young scholars’ conference

IITGANDHINAGAR

HistoryofMathematicsinIndia(HoMI)Project

YOUNGSCHOLARS’CONFERENCE

on

HISTORYOFMATHEMATICS17&18April2021

BOOKOFABSTRACTS

Conference Programme

10:00—10:15am Inauguration10:15—10:45am K.Mahesh

Nīlakaṇṭha's commentaryontheGaṇita sectionofĀryabhaṭīya

10:50—11:20am G.RajarajeswariIterativeprocessforobtainingtheInfiniteseriesfortheSineandCosinefunctionsintheKeralawork,Mahājyānayanaprakāraḥ

11:25—11:55am Athira K.BabuAlgebraicoperationsofsurdsinBījapallava,auniquecommentaryonBījagaṇita

12:00—12:30pm M.DevanandMallayyaMathematicalapplicationofMādhava’s Seriesapproxim-ation andconvergenceaccelerationtechniques

10:00—10:30am KeshavMelnadAglimpseofastronomicaltabletextthroughHaridatta’sJagadbhūṣaṇa

10:35—11:05am AkhileshKumarV.&ShailajaD.SharmaAsurveyofcomputationalmethodsinKolam

11:10—11:40am Sahana CidambiReconcilingerror:MathematicalinnovationsofGaṇeśa Daivajña

11:45—12:15pm Manosij GhoshDastidar &Rohitashwa SarkarAbriefhistoryactualinfinity:Aristotle,Leibniz,Cantor

12:20—12:40pm ConcludingDiscussion

Day 1: 17 April 2021

Day 2: 18 April 2021

Notes: 1. Every presentation of 20 minutes will be followed by a discussion of 10 minutes with our experts and audience.

2. All timings are in Indian Standard Time.

3|HoMI|YoungScholars’ConferenceonHistoryofMathematics2021

K. Mahesh Research Scholar, IIT Bombay

Dr.K.Mahesh,aResearchScientistinIITBombay,hasbeenworkingintheareaofhistoryofmathematicsandastronomywithafocusonsourceworks in Sanskrit. He did his post-graduation (Ācārya in SiddhāntaJyotiṣa)attheRashtriyaSanskritVidyapeetha,TirupatiandthendidPhDinIITBombay(2010)withProf.K.Ramasubramanian.Hedidhispost-doctoralresearchatCNRS&UniversityofParis.Anumberofhisresearchpapershavebeenpublished inpeer-reviewedjournals.His twobooks,CriticaleditionofKaraṇābharaṇa,andStudiesinIndianMathematicsandAstronomy (co-edited), have been published recently. The IndianNationalScienceAcademyhasconferredtheprestigiousYoungHistorianofScienceAwardonhimin2015inrecognitionofhissignificantresearchcontribution.

Nīlakaṇṭha's commentary

on the Gaṇita section of Āryabhaṭīya

The Āryabhaṭīya is one of the valuable works on astronomy and mathematics,authored by Āryabhaṭa at the end of 5th century CE, that received favourablythroughoutIndia.Asanevidenceofitsreputation,wefindseveralcommentariesonitcomposedbylaterastronomers.Apartfromthatweseemanyindependentworksandkaraṇatextsbasedonitwhichwerealsopopular.Amongthecommentariesonthe Āryabhaṭīya that have been authored so far, the Āryabhaṭīya-bhāṣya ofNīlakaṇṭhaSomayājī(1444–1544CE)isbyfarthebestandmostelaborateone.Hepresentsmulti-foldreasoningtotheenunciationsofĀryabhaṭaalongwithanumberofcitationsofauthority,illustrationsandvariousrelatedtopics.Someofitsdistinctfeatures found in the Gaṇita section of it would be discussed, during thepresentation.

HoMI|YoungScholars’ConferenceonHistoryofMathematics2021|4

G. Rajarajeswari Research Scholar, Department of Sanskrit, University of Madras

G. Rajarajeswari, is a Research Scholar at University of Madras,DepartmentofSanskrit.ShecompletedanM.Phil in IndianAstronomyandhasworkedonMahājyānayanaprakāraḥ,amanuscriptonderivationofsineseries,underthesupervisionofProf.M.S.Sriram,PresidentoftheProf.K.V.SarmaResearchFoundation,Chennai.SheiscurrentlyworkingonTriprasnadhyayainIndianastronomy.

Iterative process for obtaining the Infinite series for the Sine and

Cosine functions in the Kerala work, Mahājyānayanaprakāraḥ

Itiswellknownthattheinfiniteseriesexpansionforthesineandcosinefunctions

werefirstdiscussedintheKeralaworksonastronomyandmathematics,andare

invariablyascribedtoMādhavaofSangramagrama(14thcenturyCE).Thefullproofs

of theseare to be found inGaṇita-yukti-bhāṣāof Jyeṣṭhadeva (composedaround

1530CE).However there is aKeralaworkcalledMahājyānayanaprakāraḥwhich

describes the infinte series for the jyā (Rsin 𝜃) and the śarā (R(1-cos 𝜃) ) and

providesashorterderivationofthem.ThiswasdiscussedinapaperbyDavidGold

andDavidPingreein1991.However,thatpaperdidnotexplainthederivationofthe

infiniteseriesinthemanuscript.Inthispaperweprovidethederivationcompletely

based on the upapatti provided by the author of the manuscript himself. This

derivationisverysimilartotheoneinGaṇita-yukti-bhāṣā,butdiffersfromitinsome

respects. Both the derivations are based on the iterative solutionof the discrete

versionoftheequations:

𝑠𝑖𝑛𝜃 = ∫ 𝑐𝑜𝑠𝜃*𝑑𝜃* = 𝜃 −∫ (1 − cosθ′)𝑑𝜃′56

56 ,

1 − 𝑐𝑜𝑠𝜃 = ∫ 𝑠𝑖𝑛𝜃*𝑑𝜃*56 ,

andmakeuseoftheresult:∑ 𝑗9 ≈ ;<=>

9?@ , forlarge𝑛.;I@

JK6 Thisresultiscrucialfor

obtainingtheinfiniteseriesfor𝜋also.However,noiterativeprocessisinvolved

there.


Athira K. Babu Research Scholar, Department of Sanskrit Sahitya, Sree Sankaracharya

University of Sanskrit, Kerala

Athira K. Babu is a research scholar at the Department of SanskritSahitya,SreeSankaracharyaUniversityofSanskrit,Kalady,Kerala,andpursues her PhD on “Rationales in IndianMathematics: A StudywithSpecialFocusontheSanskritCommentariesofBījagaṇita”.HerresearchmainlyfocusesonIndianmathematics,theKeralaSchoolofMathematics,andancient Indian chemistry. She completedhergraduation,BEdandpost-graduation in mathematics from Mahatma Gandhi University,Kottayam,Kerala and also a post-graduation in Sanskrit Sahitya fromSreeSanskracharyaUniversityofSanskrit,Kalady,Kerala.ShecompletedanMPhilfromthesameuniversityonthetitle“AReviewonVeṇvārohabyMādhavaofSaṅgamagrāma”.

Algebraic operations of surds in Bījapallava,

a unique commentary on Bījagaṇita

The Sanskrit term karaṇī is used for surd, the irrational root of an integer. InSiddhānta Śekhara, Śrīpati defines: “The number whose square root cannot beobtainedexactly is said to forman irrationalquantitykaraṇī (!ा#यंनमूऱंखऱुय,य

राशे,त,य1तत2ठंकरणीततनाम।)”. It canbe seen that theelementary treatmentsof

surds,particularlyaddition,subtraction,multiplication,separationandextractionofsquarerootofsurds,compoundofsurdsandthelikeoccursinthealgebraicworkson Indianmathematics. The present article gives an account of the treatment ofsurds inBījapallava, a unique commentary onBījagaṇita, algebra in Sanskrit byBhāskarācārya.

TheSanskrittermGaṇitaśāstra,meaningliterallythe“scienceofcalculation”,isusedformathematics. According to Vedāṅga Jyotiṣa, “Like the crests on the heads ofpeacock, as thegemson thehoodsof cobras, so ismathematicsat the topof allscience.” This statement reveals the importance given tomathematics in ancientIndia.InSanskrit,algebraisknownasavyaktagaṇitaorbījagaṇitaanddealswiththe determination of unknown entities, while arithmetic (vyaktagaṇita orpāṭīgaṇita) deals with the mathematical operations with known entities.


Bhāskarācārya’sBījagaṇita,aSanskritclassiconalgebra,isthesecondchapterofhismonumentaltreatiseonmathematics,Siddhāntaśiromaṇi.

BījapallavaisafamousSanskritcommentaryonBījagaṇitabyKṛṣṇaDaivajña(16thcentury).ThenameBījapallava is a compound formedby the compositionof thewords bīja, meaning algebra, the science of analytical calculation, and pallava,meaning ‘sprouts’. Bījapallava thus means “the sprout of algebra”. In theintroduction of the edition ofBījapallava, T.V. Radhakrishnanwrites, “When theadventofspringwasvisiblebythesproutsonthetreesSriKṛṣṇaDaivajñarealisedthatthetreeofalgebraalsoshouldhaveitssprouts.Sohewrotethiscommentaryand called it the sprout of algebra or Bījapallava, announcing to the people allaroundthatthisknowledgealsowasboundtohaveabetterrecognition.”

RadhakrishnaSastri,theeditorofBījapallava,alsostatedthatintheintroductionofthe work, the author of the original text gives only the general enunciations inoriginal text (the mūlagrantha). A commentary consists of the explanatorystatements and demonstrations of the general enunciations. Usually, thedemonstrationsaremerelyverifications(byexamples)inordertounderstandthetextcorrectly.HereistherelevanceofthestudyofthecommentaryonBījagaṇita.

TheotherSanskritcommentariesduringthemedievalperiodare:TheSūryaprakaśaof Sūryadāsa, Bījavivaraṇa (1639 CE) of Vīreśvara, Śiśubodhana (1652 CE) ofBhāskaraofRājagiri,Bījaprabodha(1687CE)ofRāmakṛṣṇa,Vāsanābhāṣya(before1725CE)ofHaridāsa,Bālabodhinī(1792CE)ofKṛpārāma.

TherearetwoeditionsofBījapallavamofKṛṣṇaDaivajña:(1)byDattatreyaApte,entitledBhāskarīyabījagaṇitamwiththeVyākhyāNavāṅkuraofKṛṣṇa,publishedasASS99,Poona,1930; (2)byT.V.RadhakrishnaSastri, entitledBījapallavamwithintroductionbyT.V.RadhakrishnaSastri,publishedasMadrasGOS67,TSMS78,Thanjavur, 1958. The work Bījapallava of Kṛṣṇa Daivajña: Algebra in Sixteenth-CenturyIndia,acriticalstudybySitaSunderRamin2012isarecentstudy.

Bījapallavaisdividedintothirteenchapters(adhyāyas)whichcontainthesix-foldoperationofpositiveandnegativequantities,zero,unknownsandsurds(karaṇī),the indeterminate equations of the first degree (kuṭṭaka), and second degreeseparately; linear and quadratic equations having more than one unknown;operationswithproductsofseveralunknowns;asectionabouttheauthorBhāskaraandhisworks.

ThispapertriestodemonstratethereflectionsofelementarytreatmentofsurdsinthemathematicaltraditionofIndiawithrespecttothecommentaryBījapallavaonBījagaṇita,especiallyunderthebackgroundofmedievalIndia.


M. Devanand Mallayya Pursuing Integrated BSMS, IISER-TVM, Thiruvananthapuram, Kerala

M.DevanandMallayyawasborninThiruvananthapuramin2000asthesecondsonofMrs.AnithaMallayyaandDrVMadhukarMallayya.Withanardentdesiretopursuehigherstudiesandresearchinmathematics,aftercompletingClassXIIhejoinedtheIntegratedBSMSProgramoftheIndian Institute of Science Education and Research (IISER–TVM) atTrivandrumin2018,andisnowinhissixthsemesterwithMathematicsMajoralongwithPhysicsandDataScienceasMinors.

Mathematical Application of Mādhava’s Series Approximation and Convergence Acceleration Techniques


Keshav Melnad Post Doctoral Fellow, IIT Gandhinagar

Fromtheageoften,KeshavMelnadunderwentthetraditionalgurukulasystemof education for twelve years, which included learning Vedas,Śāstras,Yoga,fineartsaswellasmodernscience.Duringthisperiod,healsograduatedwithtwoBachelor’sdegreesandaMaster’sdegree.Later,having been awarded a JRF, he completed his research in the field ofHistoryofIndianAstronomyandMathematicsfromIITBombayandwasawardedadoctoraldegreeforhisthesis,"ACriticalStudyofHaridatta'sJagadbhūṣaṇa".Currently,KeshavworksasaPost-DoctoralFellowintheproject“HistoryofMathematicsofIndia”atIITGandhinagar,Gujarat.

A glimpse of astronomical table text through Haridatta's Jagadbhūṣaṇa

Tracing the development of Indian astronomy throws valuable insights on thesocioculturalandintellectualinteractionswithothercultures,ascanbeseenintheastronomical literature of Siddhāntas, Karaṇas and Tantras, and particularlyKoṣṭhakas/Sāraṇīs(tables),whicharetheprimaryfocusofthistalk.

Ancient astronomical tables are perceived as a continuous descent withmodification and advancement, spreading through the Babylonian, Greek, andIslamictraditionssuccessively,eventuallyleadingtothemanifoldtablesofmodernastronomyinitspre-computerperiod.TheoriginoftheIndianastronomicaltablesappearstobecollaterallyrelatedtothislineage,butwithasubstantialamountofindependentdevelopment.Theirmajorcontentsincludemeanandtruemotionsofthe luminaries, synodic phenomena, eclipses, trigonometry alongwith necessaryparameters for computation. At the times when algebraic notation andmathematicalgraphingtechniqueswereabsent,thetabularformatwasconsideredthe best way to transmit precise information to the user. As far as the Indiantradition goes, one of the main advantage of such tables would be to achievesignificantsimplificationinthecomputationalprocess;theybecamethestandardsupplement for astronomers as well as astrologers, who through their practicehelped compose additional calendrical, astrological texts like almanacs andephemerides(Pañcāṅga).

Thistalkfocusesononeparticulartext,theJagadbhūṣaṇaofHaridatta,whichgivesasimplifiedmethodforcalculatingplanetarypositions.


Akhilesh Kumar V. Graduate student at IIIT, Bangalore

AkhileshKumarV,21, completedhisBachelor's inPhysics fromAzimPremjiUniversityin2020andiscurrentlypursuingMScDigitalSocietyattheInternationalInstituteofInformationTechnology,Bengaluru.Hisresearch interests include Applied Mathematics, Human-ComputerInteraction, Data Analysis, ICT Regulation and Policy Research,QualitativeandQuantitativeResearchAnalysis.HeiscurrentlyworkingasaninternaspartoftheMathHeritageInitiativeatNIAS,Bengaluru,aplatformestablished in2020 topromote the studyof andexplorationintomathandcomputationalaspects,withafocusonyoungscholars.

with

Shailaja D Sharma Professor, NIAS Mathematics Heritage Initiative

A Survey of Computational Methods in Kolam

Womeninmanyhouseholds insouthernIndia followa long-standingtraditionofdecorating the threshold of their home at dawn every day by drawing stylizedgeometrical patterns on the floor, using rice powder, powdered soapstone, or apasteofriceinwater.Thesetraditionaldrawings,calledkolamsinTamilNadu,seemto have ancient origins. Kolams are enigmatic patterns, and have attracted theattention of social anthropologists andmathematicians alike. Although they are‘performed’withoutreferencetoatextbookorhandbookofpatterns,thatistosay,they are drawn from so-called muscle memory, they have intrigued onlookersbecauseoftheirinterestingandcomplexpatterns.Kolamshavebeenstudiedusingdifferent computational andmathematicalmethods, although they have enteredinto scholarly literature only since the 1970s. Lindenmeyer language, turtlegraphics, array grammars, representation using modular numbers and othercomputationalmethodsarecoveredinthepresentsurvey.Wehaveundertakenacomprehensive reconstructionof severalKolamsusingmultiplemethods. Suchacomprehensiveandcomparativestudy ismissing in the literature,whichmaybepartlywhythesehighlyinterestingpatternshavesofardrawnalimitedamountof


attention. The present study is restricted to Kolams drawn on a rectangular orslantinggridofdots,withlinesgoingaroundthedots(pullikolam).

Incomputerscience,aformallanguageisspecifiedbyagrammar,whereagrammarconsistsofasetofsymbols,alongwithrules forproducingwordsandsentences.Theproductionrulesarealsocalled‘re-writingrules’andreplaceagivenstringwithanotherstring.Picturelanguagesisaformallanguagewherethestartingstringisasymbolstringandre-writingrulesarepictures.KolamsgeneratedbysuchmethodsarediscussedbyMariaAscher(2002).

Animportantshortcomingoftheabovemethodistheabsenceofreferencetothegridofdots,which is the startingpoint for theKolamsunderdiscussion. SeveralmethodsofgeneratingKolampatternsovercomethisshortcoming.Arraygrammarsdealwithtwo-dimensionalarraysofsymbolsinsteadofstringsofsymbolsandcanthusbedefinedonagridofdots.Siromoeyetal.(1973,1974)givearrayrewritingrules wherein, by a suitable choice of primitives, terminal and non-terminalsymbols,agivenarraygrammarcanbeseentocorrespondtoadefinedfamilyofKolams.

Inpractice,thewomenfolkwhodrawKolamslayoutthepulliarrayfirstandthenstraight lines, threads or circles are drawn connecting or encircling the dots. Iflabelleddotsaregeneratedfirstwiththelabelscontaintheinstructionsonthe‘state’of the dot, then the final kolam required can be formed by simply reading themetadata contained in the labels The advantage of this method is that a singlegrammarwithafinitenumberofinstructionscangenerateaninfinitesetofkolampatternsofdifferentsizes.

Kolamscanbeanalysedasbeing constructedonagridofdots,which is eitherasquaregrid(nerpulli)oran inclinedgridorrhombicgrid(nadupulli). Incertaincircumstances,Kolamscanbeassociatedwithauniquecombinationofhexadecimalnumbers,basedonthebinarycodingofthegriddots.YetanotherapproachmodelsthestructureoftheKolam,bydefiningcomprehensivelytherelationshipbetweenneighbouringdotsonthedotarray,consideringeachdotasavertexandtheKolamasagraph.Atreestructure,calledN-line,isconstructedbyjoiningverticeswhichhave certainwell-defined relationshipsand theKolam is seenasamedial graphcuttingacrosstheN-line.ThismethodisintroducedbyYanagisawaetal.(2007)andisusedtogeneratelargefamiliesofcomputergeneratedKolams.

Nagata(2007)treatsKolamasapatternemergingovertime,throughdirectedandcontinuousmoves,similartovoicedlanguage.Tilecodingisawayofsettingupthestatusof4edgesofatilecontainingadot.I.Asetofsequentialstatusesfromthe


startingedgebetweentwoadjacenttilesiscalledachaincode.TileandChainCodescanbeusedtofullydescribeaKolam.

We thus describe a large number of different extant computational methods ofdescribingandgeneratingKolamsandlistafewadditionalapproachesnotincludedinthepresentsurvey.TheKolamssoderivedmaybefinite,extendableorinfinite.GeneratedKolamsincludefractalsandspacefillingcurves.ItisclearthatKolamsencode rich mathematical and computational content, which has been onlyminimallyexplored.Directionsforfurtherresearchareindicated.


Sahana Cidambi Research Scholar, University of Canterbury, New Zealand

SahanaCidambiisaPhDstudentattheUniversityofCanterburyinNewZealand studying Sanskrit mathematical astronomy under thesupervisionofProfessorClemencyMontelle(UniversityofCanterbury,NZ)andProfessorKimPlofker(UnionCollege,USA).Sahana’sresearchfocuses on analyzing the mathematics of select chapters of theGrahalāghava (composed in 1520 CE) of Gaṇeśa Daivajña as well ascomparing the commentarial styles and pedagogical methods ofmembers of his professional lineage, Mallāri and Viśvanātha, in theiraccompanyingcommentaries.

Reconciling Error: Mathematical Innovations of Gaṇeśa Daivajña

Astronomers’basicplanetarymodelwasoneofuniformcircularmotionwithsimpleconcentric orbits. They observed the effects of what we now attribute to theellipticityandheliocentricityofplanetaryorbits,andadjustedthisbasicmodeltobean epicycle/eccentric model that relied on trigonometry. However, in 1520Nandigrām, India, Gaṇeśa Daivajña composed his astronomical handbook theGrahalāghava, which famously claimed to reject astronomers’ dependence ontrigonometryforplanetarycalculations.Instead,heprovidedalgebraicformulaetoapproximate trigonometric computations. To account for approximation error,Gaṇeśadevisedcorrectionaltermsorevenadditionalformulae.Inthistalk,Ifocuson Gaṇeśa's unconventional inclusion of two corrective formulae to account forodditiesintheorbitsofVenusandMars,andwilldiscusspossiblemotivationsforthese verses with the commentaries of two prominent members of Gaṇeśa'sprofessionallineage:MallāriandViśvanātha.


Manosij Ghosh Dastidar Pondicherry Universitywith

Rohitashwa Sarkar Center for Studies in Social Sciences, Calcutta

A brief history of actual infinity: Aristotle, Leibniz and Cantor

Section 4 of book 4 of Aristotle’s Metaphysics introduces the argument againstinfinity.Itbeginswitharefutationofsomecontemporaryphilosophiesofdifference,notablythatofHeraclitus,whoarguesforsimultaneousbeingandnon-beingofthesamething.ForthinkerslikeHeraclitus,athingcanbothbeandnot-beatthesametime. Aristotle finds this completely absurd. A thing, for example, a man has aparticulardefinition.Whileitcanhaveinfiniteattributes,man’sdefinitionpertainstohisessence,whichiscomposedofagroupofnecessaryattributes:“manisatwo-footedmammal”.Oncethisdefinition isgiven,manbecomesthisparticular thingandnothingelse.Onecancounterthatthesamewordcanhavemultipledefinitions.Aristotle’sresponseisthataslongasthemultiplicityisfinite,wecanusedifferentwords for different definitions, thereby maintaining that each word has a fixeddefinitionandessence.

Thus,becauseeachwordisdefinedintermsofitsessentialattributes,itcannotalsobethesameasit’srefutation,whichwillpreciselynothavethesameessence.Thisistrueforthesamereasonthat‘man’isnotthesameas‘white’–eventhoughamancanbewhite.Manandwhiteastermshavedifferentessences,eventhoughtheycanbeaccidentalattributesofoneanother–likea‘whiteman’.Buttheydonotbelongtoeachother’sessence.Thisiswhytosaythatsomethingsimultaneouslybothis,say,whiteandnot-white,istomix-upthething’sessence–everythingasaresultlosesitsindividualbeingandinaworldwithoutthestabilityofessence,everythingbecomeseverythingelse.ThisiswhereAristotlelocatesinfinity.Hehastodismissit’s actuality because for him allmodes ofmeaning-making and communicationbreakdownifinfinityisreal.

Thelawofnon-contradictionisthegroundofthisentireedifice,theurlogicalshieldagainst the chaos of infinity. Infinity cannot be an actual object in this universebecause,asisevidentfromtheprecedingdiscussion,theveryconditionforinfinityis violation of non-contradiction (something both is and is not) Infinity is theincludedmiddle that splits apart thebinariesofAristotelian identity.Within thisparadigm, it thus remains out of reach, ineffable, the point atwhich the systemcollapses,butalso,itstandstoreason,itisthepointofitsobscureorigin.


Forus,whatthismeansisthatthewholestructurequiltedbythelawsofidentity,non-contradictionandexcludedmiddlesandthatreliesprincipallyonthenotionsofsubstanceandessence for thedefinitionofallobjects,produces infinityas theaporetic limit of its thought – the limit that is really ground. The truth of theAristotelianuniversethusturnsonitsheadbecauseinfinityisintruthit’sabsolutereference,itssovereignsourcethatisoutsideitbutsanctioningeverythingthatishappening within. The problem with the Aristotelian system is that it willinglyforgets it’s origin and true being, a forgetting that engenders it’s discourse onidentity.Throughoutthishistory,wewilltrytoisolatetheepistemicbreaksintheseformalstructures–Aristotlegiveswaytoascholastictheoryofgrammarwhichisnonetheless quite derivative of his own work, then Leibniz creates an actualsyncategorematicinfinite–thelatterisproperlyoftheorderofabreakorrupture.Forus,thequestionwillbe,whataretheepistemicshiftsfromaclassicAristotelianunderstanding that allow for the preparation of conditions that can generate anactual infinity? The historywe are tracingwill attempt to answer this questionthroughtheshiftsinitiatedbyLeibnizandCantor.

For a large part, medieval theologians inherited Aristotle’s views on the actualinfinite – especially influential was his refutation of Zeno inBook III of Physics.Aristotle had said that Zeno’s infinity is not an actual entity because one neverreaches it, there is always more to it. It is incompossible with the concept ofwholeness,andwhatisnotwholeorwhatisneverfullyitselfcanneverbeanything–itisalogicalmisnomer.Theinfinitereferstothewhole,butinrealityweonlygeta part of the whole – finitude. At the level of actuality, it is always finite. Formedievalists,themainconcernforinfinitywasasadivineattribute–infinitynotonly as the largest quantity, but as the highest perfection. The real task for themedievalistswastorigorouslydefineaninfinitebeing–unliketheneo-Platonicone-infinite,thiswasdivinebeingwasnotoutsidetherealmofbeings,ratheritwasthesupremeBeing.Moreover,itsexistencecouldnotbedisputed.Sowhileatthelevelofphysical things, infinitywasneveractual, itwasa littlehardtosaythatdivineinfinitywasmerely potential. Itwas a completedwhole, a totality and thereforeabsolutely real. Some, like Duns Scotus, even argued that this infinity wasquantitative.Hisargumentisalmostproto-Cantorianinitsimagination–heacceptsthat as long aswe think of infinity as a process that is always heading towardscompletionbutneverreachingit,infinityremainspotential.Butwecouldthinkofan infinitely large quantity as actual if all itspartswere available as a completewholesimultaneously.Onedoeshavetowaitforthefuturetofindthecompletedinfinity if one can think of infinity as already completely given, as a particularquantity.Thus, it is true that longbefore theLeibnizian infinitesimalwehave inDunsScotusanapprehensionofinfinityasanactualquantity.

Forthelargepartthough,despitetheobviousantinomiesthatconcernanotionof‘infinite being’, Aristotle’s views were gospel: till the dawn of calculus in theseventeenthcentury,infinitywasdeemedmerelypotential.AlthoughGodwasreal,


there couldnonaturalor logicalobject that could reallyencompass infinity.Themiddle-agesdonotconstituteanyrealbreakfromAristotelianthought.TherevivalofGreeklearningacrossEuropeinthe15thand16thcenturiesleadtoaflourishingof mathematical thought – which is why in the century before Leibniz, we findapprehensionsofanotionsthatwouldrigorouslydefinedincalculuslater:Fermat’smaxima and minima, Cavalieri’s indivisibles. The Leibnizian infinite reallyradicalizesthisentirehistory,whichiswhatwewillpresentlyfollow.

The first point of entry for us is the distinction between the categorematic andsyncategorematic. Initially a medieval theory of logic, the distinction is utilisedexpertly by Leibniz to create amajor change. Thiswill require someunpacking.Everypropositionorjudgementhasalogicalconstant,apartfromhavingcontent.Takethesentence:

Everybaseisanalkali.OR

NoChristianpraystosomepagan.

The terms every (universal affirmative), some (particular) and no (universalnegative) are logical constants, whereas ‘boy’, ‘girl’, ‘christian’, ‘pagan’ aresubject/predicatetermsthatsupplycontenttothepropositions.Theconstantsgiveformtoanenunciation,whereasthecategoremaricsubject/predicatetermssupplythematter that is organized by the formal constants. The constants are termedsyncategorematic.

Thecategorematicterms,assubjectsorpredicates,havemeaningontheirown,butthesyncategorematictermsgetmeaningcontextually–‘if’,‘then’,‘every’,‘some’aretermsthatonlyproducemeaningthroughattachmentwithcategorematicterms.Inthe caseof the infinite, formostscholastics, infinitywas likea syncategorematicterm,whosemeaningonitsownisindefinite.NormalcategorematictermscouldnotbeattributedofGod–ifthatwaspossible,Godwouldbelikeallothercreatures.Itisonlyinfinityasasyncategorematicattributethatcanbe,inasense,‘predicated’ofthe Creator.Now this syncategorematicwas accepted as a potential infinite – aswe’ve made clear, with a few notable exceptions, infinity as actual was acontradictory concept for the scholastics. Leibniz takes an entirely heterodoxposition:thesyncategorematicinfiniteisactual.Thismeansthatheisnotdisputingthedescriptionoftheinfinity–infinityremainsaprocess,anindefiniteincompleteentity,butcompletenessorwholenessisnolongernecessaryforactuality.ItisatthisprecisejuncturethatLeibnizmakeshisintervention–somethingcanbeactualwithout being a unity, the actual is fragmented, partial, not composed of atomicwholesbutinfinitesmals.

LetusremembertheprecisenatureofLeibniz’soppositiontoatomism.ForLeibniz,nature isacontinuousplenum,notawholecomposedofseparableatoms.Everypointoftheplenumcanbefurtherdecomposed,everyunitybreaksdownfurther,thereisalwaysmoreorless,alwaysanoutside.Theactualisalwaysaprocess,never


aunity.But thepartsofnatureneverseparate. It is likethesamethingtwisting,turningandexpandingorcontractingwithoutbreaking:everythingistopologicallyhomeomorphic.Theunityisthereforeatthelevelofhomeomorphiccontinuity.Itisall One. But at the same time, internally everything is in permanent flux andbecoming.TheLeibnizianactualissimplytheunityofbecoming.

What is crucial for us is how he creates modern mathematical infinity byreconceptualizingactuality/existence.Itistruthsofexistence,asopposedtotruthsofessence,thatareextendedontoinfinity.Letusrecallthefamous“CaesarcrossedtheRubicon”example.Thecrossingleadstowar,whichleadstodeath,whichleadstoseveralothereventsandsoonandthechainneverterminates.Thisseriesisthesyncategorematicinfinite.Whatitdoesisencompassanentireworld.NowwearereallygettingintotheheartoftheLeibniziansystem:everyactionenclosesanentireworld.Thereareinfiniteworldswithaninfinitenumberofevents.Everyworldisapointofview,and,tousethemorewell-knownexpression,amonad.

Thus,existence/actuality is theunfoldingof thesyncategorematic infinite.This ishow Leibniz makes the syncategorematic actual – and how he creates a newconceptionofactualinfinitythatpushesAristotelianismtoitslimits.Ifinfinitywasthelimit-pointoftheAristotelianworld,ifitwassimultaneouslyit’sterminalpointas well as it’s transcendental beginning, here we a system that completelyinternalizesit.Itisarealobjectinthisworld.Itcanbeshownindetail,thoughitwouldbeoutsidethedirectpurviewofthiswork,thatwhatmakesthispossibleishisradicalrehashingofthePrincipleofIdentity,andtheaddedsupportoftheLawofContinuity,thenotionofcompossibilityandhisreadingoftheanalyticapriori.ItistheircombinationthatrupturesthereceivedhistoryofAristotelianlogic.

Itisonceagainaruptureattheleveloftheconcept‘actuality’thatwewilltraceafterthis.GeorgCantorgivesusatrulymathematicalnotionofactuality,definedintermsofcardinality.Whatmakesalephnaughtandalephoneactualisthattheycan,inasense,bemeasuredascardinality.Bymakingcardinalitythegroundforactuality,hecreates what is perhaps the first properly mathematical and non-philosophicaldefinitionofactuality.WhatfollowsisashorthistoricaldemonstrationofhowthisideafirstdevelopedinHumeandwentfromGalileo,LeibniztoCantor:

InAtreatiseofHumanNatureHumewrites:

“I have already observed, that geometry, or the art, by which we fix theproportions of figures; though it much excels both in universality andexactness, the loose judgments of the senses and imagination; yet neverattainsaperfectprecisionandexactness.It’sfirstprinciplesarestilldrawnfromthegeneralappearanceoftheobjects;andthatappearancecanneveraffordusanysecurity,whenweexamine,theprodigiousminutenessofwhichnatureissusceptible.Ourideasseemtogiveaperfectassurance,thatnotworightlinescanhaveacommonsegment;butifweconsidertheseideas,weshall find, that theyalwayssupposeasensible inclinationof thetwo lines,andthatwheretheangletheyformisextremelysmall,wehavenostandard


ofastraightlinesopreciseastoassureusofthetruthofthisproposition.Itisthesamecasewithmostoftheprimarydecisionsofthemathematics.Thereremain,therefore,algebraandarithmeticastheonlysciences,inwhichwe can carry on a chain of reasoning to any degree of intricacy, and yetpreserve a perfect exactness and certainty. We are possest of a precisestandard,bywhichwecanjudgeoftheequalityandproportionofnumbers;andaccordingastheycorrespondornottothatstandard,wedeterminetheirrelations, without any possibility of error. When two numbers are socombined,asthattheonehasalwaysauniteansweringtoeveryuniteoftheother,wepronounce them equal; and it is forwant of such a standard ofequality inextension, thatgeometrycanscarcebeesteemedaperfect andinfalliblescience.”(Thewordnumbershererefertoacollectionsincethewordsetwasnotinusageatthetime.)

Itiscrucialtomakeanotehereofthekindoflanguagethathasbeenused,-‘...aunite answering to every unite of the other’. This language foreshadows themathematical conceptof aone-to-onecorrespondenceorbijectionbetweensets.TheinterestingpartisthatHumechoosesto‘pronouncethemasequal’.

IfLeibnizhadmadesimilardeliberationsinhismind,itwouldputhiminanuneasysituation because itwould violate his seminal principle called Indiscernibility ofIdenticals.Leibnizputforwardthisontologicalprinciple(alongwiththeIdentityofIndiscernibles)whichbecamesowidelyusedandagreeduponthatitlatercametobeknownasLeibniz’law.Simplyput,itstatesthatifentityAisidenticaltoB,thenAandBhavethesameproperties.Leibnizcouldseequiteeasilythatonecouldputthenaturalnumbersandthesetofevennumbersinaone-to-onecorrespondence.

1→2

2→4

3→6

4→8

Andsoon...

Buttosaythatthesetofnaturalnumbersandthe‘seeminglysmaller’setofevennumbershadexactlythesamepropertieswasanideaLeibnizwasnotabletoaccept.Thisisaplausiblereasonwhyinspiteofhisconvictionontheexistenceofthe‘actualinfinite’hecouldnotacceptthepositionCantorwouldtake.

(Itwouldbenegligentat this juncture ifwedon’tadmit thatHume’swritingandLeibniz’deliberationswereafewdecadesapartbutwecanconvincinglyarguethattheessenceofHume’swritingswascompletelycontainedinthedeliberationsontheGalilean paradox and Euclid’s fifth axiom (part-whole axiom) and Leibniz hadwritten on these topics at length. Leibnizhad indicated that hewas reluctant tobreakawayfromthetraditionsetbyEuclid’sfifthwhichstates that thewhole is


greaterthanthepart;thereasonsforthatreluctancecanbeexactlytheonewehaveoutlinedabove.)

WeshallrevisitLeibnizandhisviewsontheinfiniteandhowtheyhelpedshapehistheoryoftheunconsciousviahislawofcontinuity,butfornowweshallturnourattentiontoCantor.

InspiteofhisrespectforLeibniz’metaphysics,Cantor,unlikeLeibniz,didnotfeeltheneedtoberestrictedbythem.Hisideaofinfinitywasremarkablydifferentfromwhat anyone could conceive of at the time and the cultural resistance to his settheorywasingloriousandoppressing.ThetrulyradicalideainhistheorywastherejectionoftheEuclid’sfifthaxiom(thewholeisgreaterthanthepart).Itstemsfromtheideathatsincethatcomparisonwasonlymeaningfulwhilecomparingfinitesetsofobjects,itisatbestacontingenttruthandcaneasilynotbevalidforinfinitesets.InsteadoffindingwaystosubvertGalileo’sparadox,Cantorembracesthespiritoftheparadox.

(Galileo’s paradox could be restated as: There is a one to one correspondencebetweeneachnumberanditssquares.

1→1

2→4

3→9

......

And so on.Which convincedGalileo that since there is for every square there isexactlyonepositivesquarerootsoitwouldseemonesetcouldnotbegreaterthantheother.Heconcludedthatthewordsless,greaterandequallosessignificanceinadiscussionofinfinitesets.)

Insteadofbeinguneasyaboutthenatureofthisproblem,Cantorchoosestomakethisthebedrockofhistheory.HefollowsDedekind’sdefinitionthataninfinitesetispreciselyonewhichhasaonetoonecorrespondencewithapropersubset.Whenitcametoinfinitesetshehadtwolaws.Thefirstwasthatifthereexistedabijection(one-toonecorrespondence)betweenthemtheywereequivalent.Heascribedtothemthesamecardinality.Thesecondwasthatthesetofintegershadaninfinitenumberofelementswhosecardinalityhedenotedby0.Thecardinalityofpowersetsofevery infinitesetwasofagreatercardinality thanthecorrespondingsetswhichinitiatedthestudyofcardinalnumbersandthetransinfinite.WehavethussuccessfullydemonstratedhowCantorappliedcardinalitytotheconstructionofamathematical actuality, thus completing his break fromLeibnizian infinitywhilealwayspayinghomagetoit.

Cantor’s theory had gained almost universal acceptability inmostmathematicalcommunities(althoughmanyalternativeformalismsexist).However,therewereafewmathematicalquestionsthatneededtoberesolvedlikeRussell’sparadoxand


soonwhichledtorefinementslikeRussell’sTypetheoryandtothemuchusedZFC(Zermelo-Fraenkelaxiomswiththeaxiomofchoice).

AtthisjunctureweshouldalsomentiontheformulationoftheSchroder-Bernsteintheoremwhich states that given two infinite sets A and B such that there is aninjective(one-to-onemapping)fromAtoBandalsoaninjectivemappingfromBtoAthentherealsoexistsabijectivemapfromAtoB.OrinCantor’sformulationtheybelongtothesameclassofinfinity.Thistheoremgivesusanextremelyimportantclassification enabling us to compare and in some way equate between thecardinalitiesofdifferent infinitesets.Weshouldalsomentionthat inspiteof thenaming of the theorem, the result was discovered by Cantor and the first proofavailablewasgivenbyDedekind.

FromaphilosophicalstandpointthistheoremissignificantbecauseitshowsusthateveninCantor’sformalismof ’actualinfinity’theconstructionisnotabsoluteandwithhisformalismwecancompareandclassifydifferentkindsofinfinitesets.ButevenCantorrealisesthattheiterationsofsubsequentalephsandtheirformulationofbeingthepowersetofoneaftertheothercreatesadifferentparadigmofinfinitywhichcannotbecapturedintheparadigmhecreates.

InCantor’sownwordstoaletterwhichhewrotetoRichardDedekindhesays,“IhaveneverproceededfromanyGenussupremumof theactual infinite.Quite thecontrary,IhaverigorouslyprovedthatthereisabsolutelynoGenussupremumoftheactualinfinite.WhatsurpassesallthatisfiniteandtransfiniteisnoGenus;itisthe single, completely individual unity in which everything is included, whichincludestheAbsolute, incomprehensible to thehumanunderstanding.This is theActusPurissimus,whichbymanyiscalledGod.”

InfinityintheIndiancontext

IfweturnourattentiontoIndiancontext,wewillseethatinterestingdevelopmentshadbeentakingplaceforcenturieswhichrequirecloserscrutiny.

InhisbookTheCrestof thePeacock:Non-EuropeanRootsofMathematics,GeorgeJoseph notes how as early in the sixth century BCE, Jain mathematicians haddelineatedtheinfiniteintofivecategories:infiniteinonedirection,infiniteintwodirections,infiniteinarea,infiniteeverywhereandperpetuallyinfinite.Whilefromthe categorization it is clear that theirunderstandingof infinitystemmed fromaspatio-temporal paradigm, we must also recognize the fact that they were alsowilling to transcend that through their conception of ‘infinite everywhere’. Thisconceptionbeingquiteclosetotheideaofanactualinfinitywhichwehavediscussedpreviously.Now,wehaveseenhowLeibniz’srejectionoftheAristotelianorthodoxyregarding infinity led him to cement his theory of calculus. Let us stop at thisjuncture to analyse the work of Madhava, the founder of the Kerala school ofAstronomyandMathematics.Hehadformulatedtheinfiniteseriesrepresentations


of trigonometric functions which would only appear in western mathematics acoupleofcenturieslater,usingformalmethodsincalculuslikeTaylor’sexpansionandsoon.ThisleadsabranchofhistorianstopointoutclaimsofeurocentrisminoutlooktowardsmodernmathematicsandmaketheclaimthatearlyformulationsofcalculuswerethusmadeinIndiamuchbeforethewest.Insteadofchoosingtoinvest in that line of questioning we take a far more interesting look at thesignificanceofwhatMadhavahadachieved.HisresultshowsusthatitispossibletoconceiveofbasiccalculuswithanAristotelianframework.ItisnotnecessaryforustoadoptLeibnizphilosophicaltakeoninfinityinordertoestablishbasiccalculus.Infact,Madhavaandhissuccessorswouldgoontofindinfiniteseries,expansionsforPiaswellbasedontheirknowledgeandtreatmentofinfinitecontinuedfractions.Infactthequestionoftheinfluenceofinfinitecontinuedfractionsinthedevelopmentofmathematicalideasinthefarsouthisaninterestingquestionthatshouldalsobetakenupingreaterdetailaswecanseethatinfluencepersistthroughthecenturiesto influenceevenRamanujanwhomaderemarkablecontributions in this fieldofenquiryandcompletelychallengedourpreconceivednotionsofinfinity.

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