A Genetic Interpretation of Neo-Pythagorean ArithmeticCahiers du Centre d’histoire des Sciences et des philosophies arabes et Médiévales,

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M.-L. DESCLOS, Theogonie hesiodique et philosophie pΙatonicienne de dans le mβe du ΡoΙitξue......-................ ..:......-..-..:..-.:-......... Ι. VΑNDOULΑΚΙS, Α Genetic Ιnteφretation ofNeο-ξlthagorean B. vΙTRΑc, Mecaniφe et math.matiques λ ΑΙexandrie : Ιe cas de H6ron. Ν. EL-BΙZRΙ, Ιbn aΙ-Haytham et Ιe problδme M. BLΑY, L,histoire des ph6nomenes de la couΙeur................ ...... de la couleur entre Ιumiδre etpigme Kepler J.v. FΙELD, Τhe reaΙism of C.opemicus and A.E.L. DAVΙS, ΚepΙer,s concept ofan ortit.... Sommaire des numdrοs Αrithm 1,2'3,4,5,6,7.......'

INTERPRETATION OF NEO-FVIHAGOREA ARIIAMENC" DANS M VANDOULARIS Corio Een. Cor) “The (Neo Ptr ivi ro represents an poto ate stc cure, characterised by an apganach fe ce eme in of aritmie, ea. tive 1 lal of the Euclidean Clemens. We te this ca Lo embrace the following picecs of aithmeti tory I. The Using af eve and ol amber LL The sony ofart umber; IIe The tary of ancien sos; (The thon otigured numbors and Ve Tie deory ea. For the later thee pieces, Novem. some pieces of "The ron gape is are aro 10 malpancn in Fold" Etemons. te theory of pulyeona! nuubere is fumi] in tem sch ary comet at CTS, mf a ‘Bae to eG Len ei Jr the sear 195652 My ein, M Comm La er, 190 eed 0 piva Te ur pes cee praia per ie Masao 105 nn | All Ike eed i tak eas Pre Bate rc arr hoe Si al i ok LG Boda, nie ser Elta rst Jen Cm, ca sed B VE Ge ok Speci ncn dev Greg auto nin the Sens En sonde Arabes et eins (CESAR 7069) in Vi, where an ea FOREN ef Lin wo.

AGTNITICINTTRIRETA SION OF NECPYLUAGOREAN ARITHMETIC Divpkaacor" weise On Pofonal Aumbers wich follows te ile at dag thot Ihe Noc-Pythaoreun antic mug lave Boon developed in a seal, scicontairod winner as a simple unes inti: objects The furor ta Arihoaris by Nicomachns of Goa E" ott AD <The Exposition of Cf Mathematical Knowledge fr die Rearing of Pag dy Toon of Syma (2° ceutiry AD}, -lanilichus® Commentary on Micmac’ entry AD), whic follows Intern (0 dither” the one and he soma of Nicomache rc byt tines omits comia! or adds new material” (ys se, The Mamet of ritmi Intro by Dominus of Lara (5° cenmy ADI The Commerce dici te Arias by Assis of Tales by (osano, een of coming over a domain ot lesignated by foxes: sigas. Tis approsoh can ho cem ar realised without arpealiog to ssumoione nfaxiomaro charger, bit. selying upon sou “genetic” constructivas intente to he served ne by means ofthe designate ies [-sewtorics Nieunachus’ concept of number in che context of ancient semantic traditions oomackus gives in his Inoue so Aritmetis a genera: semictie chere for the symbolic representation of numbers. He rotes tat the designation of subs The Commentary on NicomasiusTM“harsactn to Ariela" 115 hy means ef less is convermione! hie), nor manure aerate drá 00 doit The name somos pe manto Pilepons (6° century AD) numbers 15 ay means of the representation of rho ars eompnsing 1 number, The later tt que beside the athe. ae ls fam of scotia to he text of icomachs Ihe sete oF metio in cho gestes the Neo-Pyrhagoeran authors à strikingly «eroe Won thaltke Klemens Namely, & characterised by the absense 0° prou in he Duelidwn sense and a socie gel pprich m the constuction of ati het we ce goto describe in ovr pope. Lich mathematical son has lc cera bistr to consider ine his oe Y sf meules 09 4 Cetur uf eoeedense of mathemati in this period Usanery 1887, 13-12; Heath 1821 t 97-99)" The alle nbsenee of wri ity ln these works ha also given grands to believe that “the aile present in hese worbs derives susa om an anciem. prune cage of Cyagure arithmetic” ed ca uso thm as an nr of the character ni aime cionci he ctu [Kae 1995, 432) AS pape, ve Lee Nicomachis (tratt {a inet oe pot ol depanurs, becouse itis he riches and wes wel orga tie reps ing Mis ualitien [wäre 1994. Il 474475], Homeser, me al be [no scout Ue were of other Nexesatesoren aus We ve guing Io tease a war ent hae aan de de wi of Zaid Vol fwd Sei 090, 82 197) ka kal she aces Fes, however, ve mest rocoso that act leer by which wo designate a rue sl 1 nin 0-10, pa fe 3, ug ov BO si Ba nacer by mas comscoiza nd agreement not by nets. On tae other Pad, te al, siii, me hehe sin st designati uP ears ‘auld Ys the sting forth one beside the ofr af te units cond in each Thre concepls of conventi tics) Anl ature Gora) go back to Gs pre Sueruies (Pythagoras, Demeeri:us) std the Soplisis. tion between onnvention and nature was Fur the later, te distinc central element in thei philosophy. Proglus asribes in Democritus tie view tha die eaion beiswen names amd things named ve conventional, aller then nan. Te supposed te huve sel oth four argoncents, called poisson (nei, equipance (aspromev) Avion, surtaziions Arctica, vi. 2. lache; Doa, 1926, 852.

ICANNIS M. VANDOULAKIS mercapme TÍAS, and. many; (uc), METATION OF NEO-PYTHAGOREAN ARITIIMETIC respectively, 1 support af bis vitowpoint Nawely ‘he occurronos a homanymas for different rings, ile existence of synonyms for one und the same thing, the possibil i of renaming the some thing, and tho ooewrenes of imegulartie in che words formation. First, the description of a generting process, the setting forli one alpha beside the cier (mapä&kAnAos Ex0eors) Thus, the Lal iv the setting for of one alpha; the nunnber nwo a the setting Forth oftwo alphas one beside the other; the nueober three is the setting forte of lee alphas ane bey ythagenss ropurtediy holds that the agsgnment of wanes o things I not au so forth (see the nex! section), arbitrare operation, but is imposed upon things by some kind of natural apres Gears is adoquvy between the namos and de things, so thar arrangement ofunits Le à man who vonfemplates the he mcr giver can only TIP the other, and Asclepins of Thralles expa.ns us thet by not meaul Ins representation in n straight lino, but Ihe one beside the her,” mind and the nature of being”? Pythagoras: Second, the description af a process of designsting the ebjects counted. views are the caries reported in favour of2 particular senvatic theory This process of Iransformaliom of some initia] colfection of objects to a Tonblichus azoribes aba 10 Pythagoras “the symboli c and delachai use of snathematical words” Cru muohuche Kai ancEeveoueeny pion Tew schematic pattern, in principle, comar be free frau acls Of arbitrary choice. In veGauartisciv Aifos) and considers ther he was the fis] who nied siewpoint, in conzast to Pythagoras. The essential point lies in the organizato ep the naturlisic senuentie viewpoint to mattemetics this sense, Philolaus and Nicomachus are rigl ta support the eonventionalistic tin ofthe The original Pychngercan designata into readily idenzifinhle wholes, so dust when one is ficed with « pattern of alphas £o re able 16 cezayize one uf the defined pauterns art ‘naturalistic’ viewpoint seems lo be shared by Tamblichus, who accuses Philolaus for having adopled ihe “conventi onal tio associate it with the appropriate mum. ‘ew pint thas was not shared by Pythagoras himself I hs report is there is no pattern available to express the idea of “any number", tun 10 allow certain divergence of the fre, we Neo-Pythagareans from Pythngovas hint ni concerns mater of semantcs the pattems defined to represent munbers should have imernad associations, vamely to be clear hew it The opposition of the ‘naturisti’ and she Yennventionalistie” viewpoint is the point of ieparture in Third, Pluto’ Crazylus, where the semantic viewpoint is miributed to Hormogenzs, s “conventionalisti»* while the ‘naturalistc® one is It should be noted, however, that pessible to piss from ore paltern to Une west ene by performing a certain operation. This entails thai each partera should be contained :n its successor The unit (uovérs) in tho Pythagorean arifimetica; tradition is a “minimal Supported ly Cratylus? Therefore, it seems Bie: Nisomachs” conventionalis das is roots in the semantic tradition of the 5" century, yet must probably anti" “indivisible by mature and serves as the “asturel beginning of numbers"? dint is, it precedes the concept of number. I is “potentially all the cannot numbers". be roduced directly in Pythagoras himself. historians of mathematics up to date have neglecied ft shecld be noted that this relation ofarithmetic to semantics” Number in Nicomachus pessesss internal structime Eux - ‘arrange men). U is a (finite) “suite” (or a achernatie patzeru) ai’ signs, unbeundnl in the direution al increase and banded below by the menas a the direction of Anulysîs of the Nicomuchesn semiotic scheme Ono can distinguish for number che following aspects of the Nicomachein semi devrease. ln 1 heon’s words, ‘armbar is à coll Imponobioues) of mllitade beginning ion of units, ora progression ‘rom a unt and a relrogressiv scheme: Èroot. Je Mates Crau mation stent, Lu te Miramgek desorientación, CE Arioli De 7, 1, Pos 19,23, Pill inergresascae See, for instance, Cage" teatmen: af nur >in Micomachus, where si is completely averse © Agelegins of Tire es, Comentar do Smash” Sum oct to Arion’ I somunsrtora, 265) 3,25, Pasquali ai D comma s, [P'Ooge 1928, 1:1-12) CI in Tamblchus, ia Vcore Art icons, éd Mb kurcaenions Arten ed, Host. avi, Bo nn btraanstion 11,1, Pinel eae i, it. 4 Mache,

IOANNIS A GENETIC VANDONI.AKIS KGvancBiowds) ceasing at a wait"! Nicem achus describes number 95 “a low ofquanlity made up of units” (mocsTreros ye ix kaue Al these descriptions sugg est an notion of interna “dices?” OF NUG-PYTHACOREAN ARITUMETIC 119 Number aud iteration n cuyo) order struct far thy number Lanblisins INTERMRSTATION weites Chat the notion of number xs ‘eollestion of mis bak ro Thales wha foll ows the Ceypiian paler! un, Gondor aug sc Dis point whereas the tio n muta an “dete multad o” (AG SIPIGUEVOV) oa in er byoe Eudoras. d Nevertheless, (he arac terisation sé auinher by means af such tear s like “Tawi” (mépes) of “limi ting quant? (pate most) — the Inter is ascribe by lamblächun co Trad thes !” seems to Lally with the Pytb agorcan plil particularly, Philalaus” osophical tradition ofth dectrine ofthe Limit ad Further, the Finite sequence Me sequence of the so call GB the Umlimitet (mp of such simple suites can be cons tructed, ie ed “properly ordered” (edr 8 3.0. 1,2,3.4,5,6.7,8,9,10, exemplilying the inode of au Wold like 10 add a pebble oro remore 1 o You hin th sell roman the came? ane easel” figura! numbers. The basic property of the unomon is that its application is in itis) 13,14, 15, suits © garnets ito an od aber or tar eve, The “gnomon” on tho other hurd, has ¢ similar furetior. inthe theory ef we doubts by unfunal members (1,12, which is called the nano! pater éverous} mimhens 10.8, 03.18, ‘that zo sequence (which As e 5" cemur, Knorr 195] as an early eviden closed with respect 10 the kind of figured number (he kind of polygonst ges) by Nicomactu! and consimctian considere in sch cass, Nicoinechus makes use of of the kind serves of number Although his concept is introduced in Book I it alraıly in Rook Y. The natu ral suite should nt bo confised with the naniral series . ly iccunschus, it is u finit e constructional Sheet. The concept of ihe natural suite is intrinsi cally connected with he notion of “properly o:der” In Nicomachms, the natural suite is always a suite of irons ordered numb ers, hat is it onsbodies the specific according la which & is constructed, ruiahee). This property is clearly stated, for instance, by Luublichus: “It is called gmomon the aumento [because] woen added it preserves the same kind of each of che polygonal {wembers,"2" Most of the available general Aefinitions wl gromon aro based en the property of similrily of the wholes after and hefore its placement regularity or re Ncomachus,rrachctons Ari ieue, I vil, 3, Hocke; D'Or 1936, 8 Diets. Fragni, bricharanas, 0.2 torte de rene il cate rv say aie ea UN "Than, Asset era ma Ue; Dopuis i962, 28-25, porn inge Pies 11 Sor aka [Herodcmus, Hiugriae. 4.16, L1R-20, ude? SE Tarıbichus, Moment, Nicomachs, ira cinetica. vi. ered ét, Li . Titor also calle “yuo and “gRomozis si Peet, Opus 1956, 62-85. rubo” (cons Matane

Soa M VANDOULARIS erative process and the linite cannot expect eshuibit cho generative power uf the monad”. Nevertheless, one he ides of iteration eras the possibilit; of unlimited confina of process of addition by 2 rit. In Thoom, ws find the iterative operation of die by a unil in x content vividly expressing the endless chavavter of this process: "The unit can increase by addition up to che infinite". Inthe same fashion, Nicomachus stay tha: “urily à the Peginzing of all upmbr that advances unit hy unt in one direction” Further Nicomachus states that “the to acturnceTM® an Imblichus that “the mulltudo can “The scope of arifhmetie and the finite increase lo he infinite” * Prcclus, iu the second chapter of his fest Prologue, Nieomachus, Inuiblichns and Preclus rnnitude nover vasos can Tall under the scope of science: ‘where ho considera tbe Limit and the Unlimited i common ontolog cl principles of wnihematies sis that slough mumibes can anfiitum Gx" &mapon, véxps Go, any given uumber inerense aot foie: “number, | individuel à Arien” | of limited continusbiity ofthe process of increase hy a unit is applied (o number or to mire (dip). However, my ghen cuniber is taken always to be “finite! ‘infinite’ or ar ch our iéess nnd demonsuarons proceed does n°: ls akogsther ee dhe ink for the purposo af Snowing it, tor tke infinie anc wes ely Incanpreheusihi 1o Jajowtedge, maker it takes e yet 3he underalaoding four begínniog from the unit, can increase without limit. yet any number taken “ls, the propedty in also to agree that only fine oYjects the Gate For demanstation, hat is, asumes de infinite por for e sake oft infinite, bat Zor o suke of the iit.” “defini” wherens multado is ‘indefinite’ in ils neture and is associated by the Pythagorean auihors with the medium of te discrete Cu Bones) The dostrine of the Lunit and the Unlipited, to which Proclus and other Pydhagoreans often mention, is eather obscure, funofar no it ails also 2 cesmalugical interpretation and, inrcby, involves a cosmological interp Lion of munber end the generation of numbers. Use al ese principes in à senso relevant, in some respect, to mutherercies should be sought far, in aur view, in he eeveeprion of te Limit as delimiting principle against the back“ ground of the potential infrile malium of the Unlimited, which we ind i Tamblichus and Proves. faublichas, for (stan. ites the Limit sil defining, comfining principle against Ihe background of the advansing # infine Empása Liv emi vo Anm ét Soren & Und to mapas.” Moreover. pd. Mile * loach ico 2 kunblienus ems iow)" This viewpoint is culspakenly ascribed fo the Pythagor rectus gite ie “The Pyihagoruaus consider quantity at mag e Fate in ach caso. “oias Ceanprehe2d ins nis ar Mimet, vam it ie Impossible (o siher thea Faces Flament hiram commentary 6, [rieti Mom, 1982, 5. 3, Hocheà "Oiga, 1926 52). in Nîcomuchi Artthmericum entrodie mem 7,20, Psa commenti, 6», Page 12.5 À rambiches, De camera m: fe primi orafyi. bue For they say Mos the sciences study mo fisio ın absracion fra imite queres End manie Proche, Duro 1966, 136457, vot cet inter by Pride M 90.212 Ft. 285 24, Frein . now, sima on frredkssvmen, 3 ambi, la Nm compa albe maia elec. 7,3, Fest 15.20, foma. 26, Piselli; De acts. Liemenioren Ihren commen, 36, Castles ni Moros.

TOMAS 4, VANDOLI AGENUDAC ARIS Im anolher passage, Procl > us Proelus stresse stresses s dur the di subject matler Eyttagorean science is fuensel an the study ofthe Limited: [The Pyah of the INTERPRETATION OF NEO-PYTHAGOREAN ARITHMETIC 123 af esch term by 2. 1he construction of odd numbers is made Ly Nicoruschs (hen he proceeds 19 the sonsirnetion of the evex-times odd nuricere stiences) exclude infin ite Kom ‘The theory of selves steaightway wit h the ‘Lin tie odd und even is exposzo in considerabile detail ia the fi Book of Nicomachus' Introduction to Arithmetic, Apart Iren the even and the dd number, Nicomsvnus inlvaduces varios Kinds of numbers, vivi ure ¡Hass Since. howe all mat ve ter, aud magno are by thos ows natu oCre act y infinite... and sinn scenes ae aay sein en of lin ves things, nd mer > inns, is eecondingly ovitent that scenes dealing ethe r with mago ih malt, pe se. could never be finite, for each af he An sch nude inthe die a he soz, and nga à deetofin the tess." aa u trated by means of the cenesponding stern of the nstaral suite and aneomıpsnied hy the deseription of ar effietive peoseduce tor thelr construction. The are defined by means of te diferent basic concepts: “divi numbers kinds of ing’ Giawpetiven or ‘partitioning’ (urpLotñvas, and canins" (verpalv) Nicomachus uses the operation of dividing far Ike classification af the gecus of the even and the concept nF mexsuriog, for the classification of the odd num: hers. e lon, artt shold the subject of te (Ayg o ideo A uy of Hale “ane s ora of the ern cation ofthe genus Classifi ite an Unfinined medium. mes "even bors”. Gmes-dd”, and the “Dad tiges-ever mn. The basie concept used fer classification of the even number is the According lo Nicemachns, an evenimes oven ‘number is a number that males successive dls.sions into two esual pas that ¡lo doctri ofne the absolute quan tertninate in the unit” Theon gives au analogous definicion, but its based on tity ‘The first basic classification (ou Pf numbers is into even and odd, Nico machus gives Ares different desc riptions of the even and the odd” ‘These ied by any graphical ilsranio n or procure for Flowevex, the “even-times even”, the Kum-times even members. TRAGOREAN ARITHMETIC their construction. subdiv'écd ine three disjoine classes of numbers operation of “viding” Bio. Bla Berón) FL PRINCIPLES OF NEO.PV definitions are nur accompan ‘The even numter is farther their corstcuciion is obrisus. Turcos, for instance, can he obtai ned from tke catun suite The shite of even by multiptieat on lhe inverse concept of mutiplisation. Thus evoa-times even nucabers are those generated by the muliplication of nso even numbers, have all their pasts even, and have mu part homonymous 10 odd nez Tt is noteworthy that the definition of evsa-timus even number in the Now Pythagorzan tradicion dilfe:s significartly from ‘comes clear in Proposition IX. that of Euefid. Ihe point 34 of the Elements, where itis proved that Yo 37, Peden; or Mimi, fered À Tato Ii Hooke larabliclus, in Sierumacài Arubeceticua intro duction a, mathematica sientia, 7. 3, Fests 'Ooge 2926 7, 26. ia Pig 1? O lll De cosa , D'Ovge 1320, 14. CF Dupes 1965, 3-25 1926. ice 14. Kioomach Cables ara onen asis dre , acon te L when, e, 6 x, sche, Arai vin, Hoche: D'Onge e hin, Ml DCU 7 come, dona Artesia 33 i, A, ehe; D'Ooge 1946.81, Affiliato Wells Cescibeg by lent di “ico raed moule repito 8 ea Rando rd & dis ri 37 Tishectoomeutat [Thea poste tim mantencion 20 ga Misia lion, vi lle; Pups 1966, M It Shou be note hot petti, sie objet ni ello oF wt cn t's F1-plizeion à, sii speaking, à choc formes ur one-dimensional lps, gives e om dimensinai

À GENETIC INTERPRE! ATION OF NERS SIA A IOANNIS + is both oven-times oven and even-times add. Buckid's certain kind of man: definition isc M. VANDOULAKTS Nicomachus gives an effective procedure far the construction of all even times even mmbers: beginning Som the anit we proccod by the double tating infinity. The pattern of this corsiruorion is exeroplified by che following mat Instewl of the sign of ellipsis, Nicomachus writes in words: Kai &p' doovoty and so fort) way are the only ever-times even aumbers lat can be constructed, “sy thal nove wili escape, but all successively lal] under it”! We can distinguish two asneci in the above definition: -A statement that defines the made of construction of the even-times even sumbers =A statement asserting thot hy ibis method we can poteatilly coustruct al ke successive even-times even numbers ‘The pattern of the natural suite cennot be vonsidered by itself as an explicit even numbers for the following reason Since the set af even-times even numbers is infinite, such 2 definition through a finite segment of it would be incomplete, if noe suppleutented by a rule for calculating the successive ever-limes even numbers, This mule is always he resulting consteaction is exemplied by tke following netural suit: you wish” Gol xe e ud far as dha way, “proascc Dédtimes even numbers. The last class of even Bventimes odd numbers. The sevond class of even numbers is the ever- 5 adi numbers that is dotinod ss "2 aumber of which, though it admits of to division into to equal hélves, aller the fashion of the genus commonto i into two equal even mimber, which is “An even number which can be dividedthe of its ever: parts, whose parts also can be divided, and sometimes unizy”,? parts es prov Ihe far as Paste, but it canst carry Ihe civiston of is parts isesmore sophisticated than Ihe fore for their construction given by Nicomachus previous coustructions. Itis realised In threc stages First, the mural suite the ode numbers from 2 onwards is constructed: 3,5,7.9,14,13, 15, Next the natural suite of tae evenctimes avez numbers, beginning from constructed: two natural suites hy “The result is obtained by a kind of composition oP isthemulti plied by all the the following rule: each number of Ihe first suite by Nicu machus En (le numbers of the second suite, The outcome is presented LIT Es Cat mane Tartine qe Taz by Niconachts gives hore also an affective procedure fur ih construction ofall eventimes odd rambers. This provedone is realised in 160 Commenter 10 Nicom has” ‘Imoniutie to Arme So qe anne spon, Commentary om Micemochaus' ‘Tnrasacium to Ariel « tipe tht loa I. Hacke Dogo 1926. 13-16 u | [40 ag | 72 TN FE SI [32 | ] 758 Tos Jos In [16 J 220 | om Liga | 286 | 576 Tio DE | 108 [2806 da Las [ve 1352 [204 stages asus of Tal 4 is 4, 8 16, 32, 64, 178, 256, and the cventimes cven, the halves êre vot immediately divisible into n equal parte’ robes it tho odd-tiauss form of table. explicitly stated in wurd ti term of the suite af odd numbers by 2. in the fol “The sign of ellipsis above i expressed by Vicorouchns in weds RT a 16,32, 64, 128, 256, 512, definition of the concep: of evenstimas suite ovatnes is mantormed ine a now suite by multiplication of eae 4, 22.26, 30, 18, 6, 10.1 ral suite Moreover, Nicomachus decla:es that the numbers constructed in this the unit nd provooding with u differance of 2 o infinity M cise hy AsclepiusTM aud Phitopoaus 1,2,4,8. “the naturel suits of edd numbers is consructe, 4. heginnig ron Table LT constriction of the cdd times even manibers OF ini ems defs ain giave Bid Lx, 2, Hoste: "Dog 1926. 816. Theor. by e gai r numhe he ar deired r times sean numbe rue, 2ve mr boss. multiplicaron Feds and even

IOANNIS M VAN DOULAEIS qe mus stress sein rose hat this procedpro uniquely deu nines the construction ofthe editimes even mumbers: “how ever far YoU go you wi} get nothing but Wie odd nes sven numbers”. Classification af the gerus of secondary and composit e” subiva into “prune and incouupost e"s and "that whichis seco ndary and ouest i ofa number A gui part of unity is called the paramo umber p, Fo instance te its Patocymaus part. amaber rc hus the (ad À gth part of unity i Br p fq is dirt from p and y mes fu as called heferumpmmau port urepe. s Foc instance of sn the Number 9 as only one heteronymocs par, the third pan of usi: the mumber TS hes ue bsteronymous pars, dhe third and The second concept inchndes the third [Nesselmemi 1842, 194]. However, the example given ite in real to mother”; hi lastes t ira longer made hy means of the conzupt of “dvidigg, bat thal nf “meee ne ‘ich i also andefined in Nivoma chus Moreover he uses ar ann sb, tons: the “parenymous pat” Ca po px) and the here ON yore paof rthe number can have Seteronymous part or paris as well as puonymous Nicomacilus below suggest that the seven sell hut prime end incompos Gccpcongsev use fore, this part besides ei This ulnscification dues nat establish Gisjoint classes. i the oda Now the genus ofthe ost mute is A GENETIC INTERPRETATION OF NEO-PYIMAGORI AN ARITHME tho Rh parts of wy hye vonsept is conocivod ay characters inga nutaber, wheres the third sooms to apply rather toa pase of numbers. © Nicomachus gives a unified method of consinschiem of all Ihe afer tioned kinds of cel numbers by mosas of tho sieve of Zratosthenes (labk 21. “The woiural suite ofodd numbers is se forth, begining wilh 3, re 3.5,7,9,11, (315,17, 19, 21, 23,25, 25, 29,31, 38, 35, 37, Then, seating with une first we observe that ahe Lens i ean mesure by ¡30 places apart as far as wo proveed, socordirg 10 the following rule: it meas ures the £rs: such number ccourred, ie. 9, by Une quantity thor sand frst în the suite, Le. 3 times: it measures the next such number occurred, Fo. 15, by the quantity of the second in erder, Let. ie. 5 times; an again che next such nue 56. 21 by the quantity of tho third in order, Le, 7 timos, “and so on ad inf tam ha the same way”. Now, ve ome the terms the sucor£ number S und observe Heat it can monsu all by Four places apt, the frst, ze. 15 hy a quantity af the first inc der, de. by 3: the seven, Ze. 25 by the quantity of the secord in order, ie By ch can be measured voly by the S: the thin, io. 35 by that ef the third. Le, by 7; and so on in che same say uni. che secondary and composite num Again, the thind arm 7 measores all terns sie places par, aad the first by the Fist qeantity in order, the second. by the secend quantity. she thied, by the thine quantity, This process sii he continued without jntarrupling, so Eat Ihe numbers val szcecod lo the did 1,8, Hoche; D’Ocge 1926, 217 Agora" i seque dy 5 pae a See af Mu, Ihe Pringle of sassi ata semai mr, ‘and he sone Seul min mue ty m ocorsanoe wh their Them. ir which orme and songs e Ge Mann Pine hunters as hac e a tice rombos infinito, er by the dow hig ef the poszion in de suite aseupiod by the sneasuring and rot Of the edd number token as bla of camaosi mbes, Cunclive by the orcorly progres oF the even munbera Gom 2 la ne te soon of measure. tewev en, this clacton deve a à nice way the del os 0 ina, plane hed slat sone pes Mot measuring al separating tens measured is determeed “ine position$ re sunt One ti a ve I HH). tenis tol Do nam of times tenuis risaurod 2 tasers im Une suile fen 5 fixed oy the outer (cm, and th advance ul Ue odd

A GPNUTIC INTERPRETATIO OF NEO-PYTHAGOREAN N ARITHMETIC IOANNIS M VANDOULAK IS 129 357917 TATI The theoty of perfect numbers 3579 138 90 Numbers are classified by Nicomachus into “over perfect”, “perfect” and 3 7 E “defivion®. Over-perfoct number (apres) Is ane, which is greater than the sum of ts parts; daficsené number (MAÉ) is that which is less Man the 3 O 5 sum of its parts; peer umher (rÓNsica) iy that which is equal lo tae sum of its parts Nicomachus gives also an efficuse “neal and secure” procedure for the construction of perfect numbers, which “oeither mass by any of Ihe perfect TT numbers, oor fails to differentince any vl Ihese that are not such’. eas constructs the natural suite of the t TO = | TTT A 7 1 3 » 1 Then he adds them together one at a time and each time he considers their 3 sum. If it is prime and incemposite number, dien he multiplies it by the last number added, and the result will he perfect nminter. 3 ? eveatimes even numbers 1.2.4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 1 H First, ho and composite, instead of muliplying, 9 the He adds the next umber and considers sum; iP itis secondary ane composite le costhe incomposite, he multinlies st by if tie sum is secondary next; but if i is prime nud che tast tery edded aud the result will be perfect numbers ad 50 on au finite. Taole 2. The sieve of Eratosthenos The dottrine of the relative quantity Now the numbers that are measured by no number are site, eg. t1, 13, 17, wie. prime and incompo- Those measured by only une munbis r in accordance wih their awa quantiy will have one heteroaymous pari in addition to the Panmymous une, e.g has one third as heterontmons part in addition to the aronymous part one ninth. Those measures! 3y only one number, bul in accor dance with the quantity of some other nombcr, different trom their own, or mensured by 18:0 numbers. will have several heteranymans pars in additi on 10 the paranyimous one, eg. LS thet has Wo heteronymous parts, one third and one fifth, in addition 10 its paronynınıy one tilteenth. These will be secondary aud composite, Finally, numbers which are în itsell'sccondary sd composite but primary and incariposite in relation to another are ıhe numbers that are measured üy some prime and iocom posite number in accordance with ls quantity, ¡Cone thus constructed de compared to another of similar Egaulity and Fnegualiy. Fynetity (OTH anil insqualily {devas duced hy Miconehus as the highest generic divisions of Me relative quantity. “Any thing shea compared wi another Ling can lie either sua or secsitan non datto” These concepts are considered lo be aft absiraetion £s unity and tbe dyad fb: the absolute quanti. umequal, ‚une level ul Inequaiity is further divisled inte tro elasnes, the grenter and the less, class of the greater incíudes rueliplos, ple superparticulars, The superpartic ars, supurparticols, muli and multiple superpartients. bmees the reciprocal retics of The class of the less em the class ef the greater. They are denoted as subraulriples, subsuperparticulars, subs aperpartients. subunlliple subsuperpar vi & bide dE construc tion, For example, the number: Y une 25 that are vonstrueted rom 3 und 5 respoclively, measering their ow quantity, wher compared between wich ‘other have ne common measure except Ihe unit 1) is ire Siren ini avi a. Hore: "Onze 1926. 821 wii 23, che ARO this redire tee Sen 6 Daye tars: vicini m var par tp thin ting beside condi Faneostiesson's th anote thing enter equal or unequal. and these" [Ou 1916, #2 dridco IL|, 12, sache; IYDoge

A GENETIC INTERPRETATION OF NEO-PYTHAGORKAN ARITHMRTIC IOANSIS M, VANDOUL AKI S 131 In case equality is understand as involving e geometric pattern, the nomber suite’)3, fin insonne. canno: da ote obtained sue OF thee given sql tems, nd conversely that equity may from any given suite exemp litying a farm dures are also described by Theen, of inequality. may ‘These be proce. ed be qual to the triangular number 3, for chose numbors are mumerically identical, Bul differ in configuration, that is they ere not graphically equal. [Fence, the fundamental trickuleny of numbers into Lin cur, piane, ard solid (gee below) demarcate throe types of abjects in Neo Pythagorean arithmetic, according to ‘he dimension of Ihcir graphical representation, fined (hamopay yap csbcuiav mit Accordingly, cqualily should bo possibly assumed as applying only to he ra). Wes considered Det equality hokis whor ure of che compared Kie s hir fenceeds (Upa), ‘cher, either nor BEA short (Au) in balk, length, wigh or ny t “telat” map Ph o ihe kind af quandry Y jects of the same Iypc. Nevertheless, ché is not enougk p. 123), as well as à pentagonal number (see p 133), Consequently, with Be coming [nam] has equality, us graphical equality, can actually apply tal <9 numbers of the Same kind, thet is to numbers ofsimilar combinatorial complexity. comparso either. ‘rhe plane nunber 12, for instance, is on even number, but also an cblong number (s ning wens i Nicomachus and the other Neo-Pythagorcan authors are not explicit on that point. Lruly, they never state the contrary proposition’ thet, for instance, the oblong number 12 is equal to the pentaggal number 12. We find, however, propositions, where Nicomachus. talks about polygonal numbers 25 ‘system of numbers uf die same ra Coin éaorayévi# numbers. show? CM, lao: apabonhy i a m y fey han This venus in the definition of equality car be eliminate? hy the adopfina of the Lollowing assumpticre: any process of eourt'ng of Lie numbers of a collection is a proc ote apice stead ol ‘esoeeds" em peer GG and (Gnteofxt) and als shr lee Mco (RME Dl hus, icomecus ff ransformation of the initial pattern Inte a pattern ofa different type (presumably lincar) with the same number However, one docs nol find in the texts ofthe Noo-Pythagareans any stipulation of similar kind. The theary af figured numbers to an explanation by Tamblichus,”' the process of comparisun shoul d be consid o EVEs R À This description suggests thal in end as “ome hy ume” del MR Timing back to the inguiry of preperties of absclute numbers, Nicernaches intraduees “he concepts of “plane”, “surface”, and “solid” Thing, base an the nation of dimension ‘them will coincide with an initial sogme nt ofthe other, y configuration, = Nicomachus Farsagucaienis Arichmerlune I uil dd backe Gurblichws in icomachs Arithmecenm tmrı (he soresoondiag [quamıny] ha ottime prier or Sur Arithmerisum intr eductinnem 56, UP lars a Nrcumachi dvithavericu e BRIE re Tamil 2 (Biácmmua). Ascontingly. by linear, plane, and solid manhrs are meant those heving one-, (va, oF three-Jimensional geomerrioal sespectivoly The generation of triangular numbers. Attong plane rumbers, the Irianguinr are sonsiderel by Nicomachus the mest primary and elementary, moanber is delined es “one which, when añalysed A arsamgulur into units, shapes into ne In icone ht erinicoductioncm 4124 Sel “icones intresizenonis di méticae Mali, D'0v 1926, 856,

NETIC INTERPRETATION OF NEO-PYTIAGOREAN ARTLUMETIC IOANNIAL S VANDOULAKIS. gular Form the equilatera! placement of fs parts in 2 plan numbers ure constructed acco rding to the following peter besa, 4-529, 977216, 1649225, © These Theon adds again that due same reas ing can be continued ad infinitum 3,6, 10, 19, 21.28, (rad uexprs érraipou d aíos Ayo. Nicomachus describes the following procedure for the construction op (angular numbers. Beginning from tre neural suite By multiplication, the square numbers are obtained as products of the same successivo numbers, that is 1.2,3,4.5,6,7,8,9,10, 11, 12,13, 14,15 We (ake he fist eu and hove the potu ta ly lest 13 Then addi the next ng term, ¿e, 22-4, 96, 46216, 2) we get the actually first triangule side one unft Then by addition of the next term of (he natural suite, number 6. In the same way, former configuration, by side, number beneath # gives the second tam lar the next term, Ze. 4, armed into units, and ji a aa ala se Gale solola ale alla da lala AS e WS adding all ke succcseive tuni ‘ranged into units, and joiving e... fe. 3, aranped lo the former configura, il gives the third triangular number 10 process in the same vay by The successive configurations ol square numibers are consmructed below: angular number, which i 3) ls configuration is cons tructed by selting two units , into unit. and joined to the 333 s ofthe mural suit, therm to the preceding configuratio n. 143 SD Tr should be noted thar the metheds of episynthesis and multptication are canccived by Theon ss schemes of reasoning that may apply to different things. The larer scheme applied to the suite of odd munbers as described above generates square numbers, whereas applied to the suito of even numbers generates the suite of oblong (Erepourixcis} mambers.** Indeed, let 2,4, 6,8, 10,12, ED rezza Com Theon expleias tn this proc edure can go om ad infinitu m Ge bps The suite of even ‘The meted by episynthesis runs as folicws: Beginning from the suite of odd numbers we fer the sum of tha frst two terms; hen ths su fe ede 0 the third term: the procedure can.i nues by aiding cach new sure te the nest era Ths beginne from the suite successive oblong numbers aro 2+4=6, GH6=12, 1248620, 20+:0=30, successive nal heategonal, octagonal, emengonal and decagonal members ate construszed. As it is noted by Nicomach “tie doctrine of these number ¡sto the highest degree in accord with their geometrica! representation, and not Out of harmony with 1, that is he aritmetica! theeey deveteps in aocordanes to its combine: isl model Like the triangular ané the square numbers, which are constructed tron the terms of the narucal seine tat diflor by L55911, I snd 2 reopessivey, pentagonal Tiers are consmicied Ann torm cha differ by 1 terms of squave numbers socios Inereduenoni. Arthmetizae I. vi i Viormechus the The generation of polsgonal numbers. La analogous way, pentagonal, hexago are and oblong numbers. Ti a describes vo procedures far the construction of square numbe rs: by cpispmhesis or mat) and by matiplicaion terra oe m) Ve can form the then vonstruetel in the Zoliowing way Ci The generation of numbers; 14, 16, 18, Inronuedionis ArinAmeti Ti va, rerum moskowaneasim ad iegemtem Piarémen tibia |, P'hon Keposttio I, Hache; D:ctoge er 1026, 192 813 Roche; D’Ooge 192

A GENETICINITRPRETAMION OINLO PYTURTONEAE ARTI From Use way that the objeeis ofthe theocy, at Is numbers of various kinds sre introduced "Fora the legical technique nscd forthe development of arithmetic Ascordieg to Ihe axiomatic approach, as described, for instance, dy Aristotle, one shouid proceed Iror some initial cuncepis, which are evident nd a set of propositions (axioms), describing relariuns betwe n these objects, ‘which are tive, immediate, ose Fui, prior and causative of the theo144474..40n-2 Tinwevor, in the Neowvtkszoveau arithmetic we mest another approach, Ino point af departure of ai mich Lier by 4, hoptagonal fro m terms wish {fr by $, oetagonalnr the theory % net some init suncopts and stmencnis, but à given object ~ the monté, designated by a ler -and a generative operation (move precisely a set of generative operations) that serve 26 rales orthe constriction of new Nice (sites ur onfigmtiuns) out of given ones‘ ‘Te mother Ius us 4086 than the mer Dy or oF a guumonie nalbe es] athe ales cum bi ihe indie male, 2 in the sue , 3 un hepiagon, end 3301, of any polygona l dfx by 2 name ofthe 0 polygo aata l~ à, UD, # m de hesagon. $ in te with sims increase!” Consequent a Ärg ly ona , l number of ) LE 2 ra 2 2 Successive pnomon be designate je, thats L= {a} andy Over this stan iterative procedure of altching on alpha i adlrted. Numbers s ave: san be dolina then as les of he form OIE, ‘whore + means that iv an abbreviation or tae NL FE NRO-PYTHAG ORFAN ARITHMETICAL REASONING ‘Whe genetic From semictio point of view, the ‘aiphabe® of Neo-Pythagcrean arithmeric consists of sajely one object, ie. die roma, denied by an alpha, and taken lo sides is n STH, 142082), Die« alphahe » of Neo-Pothiagorsam arithmetic! approach ‘cousisting of 4 Ti is impariani to stress tke difference between Lis concept On the gromds of the above analysis. wo can couclude thet the Pyt lagocean app Bus proachroach to10 arit arith in tic is preda ominantl hime y nor-aonniis. sigrs. This À cepts are relative to some set cheory and tharsay depend on sottheorefie Axioms, Moreover, the reduchien of Sito: teponitee rerum matkemar Hicemachi Aruhmeriznw Inscaderılan en arllmnetie 19 set theory, as is impiemented, for ins:anen, by Frege and Dodokénd, ir volves cuantilicatión over in um ou lagendten: Pla tonem willen ui Filles Cupuse 1066 9869 Niwomachis Jntrodus tianis desthmeri cae M 4; D'Ooge, 1426, $35, A a ise alin Uessibed by Tora (Lcfesiio corn maté ne this ole o Rite about 10 Bc) of number and fe modern concoptualisation. En modarı number tteny the arithraelic con- 5 Since the set of natwral numbers s definod Set, each ol" whwoh contcins the empty set and the successor of each member of sir ımGute Piste.Pech ro Diopha li sinss as {1 70-3 e Farmer; Heath 1464 vite se, 33 the iulersection of all infinite

INTERPRETATION DE ABE-PYTHAGOR ZAR SAT D IOANNIS M VANDOULAKIS it Ta the Neo yes approach, on he contrary, the thee comccpig do rot ely on any concept of ser, but an che roti of the “SUC”, and amero depends on constevstional gouetie possiblities rer tha on ¡Genero definition of arithmetic concepts ‘The definitions ofall kinds of numbers considered in Neo-Pylhagarsın arihour common sehen mere we Et = abstraction, that à by à Moreavez, tke “suite” is only potenziali, property. whit: detines a class of numocro poscensing dra properly, Isera infinie object, whereas infinite sets, ln the coodem some, aro taken lo he ull the arithmetic predicates are accompanied by an eltective prosodire for the actualy infinite genetic corsimiatiom of the considered set ofmumers, Whenet; cho focus isnot nice of setdbiconzio character. Fuser, he natural suite can he inradases as a Sequence oF the oem 23,4 and the various kinds € users can then be eoondingt0 certain rules on the existence of cortam number, passessing certain property, bul om tha mude of the genetio construction of tre ambos. Ta this sense, Ihe specifica ecified as suites constructed definition, From this pois of siew, Neo-Tyctagoresn armee i a informa! theory Penn having a distinctive combinate tions of numbers in New-Pythayorean aritunetic cua be considered £s yeni charter, insofa a it concert The genesis rule is Musa ig à umber of initial cases and Ure ancient autor closes bis easoning by the conclusion that the construction according to this rale can go on ai anfünimen. In virtue a Ihe constructivo constructions of varons finite schemetie pnlers, las suites and contigue. meno of the objects to which Gans, tn this contest, aretino reason ts conducted as rbcoretial int Lions bear an inductive character of the feno disecssod in the next seston, sal reasoning concertina Ihe possibili fo samy cm certain genes this kind of reasenine is applied, ese Yet, in Neo Pythagorean srthineli¢ the process of comstrection docs ner consiruzione over a domain of concreto objects. Such type of arithmetical fake place in tine sensoniog adi the representation by fconfiguraions of lees as ine brithmerie, the succession of stages of coustivetion is succession En tine, and interpretation Ce lin condimalin of le atthroetical statements and develops in accordance wath is combinatorial modo, Howewer, i sheuld ts the modera corsirustive case, La ía in intuitionistic incompletabliy arises rom the Tact that tho theory is à theory af ar idoatised be Fino mind, wich is located at some point in Cine sud has avilable only what mond thar very te action is paid hy the NowPythagorcan authors tothe it bas constructed Ta the past and its lntentiocal atindes towards the Feuer, milice al representation by letra self, ut slo the “experimental” par af Thus, there is na stage at which all constreerions are complete. Such a seman the theory. n Chir works, atideneical seasoning, 22068 to be predominantly rio picture, how contemplative activi [Virao 1994, IL 482) ‘Ag we havo already note the jets ofthe Noo-Pythagurees amet i mot found in veo Pyrlsgorsan artici, along iL can be chsrocterived as arllamet of init mind {kee secan on she supe of rimes aud the finte, p. 121) are divided ints die levels depending on the diension ofhe represent don. Eneas umbers que gentated Gt the monad: plane numbers me generated from the lincar ones: solid mmbere ar generated ran the appropriate combinen of pling and ligar nai ess. Such a aient ot hjees into linen, plane al solid rumbens nukes necessa de use of deren gerseiive operations a each level. Alt level of leas aumters, iestatoe, che basic gomme operat.on 5 that of “he «ddlion hy ail gente opening. Meroe, equal au ppiy only to aos afte so level and similar combinatoria Sompleni Siegertfeatures ofones: consieuetons AU goneri cons notions of the Neo Pythagorean anthmete have the following Features "They begin fom: the sme iil object, be. the moras 2) Provided thatthe result of the application oF certain iterative operation Laden by a unit, application of the gan inerte numbers of th same Kind, now numbers an cmstraciod; Thar are sonst Hm tes cr other deivaise cpensione) begin ask dit mem Ihe u, bt Gioral sue bess cas came ofthat te. begin te been alt mern and cremona potrà, und. eh or

NETIC INTERI RICE LION OF M TOANNIM, S VANDOU LAKIS (5) Ms stated that the method oF gener ation ola particular kind of number generate all the numbers can arithmetical reasoning, inten as essential charactessties nf Neo Pytha goreay that the oiethed of construction described by Ibe tras have to be constructed. “exhausts” all die nurhers first rs) clauses, us tbe clauses (1), (2), anal (34) are ta are specifi by means of assumes the universe of al does ner. leases (1) and (2), "Then clause (34) cen constructed in the way desc In virtue of te law nl is taken to mean tha there aro no defining property except these that have However, this does not seem to be die way of thinking of she ancie authors, for whom the totali of all mumbers was nut considered as given beforchand. Ir voutd no be t far one more reason (i ibe sense of is officine cons the stat“a em numb en er t veli truction was ) for he Neo- Pythagorean aritracticians an experiments?” fact, whereas te statement “hurt are no mirnbers such that..." isact 22 “experimefal, nt Accordingly, clause (3) is sentence by Ue Neo-Pythigoreans ir We form of negative existent The objects intruucod in Neo-Pyth agorean arithmetic by genetic al def vious are infinite sequences, usuall y incompletely exemple by à neri suite or certata eninbinatoria! soufigeealun. (suites and combinatoziai contigurat considered dusofar as the objects discu msed ions) me always finite instances and their vorplarion, what is impliity a involved here is the afar Fiat of pareil fi, wat allows reasoning ubout hoves ver loos gene processes ‘The realiabilif of the genetic eunstricti ons à taken te Potential. successive combinatorial Meo-Pyrhegorean arithmetic are not ouly (potentially) infinite, but alsu intrinsiparefy extenssonel. Generative operations fusofar as mature] munbers er» consirueled Segicning from the merad by addition of a unit, the iterative operation of transition Im a numer ta ils successive plays & Zundarmenta, role in Neo Py-tiagorean arithmetic. Detinitious of arithmetical concepli are reduced 16 the demonstration ow a definite considered Sind ofnumber. Arithmetic thoaribed by (1) and (2). never please of The possibility Yor an objosy objects 3 given beforehand. objects in dhe set determined by the that..." construction combinatorial rale works, when one posses fiom a nuniber 10 its suecessar, in excluded mile, every cbjac of the universe either possesses the Hetiing ov of to define » set of objects that fe posses the fina property is based on clause (37). This clause implica Property process ewe clauses Special attention should be paid on the formulatio n af the third clause. In mod em gencia dctinitions, the analogous claus e is usually ph-ased as follows 8°) There ae mo ather objects except those generated by the appli cation ofthe First wo dhe Anis sense, we van say that tho Neo-Py-hegorean approach to arithmetic Is not The thir clause (sumetizis ont by the Noo-Pythepim em autho Says and cally associated wich he ru fr their genetic consti. om the mona. I The first two clauses enable one 19 comet rw numbers out oF nes. wish", 139 configuraons ca conte ad fini, Motecves, the jets introduced in of the kind required, These features can bs you NEO-PYTHAGOREAN ARITHMETIC The neural suites, fr tosace ore tem s, to be extendible le «ns far jemansteation tha: the transition from a

A GENETIC INTERPRETATION OF NEO-PYTHAGOREAN ARITHMETIC IOANNIM. S VAN DOULAKIS of the Genetic construction vs. proof (infinite) set of “all natural numbers” was alien to the 141 Greek mathematical thinking.” The same view is supported by Unguru [1991, 278; It is obvious that the method outli ned above is not grounded upon the idea of 1994], who, in addition, stresses the difference between the ancient and the ¡modern conceptualisation of number. Unguru proceeds from the modern Proof (in the strict sense of the word). In the context of NeoPythagorean arühmetio, numbers are conceive d as given and any statement about them conception of the principle of mathematical induction (see the induction Hal means, IR for example, two the context of axiomatic theory of Peano arithmetic and presupposes the asserts something, which is confirmed in each instance by simple combinato. numbers are given, itis sufficient by of the construction of the corr esponding configuration (or by obser over the exemplary suite), whet correct or mot. to conf, vations her what has been stated abou t these number sis The demonstration is perf ormed by inspection over a finite Fragment of a usually (potential) infin say that the statements of Neoite object, In this sense, we are go cannot be found in Greek arithmetic. As Rashed has put it: Tf we confine ourselves to a rigorous formulation - which is essential — attempts an explicit way will be rejected as outside mathematical induction. Therefore, the foundation of Neo-Pythagorean arithmetic is not proof (nées) in the style of Fucl idean Elements and the works of other mathematicians of (he classical antiquity, but the idea of effec tive genctie construction, by means of whic h the corre realized modem concept ofset. Obviously, use of mathematical induction in this sense that do not state the argument of induction — P( n) > Pa 4 1 ) for any min Pythagorean arithmetic have: finitary meani confirmed. scheme suggested by him in [Unguru 1991, 274]), which, however, is valid in ctness of arithmetical statements is This type of arithmetical reas oning about given numbers can be without assumptions of axio matic character. The conf irmation of arithmetical statements is reali zable by a specific ‘experiment’ But as this rigour is related to a complete system of axioms - known as Peano’s system — which includes precisely the exact focmulation of the principle. of mathematical induction, all earlier formulations are necessarily naive [Rashed 1994, 77-78] However, Unguru seems to maintain a stronger thesis, namely that use of mathematical induction necessarily needs the general abstract concept of natural number (understood as independent variable), which is not true, On the other hand, Mueller [1981] admits a pre-structural form of mathematical induction. Fowler [1994] has suggested an actually finitary form The rule of mathematical indu ction The question whether mathematical has puzzled historians of mathema of mathematical induction (called by him “relaxed form of mathematical induction”), resting on a liberate passage from one case Lo its next in induction is used in Greck mathemat ics tics. B.L. van der Waerden [1954] admits ihe use of some kind of mathemat ical induction in Greek mathemat ics, yet only “essentially”. edge H. Freudenual rather cautiously ascri bes to Eucl id the knowl of only a “quasi-aligememer [our emphasis ] Induktionsschluss” [Freudenttal 1953, 28]. On the other hand, J. Itard [1961] seem s to ascribe unreservedly 10 Euclid the use of complete induction in Prop 13, 27, 36, IX 8, 9. A discussion of his claims is foun 721. A survey of various ositions VII 2, 4, d in [Vitrac 1994, 467. forms eros by the historians of mathematical induction and their of mathematics is given by Rash od [1994, 62- D.D. Mordukhaj-Boltovskoj [194 8-50], the editor of the Russian transla tion of Euelid’s Elements, excludes the use of mathematics, because the concept succession, and has argued that such a scheme could have been used by the Greeks. Finally, Acerbi [2000] has suggested regarding the Platonic passage Parmenides 149 a T-e3, as a full-fledged example of proof by A property of polygonal numbers An interesting property of polygonal numbers is found in Theon: From the multiples of the unit, that is, the doubles, the triples and the subsequent, all numbers which sue sively leave out one are square, all those which successively leave out two are cubes, and those which successively leave out mathematical induction in Greek of the infinite and, particularly, the concept complete fuetion. % Concerning Mordukhaj-Boltovskoj's viewpoint, sec [Medvedev 1990]

A GEN MIC INTERE PATIÓN OF NeO-PYTHAGOREAN ARITMIE IOANNIS M. VANDOULAKS Five are cubes sod squares at once; and they have sles square mer, boing cubes, and big square have cubo sec.” ‘This statement is especially interesting, because it is dhe Neo Pythagereag que rule of mathematical induction Lot us elueidate how mathematical induction could be enderstood in the cuntext of Nev-Pyihagorean arithmetic, Los statement P states something about numbers. As we bave seen the Neo-Pythigorean arithmeticians, beginning, in called fom the unit eumfinned seat Ihe statement P holes forthe first case. Assia, fivitary form of mathematical induction" Jhe piaof of Theon runs as is confirmed for the sccom case. Again, provided thal it holds for he second analogue of te Propositieu IX, 8 of Euclid's Efemenss, Beth propositions. Troan and in Euclid, ore proved by a morte pf rensoniug that we have follows: Vat ro the ronzio manners those whieh hagimng fz the mil leave our vue fre square, due leave out Lu bes, and bose lee ut five ces end square al one Is clcar By nz following: ser Forth e spite of double racers, 4,8, 16,32, 54 128, 256 im tis site the Bal double is 25i Gallows4 which is squares then follows & eue: thoa 16 Wish is ssuare; then 32; ator i 64 asbl is squece sad ‘cute; then 128; then 286 which ic scuare and be same ressonige cl itn Set Forth she (sue af] pie momies, tt is In this suite Ike [aumbers] leave mut one are square, and rn the [sie of] Sve lo ke manner, [suites off multiples il willbe Found Gat all Use Javi ou two ae cubes, those Tezze out five are cubos and square torce LU show! be noted that Theon does wat give a npilisd meted of prao! for all the multiples of the unit. He does nat refer to the Fuclidean concepts of ratio" and “continuous propertion”. Insicad. ho constructs the correspon Suites: che suite of Uouhles, the suite af triple, ml 40 Nth und confirms the required property for each sue separately. Uist the require? property holds for continued a infinitum, What we actually encounter here is net the applicarion of an abstract fie fact that the corresponding constructions in the successive cases follow cerain rogelaity. Therelare, the ostablishinent of « general propeny oF numbers is reduced (afler its confirrastin thar the mona possesses the considered propery) to the demonstracion chat the vensidered property is “inheritahte" when passing Jim a number to its successor. ff every number transmits # 1,3,9,27,81, 243, FD, Fold numbers ara dre ensequent [suits of] multiple. aso, they confirmed it for the mex! case by Ihe construction of the next member. Fineliy, il is concluded im one or another way that Ihis process can he jal law, but a type of reasoning that esteblishes a conclusiun as drawn from as which thal it belds for fte First use, the next number is constructed and this statement Un these grounds, he comeludos ey suite of multiples, Thus, indus used: On the sequence of the suites of multiples, To estalish that the requires property holds in sach suite certain property to the next number, imespective ofauy particular quality af the numbers themselves but in virtue of the rule of cheir conscruction, then all we are going fu call numbers will have tbe property. This mode of reasonmg iia foro of mathematical induction or rule of morhematieol induction. informally speaking. such a cale says Wat if some initial okjects have some properly and this property is Steruditaty”, construction, then any object Bus this ¿go Mr: ar a discucion oF te ro mv FQ), in u uniform manner thar does uot depend on the actual stage of lhe procedure, In modern terms this type of reasoning cun be proserted by the latlowing Sche of inference: me For the fist hematscarei va fegendun Platanen autre | and ya on cel finit Durals "966, 86.39 È jan proporiion, so Landau 145, 172-1 property. The property af numbers is established by a procedure af genetic construction which ooastruces PD, For the tiré case can Fngasitio voran m because of a certala rule of Par: za PS PO

be rarak SA TWIN OFS O-PYTHACORSAN ARITIIMETIC ACIDIC 15 Roanne ar VARIOUN ARES IL is clear thatthe inferential schecue that is repose in all Mess cases (à wird AGyog) isthe following PS) ‘here stands for an avbiteary (piven) number (Suite) aud S7 its successor. This inforence scheme states that on die sure Mar the statement 2 is firmed for any given number 4 it cr be confirmo for the moxt number Sas sell. Consequently, the statement P can be understood as asserting something of mp (reeves) given number Im comparison lo the madem principle axlomatle systera of Peano arihmetie, oF mathematical juduotiau in the wo would like to stress the following differences Fire, (he moe form af mathemarice] induction is understoni in Ihe conte: ofthe axiomatic system of Peano arithmetic, Ihn presupposes the ‘cept of set (in particular. the set ofnatural numbers), eon On the contra the Neo- Mythagovean Form weilher i stated in the context of any axiomatic system af arithmetic, ner the totlity of sura! numbers in presupposed us given before hand, In the modern sa un each syste of theory which contairs number theory, the principle sra becomes à sectheorets of prrciplo derivable from the asqoras of fie spste and wacthor induccion Is applicable to a certain sentence ofthe. demon both on the the rimani scemi ofthe axis of the symm and the definitions for mocos [Wang 1970, 468] Second, in contrast 1 the mode fr of rethomacal induction the Neo yıhaporcan form docs ot ee variables of any kind (number in NooePytaagoveaa anes, hat % The quamirisatoml werds nota visable always given), Accordingly, thee i 10 use of any quantifier ogin cer the ct) infinite sc pf numbers. in the expositions of Nuo-Pythaguccan ariete, one des not find emmeintion of PSI) PUK PO Generali all stra used, for example, fy Theon, do 55 universal theorens, this formal statements beginning with quantificar tiva words ofthe type “all, “even” eta, The problezn primiily concerns she ase when 2 general propeszy ls asserted of an infrite in the sense described Above) domain ef objects. In these cases the genoral property is established by ndnctien, chat is by means of certain constructions of such a character that the so tb te implement the corresponding, possibility to repeat a sila reasoning Construction for any other giver: member of the sume kin is evident. On that round que can conclude that whatever nunber of the Kind might have been ie, it's possible to vonfim (by analogo ine of reasoning) th this um. er has the property in question, is A generally inferential scheme ths might have boon used in vrilhmetie described in Posterior Analytics by Arisatle, ‘an attribute belongs lo a subject unlversaTy whe it can be shown to Belong to. fan aubiteary and primary instance of that subject ¿ronda Usos Tod nes apa raven The Ces ruxbv in this passage usually rendered ws “chance instance” ar analysis “random iustarce”. This rendering is misguided hy the ciymlogiad axés in ofthe word, which derives from the word 1031) However, “he useg ofis also sug this context should be understood as "arbbrary”. Sach a readin ecu by its use in Eussid’s Elements”? rhe focus is on the individual subject chosen, rather than on the candor character ofthe act of choice {he use af such a rule of gensality is necessary for the devolopment of Noo-Pythagorean arthmatic, in absense of specilie quantification. In his way. the stalemeats of Neo-Pyibagocean aritlaneric van be understood as general écclamajons of oor capability lo implement certain constructions for any given number of certs Kind. The belief that we can implement dhe required sonstrnetion for any given asserts P hetually act as quantifiers, but simply oxpress the fact thata statement something ef ap given munbo:. Consequontly the Nog Pythagorean fun of number (combinsiccial pattern) might have been rooted ın the experience mathematiealinCuetion is, essentially, a quannifcr-free “expeiiuentation” it hs form ef induchen. hind, jn the Neg- Pythagorean form af tatheruaticat Indheniona {ined from the readization w such constructor. As 2 result of this Kind of ‚nes clear how one has lo proue in exch cast, ic. a sateen P abboni members is never asserted of “he complete tots ity nf ostra numbers Y rise Hosier Anali tic 78925 ra puoi (roche ton) and edita Naeh Ire re a0 Of Me Versi,“rbt and 1 5, osi ee cosà pid lea

LORRI 34 PARDON ach casa i is car what we have ro do. A GRENIER LAS In ths sense, die statement ci Fished in Neo Fyth arc ag tic ave or general, en hat n they eal ‘uy mbr of tana bind the outcome of the comsimer ica définie rele, is a mamber poss ossing rhe requiced property Negation and tertium nom datar à MT ram fron tre comparison of the ja sphical representations of nunbers of the ht fy accordine 1 TION UF NFO-PYTUAG ONAN ARITEMICTIC sure OTe. Let two numbers of the same Kind he gica. y Soh of them begin som ths nit, TT sve begin to const thom carrying our the operation of addition by a unit or by placiug tbe guomon), ther (he Axo poaveduyes will continuo inthe sue way and wo cases am possible ‘The two proscsses will stop at the sente time; „One process will continue while che cher will sop, In the first caso ths serrespanding e suites ur con iguratians of nunkers van yet Into coinsidenes, whorces in the Second case the coutlguration ul’ one umber can be got ire esincidence enly with an initial segment of the othes Acvordirglv, La the former case the numbers are equal, while in the late, dre omequal CONCLUSIONS In our understanding. Lagoon the Neo-Py-hanowan avthmetic is & positive duitary <> acthunets har can be developed withowt assumeticns uf axiowialie character, nr as inforzal theory of countung over a dona of concrete finire objects of combinatonal aharacter These objects ars intcinsieally associsted ‘wit a cortain rale for their constructor. and characterised by a specifi modo of organisation of a desigrated entity, akon for unit into wholes possessing intemal sruchurs The role of these objects for the olaboratlon 9° the aritmetica theory is double: Firstly, they are cembinaterinl parte, that is they obey coruin reguTartes of combinatoria! Charuster. Seven, 50 do aay, a inode of existence of they are Tenetiemal, warily, hey “uber: aunıbers Nave nol been get akstacted Fr thet combinatorial made! The Uetiuitiun of aritmetica! <onsepts and the establishment of themes in this type of aulhmaetie is reduced 10 die iepiementarion of certain goncrio Censtuctions that are carieé out, sting from tac moasd and foliowine 9 definite couibinatoral rule describe i weeds. Incofar 5 the objects involved în arithmetical easoring are irf.ane. ths corresponding constmetions incamplere and imply assumed petentially celitzbie. ane To this offoc, a fini lary quantifier ree Term of reduction is applied When à general property is asserted of “all” nurcbers ul'a certain kind, This sposafie form oF indasiina

VANOS A. AGENTE RN eo ay compatible with tke type of aritm in develaned by the Pythagoras snd the structure uf their universe, bot also sufficient forthe requirements artnet sep. 12 vola Asclepins ofvolta: Do is important to stress that the Archie mathemuticians perceived y difference in anpmach between Nicornaciaus and Buclid bn sl-Haydarı sates 1. Tarim (81), Commentary fo Micamachu ches' athınene proseeds by induetive rersoning, Buelid's version is devel pod by dechire reasoning. Diephan LG. Bastmaluna Properties uf toas are aio ise ways Diophantos. Gest ey incon, si pharma [sivunachus of Geren. a. so proper and to Find he number This is Moma Te cer "n a book où atari TL. say of showing properties 0° mus Use whieh refer fo sn The Nicamachean type of arithmetic was galled abarichmatigr, phonetic Arabic transcription ul the Greek term GpiBprrtazi, with colaboración Gin ond Houk 461, Diophamos of Greek Algebra, Cambridge, Eng, ‘Canibridge, Eng,, 1910, prosoeds hy ro and deduzione. All properties of number grasped by Tron are contained in these br ok [€ UST] or in (6d) Arihmetica the Book IN. Veselnveky) of Pelygonal Numbers. aiar Apndoterma). Inmodzetion and Commentary, Moscow, (994 ‘ke The numbers anc by ore and it we isis between ther, Wo Tini y Aistingoishing and consdering Tabrodcton to Arithmetic’, hiladetphia, 1965. explicitly te disiaction between the Lo versions of ecmetic: while Nicoma sewonaria: A Study in the Tutor of 1689, 2° 6d, Divino ofllesemdeic, impr. Dover New York, 1964. of the Ancient B.S. Stumstls (éd,Digphamn' Avithmesica, The algebra on AA ain, Hee Fei étape tar whichis 4 y whereas the Euclidean variety of ariumetic wes cniled ‘fm ai-tad. thar is “the science al Dire, Athens, 1963, P. Tannery, (68 eb tr), Ditehomi Alexandr opera ommnia com Craccis numbers" Rashed 19, 246] commenturis, LLL 193,11, ii 1895, vep.Stntgen, 19%. P. Ver Becke, Dipinto dtlewandria, Bruges, 1926, 1éimpr. Paris, 1999. BELO RAPILY Futa Sources 1. Avistate L Bekker (80). Aristofels Opera Omnia, Berita New Ein, 1821-1 ©. Gion (4, 1960 LH Freese ec at. Camibrideo, JP JA. Steh, D: Ross (644), The Works of dristarlo, Oxford, “418-1952 and of elaborated. way ds sik LIWETATION OF NEO FVISACORTAN ARTIGO VANDOULAKIS (635), Aravorie (Lieb), London, 1926-1970 and cop 1909,23 vuls, (Opts, Thesis recension ofthe Option and Catopirics si bol 1895) 1 ViliPhennomana and rausical writingsh 9 6 Mass. 22 vol M.Unyduck etal, (685), Commentare in Arisiteiemi Cirueca, Berlin, Heiherg (éd), Enefides opera oneri, rg, 1916-1 (Element Si, soi 18, 1583, Elements LSBU (Element ) 1885 IV Clemens critica), prolegomens with Elements the to aivavsciolia LV ¿Elements 18%, € VII 1888 LVL (Dara with ths comment ol Mariues ara scho: 1852 in 29 H Meno (64), Eredi opero omnis, Le 18831916: LI (Elements 110, 1886 LIV [elements ma, à) Elune 1884, vo ISS LH (lero 1885, (hmm xy avscholia te the Element veut prolegemena critica) 188, LVL Ge wich the carmentars SI Merinus and schol), 1896 I Tha abs, Sharh end Ku LA (Commenter, cx the Premi net's amen). Sv, ara. MS 159, 1.293 Quote no Past Loti, 190% Theo recension ofthe Optics. ae Catapirios with chat. 189%, LVM Persona aad musica wings), 1916

SANTA VANDONZARIS S Stamatis (éd), Eielädis Ekmenia, revised ed. of Hsiheig's Grevk text, Teubner. 1969-77 1. post JI. Heiberg, L (Elements In), A 1969, HN (Elements y in}, 197U, LAN (Plamener xd, 1972, LIV (Plements xi. i), 1973, LV (pacts and 2 alex - proleganoena Critica, ets), 1977 DRE TON GENES PVIMAGORCAN ARNO 1956, D D. Mondukhaj Bottovskaj, Euclid' Elements (Hasan Emma), (. LIL, Mosca Temingrad, 1948, 3949, 1950, With the collabo-ration of M. Ja Vyucdski and LN. j Veselovshy) 151 Nicomachas of Ceres I Hoche (éd), Micomachi Geraseni Pyuhazerer iuroductinis Ariane Libri I, Leipzig, 1866 M.LD’Onge (6d), Mizomochu of Geraso: ‘TL Heath (6d), the Pisricen Books uf Buetis Flemems. Translated fiom Ue feat of org with Äuireduelion end commentary, 1. il Cambr | Dover idge / New York, 1908, 1926, AGENETIC Inmorluetion ro Arithneric translated into English, New York, 1866 Provtes G. Friediein (ed, Procii Dawischt on prima Buches Elomenturun yum commenter, Leipzig, 1973 GR Mowow (84), Prius: À Commentary on the First Book of Enchd' Elements. Translate with intracuction und motes, Princeton, 1970. B. Vitre (&d), Paoli: Les Éléments,= LIV, Paris, 1990- W Krol) Heroderus (84), Prucii Diadseh x Platonis Tumacum Commentaria Leipzig, 1899-1901, K. Hide (60, Historian, £1, Books TV; I Books V-IX, Oxford, ©. Pasquali (6d), Bron of Alexandra Ve, taria. Schunädt (6d), Heromis Alecandrini opera quae supers crm 5 vols and Supplement, Leipzig, 1899-4914: 1.1. 8. W. Schmid, neumuiic a er Automatica 1899; Supplement lo CI, W Schmidt, Die Geschichte der Teaubertisforung. Gwiechisches Warwegister, 1899; LR, Fase, mot published: Commencation LIT, dd disptriea, Defuitiones cum varlis Geometries, 10% LV, H. Schone, 1903 LIN, Ratones éd collections, 8. LI. Ad Heronir Heiberg, dimeendi Heiberg, Aeris quae 2 of Zleronis forwir quae feruntur Stercomenrics el De mensuris, 1914. Cratylum commen- Bsfhugors Diels-Krane (6, Notices, Diels Kran? 14 (1.1, np. 96-105) These of Sora ti Hiller (60), Theanis Snyrmaci Phliovoghi Platonic’ Esports rerum mathamaticarun ud leyera Platoncar wit. Leipzig, 1878, 1. Dapuis, Théon de Smgene Philusephe Plavonicwen Expenition Books N. Festa (0.1. De comuni mathematica sctentia, Leipzig, 180 icona Arihmeeunn rechnen, Leipzig, 18%. Inco ho mes pos Hoche (Ed, Commentary un Nicuorachas" Warn 1864-67 Platames Comaimences Mathémitiques Utiles, Pais, 1892: Bruxelles 1966, Jamblichus N. Pistol Sd), Presti Dredoció in Leipzig, 1908. Th L, Heath, 4 History of Greek Mashematies 1.11, Onferd, 1921 Mothemaaics m Arete, Oxliud. 1970. (7 éd. 1948. I. Tas, Les Livres zrithmerignes "Et, Patsy 1961, “troduction to Arihmeric WR. Kno, Phe Aiuti uf the Ruckideon Hemenss, Dordrecht, 1975

A GENETIC INTERPRETATION OF NEO-PYTHAGOREAN ARITHMETIC 153 IOANNIS M. VANDOULAKIS F. A Medvedev, On the mathematical infinity in Ancient Greece in the interpretation of D.D. Mordukhaj-Boltovskoj. (O Matemarwse-ckoi Gecxonemmocra 8 JIpeseii T'peuma 8 ronxosanmu JJ. Mopayxaii. Bonrosckoro), Moscow, Preprint N° 37, 1990. 1. Mueller, Philosophy of Mathematics and Deductive Structure in Euclid's “Elements”, MIT, 1981. G.H.F Nesselmann, Die Algebra der Griechen, Berlin, 1842, réimpr, Frankfurt, 1969. Mathématiques Arabes, Paris, 1984. Archive for History of Exact Science, 55, 2000, p. 57-76. 1. O Bashmakova, “On Ancient Greek Mathematics of the First Centuries AD. (06 amruunoli Matematuxe mepabix peros Hauleit apa)”, Hcmopuxo-mamemamuieckue uccnedosanun, 14, 1961, p.473-490. O. Becker, “Lehre vom Geranden und Ungeranden im Neunten Buch der euklidischen Elemente”, Quellen und Studien zur Geschichte der The Development of Arabic Mathematics: Between Arithmetic and Boston, 1984, London, 1994. P. Tannery, La Geometrie Grecque. D.H. Fowler, “Could the Greeks have used mathematical induction? Did they use it? Critical Remarks on an Article by S. Unguru”, Physis, Algebra. Translation of R. Rashed by A.F.W. Armstrong, Dordrecht XXXI, 1994, p. 253-265 H. Freudenttal, “Zur Geschichte der vollständigen Induktion”, Archives Internationales d'Histoire des Sciences, 6, 1953, p. 17-37. Histoire generale de la géometrie élementaire, Paris, 1887. V. A Smirnov, “The Genetic Method of Construction of a Scientific Theory”, (Tenetuuecknfi merom MOCTPOCHMA HAYUHOR TEOpHH). B. L. van der Waerden, Science Awakening, Groningen, Noordhoff, 1954, English translation of Ontwakende Wetenschap by A. Dresden, with additions by the author, Groningen, 1950. HG. Zeuthen, Geschichte der Mathematik im Altertum und Mittelalter. translation Moscow: USSR Academy of Science Publishing, 1962, p. 263- of H. Stein, “Eudoxos and Dedekind: on the ancient Greek theory of ratios and its relation to modern mathematics”, Synthese, 84, p. 163-211. edition with a Postscript by the author, Dover. German Dunocogerue sonpocsi cospemennoù dopmanouoli noeuxu, 284. H. Wang, Popular Lectures on Mathematical Logic, New York, 1993; 2% 1896. F. Acerbi, “Plato: Pamenides 14973. A Proof by Complete Induction?”, Mathematik, Astronomie und Physik 3, 1936, p. 533-553. R. Rashed, Entre Arithmétique et Algèbre. Recherches sur l'Histoire des Copenhagen, Articles Forelaesning over S. Unguru, “Greek Mathematics and Mathematical Induction”, Physis, XXVIII, 1991, p. 273-289. Mathematikens Historie: Oldtig I Middelalder, Copenhagen, 1893. “Fowling after Induction. Reply toD. Fowler'sComments”, Phpsis, Histoire des Mathématiques dans l'antiquité et le Moyen Age, Paris, XXXI, 1994, p. 267-272. 1902. Gauthier-Villars, French translation of Forelaesning over Mathematikens Historie: Oldtig I Middelalder, Copenhagen, 1893. 1. M. Vandoulakis, “Was Euclid’s Approach to Arithmetic Axiomatic?”, Oriens-Occidens, 2, 1998, p. 141-181. “On the Style of Neo-Pythagorean Arithmetic Thinking” (O crane neonéaropelickoro apHDMETHMECKOTO MBILINCHMA). éd. A. G. Barabashev, Cmuru 6 Mamemamuke: coyuorynomypnan Durocobus mamemamuxu, Saint Petersburg, 1999, p. 324-329.