they re tied to fts nd prolems elonging to other mthemtil resD rithE metiD geometryD nlysisD topologyD etFD or elonging to n erlier stge of the lgeri trend itselfD s it isD eFgFD the se for the onept of groupF sfD on the other hndD n individul thing is understood in its ojetive links with other things forming oherent networkD tht mens tht it hs een understoodD relizedD knownD oneived concretelyF sn suh perspetive we n underE stnd how we my hve concrete knowledge of highly strt oneptD s it hppens espeilly in modern mthemtisF hus n strt onept eE omes onrete not only through its instntitions @reliztionsD modelsAD ut lso through the theories in whih it plys roleD iFeFD through the theoretil or tehnologil pplitions following from itD nd still through the theories to whih it gives irth y eing inluded in more generl strt oneptF por instneD the lgeri onept of group is mde onrete IA through its emodiment in rithmetil nd geometril modelsD PA through its use to repE resent symmetry in physis nd to lssify rystl strutures in hemistryD nd lso QA through the tegoril onstrution of GrpF 3 Descriptive analysis of the fundamental thinking processes underlying the main ways of getting abstraction he title of this setion seems mitiousF roweverD s must sy tht sine s m no expert in ognitive sienesD s m essentilly relying on more or less diret nlysis of tul mthemtil proedures omined with informtion got in ognitive sientists9 redingsF gognitive sientists nme tegoriztion ny kind of tivity tht involves ssoitionD omprisonD nlogyD nd orresponE dene etween two or more thingsF s will detil the tions performed in suh n tivityD whih is in ft the tsk of getting strt idesD from the most simple to the most sophistitedF

3.1.
estrting is result of severl overlapping or intertwined thought opertions tht s desrie nowF ! gonsidering thingsD not neessrily physil onesD not neessrily loted in spe nd timeF ! gompring things not in themselves ut sub specie generalitatisD iFeFD ompring them s possile smples of something elseD something whih is not neessrily lredy known ut only glimpsed nd still reltively vgue or fuzzyF reision omes lterF ! eleting one or severl spets @qulitiesD propertiesD preditesA in the things sumitted to omprison nd presumed to hve something in ommonD then presumed to e classed @susumedA under some conceptF ! veving side or disrding ll other spetsD espeilly spei( sustnE til or speEtime spetsF his opertion hs een lled ideliztion euse it omes down to extrting form from sundry situtionsY it hs een espeilly promoted in the eginning of the PH th entury y strt lger nd strt topologyD whih mde fmilir the study of strutures not qu eing ssoited with ny spei( instneF sdeliztion follows from seeing or guessing some invariant si properE ties tthed to plurlity of pprently heterogeneous situtions nd it leds to unifying view of the di'erent domins on whih we perform the sme type of opertionsX ountingD dditionD sutrtionD ompti(E tionD etF sdeliztion hs lso heuristic role in suggesting possile

Facets and Levels of Mathematical Abstraction 95 or unexpeted onnetion with sitution not hving een onsidered t (rstF he extrted form is not rigidY it my e 'eted y some onE trolled vrition in pssing from ertin type of situtions to nother oneX the ddition of two vetor suspes di'ers from the ddition of let us sy two rel numersF ! ssolting some property or some set of properties of the opertion@sA unE der onsidertion nd viewing them on their ownD iFeFD trnsforming the seleted onjuntion of predites into thoughtEojet @prege9s rdiE l seprtion etween onept nd ojet does not (t mthemtil prtieAF eire lled this kind of trnsformtion re)etive or hyE postti strtionD russerl lled it themtiztionF gvillès popuE lrized the term themtiztionD t lest mong prenh philosophersF hemtiztion is espeilly importnt in onsidering s whole n innite olletion of thingsY it plyed fundmentl role in the emerE gene of set theoryD nd it hs een onsistently odi(ed within di'erent frmesX ussell type theoryD ermeloEprenkel system @pA nd uine9s system @xpAF hemtiztion is essentil in pssing from set S of elements to its possily mny strutures nd from the study of struture Σ on the set S to the study of the struture Σ in its own rightD iFeFD to the study of lss of homomorphisms etween strutures of the type ΣF he stndrd exmple is given y the pssge from hedekind9s xiomtis for numers nd rilert9s xiomtis for geometry to immy x÷ther9s style of studying lsses of group9s homomorphismsD lsses of ring9s hoE momorphismsD etF ettention is pid to homomorphisms rther thn to the sets tht re respetively soure nd trget of themF st is tht ttitude tht hnged the fe of lger @see ertin IWTPD SSS nd eyl IWQSD RQQA opening up wide domin of reserh nd new stu' for developing new insights nd new proedures typil of the begrifiche Mathematik D 14 whih ws understood s the study of lgeri or topologil strutures onsidered in nd for themselvesF hemtiztion plys lso role in trnsforming n strt ojet @prediteD onept mthing mny items possessing similr struturesA into onrete ojetD whih eomes element of some lrger lssD eFgFD the struture of elin groups viewed s n element of the tegory of groupsF hemtiztion is still involved in nlyzing onept y reking down its glol unity into omponents tht were formerly tightly onnetedF enlysisD in this hemil senseD omes out t ideliztion nd themE tiztionY it is dismigution of mening y dissoiting nd studying seprtely hrtersD whih hve een intuitively ssoited during enturiesF st wsD eFgFD the se when iemnn showed @IVSRA tht not

96

Hourya Benis Sinaceur every spe is metri spe or when hedekind @IVUP or even soonerA showed tht not every spe is ontinuous speF hus the onept of spe eomes very generlD divested from ny prtiulr propertyD nd simultneously sujet to di'erent spei(tionsF fy spe we unE derstnd now ny set of elements tken s sustrte for some seleted reltions nd funtions nd s spei(tions we get new subclasses of oE jetsD in our exmples the sulss of metril spes nd the sulss of ontinuous spesF wore generllyD y iterted themtiztion one pushes further mthemtE il oneptul onstrutionsD s it is well illustrted y tegory theoryD whih is theory of systems of struturl theoriesD treting the notion of struture in uniform mnnerX eFgFD sets nd usul funtions form the tegory of sets @SetAD groups with groupEhomomorphisms @whih preserve the groupEstrutureA form the tegory of groups @GrpAD topoE logil spes nd ontinuous funtions @whih preserve the topologil strutureA form the tegory of topologil spes @T opAF estrting ginD funtors re strutureEpreserving mps etween tE egoriesF puntors @rrowsA re the very ojets of tegory theoryY they elong to higher level of strtion thn morphismsD whih in their turn re on higher level of strtion thn mpsF fy studying tE egories nd funtorsD we re not just studying lss of mthemtil strutures nd the morphisms etween themY we re studying the relationships between various classes of mathematical structuresF his is fundmentl ideD whih (rst surfed in lgeri topologyF erhing for generl invrints mkes up the dynmi onstrution of new lyers of sophistited strtion proessesF estrting yet ginD ntE url trnsformtion provides wy of trnsforming one funtor into nother while respeting the internl struture @iFeFD the omposition of morphismsA of the tegories involvedF 15 reneD nturl trnsformtions 15. If F and G are functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism η X : F (X) → G(X) between objects of D, called the component of η at X, such that for every morphism f : X → Y in C we have:

η Y • F (f ) = G(f ) • η X
This equation can conveniently be expressed by the commutative diagram:

F (X) F (f ) -→ F (Y ) η X ↓ ↓ η Y G(X) -→ G(f ) G(Y )
The notion of a natural transformation states that a particular map between functors can be done consistently over an entire category. Informally, a particular map, let us say an isomorphism between individual objects (not entire categories) is referred to as a natural isomorphism, meaning implicitly that it is actually dened on the entire category, and denes a natural transformation of functors; formalizing this intuition was a motivating factor in the development of category theory.

Facets and Levels of Mathematical Abstraction 97 n e onsidered to e morphisms of funtorsY they yield the usul hoE momorphisms of strutures in the trditionl set theoretil frmeworkF end so onFFF he point is the endless dynamical ontention of polysemous symols nd symoli opertionsF yf ourseD this ontention is not neessrily linerY it forms kind of tree with interweved rnhes t the sme level nd from lower to higher levelsD orD s sid oveD omplite nd dense networkF ! enlogies re onurrent with ideliztion nd themtiztionF etting upD guessingD or looking for nlogy 16 etween sundry situtions is min wy to ring to light similritiesD di'erenes nd possile reltions etween two or severl thoughtEojetsF gomined with ideliztion nd themtiztionD nlogy is si onstituent of strtionF yne mkes sometimes distintion etween nlogy nd strtionF qrounding on the emergene of strt group theory from IA the theE ory of lgeri equtionsD PA numer theoryD nd QA geometryD nd on the oneption of modern lger s the study of lgeri strutures whih me after IA strt group theoryD PA strt (eld theory nd QA strt ring theoryD tenEierre wrquis rgues tht it is n emE piril ft tht nlogy onerns two thingsD while strtion omes only when three or more things re onsidered wrquis forthomingD S!TF esoning y nlogy is indeed trnsferring informtion or menE ing from prtiulr sitution to nother prtiulr situtionF e good exmple is given y tFEF wrquisD nmely hedekind9s nd eer9s work on lgeri numer theory nd lgeri funtionsF enother exmple is the trnsfer of lgeri lws nd tools to logi in the works of qF fooleD eF de worgnD iF hröderD etF estrtion omes in ply when severlD nd not only twoD domins of entities or severl lsses of strutures re a priori in questionF sndeedD t (rst step theory is strt when it hs a priori a plurality of modelsF he plurlity riterion is indeed ommonly used to distinguish etween onrete or mteril xiomtis nd strt xiomtisD eFgFD etween iulid9s geometry nd rilert9s xiomtiztion of grtesin geometryD 17 whih permits to onstrut di'erent geometri models y seleting di'erent sets of xiomsF et seond stepD domins of entities re negletedD while one onsiders a priori plurlity of strutures long with their spei( struture preserving morphismsF futD even in the erlier stge of onsidering similritiesD di'erenes nd reltions etween only two situtions elonging to the sme domin @or only two domins of di'erent entities or only two struture typesA is 17. [Hilbert & Bernays 1934-1939, 20]. In Hilbert's and Bernays' terms the distinction is between inhaltliche und anschauliche Axiomatik and formale Axiomatik.

98 Hourya Benis Sinaceur involved the impliit ssumption tht it must e some strt frmeE work in virtue of whih the trnsfer from one sitution to the other or from one domin to the other or from one struture type to the other is possileF enlysing how nlogy worksD renri oinré writes tht the mthemtiin must hve diret insight of wht mkes the orE gni unity of sundry situtionsF 18 enlogy s guide for mthemtil invention nd for gret produtivity with eonomy of thought is the hief theme of oinré9s tlk t the IWHV snterntionl wthemtil gongressF eording to oinréD the ruil step is the pssge from mteril to forml 19 nd from diversity to uni(tionX nlogy etween mterilly di'erent entities or proedures ppers when one seesD onE strutsD or supposes forml similrity etween those entities or proE eduresF 20 porml similrity hints to unique mouldD whih my serve for prediting or (nding out unexpeted nlogies with new items nd whih my thus led to more preise view of the rhiteture of the whole ody of mthemtisD s it hppened with the onept of groupF husD serhing fter nlogies involves n strting mindD if not yet systemti use of the strt methodF 21 here is tully kE ndEforth ply etween nlogy nd strtionX setting up nlogies leds to oneive of n strt theory ndD one n strt theory is t hndD it is used to unerth more nd deeper nlogiesF 22 18. [Poincaré 1900, 127128]. Notice that Poincaré used unity rather than identity.

19. Poincaré is using the term formal as the opposite of material and he underlines the important role of language in discovering new analogies between domains sundry at rst sight, but he does not mean a logically formal language, as it is meant in Hilbert's and Bernays' Grundlagen. 22. Saying analogy or similarity is not saying identity. While mathematicians are using analogies to set up isomorphisms between sets or equivalence between categories, some cognitive scientists are using the mathematical concept of isomorphism for giving a theoretical explanation of analogy (see e.g., [Gentner 1983, 155170]).

Facets and Levels of Mathematical Abstraction 99 3.2. elthough s hve tken exmples minly from modern mthemtisD it must e stressed tht strtion is there from the very (rst eginningF iven the most elementry notions of mthemtis re strtX the notions of numE erD of retngle or tringle or irleD etFD re strt notionsD iFeFD produts of strting proessesF por instneD whole positive numers result from severl strting proessesX ssoiting symol with olletion of tul thingsD dissoiting this symol from this prtiulr olletion nd ssoiting it with ny olletion of the sme numer of thingsD then estlishing oneEtoEone orE respondene etween mny di'erent olletionsD omining this symol with other symols similrly generted in order to perform opertions like dditionD multiplitionD nd so onF st is only through long hit tht we onsider positive integers s given intuitive concrete ojets nd geometril (gures s onrete sptil visuliztions supporting the proof proessF estrtion is lwys there nd is n ongoing proessD eoming more nd more sophisE titedF es ghF F eireD iF russerl nd tF gvillès rguedD strtion is onstitutive of mthemtil thinking nd it n e repetedly exempli(ed in the proesses of idelizingD themtizingD extrting invrintsD nd setting up nlogiesF he more dvned the strtion proessD the more onrete the strt ojets eome"lssesD struturesD opertions s suhD funtions s suhD morphismsD tegoriesD etF hus it is not prdox to think thtD in mthemtisD higher levels of strtion produe more nd more onrete thoughtEojetsD onrete in the doule sense tht they re omplexD individE uted ojets with vrious determintionsD nd tht they eome onretely known nd mnipulted through symoli formultionD preise digrms or even skethy drwingsF gonrete mens simultneously polysemous 23 nd dily hndledF 3.3. esent towrds strtion is not limited to the logil proess of subsuming prtiulrs or prtiulr ides under more generl oneptF xotly prege rejeted the eristotelin ἀφαίρεσις s eing not the only sort of logil strtion 24 nd he dissented from the trditionl view on oneptsY he used mthemtil toolsD nmely funtionl reltion nd n eqution for stting puttive logil de(nition of the onept of numerF sn most elementry ses indeed mthemtil onept enompsses more thoughtEproesses thn only the logil susumptionD to whih orresponds the setEtheoreti opertion of inlusionF sn prtie mthemtiins re deling with mny sorts of operE tions nd lultions nd mny sorts of reltionships etween strutured sets @oneEtoEone orrespondene nd equivlene reltion s in the so lled rume9s prinipleD 25 liner trnsformtionsD group homomorphismsD morphismsD etFAF 23. On the polysemy or ambiguity of axiomatic concepts, see e.g., [Benis Sinaceur 1991, 191196].

24. Actually, Frege thought that Aristotle's analysis was psychological.

25. The name Hume's principle was coined by George Boolos. This principle plays a central role in Frege's denition of numbers, and it says that the number of Hourya Benis Sinaceur usumption is fundmentl level of lssi(tionD ut in mthemtis fruitE fulness nd new insights result from omining it with other strtion proE essesD s prege mde ler in his seminl reform of logi nd s s shll illustrte elow y some mthemtil exmplesF s m not sying tht the sent toE wrds strtion is not logil sent from step to stepF s m just sying thtD in mthemtil prtieD t ny stepD genuine mthemtil stu' (lls the logil moveF his is why s hve stressed hereinove tht the sent is t one semnti nd synttiF wthemtil strtion is mnyEfeted nd multiEleveled proess nd it leds to sophistited nd rnhed hierrhy of mthemtil onepts nd opertionsF woreover it is not lwys the se tht the more strt onept is the more undetermined it isF por instneD with just generl onept of set s olletion of ny things one does not go frF sf one wnts tul nd e'etive workD one must egin y menE ing determintionD iFeFD y setting up the xioms ruling onsistent usge of the oneptF st hppens often tht the more strt is struture the more overdetermined nd strti(ed it isX xiomtis nd tegory theory give mny exmplesF he mthemtil rnhing of onepts is simultneously ompliE tion of onepts tken in isoltion nd lri(tion of their mutul linksX ringing to light new nd new strutures gives more nd more power to solve prolems not one y one depending on their prtiulrities ut uniformly in one go grounding on the generl struture (tting ll of themF 3.1 Abstraction and axiomatization e rpid look t the history of mthemtisD espeilly of modern mthemtisD shows tht strtion is losely tied up with symoliztion nd xiomtizE tionF wthemtil thinking is thinking with nd on symols nd digrmsD my they e onsidered s representtions or s themselves mthemtil oE jetsF enywyD retive mnipultion of symols nd digrms does not dwell only on their drwingsY it pertins their menings nd meningful onnetions with other symols nd digrmsF estrt onepts @strt struturesA re usully de(ned y (nite set of xioms tht stte the reltions to e stisE (ed y ndidtes for eing models of those strt oneptsF fut strt onepts need not to oinide in every respet with their less strt ounterE prtsY the mening hnges in etweenD 26 it eomes more shrply determined nd yet more miguousX not every group is elinY the multiplition of F s is equal to the number of Gs if and only if there is a one-to-one correspondence (a bijection) between the F s and the Gs. Facets and Levels of Mathematical Abstraction 101 integers is symmetriD omposition of permuttions of three ojets is notY in tegory theory the term struture hs not extly the sme mening s it hs in set theory nd in model theoryX strutures of strutures do not lwys redue to strutures of elements @see ewodey IWWTAF here re relly di'erent levels of strtionD even if there re onneting pths etween levelsF wthemtiins di'erently oriented hve reognized xiomtiztion s n e'etive tool for understnding nd inventionX hedekindD rilertD immy x÷therD nd imil ertinD ut lso oinréD eylD nd frouwerD 27 who did not rejet the use of the xiomti method ut rther the view tht it might provide foundtion or dispense with lultion nd lgorithmi proofsF yne must distinguish etween xiomtis s fruitful mthemtil method nd xiomtis s puttive foundtion or useless mthemtil ideologyD whih is n epiphenomenon hrmful in tehingF sn prtieD it would e surd to go without the xiomti ontriutionsX for instne qlois9 theory hs een deeply nd e'etively understood only fter hedekind9sD eer9s nd ertin9s xiomti presenttionsF orking with xioms develops new insights nd idesX notly the study of tegories is n ttempt to axiomatically pE ture wht is ommonly found in vrious lsses of relted mathematical structures y relting them to the structure-preserving functions etween themF e systemti study of tegory theory then llows us to prove generl results out ny of these types of mthemtil strutures directly from the xioms of tegoryF wthemtis is lwys iming t more nd more generl results out more nd more omplited struturesF elthough xiomtiztion plys now n indispensle role in mthemtil prtieD it is not the only wy to mke mthemtil proedures strtF s will now give nonEexhustive list of other mthemtil strtion proesses tht interply in mthemtil thinking nd tully illustrte the unesing itertion of intertwining proesses of setting up invrintsD idelizing entities nd proeduresD trnsforming opertions into ojets @themtizingAD ringing to light nlogies etween setsD struturesD tegoriesD etF 4 Various samples of mathematical abstraction processes 1.

epresenting n innite numeril sequene y its law of reurreneF yne gets the lw y disrding onrete calculation nd retining only how one psses from any element n to its suessorF yne does not tully know ll the elements of the sequene ut one knows how to generate the sequeneF rere it mtters of (nding out rule of lultionD not oneptD ut the rule 27. See e.g., Poincaré's praise of the concept of group [Poincaré 1908]; Weyl, [Weyl 1932, 349] and [Weyl 1951, 464]; Brouwer's conception of geometrical method [Brouwer 1909].

Hourya Benis Sinaceur dispenses with enumerting ll the elements of the sequene like the onept of even integers dispenses with enumerting ll the multiples of PF 2.

hisrding the spei( nture of the elements forming sequeneD eFgFD the sequene of positive integersD so s to hrterize the order type of the sequeneF por tht hedekind invented the onept of chainF 28 ht mtters here is neither the integers themselves nor even their genertive lw y itselfD ut the ordering generted y this lw @liner disrete orderAF he level of strtion is higher thn in the exmple ID euse we re not onerned with prtiulr lultion lw vlid for one prtiulr sequene ut with lw type generting n order struture suitle for integers nd for sequenes of unspei(ed elements s wellF hedekind9s de(nition shows tht integers re prtiulr instntition of generl strutureY it indites one wy of linking strtion nd generliztionF 3.

gomining opertions @+D ×D etFA nd seleting properties of these opE ertions @ssoitivityD ommuttivityD etFA in order to form di'erent kinds of mthemtil strutures @oneptsAX groupsD (eldsD ringsD idelsD lttiesD vetor spesD tegoriesD etFD tht onnet models originting from di'erent mthemtil res or di'erent struturesF estrt onepts re multiply in-stantiatedD they de(ne not one single model nor single strutureD ut classes of models and classes of structuresF his kind of strtion is relly modE ernD in ontrst with iulid9s xiomti system for geometryD whih onerns one single model @the rel threeEdimensionl speA of one single struture @the struture of iuliden speD relized for instne y the vetor spe R n with the stndrd inner produt nd y the vetor spe of rel polynomils of degree ≤ n with onvenient inner produtAF sn the spirit of rilert9s nd fernys9 distintion one sets iulid9s mteril system in ontrst with rilert9s system in Die Grundlagen der Geometrie @IVWWA or hedekind9s sysE tem for rithmeti hedekind IVVVD whih re strt systems @informlly presentedAY moreover one mkes di'erene etween hedekindGrilert9s style nd immy xoether9s style of strtionF sn the perspetive of strt set theory we re usingD for instneD the following termsD whih mostly ppered in the IW th enturyX ! strt setD whih surfes in gntor9s mtured theoryD ! strt groupX hedekind reognized similrities mong vrious mthE emtil struturesD like rottions nd quternionsD nd identi(ed them s instnes of the strt notion of group hedekind IVSSEIVSVD RQWF reinrih eer gveD in IVVPD xiom systems for groupsD nd lter on 28. A chain is the minimal closure of a set A in a set B containing A under a function f on B (where being minimal is conceived of in terms of the general notion of intersection).

Facets and Levels of Mathematical Abstraction 103 these xiom systems hve een formlized nd investigted in their own rightF ! strt numerD whih ws usedD for instneD y folzno in the sense of numer s single entity nd in ontrst with onrete numerD whih is numer ssoited to the things eing ounted folzno IVSIF e owe the strt xiomti hrteriztion of the sequene of positive integers to hedekind through the de(nition UQ of hedekind IVVVX sf in the onsidertion of simply in(nite system N set in orE der y trnsformtion φ we entirely neglet the speil hrE ter of the elementsY simply retining their distinguishility nd tking into ount only the reltions to one nother in whih they re pled y the orderEsetting trnsformtion φD then re these elements lled nturl numers or ordinl numers or simply numersD nd the seEelement I is lled the seEnumer of the numerEseries N F ith referene to this freeing the elements from every other ontent @strE tionA we re justi(ed in lling numers free retion of the humn mindF he reltions or lws whih re derived entirely from the onditions αD βD γD δ in @UIA nd therefore re lwys the sme in ll ordered simply in(nite systemsD whtever nmes my hppen to e given to the individul elements @ompre IQRAD form the (rst ojet of the siene of numers or rithmetiF ! strt (eldX this struture hs een de(ned y teinitz teinitz IWIHF ! strt speX it surfed in iemnn9s fmous pper iemnn IVSRD where topology nd metri for spe E is de(ned before de(ning the funtions hving their rguments nd vlues in EF prom IWIR onE wrds @rusdor' IWIRA it ws known tht topologil spe ws set strutured y lttie of open susetsF fut it ws not until the middle thirtiesD with the work of wrshll tone @IWHQEIWVWA on the topologil representtion of foolen lgers nd distriutive ltties tht this onnetion etween topology nd lttie theory egn to e exE ploitedD nd it eme ler tht it is possile to onstrut topologically interesting spes from purely algebraic dataF ! sn the tegoril perspetive we re using morphismD whih is the strt generliztion of strutureEpreserving mppings etween two mthemtil struturesF sn set theoryD morphisms re funtionsY in liner lger they re liner trnsformtionsY in group theoryD they re group homomorphismsY in topologyD they re ontinuous funtionsD in mnifold theory they re smooth funtions @funtions hving derivtives of ll ordersAD nd so onF Hourya Benis Sinaceur 4.

glssifyingX his tion my e diret s when one ollets elements in set"y the wy it is epistemologilly meningful tht for olleting uroneker sid begriich zusammenfassen D expression tht might hve een ome under hedekind9s penD while gntor used the presumptive ontologil zusammensein"D or when one ollets mteril @interpretedD emodiedA strutures under the hed of n strt struture of whih they re modelsD or when one ollets strt strutures in tegoryD or when one rnks tE egories under di'erent typesX elin tegoriesD grtesin losed tegoriesD omplete tegoriesD toposD etF e more strti(ed tsk onsists of dividing set in lsses of equivlent elements 29 nd mking up the identity of lss from the equivlene of its memersX quotient groupD quotient ringD quotient (eldD etF iquivlent guhy sequenes of rtionl numers re identi(ed for de(ning the onept of rel numerF imilrlyD prege used the proess of forming equivlentD nmely equinumeril lsses for dening positive rdinl numersF ussell nmed this kind of de(nition the strtion prinipleY it is the sujet of mny philosophil re)etionsD ut in mthemtis even though it is systemtilly usedD it is only one strtion prinipleD only one way to perform strtionD nmely forming quotient struture of some given strutureF sn prtiulrD this wy must not e onfused with those listed in PF @order strutureA nd QF @lgeri strutureAD whereD onsidering mteril strutureD we do not strt y de(ning n equivlene reltion on the underlying set of elementsD ut we onsider the shemti struture itselfD independently of the mteE ril elementsD nd exmine whih omptile reltions my e mthed for hrteriztionF uh struturl de(nitions were not welome in prege9s oneptionF prege9s strtion priniple ws not mthemtil noveltyY the novelty lied in introduing typil mthemtil reltionD the oneEtoEone reE ltionD within the sope of logi nd presenting this reltion s logil tool for de(ning oneptF wore generllyD the equivlene reltion is involved in lssi(tion theoE remsD whih nswer the questionX ht re the ojets of given typeD up to some equivlenec ixmpleX the edderurn theorem @IWHVAD whih sttes tht every simple ring tht is (niteEdimensionl over division ring @ simple lgerA is mtrix ringY it is wy to unify the rel numersD the omplex numersD the quternions nd the squre mtries under the sme strutureF imil ertin lter @IWPVA generlized this result to the se of ertinin rings @rings stisfy the desending hin on idelsAF everl levels of strtion re rossed from the strt onept of ring to ertin9s theoremF enother fmous exmple is the lssi(tion of (nite simple groupsX every (nite simple group elongs to one of four lsses @yli groupsD lternting groupsD lssil vie groupsD spordi simple groupsAF sn tegory theory equivlene is very essenE tilX one resons on equivlent tegoriesD iFeFD tegories relted y funtor 29. Elements of an equivalence class satisfy a relation, which is reexive, symmetric, and transitive.

F D whih hs n inverse GD ut the omposition of F nd G is not neessrE ily the identity mppingY thus equivlene of tegories is less restrited thn isomorphism of tegories nd llows to trnslte theorems etween dierent kinds of struturesF 5.

glssi(tion is down top proessF qoing top downD the onverse tion is lso wy to show the struture of n entity or proedure y reking it up into simple pieesX eFgFD reduingD ftorizing numerD polynomilD n idelD in order to unerth the sing uilding loksF qenerlly ftoriztion nd lssi(tion lend together for produing wht ws nmed struture theorems in the IWQHsF 30 por instneD uroneker proved tht every (nite elin group is uniquely presented s diret produt of yli groups of prime power orderY this theorem pplies to qlois9 theoryD to numer theoryD nd to other theoriesY it is generlized to (nitely generted elin groups nd to (nitely generted modules over prinipl idel dominY 31 in the ltter se the struture theorem roughly sttes tht (nitely generted modules n e uniquely deomposed in muh the sme wy tht integers hve prime ftoriztionF ht shows deep onnetions etween rithmeti nd lgerX historilly tht ws result of the projetD shred y uronekerD hedekind nd eerD to rithmetize lgerD iFeFD to ring to light the nlogy etween divisiility of the integers nd divisiility of idels in ringF 6.

hinking in terms of functional reltionD so s to mke room for esE tlishing other identity reltions thn equlity of elements of some setD or equinumeriity etween di'erent setsD or isomorphisms etween distint modE els of this or tht strutureF sn set theory one ssoites frequently n element a elonging to set S to n element α elonging to set ΣD nd one resons on α s representtive for aF elthough one my desrie this proess y syE ing tht it onsists in seeing a as n αD one must undersore tht wht is t stke is not the mentl ontent of n ide VorstellungD whih would onsists in psychological ssoition of α with aY wht is t stke is the presenttion Darstellung of something s something di'erent ut similr in some respetD more extly the functional ssoition of α with aD whih mkes α = f (a)F st my hppen tht it is muh esier to get results y resoning on the imge α rther thn diretly on the soure element aD nd then to ome k to a djusting the otined resultsF hedekind sw very fundmentl wy of mthemtil thinking in the ility of the mind to relte things to thingsD to 30. This expression was commonly used; one can nd it for instance under Helmut Hasse's pen [Hasse 1931, 496] (see [Benis Sinaceur 1991, 187191]).

31. Principal ideal domains (PID) behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor, although it may not be possible to nd it using the Euclidean algorithm.

Hourya Benis Sinaceur let thing orrespond to thingD or to represent thing y thing FFFDinge auf Dinge zu beziehen, einem Dinge ein Ding entsprechen zu lassen, oder ein Ding durch ein Ding abzubilden F sndeedD rel numer x is ssoited with ertin lss of equivlent guhy sequenes (xn) of rtionl numersD rtionl numer p/q my e identi(ed with the equivlene lss of the orE dered pirs of integers (p, q) with q = 0D modulo the reltion (p, q) (p , q ) i' pq = qp D etF wore generllyD when struture A is emedded in nother struture B y n injetion f D every element a of A is identi(ed with its imge f (a)D in BF f (a) is nother wy to present aD whih then hs multiple identity orD more extlyD we hve for a severl distint representtives tht we identify s referring to the same entityF hen f is ijetionD a nd f (a) re distint ut ehve in the sme wy in the struture A nd the struture B respeE tivelyD A nd B eing isomorphiF his proess turned out to e essentil in tegory theoryF es spotted y tFEF wrquisX here is no uniqueD glolD nd universl reltion of identity for strt ojetsF FFF estrt ojets re of di'erent sorts nd this should menD lmost y de(nitionD tht there is no glolD universl identity for sortsF ih sort X is equipped with n internl reltion of identity ut there is no identity reltion tht would pply to ll sortsF 32 sn mthemtisD one looks permnently for new presenttions of the sme enE tity @or tken to e the smeAF he onept rel numer is thought through di'erent presenttionsD tul @guhy9s sequenesD hedekind9s uts mong othersA or possileD ut it must not e onfused with nyone of themF sn good sesD di'erent presenttions for the sme entity re provly equivE lent in the sense tht the mening of theorems vlid in one se is preserved y theorems vlid in the other seF he question of the smeness of referent through di'erent presenttions or de(nitions poses di0ult epistemologil prolemF wthemtis fes this prolem onstntly nd solves it prgmtE illy y showingD in se it is possileD n equivlene reltion etween the entities under onsidertionF por instneD topologil spes n e de(ned in mny di'erent wysD eFgFD vi open setsD vi losed setsD vi neighourhoods @rusdor'AD vi onverE gent (ltersD nd vi losure opertionsF hese de(nitions desrie essentilly the sme ojetsD wht gtegory theory expresses vi the notion of concrete isomorphismF 7.

roly the most fundmentl tion is thinking in terms of invrineY it opertes in ny mthemtil re nd orresponds to the tsk of isolting intrinsic or stable properties of the ojet under studyF yne wnts indeed to 32. See [Marquis 2012, 9, fn 20]: Each sort of abstract entity, for example, monoid, group, ring, eld, topological space, partial order, etc., has its criterion of identity.

It is certainly a nice feature of category theory that it provides a unied analysis of these criteria of identity as being isomorphisms in the appropriate category.

Facets and Levels of Mathematical Abstraction 107 study not only the struture of some entity ut lso how it ehves under trnsformtionsF e few exmples re elowD tken from rithmetiD geometryD lgerD topologyD lgeri topologyD nd tegory theoryF ! he rdinl numer of set is invrint under the proess of ountingD ngles re invrint under slingsD rottionsD trnsltions nd re)eE tionsY for ny irle the rtio of the irumferene to the dimeter is invrint nd equl to π.

! pelix ulein hrterized geometry y set of geometri invrints under given group of symmetriesY eFgFD lengthsD ngles nd res re preserved with respet to the iuliden group E(n) of isometries @iFeFD re)etionsD rottionsD trnsltions nd omintions of these si operE tionsAD while only the inidene struture nd the rossErtio re preE served under the most generl projetive trnsformtionsF ! ylvester lw of inertiX ertin properties of the oe0ient mtrix of rel qudrti form @homogeneous polynomil of degree P in numer n of vrilesA remin invrint under hnge of oordintesF ixpressed geometrillyD the lw of inerti sys tht ll mximl suspes on whih the restrition of the qudrti form is positive de(nite @respeE tivelyD negtive de(niteA hve the sme dimensionF ! sn rilert9s invrints theory the (nite sis theorem sttes tht evE ery idel in the ring of multivrite polynomils over x÷therin ring is (nitely generted @invrine omined with redution to sisAF rnslted into lgeri geometry tht mens tht every lgeri set over (eld n e desried s the set of ommon roots to (nite numer of polynomil equtionsF ! he norml sugroups of ertin group G re the sugroups of G invrint @stleA under the inner utomorphisms of GF ! he dimension of topologil spe is invrint under homeomorphismF ! elgeri invrints re used for lssifying topologil spes up to homeomorphism orD more usullyD to homotopy equivleneX 33 given two spes X nd Y D we sy they re homotopyEequivlent or of the sme homotopy type if there exist ontinuous mps 108

f : X → Y nd g : Y → X suh tht g • f is homotopi to the identity mp idX nd f • g is homotopi to idY F
Hourya Benis Sinaceur homotopyEinvrint if it n e expressed s funtor on the homotopy tegoryF hese exmples show tht the ides of funtionl reltionD invrineD equivE leneD lssi(tionD ftoriztion nd some others re working together nd re using tools from one re @eFgFD rithmeti nd lgeri tools respetivelyA for hrterizing entities elonging to nother re @eFgFD lgeri numer theory nd topologil spes respetivelyAF 5 Conclusion wthemtil strtion onsists in vrious proesses inresing knowledgeY so s hve onsidered it from its epistemologil spet rther thn from its logil or ontologil spetF he question whether the strtion proess is logil or psyhologil gives rise to rgumentF s think the proess hs evidently logil side nd psyhologil sideD the ltter eing y now very muh investigted y ognitive sientists nd neurosientistsF prom the point of view of mthemtil prtieD strtion is n indispensle tool of work nd produtionF s hve een interested here y the multiple wys of onstruting nd developing mthemtil strt ojetsD nd s hve tried to show whih permnent tions re involved in ll those wysF estrtion is very often linked with generliztionY nevertheless there re strt nd non generl ojetsD suh s hedekind9s integersD whih re prtiulr model of generl strutureD nd there re onepts tht re eqully strt ut hve di'erent degree of generlityX eFgFD the onept of group is s strt s the onept of (eld nd it is more generlF s hd personlly no exmple of generl proedure or entityD whih would not involve strtion t some levelF tFEF wrquis gives the exmple of pssing from the notion of ontinuity of funtion f : R → R t point to tht of ontinuity over rel intervl wrquis forthomingD IUF ht leds me to think tht generliztion n sometimes e mde without using strting proessesD while ny proess of strtion involves generliztionF wthemtil strtion hs more thn one wyY it is not limited to eristotelin onept formtion even though oneptuliztionD tht is to sy forming onepts y vrious proeduresD is one essentil wy nd is very hrE teristi of modern mthemtisF woreoverD di'erent wys re simultaneously used in onstrutions of higher nd higher levelsF ome wys re known from the eginningsX ideliztion @geometril shpesAD invrine @invrint rtio etween lengths or integersAD ftorizing @integersAF yther wys re more spei( of modern mthemtisX mking whole from n in(nite numer of unspei(ed elementsD

8.

  Poincaré called that qualitative analysis [Poincaré 1928, XXIXXII]. 9. See [Krömer 2007, especially, 8184]: commutative diagrams play a central role. 10. [Tammet 2005]. When D. Tammet multiplies two numbers, he see[s] two shapes. The image starts to change and evolve, and a third shape emerges. That's the answer. It's mental imagery. It's like maths without having to think.

  16. I have analyzed dierent aspects of the fundamental role of analogy in the progress of mathematics in[Benis Sinaceur 2000].

  20. En mathématiques, [...] des éléments variés dont nous disposons, nous pouvons faire sortir des millions de combinaisons diérentes; mais une de ces combinaisons, tant qu'elle est isolée, est absolument dépourvue de valeur ; [...] Il en sera tout autrement le jour où cette combinaison prendra place dans une classe de combinaisons analogues et où nous aurons remarqué cette analogie; nous ne serons plus en présence d'un fait, mais d'une loi. Et, ce jour-là, le véritable inventeur, ce ne sera pas l'ouvrier qui aura patiemment édié quelques-unes de ces combinaisons, ce sera celui qui aura mis en évidence leur parenté. [...] Si un résultat nouveau a du prix, c'est quand en reliant des éléments connus depuis longtemps, mais jusque-là épars et paraissant étrangers les uns aux autres, il introduit subitement l'ordre là où régnait l'apparence du désordre. [...] ce n'est pas seulement l'ordre, c'est l'ordre inattendu qui vaut quelque chose [Poincaré 1908, 168170]. 21. See J.-P. Marquis' ne decomposition of the abstract method into four components in [Marquis 2012, 910].

  Boolos and other logicians as well have recognized that Hume's principle is not a logical truth, but from it we can logically deduce what we now call second-order arithmetic. See [Boolos 1998] or [Zalta 2013]. 26. The fact is stressed by J.-P. Marquis in the case of the passage of set and mapping to abstract set and arrow. Marquis writes : Abstract mathematics, like the concept of mathematical structure, is open, in the sense that it denotes changes with respect to the theoretical tool used to interpret and illustrate the concept [Marquis 2012, 2, 11 sqq.].

Facets

  digrmsD sets of xioms s beingD rther thn expressing mthemtil ojetsD setting up nlogies etween pprently di'erent ojetsD setsD struE turesD theoremsD etFD nd orreltively deling with classes of strutures nd theoremsD onsidering funtionl reltions or orrespondenes etween elementsD struturesD funtorsD themtizingX viewing opertions of one level s ojets of the suessor levelD deling with strt strutures nd proving struture theorems with the help of strutureEpreserving mpsD onsidering equivlent lsses of elementsD of struturesD of morE phismsD etFD nd proving lssi(tion theoremsD trnsferring theE orems etween tegoriesD etF he vrietyD wide enoughD of the exmples s hve relled shows tht the notion of mthemtil strtion is plurl nd )exileF he strtion proess is openX new steps towrds higher levels yielding more strtD more sophistitedD nd more enompssing onepts re to e expetedF Bibliography AlexandroffD vel IWQSD immy x÷ther9s oituryD inX Gesammelte AbhandlungenD edited y xF tosonD pringerEerlgD I!IID IWVQF Aristotle IWQQD Métaphysique µD risX rinD prenh trnsltion tF riotF "" IWTPD Seconds AnalytiquesD risX rinD prenh trnsltion tF riotF ArtinD imil IWTPD ur rolemlge der wthemtikD inX The Collected PapersD edited y F vng 8 tF teD edingD wssFX eddisonEesleyF AwodeyD teve IWWTD truture in mthemtis nd logiX e tegoril perspetiveD Philosophia MathematicaD R@QAD PHW!PQUF BenacerrafD ul IWUQD wthemtil truthD The Journal of PhilosophyD UH@IWAD TTI!TUWD reprint inX fenerrfD F nd utnmD rFD Philosophy of Mathematics. Selected ReadingsD Pnd ednD gmridge niversity ressD IWVQD RHQ!RPHF

  qoing furtherD one de(nes the homotopy tegory s the tegory whose objects re topologil spesD nd whose morphisms re homotopy equivalence classes of ontinuous mpsF wo topologil spes X nd Y re isomorphi in this tegory if nd only if they re homotopyE equivlentF hen funtor on the tegory of topologil spes is 33. Two continuous functions from one topological space to another are homotopic

i one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions.

  " PHHHD he nture of progress in mthemtisX he signi(ne of nlE ogyD inX The Growth of Mathematical KnowledgeD edited y iF qrosholz 8 rF fergerD fostonX uluwer edemi ulishersD Synthese LibraryD volF PVWD PVI!PWQF "" PHHVD Richard Dedekind, la création des nombresD wthesisD risX rinF BolzanoD fernrd IVSID Die Paradoxien des UnendlichenD veipzigX pelix weiner erlgD rmurgD IWUSY prenh trnsltion y rF fenis ineurD Éditions du euilD IWWQF BoolosD qeorge IWWVD Logic, Logic, and LogicD gmridgeD wssFX rrvrd niversity ressF BrouwerD vuitzen iF tF IWHWD Het wezen der MeetkundeD emsterdmX glusenD prenh trnsltion in vrgeultD tF Intuitionisme et théorie de la démonstrationD risX rinD wthesisD IWWPD PS!QUF BurgessD tohn F 8 RosenD qideon IWWUD A Subject with no ObjectD yxfordX yxford niversity ressF ClearyD tohn tF IWVSD yn the terminology of strtion9 in eristotleD PhronesisD QH@IAD IQ!RSF DedekindD ihrd IVSSEIVSVD eus den qruppenEtudienD inX Gesammelte mathematische WerkeD frunshweigX iewegD volF sssD RQW!RRSD IWQPF "" IVVVD s sind und ws sollen die hlencD inX Gesammelte Mathematische WerkeD edited y F prikeD iF xoetherD 8 yF yreD frunshweigX iewegD volF sssD QQS!QWID trdF frF fenis ineur PHHVD IQI!PPHF DehaeneD tnisls PHIID The Number Sense: How the Mind Creates MathematicsD xew orkX yxford niversity ressF FregeD qottlo IVVRD Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begri der ZahlD fresluX ilhelm uoenerD prenh trnsltion y glF smertD Les Fondements de l'arithmétique : recherche logico-mathématique sur le concept de nombreD risX Éditions du euilD IWTWF GentnerD hedre IWVQD trutureEmppingX e theoretil frmework for nlogyD Cognitive ScienceD UD ISS!IUHF HasseD relmut IWQID Üer pEdishe hiefkörper und ihre fedeutung für die erithmetik hyperklomplexer hlsystemeD Mathematische AnnalenD IHRD RWS!SQRF HausdorffD pelix IWIRD Grundzüge der MengenlehreD veipzigX on eitF Facets and Levels of Mathematical Abstraction 111 HegelD qeorg ilhelm priederih IVIPEIVITD Wissenschaft der Logik, I-IIID xürnergX tF vF hrgD new edition y qF vssonD veipzigX pelix weiner erlgD IWSIF vX http://www.deutschestextarchiv.de/book/ view/hegel_logik02_1816?p=9F HilbertD hvid 8 BernaysD ul IWQREIWQWD Grundlagen der MathematikD pringerEerlgD Pnd ednFD IWTVEIWUHF KantD smmnuel IVHHD Logik: ein Handbuch zu VorlesungenD uönigsergX priedrih xioloviusD inglish trnsltion y tohn ihrdsonD vondonD IVIWF vX http://books.google.fr/books?id=2CFNp0S753sC&printsec= frontcover&hl=fr&source=gbs_ge_summary_r&cad=0#v=onepage&q&f= falseF KreiselD qeorg IWVSD wthemtil logiX ool nd ojet lesson for sieneD SyntheseD TP@PAD IQW!ISIF KrömerD lf PHHUD Tool and Object: A History and Philosophy of Category TheoryD firkhäuserF LandryD iline 8 MarquisD tenEierre PHHSD gtegories in ontext X ristorilD foundtionlD nd philosophilD Philosophia MathematicaD IQ@IAD I!RQF LewisD hvid IWVTD On the Plurality of WorldsD yxfordX fsil flkwellF LockeD tohn ITVWD An Essay Concerning Human UnderstandingD volF sssF vX http://www.gutenberg.org/ebooks/10615F MarquisD tenEierre PHIPD gtegoril foundtions of mthemtisD or how to provide foundtions for strt mthemtisD The Review of Symbolic LogicD T@IAD I!PSF "" forthomingD wthemtil strtionD oneptul vrition nd idenE tityD hrft ynlineF ParsonsD ghrles PHHVD Mathematical Thought and its ObjectsD xew orkX gmridge niversity ressF PoincaréD renri IWHHD hu rôle de l9intuition et de l logique en mthéE mtiquesD inX Compte rendu du deuxième Congrès International des MathématiciensD risX quthierEillrsD IIS!IQHD IWHPF "" IWHVD v9venir des mthémtiquesD inX Atti del IV Congresso Internazionale dei Matematici (Bologna)D omX edemi dei vineiD ITU! IVPD IWHWF "" IWPVD ×uvres ID risX quthierEillrsF QuineD illrd vF yrmn IWSID wo dogms of empiriismD Philosophical ReviewD THD PH!RQD reprint in From a Logical Point of ViewD Pnd edn revFD rrper 8 owD IWTPD PH!RTF "" IWTHD Word and ObjectD xew orkX tohn iley 8 onsF "" IWWHD Pursuit of TruthD gmridgeD wssFX rrvrd niversity ressD prenh trnsltion y wF glvelinD La Poursuite de la véritéD risX Éditions du euilD IWWQF RiemannD fernhrd IVSRD Üer die rypothesen welhe der qeometrie zu qrunde liegenD inX Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu GöttingenD edited y F hedekindD volF IQD IQQ!ISHD IVTUF vX http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/F RosenD qideon PHIPD estrt ojetsD inX Stanford Encyclopedia of PhilosophyD (rst pulished PHHIF SteinitzD irnst IWIHD elgerishe heorie der uörperD Journal für die reine und angewandte MathematikD IQUD ITU!QHWF vX http://eudml.org/doc/ 149323F SzczeciniarzD tenEtques IWWWD estrtionD inX Dictionnaire d'Histoire et de Philosophie des SciencesD edited y hF veourtD risX resses niversitires de prneD R!WF TammetD hniel PHHSD e genius explinsD The GuardianD IP peFD QRF WeylD rermnn IWPID Üer die neue qrundlgenkrise der wthemtikD Mathematische ZeitschriftD IHD QU!UWD inglish trnsltion in wnosuD FD From Brouwer to HilbertD yxford niversity ressD IWWUD VT!IIVF "" IWQPD opologie und strkte elger ls zwei ege des mthemtishen erständnissesD Unterrichtsblätter für Mathematik und NaturwissenschaftenD QPD IIU!IVVD reprint in Gesammelte Abhandlungen III @rrsgF uF ghndrsekhrnAD pringerEerlgD IWTVD QRV!QSVF "" IWQSD immy x÷therD Scripta MathematicaD QD PHI!PPHD reprint in Gesammelte Abhandlungen IIID IWTVD ferlinY xew orkX pringerEerlgD RPS!RRRF "" IWSID e hlf entury of mthemtisD American Mathematical MonthlyD SV@VAD SPQ!SSQD reprint in Gesammelte Abhandlungen IVD RTR!RWTF vX http://www.jstor.org/discover/10.2307/2306319?uid= 3738016&uid=2129&uid=2134&uid=2&uid=70&uid=4&sid=21102906564633F ZaltaD idwrd IWVQD Abstract Objects. 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	110	Hourya Benis Sinaceur
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I am simplifying the very suggestive although intricate developments of Hegel's drittes Buch of [Hegel 1812-1816].