Abstract
Since the beginning of the 19th century, Schleiermacher pointed out the presence of a circle (hermeneutic circle) in the comprehension: A particular element can be considered only taking into account the universe to which it belongs, and vice versa. Heidegger reconsidered the mentioned issue, so the comprehension is not understood according to the model of textual interpretation; it is based on the relationship between human beings and the world. When we consider a “historical” piece of knowledge (both an artistic masterpiece and an ancient mathematical idea) we can try to frame it in its historical period; nevertheless we can read it from our present, actual viewpoint. As regards mathematical knowledge, for instance, it is not really meaningful to look for “general rules” of mathematical evolution. Every culture clearly influenced the development of its own mathematics, for instance, by using artefacts and semiotic representations (Peirce). Both historico-cultural approaches (Radford) and anthropological approaches (D’Amore and Godino) ask us to investigate how cultural contexts determined mathematical experiences. Therefore art and mathematics can be linked by a hermeneutic approach: they are human enterprises and they are not to be considered as paths towards a form of truth “out there” (in Richard Rorty’s words).
Sommario A partire dall’inizio del XIX secolo, Schleiermacher segnalò la presenza di un circolo (circolo ermeneutico) per il quale il particolare può comprendersi soltanto partendo dall’universale di cui esso stesso è parte e viceversa. Il problema fu ripreso da Heidegger: dunque la comprensione non viene più ad essere orientata sul solo modello della spiegazione teoretica dei testi, bensì sullo stesso rapporto che gli esseri umani hanno con il mondo. Quando ci accostiamo ad un’opera “storica” (un capolavoro dell’arte di alcuni secoli fa ma anche un contenuto matematico antico) possiamo osservarla cercando di collocarla rigorosamente nel proprio periodo storico, ma anche leggerla con i nostri occhi. Per quanto riguarda la matematica, non ha ad esempio molto senso tentare di individuare le “regole generali” che avrebbero determinato l’evoluzione della matematica. Ogni cultura ha chiaramente influenzato lo sviluppo della propria matematica, ad esempio mediante l’uso di artefatti e di rappresentazioni semiotiche (Peirce). Sia gli approcci storico-culturali (Radford) che quelli antropologici (D’Amore e Godino) ci chiedono di stabilire come i contesti culturali abbiano determinato le esperienze matematiche. L’arte e la matematica possono dunque legarsi ad un approccio ermeneutico: sono costruzioni umane, non momenti di accesso ad una qualche Verità “là fuori” (nelle parole di Richard Rorty).
To Richard Rorty, 1931–2007
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsBibliography
Artigue, M. (1998). L’évolution des problématiques en didactique de l’analyse. Recherches en Didactique des Mathématiques, 18(2), 231–262.
Arzarello, F. (2000). Inside and outside: spaces, times and language in proof production. Proc. PME-24, 1, 22–38.
Atkin, A. (2006). Peirce’s theory of signs. http://plato.stanford.edu/entries/peirce-semiotics
Bachtin, M. (2000). L’autore e l’eroe. Torino: Einaudi (Estetica slovesnogo tvorčestva. Moskva: Izdatel’stvo Iskusstvo, 1979).
Bagni, G. T. (2006a). Some cognitive difficulties related to the representations of two major concepts of Set theory. Educa. Stud. Math., 62(3), 259–280.
Bagni, G. T. (2006b). Linguaggio, storia e didattica della matematica. Bologna: Pitagora.
Bagni, G. T. (2007). Rappresentare la matematica. Roma: Aracne.
Bagni, G. T. (2008). Richard Rorty (1931–2007) and his legacy for mathematics educators. Educ.Stud. Math., 67(1), 1–2.
Bartolini Bussi, M. G. and Boni, F. (2003). Instruments for semiotic mediation in primary school classrooms. For the Learn. Math., 23(2), 12–19.
Bartolini Bussi, M. G., Mariotti, M. A., and Ferri, F. (2005). Semiotic mediation in primary school: Dürer’s glass. In M. H. G. Hoffmann, J. Lenhard, and F. Seeger (Eds.), Activity and sign. Grounding mathematics education. Festschrift for Michael Otte (pp. 77–90). New York: Springer.
D’Amore, B. (2005). Secondary school students’ mathematical argumentation and Indian logic (Nyaya). For the Learn. Math., 25(2), 26–32.
D’Amore, B. and Fandiño Pinilla, M. I. (2001). Concepts et objets mathématiques. In Gagatsis, A. (Ed.), Learning in mathematics and sciences and educational technology I (pp. 111–130). Nicosia: Intercollege Press.
D’Amore, B. and Godino, D. J. (2006). Punti di vista antropologico ed ontosemiotico in Didattica della Matematica. La matematica e la sua didattica, 1, 9–38.
D’Amore, B. and Godino, J. D. (2007). El enfoque ontosemiótico como un desarrollo de la teoría antropológica en Didáctica de la Matemática. Revista Latinoamericana de Investigación en Matemática Educativa, 10(2), 191–218.
D’Amore, B., Radford, L. and Bagni, G. T. (2006). Ostacoli epistemologici e prospettive socioculturali. L’insegnamento della matematica e delle scienze integrate 29B, 1, 11–40.
Dörfler, W. (2006). Inscriptions as objects of mathematical activities. In J. Maasz, and W. Schloeglmann (Eds.), New mathematics education research and practice (pp. 97–111). Rotterdam-Taipei: Sense.
Engestroem, Y. (1990). When is a tool? Multiple meanings of artifacts in human activity. In Learning, working and imagining: twelve studies in activity theory (pp. 171–195). Helsinki: Orienta-Konsultit Oy.
Font, V., Godino, J. D., and D’Amore, B. (2007). An ontosemiotic approach to representations in mathematics education. For the Learn. Math., 27(2), 9–15.
Grugnetti, L. and Rogers, L. (2000). Philosophical, multicultural and interdisciplinary issues. In J. Fauvel and J. van Maanen (Eds.), History in mathematics education (pp. 39–62). Dordrecht: Kluwer.
Hoffmann, M. H. G. (2007). Learning from people, things, and signs. Stud. Philos. Educ., 26(3), 185–204.
Kuhn, T. S. (1962). Die Struktur wissenschaftlicher Revolutionen. Frankfurt a.M.: Suhrkamp.
Liszka, J. (1996). A general introduction to the semeiotic of Charles S. Peirce. Bloomington: Indiana University Press.
Marietti, S. (2001). Icona e diagramma. Il segno matematico in Charles Sanders Peirce. Milano: LED.
Peirce, C. S. (1885). On the algebra of logic: A contribution to the philosophy of notation. Am.J. Math., 7(2), 180–196.
Peirce, C. S. (1931–1958). Collected papers, I–VIII. Cambridge: Harvard University Press.
Peirce, C. S. (1998). The essential peirce. Peirce Edition Project. Bloomington: Indiana University Press.
Rabardel, P. (1995). Les hommes et les technologies: Approche cognitive des instruments contemporains. Paris: Colin.
Radford, L. (2000). Signs and meanings in the students’ emergent algebraic thinking: a semiotic analysis. Educ. Stud. Math., 42 (3), 237–268.
Radford, L. (2002). The seen, the spoken and the written. A semiotic approach to the problem of objectification of mathematical knowledge. For the Learn. Math., 22(2), 14–23.
Radford, L. (2003). Gestures, speech and the sprouting of signs. Math. Think. Learn., 5(1), 37–70.
Radford, L. (2005). The semiotics of the schema. Kant, Piaget, and the calculator. In M. H. G. Hoffmann, J. Lenhard, and F. Seeger (Eds.), Activity and sign. Grounding mathematics education (pp. 137–152). New York: Springer.
Radford, L. (forthcoming). Rescuing Perception: Diagrams in Peirce’s theory of cognitive activity. In de Moraes, L. and Queiroz, J. (Eds.), C.S. Peirce’s Diagrammatic Logic. Catholic Univ. of Sao Paulo, Brazil.
Rorty, R. (1979). Philosophy and the mirror of nature. Princeton: Princeton University Press.
Rorty, R. (1989). Contingency, irony, and solidarity. Cambridge: Cambridge University Press.
Rorty, R. (1998). Truth and progress. Philosophical papers III. Cambridge: Cambridge University Press.
Savan, D. (1988). An introduction to C.S. Peirce’s full system of semeiotic. Toronto: Toronto Semiotic Circle.
Sonesson, G. (1989). Pictorial concepts. Inquiries into the semiotic heritage and its relevance for the analysis of the visual world. Lund: Lund University Press.
Stjernfelt, F. (2000). Diagrams as centerpiece of a peircean epistemology. Trans. Charles S. Peirce Soc., 36, 357–384.
Vygotsky, L. S. (1978). Interaction between learning and development. In M. Cole, V. John-Steiner, S. Scribner, and E. Souberman (Eds.), Mind in society: The development of higher psychological processes (pp. 79–91). Cambridge: Harvard University Press.
Vygotskij, L. S. (1997). Collected works. R. Rieber (Ed.). New York: Plenum.
Wartofsky, M. (1979). Perception, representation and the forms of action: towards an historical epistemology. In Models. Representation and the scientific understanding (pp. 188–209). Dordrecht: Reidel.
Zeman, J. J. (1986). Peirce’s philosophy of logic. Trans. Charles S. Peirce Soc., 22, 1–22.
Acknowledgments
The author would like to thank Bruno D’Amore (University of Bologna) and Willibald Dörfler (University of Klagenfurt, Austria) for their very valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Bagni, G.T. (2010). Mathematics, Art, and Interpretation: A Hermeneutic Perspective. In: Capecchi, V., Buscema, M., Contucci, P., D'Amore, B. (eds) Applications of Mathematics in Models, Artificial Neural Networks and Arts. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-8581-8_18
Download citation
DOI: https://doi.org/10.1007/978-90-481-8581-8_18
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-8580-1
Online ISBN: 978-90-481-8581-8
eBook Packages: Humanities, Social Sciences and LawSocial Sciences (R0)