Skip to main content
SpringerLink
Log in

Epistemic Justification and Operational Symbolism

  • Published:
Foundations of Science Aims and scope Submit manuscript

Abstract

By the end of the twelfth century in the south of Europe, new methods of calculating with Hindu-Arabic numerals developed. This tradition of sub-scientific mathematical practices is known as the abbaco period and flourished during 1280–1500. This paper investigates the methods of justification for the new calculating procedures and algorithms. It addresses in particular graphical schemes for the justification of operations on fractions and the multiplication of binomial structures. It is argued that these schemes provided the validation of mathematical practices necessary for the development towards symbolic reasoning. It is shown how justification schemes compensated for the lack of symbolism in abbaco treatises and at the same time facilitated a process of abstraction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abdeljaouad, M. (2002). Le manuscrit mathématique de Jerba: Une pratique des symboles algébriques maghrébins en pleine maturité. Septième Colloque Maghrébin sur l’Histoire des Mathématiques Arabes (Marrakech, 30–31 mai et 1er juin 2002).

  • Allard A. (1992) Muhammad Ibn Musa Al-Khwarizmi Le Calcul Indien (Algorismus). Versions Latines Du Xiie Siècle (Collection d’Etudes Classiques). Peeters Publishers, Namur

    Google Scholar 

  • Arrighi, G. (eds) (1964) Paolo Dell’Abaco, Trattato d’aritmetica. Domus Galilaeana, Pisa

    Google Scholar 

  • Arrighi, G. (1965–1967). Un ‘programma’ di didattica di matematica nella prima metá del Quattrocento (dal codice 2186 della Biblioteca Riccardiana di Firenze). Atti e Memorie dell’Accademia Petrarca di Lettere, Art e Scienze di Arezzo, nuova serie, 38, 112–128.

  • Arrighi, G. (Ed.). (1973). Libro d’abaco. Dal Codice 1754 (sec. XIV) della Biblioteca Statale di Lucca. Cassa di Risparmio di Lucca, Lucca.

  • Arrighi, G. (Ed.). (1985). Giovanni de’Danti Aretino,Tractato de l’algorisimo. Dal Cod. Plut. 30. 26 (sec. XIV) della Biblioteca Medicea Laurenziana di Firenze. Atti e Memorie della Accademia Petrarca di Lettere, Arti e Scienze, nuova serie 47, 3–91.

  • Arrighi, G. (Ed.). (1987). Paolo Gherardi, Opera mathematica: Libro di ragioni—Liber habaci. Codici Magliabechiani Classe XI, nn. 87 e 88 (sec. XIV) della Biblioteca Nazionale di Firenze, Pacini-Pazzi, Lucca.

  • Arrighi, G. (Ed.). (1989). Maestro Umbro (sec. XIII), Livero de l’abbecho. (Cod. 2404 della Biblioteca Riccardiana di Firenze). Bollettino della Deputazione di Storia Patria per l’Umbria 86, 5–140.

  • Boncompagni, B. (1857). Scritti di Leonardo Pisano, matematico del secolo decimoterzo, pubblicati da Baldassarre Boncompagni. Roma, Tipografia delle Scienze Matematiche e Fisiche, 1857–1862. Volume I, Liber Abbaci.

  • Burnett C. (2006) The semantics of Indian numerals in Arabic, Greek and Latin. Journal of Indian Philosophy 34: 15–30

    Article  Google Scholar 

  • Calzoni, G., Cavazzoni, G. (eds) (1996) Tractatus Mathematicus ad Discipulos Perusinos. Città di Castello, Perugia

    Google Scholar 

  • Cajori, F. (1928–1929). A history of mathematical notations (2 Vols). La Salle, Il: Open Court Publishing. (reprinted by Dover, 1993).

  • Cardano, G. (1545). Ars Magnae, sive, De regulis algebraicis lib. unus: qui & totius operis de arithmetica, quod opus perfectum inscripsit, est in ordine decimus. In N. Johann Petreius (Ed.), (English translation by Witmer R. T. (1968) Ars magna or the rules of algebra. Cambridge, MA: M.I.T. Press. Reprinted by Dover Publications, New York, 1993).

  • Cassinet, J. (2001). Une arithmétique toscane en 1334 en Avignon dans la citè des papes et de leurs banquiers florentins. In Commerce et mathématiques du moyen âge à la renaissance, autour de la Méditerranée(pp. 105–128). Actes du Colloque International du Centre International d’Histoire des Sciences Occitanes (Beaumont de Lomagne, 13–16 mai 1999). Éditions du C.I.S.H.O.,Toulouse.

  • Chemla K. (1997) What is at stake in mathematical proofs from third-century China?. Science in Context 10(2): 227–251

    Article  Google Scholar 

  • Chemla K. (2003) Generality above abstraction: The general expressed in terms of the paradigmatic in mathematics in ancient China. Science in Context 16: 413–458

    Article  Google Scholar 

  • Chemla K. (2005) The interplay between proof and algorithm in 3rd century China: The operations as prescription of computation and the operation as argument. In: Mancosu P., Jorgensen K. F., Pedersen S. A. (eds) Visualization, explanation and reasoning styles in mathematics. Springer, Heidelberg, pp 123–145

    Chapter  Google Scholar 

  • Chemla, K., & Shuchun, G. (2004). Les Neuf Chapitres, le classique mathématique de la Chine ancienne et ses commentaires. Traduction française avec des addenda détaillés et une édition commentée du texte chinois du livre et de son commentaire. Paris, Dunod, 2004.

  • Cifoletti, G.(1993). Mathematics and rhetoric: Peletier and Gosselin and the making of the French algebraic tradition. PhD, Princeton University 0181, Dissertation Abstracts International. 53, 4061-A.

  • Djebbar A. (1992) Le traitement des fractions dans la tradition mathématique arabe du Maghreb. In: Benoit P., Chemla K., Ritter J. (eds) Histoire de fractions, fractions d’histoire. Birkhäuser, Berlin, pp 223–245

    Google Scholar 

  • Folkerts, M. (1997). Die älteste lateinische Schriftüber das indische Rechnen nachal-Ḫwārizmī. Bayerische Akademie der Wissenschaften, Klasse, n.F., 113, München.

  • Franci, R. (2001). (Ed.) Maestro Dardi (sec. XIV) Aliabraa argibra. Dal manoscritto I.VII.17 della Biblioteca Comunale di Siena. Quaderni del Centro Studi della Matematica Medioevale (Vol. 26). Siena: Università di Siena.

  • Friedlein G. (1869) Die Zahlzeichen und das elementare Rechnen der Griechen und Römer und christlichen Abendlandes vom 7 bis 13 Jahrhundert. Verlag von Andreas Deichert, Erlangen

    Google Scholar 

  • Goldthwaite R. A. (2009) The economy of renaissance Florence. The John Hopkins University Press, Baltimore

    Google Scholar 

  • Heeffer A. (2008) Text production reproduction and appropriation within the abbaco tradition: A case study. Sources and Commentaries in Exact Sciences 9: 211–256

    Google Scholar 

  • Heeffer A. (2009) On the nature and origin of algebraic symbolism. In: Van Kerkhove B. (eds) New perspectives on mathematical practices, essays in philosophy and history of mathematics. World Scientific Publishing, Singapore, pp 1–27

    Chapter  Google Scholar 

  • Heeffer, A. (2010a). Algebraic partitioning problems from Luca Pacioli’s Perugia manuscript (Vat. Lat. 3129). Sources and Commentaries in Exact Sciences, 11, (to appear).

  • Heeffer, A. (2010b). The body in renaissance arithmetic. In: A. Pease, M. Guhe & A. Smaill (Eds.), Mathematical practice and cognition, proceedings of the international symposium on mathematical practice and cognition (pp. 21–22). AISB 2010 convention, 29 March–1 April 2010, De Montfort University, Leicester.

  • Heeffer, A. (2010c). The symbolic model for algebra: Functions and mechanisms. In L. Magnani, W. Carnielli, & C. Pizzi (Eds.), Model-based reasoning in science and technology, abduction, logic, and computational discovery, studies in computational intelligence (Vol. 314, pp. 519–532). Heidelberg: Springer.

  • Heeffer, A. (2011). On the curious historical coincidence of algebra and double-entry bookkeeping. In K. François, B. Löwe, T. Müller, & B. van Kerkhove (Eds.), Foundations of the formal sciences. Bringing together philosophy and sociology of science (pp. 111–132). London: College Publications.

  • Høyrup J. (1994) In measure, number, and weight, studies in the mathematics and culture. State University of New York Press, Albany

    Google Scholar 

  • Høyrup J. (2004) Leonardo Fibonacci and abbaco culture a proposal to invert the roles. Revue d’Histoire des Mathématiques 11: 23–56

    Google Scholar 

  • Høyrup, J. (2007). Jacopo da Firenze’s Tractatus Algorismi and Early Italian Abbacus Culture. (Science Networks. Historical Studies, 34). Birkhäuser, Basel.

  • Høyrup, J. (2009). How to transfer the conceptual structure of old Babylonian mathematics: Solutions and inherent problems. With an Italian Parallel. Beitrag zur Tagung Zur Übersetzbarkeit von Wissenschaftssprachen des Altertums Johannes Gutenberg-Universität Mainz, 27–29 Juli 2009.

  • Høyrup J. (2010) Hesitating progress—The slow development toward algebraic symbolization in abbacus and related manuscripts, c. 1300 to c. 1550. In: Heeffer A., Van Dyck M. (eds) Philosophical aspects of symbolic reasoning in early-modern mathematics. Studies in Logic 26, College Publications, London, pp 3–56

    Google Scholar 

  • Levy, M. (Ed.). (1965). Kūshyār ibn Labbān, Principles of hindu reckoning, (M. Levy, & M. Petruck, Trans.). Madison: University of Wisconsin Press.

  • Mancosu, P. (eds) (2008) The philosophy of mathematical practice. Oxford University Press, Oxford

    Google Scholar 

  • Nesselmann, G. H. F. (1842). Versuch einer kritischen Geschichte der Algebra (Vol. 1). Die Algebra der Griechen. Berlin (Reprinted Frankfurt, 1969).

  • Rappe S. (2000) Reading neoplatonism: Non-discursive thinking in the texts of Plotinus, Proclus, and Damascius. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Rouse Ball, W. W. (1960). A short account of the history of mathematics. New York: Dover Books.

  • Sesiano J. (1985) The appearance of negative solutions in mediaeval mathematics. Archive for History of Exact Sciences 32(2): 105–150

    Article  Google Scholar 

  • Sigler, L. (2002). (Ed.) Fibonacci’s Liber Abaci. A translation into modern english of Leonardo Pisano’s book of calculation. Heidelberg: Springer.

  • Ulivi, E. (2002a). Scuole e maestri d’abaco. In E. Giusti (Ed.), Un ponte sul mediterraneo: Leonardo Pisano, la scienza araba e la rinascita della matematica in Occidente (pp. 121–159). Firenze: Edizioni Polistampa.

  • Ulivi, E. (2002b). Benedetto da Firenze (1429–1479) un maestro dabaco del XV secolo. Con documenti inediti e con un Appendice su abacisti e scuole dabaco a Firenze nei secoli XIII-XVI”. Bollettino di storia delle scienze matematiche, 22(1) (2002) (full issue).

  • Ulivi, Elisabetta (2006). Raffaello Canacci, Piermaria Bonini e gli abacisti della famiglia Grassini. Bollettino di storia delle scienze matematiche, 24(2) (2004) (stampa 2006), 125–212.

  • Van Egmond, W. (1977). New light on Paolo dell’Abbaco. Annali dell’Istituto e Museo di Storia della Scienza di Firenze 2(2), 3–21.

  • Van Egmond, W. (1980). Practical mathematics in the Italian renaissance: A catalog of Italian abacus manuscripts and printed books to 1600. Monografia, 4, Istituto e Museo di Storia della Scienza, Florence.

  • Van Kerkhove, B., & Van Bendegem, J. P. (2007). Perspectives on mathematical practices: Bringing together philosophy of mathematics, sociology of mathematics, and mathematics education. Logic, Epistemology, and the Unity of Science (Vol. 5). Dordrecht: Springer.

  • Vogel, K. (1977). Ein italienisches Rechenbuch aus dem 14. Jahrhundert (Columbia X 511 A13). (Veroffentlichungen des Deutschen Museums fur die Geschichte der Wissenschaften und der Technik. Reihe C, Quellentexte und Ubersetzungen, Nr. 33). Munchen.

Download references

Author information

Authors and Affiliations

  1. Centre for Logic and Philosophy of Science, Ghent University, Ghent, Belgium

    Albrecht Heeffer

Authors
  1. Albrecht Heeffer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Albrecht Heeffer.

Additional information

An earlier version of this paper was presented at the conference From Practice to Results in Logic and Mathematics. An International Conference on the Role of Practices in Shaping Results in Logic and Mathematics, June 21–23, 2010, Salle Internationale, MSH Lorraine, Nancy, France and benefited from valuable comments from Jens Høyrup, Karine Chemla and Jeff Oaks.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Heeffer, A. Epistemic Justification and Operational Symbolism. Found Sci 19, 89–113 (2014). https://doi.org/10.1007/s10699-012-9311-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10699-012-9311-x

Keywords

Search

Navigation