Epistemic Justification and Operational Symbolism

Found Sci (2014) 19:89113DOI 10.1007/s10699-012-9311-xEpistemic Justication and Operational SymbolismAlbrecht HeefferPublished online: 2 November 2012 Springer Science+Business Media Dordrecht 2012Abstract By the end of the twelfth century in the south of Europe, new methods of calcu-lating with Hindu-Arabic numerals developed. This tradition of sub-scientic mathematicalpractices is known as the abbaco period and ourished during 12801500. This paper inves-tigatesthemethodsofjusticationforthenewcalculatingproceduresandalgorithms.Itaddresses in particular graphical schemes for the justication of operations on fractions andthe multiplication of binomial structures. It is argued that these schemes provided the valida-tion of mathematical practices necessary for the development towards symbolic reasoning.Itisshownhowjusticationschemescompensatedforthelackofsymbolisminabbacotreatises and at the same time facilitated a process of abstraction.Keywords Abbaco Fractions Justication Symbolism1 Scholarly and Sub-scientic TraditionsBytheendofthefteenthcenturythereexistedtwotraditionsofmathematicalpracticewith little interaction between each other. On the one hand there was the Latin tradition astaught at the early universities and monastery schools in the quadrivium. Of these four disci-plines arithmetic was the dominant one with De Institutione Arithmetica of Boethius as theauthoritative text. Arithmetic developed into a theory of proportions as a kind of qualitativearithmetic rather than being of any practical use, which appealed to esthetic and intellectualaspirations. On the other hand, the south of Europe also knew a ourishing tradition of whatHyrup (1994) calls sub-scientic mathematical practice. Sons of merchants and artisans,An earlier version of this paper was presented at the conference From Practice to Results in Logic andMathematics. An International Conference on the Role of Practices in Shaping Results in Logic andMathematics, June 2123, 2010, Salle Internationale, MSH Lorraine, Nancy, France and beneted fromvaluable comments from Jens Hyrup, Karine Chemla and Jeff Oaks.A. Heeffer (B)Centre for Logic and Philosophy of Science, Ghent University, Ghent, Belgiume-mail: [email protected] 390 A. Heefferincluding well-known names such as Dante Alighieri and Leonardo da Vinci, were taught thebasics of reckoning and arithmetic in the so-called abbaco schools in the cities of North Italy,the Provence, and Catalonia. The teachers or maestri dabbaco produced between 1300 and1500 about 250 extant treatises on arithmetic, algebra, practical geometry and business prob-lems in the vernacular. The mathematical practice of these abbaco schools had clear practicaluse and supported the growing commercialization of European cities (Heeffer 2011). Thesetwo traditions had their own methodological and epistemic principles and therefore stoodwidely apart.2 Epistemic Justication of Sub-scientic PracticesWhile argumentation, demonstration and proof have been relatively well studied for the schol-arly traditions in mathematics, forms of epistemic justication have mostly been ignored forthe sub-scientic mathematical practices. With Van Kerkhove and Van Bendegem(2007) andMancosu (2008) the historical epistemology of mathematical practices has become an inter-esting new domain of study. Such an approach favors a strong contextualization of mathe-matical knowledge, its development and its circulation, by studying material and cognitivepractices of mathematicians within their social and economical context in history. The abbacoperiod on which this paper focuses is the one preceding the scientic revolution and there-fore a gratifying subject for research. We characterize the sixteenth century as a transitionperiod of the epistemic justication of basic operations and algebraic practices. While theabbaco tradition draws the validity of its problemsolving practices fromcorrectly performingaccepted procedures, the humanists of the sixteenth century provided radical new founda-tions for algebra and arithmetic based on rhetoric, argumentation, and common notions fromancientGreekmathematics(Cifoletti1993).Despitethelackofargumentativedeductivestructures in abbaco treatises, epistemic justication is of crucial importance to this tradition.We discern three factors that motivated abbaco masters to include schemes of justicationfor the correctness of basic operations of arithmetic and algebra in their treatises: the lackof any existing authority to rely on, the preoccupation with education and the absence ofsymbolism.2.1Lack of Traditional AuthorityThe Boethian tradition depended on scholastic authorities such as the Arithmetica by Nicho-machus of Gerasa from which much of De Institutione Arithmetica is derived. The abbacotradition could not rely on any accepted authority. Even worse, the Eastern origin of calculat-ing with Hindu-Arabic numerals was met with skepticism and resistance from authorities.1The gradual replacement of Roman by Hindu-Arabic numerals necessarily induced a trans-formation of material and conceptual means of computation. Counting boards and the abacuswere replaced by dust boards and pen and paper and newalgorithms for addition, subtraction,multiplication and division had to be taught and learned. In contrary to the popular beliefthat the printed book was the most important vehicle for the acceptance of Hindu-Arabic1OftencitedexamplesaretheedictatFlorencefrom1299whichforbadebankerstouseHindu-Arabicnumerals and the dictate by the University of Padua of 1348 that prices of books should be marked nonpercifrassedperliterasclaras(RouseBall1960,p.186).However,thisdoesnotnecessarilypointtoawidespread resistance from authorities. Jens Hyrup proposes the explanation that Hindu-Arabic numeralshad to be avoided on legal documents because they could be easily falsied (personal communication). Theresistance would thus be limited to its use in nancial and ofcial documents.1 3Epistemic Justication and Operational Symbolism 91numeralsinEurope,itwastheexistenceofthescuolodabbacoandmerchantpracticeswhich caused an irreversible dissemination of the new methods. Based on the archive workof Ulivi (2002a,b, 2006), we know that about twenty abbaco schools were active in Florencealone between 1340 and 1510. Around 1343 there were no less than 1,200 students attendingabbaco schools in Florence. Boys between ten and fourteen were sent to abbaco schools afterthey mastered writing at a grammar school. They were taught the essentials of calculatingwith Hindu-Arabic numerals, rules of merchant arithmetic and the basics of units of mea-surement and the value of coins. When they became fourteen they started as apprentices inthe trade and further learned about double-entry bookkeeping, insurance and banking prac-tices. All these activities and the growing importance of mercantilism depended on the basicnotions of arithmetic taught at abbaco schools.2Precisely because the abbaco tradition wasmissing the argumentative principles, as we know them from Euclidean geometry, it reliedon a strong foundation for its basic operations and practices. Two of the three earliest extentabbaco treatises, the Colombia algorism(Columbia X511 AL3, c. 1290; Vogel 1977), Jacopada Firenzes Tractatus Algorismi (Vat.Lat. 4826, 1307; Hyrup 2007) and Trattato di tuttalarte dellabbaco by Paolo dellAbbaco (1334) and Arrighi (1964)3, spent much attention inintroductory chapters to justify the validity of basic operations on Hindu-Arabic numerals. Amissing rst part of the Columbia algorismmay have contained such section. Some examplesof operations on fractions in these treatises will be discussed below.2.2Didactical ModelsThe abbaco masters earned their living from teaching in a bottega. Some were employedby their city; others operated on a private basisand lived fromstudent fees and possiblyrenting rooms to students. Earning a reasonable living, they belonged to the middle class.The profession often depended on family relations. The Calendri family is known to consistof ve generations of abbaco masters. With teaching as their core business, their concernswith the production of abbaco treatises were primarily didactical. While some subjects thatwere covered in these treatises, such as algebra, most likely transcended the curriculum ofan abbaco school, the introductory chapters probably reected very well the material taught.One rare manuscript of the fteenth century explicitly deals with pedagogical procedures ofan abbaco school and provides some evidence on the material treated (Arrighi 19651967).A typical program consisted of seven mute or courses: (1) numeration, addition, subtractionand the librettine, or tables of multiplication, (2) to (4) on division with increasing complex-ity of number of decimals, (5) operations on fractions, (6) the rule of three with businessapplications and (7) the monetary system and problems of exchange. These subjects are alsotreated in almost every extant abbaco treatise and precisely the elementary ones are concernedwith the epistemic justication of operations. While the schemes of justication somewhatdiffers between the texts, the layout of the different elements all follow the same pattern. So,the justication schemes not only acted as validations for the methods of operations, theyalso functioned as didactical models. The schemes were devised for the justication of the2Bookkeeping record, ledgers, personal memoranda (ricordi) and debt claims all show evidence of abbacopractices. Concerning debt claims alone, Goldtwaithe (2009, xiv) writes that an astounding number of vol-umesalmost ve thousand from the years 1314 to 1600survive. This indicates that calculating practiceswith Hindu-Arabic numerals were much more common than what is testied by Latin scholarly works of thatperiod.3The Catalogue by Van Egmond (1977, 19) attributes this text to Paolo dell Abbaco and lists the date 1339.However Cassinet (2001) has shown that the date should almost certainly be 1334. The authorship by Paolodell Abbaco is contested by Cassinet (2001) and Hyrup (2007, 5455, note 144).1 392 A. Heefferoperations but at the same time acted as a model for the concrete actions that were taught tostudents in abbaco schools. The justication schemes also depended on the tangible aspect ofcarrying out the operations on a drawing board. This agrees with other instances of cognitiveembodiment of mathematical procedures and rules during that period (Heeffer 2010b).2.3Lack of SymbolismA third motivation to employ justication schemes is the lack of any form of operationalsymbolism in the thirteenth and fourteenth century. As I will argue below, it was epistemicjustication of elementary operations which provided an essential condition for the devel-opment of symbolism in the late abbaco period and the beginning of the sixteenth century.Imagine the situation of the thirteenthcentury in which existing practices of calculationsand representations of numbers were replaced by methods from India and the Arab worldwhich were completely foreign to the Latin Roman tradition. In Fibonaccis Liber abbaci,there are several instances of these foreign conventions such as writing mixed numbers withthe fraction at the left as in341 or combined or continued fractions from right to left ase c afd b(forab +cbd +ebd f) following the Arabic way of writing. Not only for our modern eyes issuch notation difcult to understand but there are many instances in which some scribes didnot understand the notation at all.4Also practitioners of the early abbaco period show littlefamiliarity of Arabic conventions. One of the earliest treatises, the Columbia Algorism(Vogel1977) writes continued fractions from left to right.5So, early abbaco masters devised theirown representations and justications. Our modern conception of arithmetic and algebra isso much inuenced by symbolism that we cannot even think without it. In order to followand check the reasoning of a problem solution from abbaco algebra we have to translate itto modern symbolism to see how it is done. Once we have it written in symbolic algebra weunderstand and we can judge the validity of its reasoning steps. Early abbaco masters had nosuch representations at their disposal. In order to understand, explain and teach they devisedgraphical schemes to accompany their discursive explanations. We nd them in the marginsthroughoutabbacotreatises:onmultiplication,onoperationsonfractions,ontheruleofthree, on the rule of false position, on the multiplication of binomials and so on. I will furtherdiscuss two types of schemes for the operations on fractions and the crosswise multiplica-tion of binomials. These operations could be explained and justied in a discursive manner.However, the fact that these schemes of justications are supplementary to the text seems tosuggest that they provided an additional justication to the audience of these treatises. Wewould like to argue that these schemesat least for the early abbaco periodcompensatedfor the lack of symbolism and ultimately led to the emergence of symbolism.3 Justication Schemes for Operations on Fractions3.1The Roman TraditionIn order to illustrate, for the modern reader, how revolutionary the introduction of Hindu-Arabic numerals was during the twelfth and thirteenth century, I will rst briey sketch theexisting practices of operating on fractions during the two centuries preceding this turnover.4Hyrup (2010) gives the example of a compiler who corrupts Fibonaccis continued fractions through a lackof understanding.5Hyrup (2010) assumes an inuence fromMaghreb way of writing through the Iberian Peninsula, and arguesthat this treatise does not depend at all on Fibonacci.1 3Epistemic Justication and Operational Symbolism 93Fig. 1 Multiplying fractions on aGerbertian abacusBy the end of the tenth century, the work on arithmetic by Gerbert provided the dominantmodel for operating on fractions.6In the same way that Boethian proportion theory can beunderstood as a qualitative arithmeticusing names rather than gures for proportionssodid Roman fractions all have specic names which were derived fromthe systemof monetaryunits and units of weight. One as was divided into 12 unciae,7one uncia into 24 scripuli, onescripulus into 6 siliquae and one siliqua into 3 oboli, alternatively one scripulis could alsobe divided into 8 calci. Fractions of 12 parts had names such as a dodrans for 9/12 and thusequivalent to 3/4. Also separate symbols were used for the most common fractions. Evidently,the number of fractions possible in such a systemis nite, but combinations of fractions couldcover all practical needs. Operations on fractions were performed on the Roman abacus, abronze device resembling the Chinese abacus. However, Gerbert introduced his own abacusin which operations were carried out on a table marked with columns (as shown in Fig. 1).8For example, the multiplication of 12 5/6 by 2 1/2 is described by Bernelinus as XII dextaemultiplied by II semis.9These two fractions are set out on the abacus in several columns: Xfor the decimals, Ifor the units and rfor the unciae (the columns for scripuli and calci notbeing used in this example). The symbol

represents the semis and

the dextae. The rststep is to multiply the 12 5/6 by 2, which is 24 and 10/6 or 24 and 20/12, to be split in 24, 16The Patrologia Latina contains Gerbert work (c. 996) on arithmetic as well as the follow up by Bernelinus(eleventh century). For a modern edition see Friedlein (1869).7Both the terms ounces and inches, being the twelfth parts of pounds and feet respectively, are derivedfrom unciae.8Taken from Friedlein (1869, 122). The Hindu-Arabic numerals on the abacus are used here for the con-venience of explanation only. While Gerberts text was the rst to introduce the new abacus there is somespeculation that the practice was in use in cathedral schools before Gerbert.9The semis is actually 6/12 and thus equivalent to a semunica or 1/2.1 394 A. Heefferand 8/12. So 2 is placed in the X column and 4 in the Icolumn. The remaining fraction 8/12is equivalent to a bisse, or 2/3, and is written in the unciae column as

. The multiplicationof 12 and 5/6 by1/2 is split into the multiplication of 10, 2 and 5/6 by 1/2. 2 multiplied by1/2 is 1 which is placed in the Icolumn. 10 multiplied by 1/2 is 5 which is also placed in theI column. Lastly, 5/6 times1/2 is 5/12 which we nd back in the column of the unciae.10Then the unciae column is added together, bisse and quincunx making one as and one uncia(2/3 +5/12 = 11/12). Thus another 1 is placed in the Icolumn and a one rin the unciaecolumn. Then we have the summation of the units 4 + 1 + 5 + 1 + 1 =12, with 1 addedto theXcolumn and the summation of theXcolumn 1 + 2 = 3. So the nal product is 321/12. Clearly, this method as well as the terminology is very distant from modern practiceand what would become the standard in Europe during the next centuries.An important aspect of the history of the Gerbertian method for operating on fractions isthe transformation of a material and tangible method to a semi-symbolic one. The operationswere originally performed on a Gebert abacus, but from the eleventh century onwards theyappear in Latin manuscripts as calculations by pen and paper. The tangible operation on theGerbertian abacus became represented by an illustration within a manuscript. Essential forourfurtherdiscussionisthattheseillustrations,whiledescribingthetangibleoperations,did not act as justications for their correctness. Instead, they acted as an illustration of theprocedures described in the text.3.2The Latin Scholarly TraditionAll knowledgeinEuropeabout operationsonfractionsusingHindu-Arabicnumeralsisultimately derived from the Arabs. Within the sub-scientic tradition this knowledge cameto Europe probably over the Iberian Peninsula through merchant relations. Texts explain-ing Hindu-Arabic numerals known to the scholarly tradition are commonly referred to asalgorisms or by Dixit algorismi (DA). The term algorism refers to Muhamed ibn M us aal-Khw arizm, who wrote the rst Arabic work on Hindu numerals. Later manuscript cop-iesarefurtherdividedintothreefamilies:theLiberYsagogarumAlchorismi(LY),LiberAchorismi (LA) and Liber pulveris (LP). There exist many modern editions, transcriptionsand translations but the most comprehensive work is by Allard (1992). A useful addition isthe edition by Folkerts (1997) of the oldest complete Latin translation of the DA family. Tocontrast the Latin tradition with the abbaco way of handling fractions we look at one exampleof division of common fractions. Figure 2 shows an excerpt of the manuscript copy of DAkept by the Hispanic Society of America (HC 397/726, f.22r).The fragment describes how to divide 20213313. Notice how the Hindu-Arabic numeralsappear in a separated gure, written with the fraction to be divided at the right hand side andthe divisor at the left hand side, according to Arabic custom.The text uses Roman numerals throughout and often expands numerals to words as octog-inta for XXXIX. The procedure prescribes to bring both fractions to a common form (uniusgeneris), meaning to a common denominator, and then divide one by the other. The procedureis difcult to follow as the text does not use the terms numerator and denominator. Theseterms were only introduced in the LA class of manuscripts.Although the text deals with operations on Hindu-Arabic numerals, all of the explanationsuse Roman numerals or words. This raises the question what the purpose and meaning of theillustration is. Except for the enunciation of the problem, nothing in the text of the procedurecorresponds with the gure. The gure therefore cannot even be considered as an elucidation10According to Friedlein (1869, 122), the source text breaks of the explanation of the operation at this point.1 3Epistemic Justication and Operational Symbolism 95Fig. 2 Dividing two common fractions in an early Latin translationof the procedure.11The gures change slightly in fourteenth-century Latin editions of theLA class. The text includes an additional gure, which reverses the place of two fractions.They also include a cross between 1 and 13 and 2 and 3 and expand on the original gureby including intermediate numbers, such as 786 and 130 (Allard 1992, 169172). While anevolution has taken place within generations of manuscripts over two centuries, we shouldnot understand the illustration as providing any kind of justication of the procedure. Instead,the Latin text describes the procedure of dividing the fractions after setting them out on adust board or writing them down to paper. In the abbaco period however, similar illustrationswill acquire the function of justication.3.3Fibonaccis Liber AbbaciFibonacciisoftenquotedasoneofthemainfactorsinthedistributionofHindu-Arabicnumerals but this is very doubtful. The inuence of the Liber abbaci is overestimated byhistorians who suffer from the great book syndrome. For a long time, the Liber abbaci wasthe only known comprehensive text on abbaco methods written before books were printedon the subject. Historians became aware of abbaco writings only since the many transcrip-tions by Gino Arrighi starting in the 1960s and Van Egmondss groundbreaking Catalogue(Van Egmond 1980). Jens Hyrup has argued in several publications that the abbaco traditiondoes not depend on Fibonaccis Liber abbaci and that Fibonacci himself should be situatedwithin this larger abbaco tradition which spread fromthe Iberian Peninsula over the Provenceregion to Northern Italy (Hyrup 2004). After studying the different approaches to operationson fractions we can further support Hyrups viewon this. Fibonacci treats operations on frac-tions in chapters 6 and 7 of the Liber abbaci. His approach is very different from the earliestabbaco writings and also very different from the Latin translations of Arabic works. Chapter6 deals with multiplication but covers also combined fractions, a subject which is absent inthe Latin translations, the work by al-Khw arizm or in other Arab works on Hindu reckoningastheKit abfus. ulh.is abal-hindbyK ushy aribnLabb an(2ndhalfofthetenthcentury,Levy 1965). In fact, common fractions and combined fractions are treated together for eachoperation. Also, Fibonacci often uses combined fractions when operating on common frac-tions. Chapter 7 concerns the three other operations, addition, subtraction and division of11Burnett (2006) gives the explanation that Indian numerals originally acted as pictorial depictions hors detexte. As such, they could be interpreted as non-discursive elements, as discussed below.1 396 A. Heeffercommon and combined fractions. He also uses schemes to illustrate the operations discussedin the text, but here we are not left in doubt on the purpose of the schemes (Sigler 2002, 78):231211671322product56256If one will wish to multiply 11 and one half by 22 and one third, then he writes thegreater number beneath the lesser, namely1322 beneath1211, as is shown here; nexthe makes halves of the1211 because the fraction part with the 11 is halves, which onemakes thus: you will multiply the 11 by the 2 that is under the fraction line after the11, and to this product you add the 1 which is over the fraction line over the 2; therewill be 23 halves, or the double of1211 halves; there will be 23; you write the 23 abovethe1211, as is shown in the illustration; and for the same reason you multiply the 22 bythe fraction part under the fraction line, that is the 3 that is under the fraction line afterthe 22; there will be 66 thirds to which you add the 1 which is over the 3; there willbe 67 thirds that you keep above the1322, and the 67 is the triple of1322; and you willmultiply the 23 halves by the 67 thirds; there will be 1,541 sixths which you divide bythe fraction parts which are under the fraction lines of both numbers, namely the 2 andthe 3; the division is made thus: you multiply the 2 by the 3; there will be 6 by whichyou divide the 1,541; the entire quotient for the sought multiplication will be56256, asis demonstrated in the written illustration.The illustration is thus the actual result of writing down the fractions and performing thecalculation, probably on a ghub ar or dust board. It also functions as a demonstration of thecorrectness of the procedure (ut inprescripta descriptione demostratur, Boncompagni 1857,p. 47).12We have some clues about how Fibonacci learned this procedure. In the dedicationof his Liber abbaci to Michael Scott he writes that he learned the art of the nine Indiangures in Bugia.13We may assume that calculation with fractions was one of the topicshewastaughtthereattheendofthetwelfthcentury.Therepresentationofthefractionsis different from the early Arabic texts as adopted by the Latin translations (as in Fig. 2).Fibonacci used the fraction as it was introduced in the Maghreb region at that time as in theKit ab Talkih al-afk ar al-am al bi rushum al-ghub ar by Ibn al-Y asamn (end of the twelfth12Several scholars have pointed out the thin line between interpreting demonstratio in medieval texts asdemonstratingorshowing.IwouldliketounderstandFibonaccisuseofthetermhereassomethingstronger than showing. This is consistent with the line of argumentation in this paper. Illustrations whichoriginally had the function of showing something, at some point become justications additional to the dis-cursive descriptions of the text.13Bugia is the Italian name for Algerian port of Bejaia. Sigler 2002, 1516: As my father was a public of-cial away from our homeland in the Bugia customshouse established for the Pisan merchants who frequentlygathered there, he had me in my youth brought to him, looking to nd for me a useful and comfortable future;there he wanted me to be in the study of mathematics and to be taught for some days. There from a marvelousinstruction in the art of nine Indian gures, the introduction and knowledge of the art pleased me so muchabove all else, and I learnt from them, whoever was learned it, from nearby Egypt, Syria, Greece, Sicily andProvence, and their various methods, to which locations of business I travelled considerably afterwards formuch study.1 3Epistemic Justication and Operational Symbolism 97century, Djebbar 1992; Abdeljaouad 2002). The scheme shown in the text therefore functionsas a justication of the procedure.3.4The Abbaco TraditionThe oldest extant abbaco texts are the Livero de labbecho (c. 12881290, Arrighi 1989) andthe Columbia algorism of c. 1290. The oldest archival reference to an abbaco master is from1265 (Ulivi 2002a). This separates the written evidence of abbaco practices from Fibonacciby about sixty years. This may seem short in the history of mathematics but apparently muchevolved during that time. While Fibonaccis treatment of algebra is very close to some Ara-bic texts, in particular those of al-Khw arizm and Ab u K amil, the abbaco rules of algebrasignificantly differ from these. Differences in rules, the order of rules and the normalizationof equations are the main arguments discussed by Hyrup (2004).In all probability, this deviation in the way algebra is treated in the abbaco tradition origi-nated not after Fibonacci but earlier. As for fractions, the relation is harder to determine. Thefact that Fibonacci used the fraction bar in a period whenas far as we knowonly one otherLatin text from that period uses it (the twelfth century Liber mahamaleth), seems to suggestthat he was responsible for introducing this Maghreb innovation. Hyrup (2010) however,assumes a different line of inuence, from the Iberian Peninsula over the Provence to theNorth of Italy. As the Columbia algorism treats fractions with a fraction bar and introducesschemes for operating on fractions, different from those of Fibonacci, this may well be thecase.In the following sections I will look at the way fractions are being treated in the earliestextant abbaco texts (Table 1). The Columbia Algorism is the oldest text in which a schemefor the division of fractions appears. Jacopos treatise of 1307 contains a more systematictreatment for multiplication, addition, excess, division and subtraction. Other texts also treatfractions but do not include the justication schemes.3.4.1Discursive and Non-discursive Elements in Abbaco TreatisesAbbaco algebra has been described as a form of rhetorical algebra referring to Nesselmannsthreefoldphasesofalgebraasrhetorical, syncopatedandsymbolicalgebra(NesselmannTable 1 The earliest extant abbaco textsDate Title Author Fractions Schemes1228 Liber abbaci Fibonacci 1288 Livero de labbecho (Florence, Ricc. 2404) Unknown1290 Columbia algorism Unknown1307 Tractatus algorismi (Vat.Lat.4826) Jacopo1310 Liber habaci (Florence, BNC, Magl.XI, 88) Unknown 1328 Libro di ragioni (Florence, BNC, Magl.XI, 87) Gherardi 1330 Libro di molte ragioni dabaco (Lucca 1754) Unknown 1339 Trattato di tutta larte dellabbaco(Florence, Prin. II, IX, 57)Paolo dellabbaco1344 Aliabraa Argibra (4 copies) Dardi of Pisa 1370 Tractato delalgorisimo (Florence, Plut. 3026)[copied from Jacopo]De Danti1 398 A. Heeffer1842). As demonstrated by Heeffer (2009) such a distinction should be considered a nor-mative one, providing a special status to Diophantuss Arithmetica, i.e. a distinction whichcannot be considered as phases in a historical development. Abbaco algebra, while not beingsymbolic algebra, contains aspects of operational symbolismin which operations are appliedblindly on abstract objects without taking into account their arithmetical contents. A goodexample is the rule of signs, discussed below. But looking at a typical abbaco text one wouldsee mostly words and sentences, with occasional special characters and marginal gures.That is why abbaco algebra is called rhetorical. We will further use the term rhetorical inthis limited sense of described by means of words only.However, at closer inspection one nds two distinct types of contents. By lack of suit-able alternatives we call them discursive and non-discursive contents. Discursive mathemat-ics takes up most of an abbaco treatise and consists of descriptions of problems and theirsolutions. Abbaco masters often use the term ragioni for the computation of their problemsolutions,asinPaoloGherardisLibrodiragioniorasattheendofaproblemsolutione cos se fanno le simili ragioni. This common phrase can literally be translated as andthus the similar computations are done (Hyrup 2007, pp. 1112). However, the term isderived from ragionari, meaning to reason, to discourse or even to talk. The ragioni ofabbaco masters are thus by definition discursive mathematics. While the problemsolutions byabbaco masters are rhetorical and discursive, a modern textbook on elementary algebra willcontain mostly argumentations which are symbolic and discursive. In the rest of this paperI will argue that the justication schemes as well as the early symbolism which developedwithin the abbaco tradition were originally non-discursive.Let merst explainwhat Imeanbynon-discursiveelementsinabbacomathematics(see Fig. 3).14The distinction goes back to ancient Greek philosophy and is prominent in theworks by Proclus. Sara Rappe, who devoted a study to the role of non-discursive thinkinginneoplatonismplacesitatthecenterofProclusideaofimaginationinmathematics.15Important here is the intermediary function of the non-discursive between imagination andargumentation. The non-discursive complements the level of discourse with abstract ideas bymeans of spatial realizations or renderings. Later use of the distinction within the context ofmathematics refers to the intuitive aspect of non-discursive elements. In Kants epistemologyof mathematics, the synthetic a priori is non-discursive, and allows us access to intuition.I will further avoid the termintuition but do appeal to the complementary power of explana-tion by means of non-discursive elements. These elements provide an understanding whichgoes beyond the discursive level. Or even stronger: the non-discursive acts as a justicationfor the discursive argumentation.Let us look at a concrete example in relation to our subject of operations on fractions.Jacopa da Firenze explains in one of the earliest abbaco treatises howto add two unit fractions(Hyrup 2007, 230):And we shall say thus, say me, how much is joined together 1/2 and 1/3. Do thus, andsay thus, 1/2 and 1/3 one nds in six, since 2 times 3 makes 6. And seize the half and14Justication schemes for operations on fractions from Jacopo da Firenze, Tractatus algorismi. This is asingle page reproduced from a microlm of Vatican Library, Vat. Lat. 4826 f. 14v. Similar schemes appear inother treatises of the same period. For a translation see Hyrup (2007, 229).15Rappe (2000, p. 132): [. . .] Proclus theory of the imagination [. . .] builds on the Platonic and Aristo-telian models but again strikes out in a very original direction as evidenced in the Euclid commentary andelsewhere. To briey review the central features of his theory of imagination, we nd Proclus here suggestingthat the imagination is a kind of intermediate ground between soul and intellect. [. . .] Proclus describes thisintermediary function as a faculty that is capable of reecting, by means of spatial realizations or renderings,the abstract ideas present in the discursive intellect.1 3Epistemic Justication and Operational Symbolism 99Fig. 3 Example of non-discursive elements in abbaco treatises1 3100 A. HeefferFig. 4 The scheme for joiningtwo broken numbersthe third of 6, which joined together make 5, and divide 5 in 6, from which results5/6, and as much makes 1/2 and 1/3 joined together. And in this way all the similarcomputations are made with whatever fraction it were.As usual in abbaco treatises, a general method is not provided. Instead, a problem is solvedin a particular way and it is indicated that this method applies to all similar cases (Et inquesto modo se fanno tucte le simile ragioni). This clearly is the discursive level whichdescribes how to look rst for a common multiple of the two denominators. Then one takesthe fractional part of this common multiple and adds the two together. The nal step is todivide this sum by the common multiple. The result is the sum of the two unit fractions. Inhis introductory explanation of the gures Hyrup writes This chapter explains in words themethods that were set out in diagrams in Chapter 9 (Hyrup 2007, 58). This would implythat the diagrams are presented by Jacopo as a kind of summary of the discursive proceduresin the text. But is this really the case? Firstly, the numerical examples of the schemes aredifferent from those in the text. Secondly, it is not so that each type of operations in the textcorresponds with one of the schemes and vica versa. When we look in detail at the diagrams inFig. 3, we do not nd a diagram which corresponds with any of the problems on adding morethan two fractions while the text of chapter 9 describes four such cases. Only one diagramdescribes how to join together a broken number to a broken number (see Fig. 4):16The numerical example of this scheme does not appear in the text. However it correspondswith some examples as problem 8 (Hyrup 2007, 232):Say me, how much are joined together 4/5 and 5/6. Say thus as we have said above, 4/5and 5/6 are found in 30. Now take 4/5 of 30, which are 24, and take 5/6 of 30, whichare 25. Now join together 24 and 25, which make 49, and divide 49 in 30, from whichresults one integer and 19, and as much make 4/5 and 5/6 joined together, that is, j and19, and it goes well.The scheme contains all elements of the discursive reasoning but not the order of operationsor the way they are performed. A scheme is therefore open to interpretations. It does not rep-resent a precise algorithm on how to conduct the calculations. Also, note that 172/96 in thescheme does not represent the solution but the division operation. Abbaco masters will alwayssimplify fractions, so the solution would be 1 76/96. The non-discursive scheme of Fig. 4 notonly allows for multiple interpretations, curiously, it can also be applied to other operationson fractions. This will be discussed in the next section. What can be concluded now is thatabbaco treatises contain non-discursive elements which support the discursive description ofarithmetical procedures. They appear as schemes which do not exactly represent the discur-sive procedure but appeal to a more general spatial understanding of the relations between thearithmetical elements involved. While the schemes use concrete numerical values it is clear16The actual scheme in the manuscript uses a curved line while most others use two connected line segmentsat the bottom. As there seems to be no conceptual difference I have used line segments for all the schemes.1 3Epistemic Justication and Operational Symbolism 101Fig. 5 Scheme for subtractingfractionsthat the spatial arrangement is invariant with respect to all possible operations of additionson fractions.If we compare the abbaco schemes with the marginal illustration used by Fibonacci whentreating fractions, it is clear that abbaco schemes depend more on the spatial organization ofthe elements. Fibonaccis schemes remain discursive. If read from top to bottom they corre-spond with the order of operations in the text. The abbaco schemes do not preserve the orderof the operations of the problems solutions. Neither does the problemsolution in abbaco textsrefer to the schemes during the calculation. While Fibonaccis schemes are discursive, andto some degree justicatory, the abbaco schemes are non-discursive and serve the purposeof justication.3.4.2Generalizing Justication SchemesIn modern mathematics, the operations of arithmetic are dened by axiomatization. Basicoperations of arithmetic, such as addition, have a clear and unambiguous meaning. This hasnot always been the case. Jens Hyrup pioneered the method of close analysis of lexical andsyntactic features in the study of mathematical sources for Old-Babylonian tablets (Hyrup2007). General arithmetical operations such as addition, subtraction, multiplication and divi-sion were subdivided by Hyrup in distinctive syllabic and logographic versions, each cover-inga different shade of meaningof the arithmetical operationinmodernsense. The subtractionoperation for example is subdivided into eight different lexical classes (Hyrup 2009). Someof them refer to concrete operations such as cutting off, being part of the cut-and-pastegeometrical model; others refer to the comparison of concrete magnitudes. In some cases theterminological peculiarities of these operations could be applied to groups of tablets fromdifferent periods or different areas in Mesopotamia.Also in abbaco arithmetic we nd different shades of meaning for common operations.The schemes from Jacopos treatise make the differentiation explicit between subtractingafractionfromanotherandcalculatingtheexcess;howmuchonefractionismorethananother. As also the problems treat these cases separately, it becomes clear that early abbacoarithmetic considered subtraction and excess as two different operations. However, they arecovered by the same scheme. Subtraction is shown in Fig. 5:In the case of excess, the smaller fraction is shown at the right hand side and subtractedfrom the fraction on the left hand side (Fig. 6). The text in the center shows o, Italian foror, because the solution procedure prescribes to rst determine which one is larger 12/13or 3/4.Notonlyaretheschemesforsubtractionandexcessthesame,theyarealsoidenticalwith the justication scheme for joining (addition). This can be understood because of thesymmetry between the two operations. However, more surprising is that the same scheme isalso used for division (Fig. 7).1 3102 A. HeefferFig. 6 Scheme for calculatingthe excessFig. 7 Scheme for division offractionsFig. 8 A symbolic version of thescheme for addition of fractionsThe scheme shows how to divide the fractions3/4 and 2/3, but there is no problem inJacopos text which deals with division. The fact that one common scheme, with the same sixelements and the same spatial organization, is used for the operations of joining, subtraction,excess and division is quite interesting. It shows the generalizing power of such schemes. Onesingle scheme provides a justication for the basic operations on fractions. Because of thenon-discursive character of these schemes there is no preferred left-to-right or right-to-leftorder. The schemes do not preserve the order of operations as discursive descriptions do.They are therefore more general than a procedural description.Ajustication scheme is not only silent about the order of operations but also on the preciseoperations that lead to numeric results. The diagram shown in Fig. 4 is basically stating thatwhen dividing 172 by 96 you are getting 7/8 and 11/12 joined together. The actual meaning,conveyed by the operations involved in the discursive part of the treatise, is that 7/8 partsand 11/12 parts of the product of the denominators make the result. This precise meaning isnot preserved in the justication scheme. We see the intermediate results 96, 84 and 88 andlines which connect these with the fractions but the diagram remains a non-discursive spatialorganization of relevant elements open to interpretation. If we match the diagram with theprocedure in the text and use an anachronistic symbolic rendering of the numbers we wouldarrive at something as in Fig. 8.While this may seem to be a satisfactory representation of the discursive procedure, otherrepresentations are equally valid. Instead of interpreting 84 as the a/b part of the product bd,we can also consider it to be the product ad. This would lead to a representation as in Fig. 9.The generalizing power of justication schemes therefore allow for new interpretationsbyabbacomastersandultimatelyledtonewschemes. TheschemeinFig. 6providesa1 3Epistemic Justication and Operational Symbolism 103Fig. 9 A different interpretationof the scheme for additionFig. 10 A later scheme foraddition and divisionnew interpretation of ad + bc the sum of cross-wise multiplied numerators and denomina-tors. Indeed, schemes in later abbaco treatises use a cross for addition and division opera-tions. Figure 10 shows and example from Luca Paciolis Perugia manuscript (Vat.Lat. 3129,w. 1478, f.12v; Heeffer 2010a). This scheme looks different from those of Jacopo, but withthe exception of the common multiple it contains all the elements and preserves the spatialorganization. There is little evolution in the 170 years that separate the two texts.The spatial organization of the elements, the numerators and denominators and the com-binatorial operationspossibleonthesetwo, appealstoanintuitiveunderstandingoftheoperations involved. This is further supported by a curious quote by Jacopo in the Vaticanmanuscript: We have said enough about fractions, because of the similar computations withfractions all are done in one and the same way and by one and the same rule. And thereforewe shall say no more about them here (f. 17r, Hyrup 2007, 236). Evidently, the rules foradding, subtracting, multiplying and dividing fractions are different. Jacopo therefore mustrefer here to the general scheme for validating operations.3.4.3Generalization or Abstraction?Thereisaninterestingparallelinthejusticationofoperationsonfractionsbetweentheabbaco tradition and Chinese mathematics, which allows for a short digression. The differ-ences between mathematics from the West and the East are often characterized as rigorousversus practical. Western mathematics is modeled by the Euclidean axiomatic method whileChinese mathematics would only involve procedures for solving practical problems. Recentscholarship in Chinese mathematics has corrected this simplied view and explored justi-cation and proof in Chinese classics (e.g. Chemla 1997, 2003, 2005). But also the view ofWestern mathematics as well-founded and based on rigorous demonstration is overly sim-plistic. The new arithmetic using Hindu-Arabic numerals and algebra, introduced in Europefromthe twelfth century, was missing a foundation and a rigorous systemof justication. Thisarithmetic and algebra was mostly disseminated, practiced and taught within the sub-scien-tic tradition where abbaco masters and future merchants, surveyors and craftsmen were themain actors. This brings Western and Eastern mathematics in a more comparable context. It1 3104 A. Heefferis interesting to contrast both traditions in relation to the topic under discussion. What arethe differences in justication of basic operations such as addition and division of fractionsbetween these two traditions? Karine Chemla has argued in several papers that Chinese clas-sics were concerned with justication as much as they were in the West. However, they placedGenerality above Abstraction (Chemla 2003). Having discussed the power of generaliza-tion of the justication schemes in previous section howdoes this compare to Chinese classicsuch as The Nine Chapters (Chemla and Shuchun 2004)? Chemla reports that commentarieson Chinese classics achieve the same justication of procedures dealing with fractions andthe same level of generality in discursive descriptions. Commentators, such as Liu Hui, relyon the general operations of equalization (tong) and homogenization (qi) to bring objectswhich differ in appearance into communication with each other (Chemla 1997). Mathe-matical objects, such as fractions, are thus instantiations of much more general categorieswhich allows that procedures for operating on fractions can be explained and justied astransformations of such general kind. Early abbaco arithmetic does not engage us to think insuch general notions for justifying operations. The vocabulary to do so will slowly developduring the next two centuries.17Instead, early abbaco treatises appeal to non-discursive ele-ments, as our justication schemes, to present operations on fractions as transformations ofa more general kind. The graphical schemes invoke a level of generality by abstraction. Thechoice for generalization or abstraction might thus be a cultural distinction which is moreimportant in characterizing the differences between Western and Eastern mathematics thanthe idea that Western mathematics is superior in rigor.3.4.4Circulation of Knowledge About FractionsSofarwehavediscussedtheschemesofasinglemanuscript only, JacopodaFirenzesTractatus Algorismi (written 1307). Was this an ideocracy of the author or do these schemessystematicallyoccur inabbaco treatises?Withonly 10% of the about 250 extant abbacomanuscripts transcribed, published or translated, it is difcult to provide a denitive opinion.From a survey of published and some unpublished manuscripts it appears that the schemesfor operating on fractions are not that common. They are found mostly in the early period of1290 to 1370. This makes sense, as argued, because operating on fractions with Hindu-Arabicnumerals was very different fromthe prevailing Roman practices during the centuries preced-ing the abbaco period. Therefore, the need for justication was most pertinent when the rsttreatises were being written. The earliest extant treatise Livero de labbecho (1288, Florence,Ricc. 2404) does not contain any schemes for the simple reason that operations on fractionare not treated in this text. The Columbia Algorism(Columbia X511, AL3; Vogel 1977), nowdated at about 1290, does contain a single scheme for the division of fractions (Fig. 11):18While the layout of the elements is slightly different from Jacopos we nd in here all thecharacteristics of a non-discursive justication scheme.Three other texts treat operations on fractions but do not contain the justication schemes:Liber habaci (1310, Florence, BNC, Magl.XI, 88; Arrighi 1987), the Libro di ragioni (1328,Florence, BNC, Magl.XI, 87) and the Libro di molto ragioni dabaco (1330, Lucca 1754;Arrighi1973). However, theTrattatodi tuttalartedellabbaco(c.1334, Florence, Prin.II, IX, 57) attributed to Paolo dellabbaco (Van Egmond 1977, 19) but contested by others17In this respect, it is interesting to repeat our earlier observation that the earliest Latin algorism in editionby Folkerts (1997) does not use the terms numerator and denominatorof fractions as later Latin texts do.18In fact, there are some other schemes, corrupted from an earlier (lost) manuscript copy of the text. Onedubious scheme for the multiplication of fractions makes Vogel (1977, 43), editor of the Columbia Algorismwonder: Was soll das Dollarzeichen bedeuten.1 3Epistemic Justication and Operational Symbolism 105Fig. 11 An earlier scheme forthe division of fractionsFig. 12 Generalization of thescheme for joining two brokennumbers(Hyrup 2007, 5455, note 144), contains many schemes. Figure 12, taken from f.39vshowshow to join 29/37 and 3/7. While the schemes do not contain the connecting lines as in thepreviously discussed ones, the spatial arrangement of fractions, intermediary and nal resultsare identical. Finally, the Tractato delalgorisimo (1370, Florence, Plut. 3026; Arrighi 1985)by Giovanni de Danti dArezzo copies the whole introduction from Jacopo and containstherefore also the schemes. An interesting deviation from Jacopos text is de Dantis explicitreference to the schemes: i rocti anno illoro per se regola cioe di multipricare, dividere,giongnere et soctrare e dire quanto e piu o meno uno che laltro vedendoli gurati (Arrighi1985, 10), the last part of the sentence referring to seeing the rules in gures. The fact thatthe author used this part from Jacopos treatise must mean that he considered it to be a suiteddidactical treatment of operations on fractions.In summary we can state with some condence that justication schemes for operationson fractions played a considerable function in demonstrating the validity of procedures ofearly abbaco practices. The schemes play a less prominent role in later abbaco treatises whenoperations on fractions became well established and became a part of the abbaco school cur-riculum. Later treatises also contained justication schemes but applied to more advancedtopics such as the rules of signs, the rule of false position and the multiplication of binomials.4 Justication Schemes and Symbolic Reasoning4.1The Parallel Between Justication Schemes and Early SymbolismPreviously, schemes for operating on fractions have been characterized as non-discursive.They serve the purpose to provide a justication for the correctness of procedures applied inthe discursive parts of a treatise. In this section it will be argued that the early symbolism inabbaco treatises serves the same purpose.It would carry us too far to go into the subtle differences between abbreviations, ligatures,mathematical notations and symbols. For the current intention, let us take a liberal stance,like Florian Cajori in his book on of mathematical notations (Cajori 19281929) who throws1 3106 A. Heefferall these together to write one global history. Interpreted this way, symbols appear alreadyin the earliest abbaco treatises and continue to play a more prominent role during almostthree centuries of abbaco practice. The fraction bar is a good example to demonstrate thatthere is a parallel between symbolism and the justication schemes. The fraction bar is asymbolic notation that we still use today. It is very well accepted in all abbaco texts and wesee it evolving from its use in numeric fractions only to its adoption as fractions of surdsor fractions of polynomial expressions. It is most likely that the fraction bar originated inthe Mahreb region and spread to Europe during the late twelfth century, not only throughFibonaccis writings but also through merchant practices over the Iberian Peninsula and theSouth of France. It is significant that the justication schemes for operations on fractions verymuch depend on the spatial organization implied by the fraction bar. The two fractions areplaced prominently in the scheme, the resulting fraction usually in the middle, intermediaryresults related to the numerator above and those related to the denominator belowthe fractionbars. So, it is fair to state that there is already interplay between symbols and justicationschemes.A second parallel is the way the discursive part of a treatise refers to the non-discursiveschemes and to the use of symbolism. De Danti makes the difference between the discur-sive treatment of operations on fractions and the non-discursive schemes of Jacopo explicit:.. and for each of these [operations] demonstrating them by normal writing (per scritura)and then doing the demonstration in the formof gures (in forma gurata).19There exists anunpublished family of fteenth-century manuscripts in which a similar distinction is madewithregardtorhetorical(perscritura)andsymbolic(guratamente)demonstrations.20While the symbols used in the symbolic part are not newsome of them also appear inprevious manuscriptsthe explicit distinction between the two methods is rather unique.I do not know of any other abbaco text in which this is made so explicit.Thesecondproblemofchapter33askstondtwonumbersgivencertainconditions.In modern symbolism the problem amounts to the equation x(x + 3) = 1 (Florence, BNF,Magl. Cl. XI. 119, f. 58v). After the problem enunciation the author asks to solve this byalgebra (Farenlo per la cosa) and also to show it symbolically (guratemente).21First,the symbolic version is given (see Fig. 13).Several remarks about this fragment of the manuscript are appropriate here. It describesthe solution to the problem in a terse format which corresponds with the solution repeated inthe text following this fragment. Firstly, for the modern reader, this may seem to be far awayfrom algebraic symbolism. However, we should consider this fragment within its speciccontext. The main text of this treatise (or at least this chapter on algebra) does not use anyabbreviations or ligatures for the unknown (cosa), the square of the unknown (censo), theaddition and subtraction operations, and the roots. All these mathematical terms for whichsymbols are used with increasing frequency during the fteenth century, are written in fullwords in the discursive part. The author therefore wants to make a clear distinction between a19f. 10r, Arrighi 22). e di ciascuno dessi dimostraremo ordinatamente per scrictura (sic) lordine loro epoi ve dimostraremo in forma gurata. Queen Annas New World of Words, or Dictionarie of the Italian andEnglish tongues, London 1611: dimostrare: to shew by demonstration.20Florence, BNF, Magl. Cl. XI. 119 (c.1433), Florence, Biblioteca Mediceo-Laurenziana, Ash. 608 (c1440),London, BL, Add. 10363 (c1440), Paris, Bibliothque Nationale, It. 463 (c1440), London, BL, Add. 8784(1442), Biblioteca Mediceo-Laurenziana, Ash. 343 (c1444). See Heeffer (2008) for a description of the manu-scripts and a partial transcription with English translation. While most of the copies are written around 1440,it can be established from internal evidence that the archetype must have written around 1417.21The terminology differs somewhat between the manuscripts, which shows that the exact meaning of theterm was not yet established. Others manuscripts use per gura, in ghura or gurativemente1 3Epistemic Justication and Operational Symbolism 107Fig. 13 The symbolic solution to a problem ((C) British Library Board, Add. 10363, f. 60r)rhetorical solutionstill the standard practice in abbaco algebraand a symbolic versionof the solutionwhich occasionally unsystematically pops up in such texts. It is a formalway to make the distinction clear between two ways of presenting the solution to an algebraicproblem. So, even if the symbolic solutions still look a lot like rhetorical algebra to us, forthe author at least there is a fundamental difference between the two.Secondly, the symbolic solution is boxed, a scribal practice to indicate that the frag-ment is outside of the text. Figure 13 shows only a line at the right-hand side of the box,but some instances in other copies show the boxes more prominently. This represents theauthors clear separation between the discursive explanation and the solution per gura.A third observation, which can be drawn from comparing the different manuscript copies,is that the symbolic solution differs greatly between the copies. The BL copy shown inFig.13 isratherconservative. Weknow thatit isat leasttwogenerationsawayfromthelost archetype. The operations pi and meno are not abbreviated as in other copies andmost likely expanded from a previous copy by the scribe. Professional scribes did not alwaysunderstand the contents and meaning of abbaco manuscripts and especially the symbolicparts are not faithful reproductions. One scribe (Florence, Biblioteca Mediceo-Laurenziana,Ash. 343) even omits these parts. This scribe did not understand the sense of writing downthe same solution twice. He did not understand the principle distinction our author wantedto make.This brings us to our main point: what was the purpose of solving a problem in two ways,per scritura and guratamente? I would like to make the parallel with the non-discursivecharacter of the justication schemes. The author considers the symbolic solution as anadditionalnon-discursivejusticationofthealgebraicsolutioninthetext. Thisbeliefissupported by a remark between the symbolic and rhetorical solution: I showed this sym-bolically as you can understand from the above, not to make things harder but rather for youto understand it better. I intend to give it to you by means of writing as you will see soon.22This quotation is very important as it shows that the author understands the two solutions astwo different approaches to present an algebraic solution. Moreover he makes explicit hismotivation to do so: to understand it better. The symbolic solution adds something tothe discursive one. If the author would consider the symbolic solution on the same level asthe discursive, it would make no sense to go through the trouble of solving all the problemsin this chapter twice. The symbolic solution adds to the understanding of the discursivesolutioninthetext.Itthusfunctionsasajusticationforthediscursivepart.Theauthor22BNF, Magl. Cl. XI. 119, f. 58r: Ora io telo mostrata gurativamente come puoi comprendere di soprabene ch lla ti sia malagievole ma per che tulla intenda meglio. Io intende di dartela a intendere per scritturacome apresso vedrai.1 3108 A. Heefferappeals to some intuitive kind of understanding in the same way as the justication schemeswere being used for the operations on fractions.The rst steps towards symbolization of algebra were being taken by the use of non-dis-cursive representations of algebraic operations. In the manuscripts here discussed the authormakes an explicit distinction between rhetorical and symbolic solutions. In later abbaco writ-ings and early sixteenth century textbookssuch as Paciolis Summa or Cardanos PracticaArithmeticaethe distinction is also being made. Here, the non-discursive elements appearin separate boxes or in marginal notations. They all serve the same function: to use non-discursive elements as an additional justication for the correctness of a problem solutiondescribed in the text.4.2From Justication Schemes to Operational SymbolismThe nal step of our argumentation is to demonstrate that justication schemes facilitatedthe transition from rhetorical to symbolic algebra, or from arithmetical reasoning to abstractoperational symbolism. Symbolic reasoning is therefore not so much concerned with the useof symbolsthese are usually introduced in the later phases of the process towards symbol-ization. Symbolic reasoning is model-based reasoning in which problems are solved usinga symbolic model. Such a model obeys the rules of arithmetic but allows applying the ruleswithout accounting for its arithmetical contents. An example of algebra in which operationsare limited by their arithmetical contents is the Arithmetica of Diophantus. Throughout thebook, indeterminate problems are solved by considering some choices for the indeterminatequantities of the problem. Often, the initial choice runs into problems, arriving at irrational(e.g. problem IV.10) or negative solutions (problem IV.27). Such solutions or intermediaryresults are not compatible with Diophantus number concept. In these cases the initial choicesare reformulated precisely to avoid irrational and negative values. The resolution of indeter-minism in the Arithmetica often depends on this mechanism. Because of the limitations ofthe number concept this kind of algebra remains pre-symbolic.The next sections will show how abbaco algebra succeeded in the process of abstract-ing the solution process from the arithmetical contents of the objects being operated upon.The validation and justication of basic operations hereby functioned as a precondition forthis gradual process of symbolization.23Accepted operations and procedures for problemsolving could be applied blindly only because there was a strong epistemic foundation fortheir correctness. Because of their appeal to generality, justication schemes played a majorrole in providing epistemic validation. The development towards a symbolic algebra duringthe sixteenth century can thus be seen as a consequence of this process of justication andabstraction. The belief in the validity of standard operations and practices ultimately lead tothe acceptance of negative and imaginary solutions and the expansion of the number conceptduring the sixteenth century.4.2.1Another Strong Justication SchemeThe simple scheme for crosswise multiplication of binomials is the most common justica-tion scheme in abbaco texts. It captures the idea that (a +b)(c d) = ac ad +bc bd.Lacking the symbolism which we here use, abbaco texts often show a simple cross and referto the scheme as multiplicare in croce. The symbols a, b, c, d, as shown here, stand forany number object, natural, rational, negative, surd, cossic or with Cardano even imaginary23This claim is argued more extensively in Heeffer (2010c).1 3Epistemic Justication and Operational Symbolism 109Fig. 14 Dardi scheme for multiplying surd binomials (Chigi, M.VIII.170, f. 4v)Fig. 15 Dardis use of ajustication scheme for provingthe rules of signnumbers. The rule is valid by its scheme of adding the four products together, irrespectiveof its contents. As far as we could establish, the scheme rst appears under the term croceand with an illustration of a cross in Gherardis Libro di ragioni of 1328. The earlier textswe have discussed above do not contain such explicit references.Now, Gherardis application of crosswise multiplication is rather surprising. He employstheschemeformultiplyingcommonfractions.Insteadofbringing121/2and151/4toacommon denominator, he considers the two fractions as binomials (12 +1/2) and (15 +1/4),each the sum of a whole number (numero sano) and a broken number (rocti):Se noi avessimo a multipricare numero sano e rocto contra numero sano e rocto, sdovemo multipricare luno numero sano contra laltro e possa li rocti in croce. Asem-pro a la decta regola. 121/2 via 151/4 quanto fa? Per diremo: 12 via 15 fa 180. Ordiremo: 12 via 1/4 fa [3], echo 183. Or prendi il 1/2 di 15 1/4 ch 7 5/8, agiustalo sopra183 e sono 190 5/8 e tanto fa 12 1/2 via 15 1/4. Ed facta.In the Trattato di tutta larte dellabbaco, originally written in 1334, the author explicitlyrefers to the two methods, one by multiplication of binomials and the other by direct mul-tiplication of two fractions (Arrighi 19651967, 28). Both methods are thus covered by ajustication scheme.The method of crosswise multiplication, the reference to croce and the non-discursive useof a conguration of elements in a cross, appears frequently in abbaco texts for the followingtwo centuries. Maestro Dardi is the rst to devise a more elaborate scheme in which the fourproducts are indicated by individual line segments. His comprehensive text on algebra, theAliabra argibra, is preceded by a separate treatise dealing with operations on surds Trat-tato dele regulele quale appartiene a le multiplicatione, a le divitione, a le agiuntione e a lesottratione dele radice. The multiplication (3 5)(4 7) is illustrated by the schemeshown in Fig. 14. It states that (3 5)(4 7) = 12 +35 63 80:24The scheme for binomial multiplication shares all the characteristics of the schemes foroperations on fractions and leads us to conclude that it serves the same function of epistemicjustication of the discursive explanations.Interestingly, in the same introduction Dardi also uses the scheme for a very differentpurpose, to provide a prove for the rules of sign (see Fig. 15).The reasoning is as follows: we know that 8 times 8 makes 64. Therefore (10 2) times(10 2) should also result in 64. You multiply 10 by 10, this makes 100, then 10 times 224The transcription is by Franci (2001, 43)) based on the Siena manuscript I.VII.17. However, this manu-script omits this scheme and simplies other schemes. They do appear in Chigi M,VIII.170 and according toHyrup (2010) also in a later copy held at Arizona State Temple University.1 3110 A. Heefferwhich is 20 and again 10 times 2 or 20 leaves us with 60. The last product is (2)(2)but as we have to arrive at 64, this must necessarily be +4. Therefore a negative multipliedby a negative always makes a positive.25The use of a general justication scheme for something as crucial as the laws of signs inarithmetic is quite significant. Firstly, it again shows the unifying power of such schemes. Thecrosswise multiplication of binomials is applied to sums and differences of natural numbers,as well as rational numbers and surd numbers. Operations on different kinds of numbers canbe justied by one single scheme. Secondly, precisely because of the belief that the operationsrepresented by such a scheme must be correct, it becomes possible to prove something asessential as the laws of signs. A negative multiplied by a negative must be a positive becauseofthevalidityofthisschemeformultiplyingbinomials.Thirdly,aprooflikethisoneof Dardi may seem trivial, but it is not. The reasons for the suitability of this scheme forproving the rules of sign go deep. If one wants to go from an arithmetic which is limitedto natural numbersas is basically the Arithmetica of Diophantusto an arithmetic whichincludes the integers, you have to preserve the law of distribution and the law of identity formultiplication. Precisely these two laws are at play in the multiplication of binomials.The justicationary power of this scheme is so strong that it not only allows to proof therules of signs but also leads to emergence of new concepts. While negative numbers occa-sionally appear in abbaco mathematics they were not accepted as isolated negative quantities.Mostly an interpretation could be given as a debt. But by considering negative solutions toalgebraic problems as a debt, abbaco masters denied the possibility of a negative quantityrather than accepting it. This attitude changed by the end of the fteenth century. In Pacio-lis Perugia manuscript (Calzoni and Cavazzoni 1996) several negative solutions appear inwhich he accepts negative quantities without any reservations (Sesiano 1985; Heeffer 2010b).The acceptance of these anomalies was made possible by a strong belief in the correct-ness of abbaco practices (argued by Heeffer 2010c). The solutions one arrived at had to beacceptedhowever strange the results may have been at that timebecause of the validityof the procedures. The epistemic condence of abbaco masters depended to a great extenton justication schemes.Half a century later, Cardano (1545, 219), who was well acquainted with abbaco prac-tices, wouldtaketheschemeformultiplicationofbinomialsonestepfurthertoinclude

5 +15 5 15

. According to schemes of crosswise multiplication this leads tothe sum of four products. The rst three products pose no problems. The rst one is 25 andthe second and third, whatever they may be, are cancelled out by their signs. The innovationlies in the fourth 15 15

= (15). How did Cardano arrive at this result? Thechapter heading of the Ars Magnae in which imaginary numbers rst appear is called Rulesfor posing a negative. Cardano nds here an elegant solution which ts nicely within therhetoric of abbaco practices. He rst poses a negative, meaning he takes the cosa to be a neg-ative quantity. This amounts to (x)(x) = (x) which is generally accepted. Thereforethe fourth product of the binomial multiplication must be 15.5 ConclusionThe abbaco tradition was concerned with teaching methods for calculating with Hindu-Arabicnumerals. Not able to rely on authoritative texts, abbaco masters devised schemes for thejustication of the correctness of procedures and algorithms. Such schemes appear as non-25For the original text see Franci (2001, 44).1 3Epistemic Justication and Operational Symbolism 111discursive elements in the texts and provide justication of discursive descriptions of theoperations. 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(Ver-offentlichungendesDeutschenMuseumsfur dieGeschichteder Wissenschaftenundder Technik.ReiheC, QuellentexteundUbersetzungen, Nr. 33). Munchen.Author BiographyAlbrecht Heeffer is fellow of the Research Foundation, Flanders (FWO) and associated with the Center forHistory of Science of Ghent University, Belgium. He teaches history of mathematics and history of sciencein antiquity. His current research is on the epistemic justication of mathematical practices in medieval sub-scientic traditions, with a special interest in cross-cultural inuences. Dr. Heeffer is currently preparing acritical edition of the popular work Rcrations Mathmatiques (1624).1 3