Educational Studies in Mathematics

, Volume 82, Issue 1, pp 97–124 | Cite as

The emergence of objects from mathematical practices

  • Vicenç Font
  • Juan D. GodinoEmail author
  • Jesús Gallardo


The nature of mathematical objects, their various types, the way in which they are formed, and how they participate in mathematical activity are all questions of interest for philosophy and mathematics education. Teaching in schools is usually based, implicitly or explicitly, on a descriptive/realist view of mathematics, an approach which is not free from potential conflicts. After analysing why this view is so often taken and pointing out the problems raised by realism in mathematics this paper discusses a number of philosophical alternatives in relation to the nature of mathematical objects. Having briefly described the educational and philosophical problem regarding the origin and nature of these objects we then present the main characteristics of a pragmatic and anthropological semiotic approach to them, one which may serve as the outline of a philosophy of mathematics developed from the point of view of mathematics education. This approach is able to explain from a non-realist position how mathematical objects emerge from mathematical practices.


Classroom discourse Conventional versus realist mathematics Epistemology Mathematical objects Onto-semiotics 



The research reported in this article was carried out as part of the following projects: EDU2009-08120 and EDU2010-14947 (MICINN, Spain).


  1. Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. In J. Kaput, A. H. Schoenfeld, & E. Dubinsky (Eds.), Research in collegiate mathematics education II (pp. 1–32). Providence: American Mathematical Society.Google Scholar
  2. Baker, G. P., & Hacker, P. M. S. (1985). Wittgenstein. Rules, grammar and necessity. An analytical commentary on the Philosophical Investigations (Vol. 2). Glasgow: Basil Blackwell.Google Scholar
  3. Balaguer, M. (2008). Fictionalism in the philosophy of mathematics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved 11 April 2012 from
  4. Barceló, R., Bujosa, J. M., Cañadilla, J. L., Fargas, M., & Font, V. (2002). Matemáticas 1. [Mathematics 1]. Barcelona: Almadraba.Google Scholar
  5. Bedau, M. (1997). Weak emergence. Philosophical Perspectives, 11, 375–399.Google Scholar
  6. Benacerraf, P. (1973). Mathematical truth. The Journal of Philosophy, 70(19), 661–679.CrossRefGoogle Scholar
  7. Błaszczyk, P. (2005). On the mode of existence of the real numbers. Analecta Husserliana, 88, 137–155.CrossRefGoogle Scholar
  8. Bloor, D. (1983). Wittgenstein. A social theory of knowledge. London: The Macmillan Press.Google Scholar
  9. Blumer, H. (1969). Symbolic interactionism: Perspective and method. Berkeley: University of California Press.Google Scholar
  10. Bunge, M. (2003). Emergence and convergence: Qualitative novelty and the unity of knowledge. Toronto: University of Toronto Press.Google Scholar
  11. Cantoral, R., Farfán, R. M., Lezama, J., & Martínez-Sierra, G. (2006). Socioepistemología y representación: algunos ejemplos [Socio-epistemology and representation: Some examples]. Revista Latinoamericana de Investigación en Matemática Educativa. Special Issue on Semiotics, Culture and Mathematical Thinking, 9(4), 27–46.Google Scholar
  12. Chevallard, Y. (1992). Concepts fondamentaux de la didactique: Perspectives apportées par une approche anthropologique [Fundamental concepts of didactic: Contributed perspectives by an anthropological approach]. Recherches en Didactique des Mathématiques, 12(1), 73–112.Google Scholar
  13. Cobb, P., & Bauersfeld, H. (Eds.). (1995). The emergence of mathematical meaning: Interaction in classroom cultures. Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  14. Dubinsky, E., & McDonald, M. A. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton et al. (Eds.), The teaching and learning of mathematics at university level: An ICMI study (pp. 273–280). Dordrechet: Kluwer Academic Publishers.Google Scholar
  15. Duval, R. (1995). Sémiosis et pensée humaine. Registres sémiotiques et apprentissages intellectuels [Semiosis and human thought. Semiotic registers and intellectual learning]. Berne: Peter Lang.Google Scholar
  16. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1–2), 103–131.CrossRefGoogle Scholar
  17. Engeström, Y. (1987). Learning by expanding: An activity theoretical approach to developmental research. Helsinki: Orienta-Konsultit Oy.Google Scholar
  18. Ernest, P. (1998). Social constructivism as a philosophy of mathematics. Albany: State University of New York Press.Google Scholar
  19. Field, H. (1980). Science without numbers. Princeton, NJ: Princeton University Press.Google Scholar
  20. Field, H. (1989). Realism, mathematics and modality. Oxford: Basil Blackwell.Google Scholar
  21. Font, V., Bolite, J., & Acevedo, J. I. (2010). Metaphors in mathematics classrooms: Analyzing the dynamic process of teaching and learning of graph functions. Educational Studies in Mathematics, 75(2), 131–152.CrossRefGoogle Scholar
  22. Font, V., & Contreras, A. (2008). The problem of the particular and its relation to the general in mathematics education. Educational Studies in Mathematics, 69(1), 33–52.CrossRefGoogle Scholar
  23. Font, V., Godino, J. D., Planas, N., & Acevedo, J. I. (2010). The object metaphor and synecdoche in mathematics classroom discourse. For the Learning of Mathematics, 30(1), 15–19.Google Scholar
  24. Godino, J. D., & Batanero, C. (1998). Clarifying the meaning of mathematical objects as a priority area of research in mathematics education. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics education as a research domain: A search for identity (pp. 177–195). Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  25. Godino, J. D., Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in mathematics education. ZDM–The International Journal on Mathematics Education, 39(1–2), 127–135.CrossRefGoogle Scholar
  26. Godino, J. D., & Font, V. (2010). The theory of representations as viewed from the onto-semiotic approach to mathematics education. Mediterranean Journal for Research in Mathematics Education, 9(1), 189–210.Google Scholar
  27. Godino, J. D., Font, V., Wilhelmi, M. R., & Lurduy, O. (2011). Why is the learning of elementary arithmetic concepts difficult? Educational Studies in Mathematics, 77(2), 247–265.CrossRefGoogle Scholar
  28. Jurdak, M. (2006). Contrasting perspectives and performance of high school students on problem solving in real world situated, and school contexts. Educational Studies in Mathematics, 63(3), 283–301.CrossRefGoogle Scholar
  29. Kaput, J. J. (1998). Representations, inscriptions, descriptions and learning: A kaleidoscope of windows. Journal of Mathematical Behaviour, 17(2), 266–281.CrossRefGoogle Scholar
  30. Lakoff, G., & Johnson, M. (1980). Metaphors we live by. Chicago: Chicago University Press.Google Scholar
  31. Lakoff, G., & Nuñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.Google Scholar
  32. Linnebo, Ø. (2009). Platonism in the philosophy of mathematics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved 11 April 2012 from
  33. Maddy, P. (1990). Realism in mathematics. Nueva York: Oxford University Press.Google Scholar
  34. Malaspina, U., & Font, V. (2010). The role of intuition in the solving of optimization problems. Educational Studies in Mathematics, 75(1), 107–130.CrossRefGoogle Scholar
  35. Morgan, C., & Watson, A. (2002). The interpretative nature of teacher’s assessment of students’ mathematics: Issues for equity. Journal for Research in Mathematics Education, 33(2), 78–111.CrossRefGoogle Scholar
  36. Norton, A. L. (Ed.). (1997). The Hutchinson dictionary of ideas. Oxford: Helicon Publishing.Google Scholar
  37. O’Connor, T., & Wong, H. Y. (2006). Emergent properties. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved 11 April 2012 from
  38. Quine, W. (1969). Ontological relativity and other essays. New York: Columbia University Press.Google Scholar
  39. Quine, W. (1990). Pursuit of truth. Cambridge: Harvard University Press.Google Scholar
  40. Radford, L. (2006). The anthropology of meaning. Educational Studies in Mathematics, 61(1–2), 39–65.CrossRefGoogle Scholar
  41. Radford, L. (2008). The ethics of being and knowing: Towards a cultural theory of learning. In L. Radford, G. Schubring, & F. Seeger (Eds.), Semiotics in mathematics education: Epistemology, history, classroom, and culture (pp. 215–234). Rotterdam: Sense Publishers.Google Scholar
  42. Resnik, M. D. (1988). Mathematics from the structural point of view. Revue International de Philosophie, 42(167), 400–424.Google Scholar
  43. Rodriguez-Pereyra, G. (2008). Nominalism in metaphysics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved 11 April 2012 from
  44. Rozov, M. A. (1989). The mode of existence of mathematical objects. Philosophia Mathematica, s2–4(2), 105–111.CrossRefGoogle Scholar
  45. Sáenz-Ludlow, A. (2002). Sign as a process of representation: A Peircean perspective. In F. Hitt (Ed.), Representations and mathematics visualization (pp. 277–296). México: CINVESTAV–IPN.Google Scholar
  46. Santi, G. (2011). Objectification and semiotic function. Educational Studies in Mathematics, 77(2–3), 285–311.CrossRefGoogle Scholar
  47. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36.CrossRefGoogle Scholar
  48. Sfard, A. (2000). Symbolizing mathematical reality into being – or how mathematical discourse and mathematical objects create each other. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms (pp. 38–75). London: Lawrence Erlbaum.Google Scholar
  49. Sfard, A. (2008). Thinking as communicating. Human development, the growth of discourses, and mathematizing. New York, NY: Cambridge University Press.CrossRefGoogle Scholar
  50. Shapiro, S. (1997). Philosophy of mathematics. Structure and ontology. New York: Oxford University Press.Google Scholar
  51. Shapiro, S. (2000). Thinking about mathematics. The philosophy of mathematics. Oxford: Oxford University Press.Google Scholar
  52. Sinclair, N., & Schiralli, M. (2003). A constructive response to ‘Where mathematics comes from’. Educational Studies in Mathematics, 52(1), 79–91.CrossRefGoogle Scholar
  53. Smith, B. (1975). The ontogenesis of mathematical objects. Journal of the British Society for Phenomenology, 6(2), 91–101.Google Scholar
  54. Tymoczko, T. (1991). Mathematics, science and ontology. Synthese, 88(2), 201–228.CrossRefGoogle Scholar
  55. Wittgenstein, L. (1953). Philosophical investigations. New York: The MacMillan Company.Google Scholar
  56. Wittgenstein, L. (1978). Remarks on the foundations of mathematics (3rd ed.). Oxford: Basil Blackwell.Google Scholar
  57. Yagisawa, T. (2009). Possible objects. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved 12 April 2012 from

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Vicenç Font
    • 1
  • Juan D. Godino
    • 2
    Email author
  • Jesús Gallardo
    • 3
  1. 1.Departament de Didàctica de les Ciències Experimentals i la Matemàtica, Facultat de Formació del ProfessoratUniversitat de BarcelonaBarcelonaSpain
  2. 2.Departamento de Didáctica de la Matemática, Facultad de EducaciónUniversidad de GranadaGranadaSpain
  3. 3.Departamento de Didáctica de las Matemáticas, Didáctica de las Ciencias Sociales y de las Ciencias Experimentales. Facultad de Ciencias de la EducaciónUniversidad de MálagaMálagaSpain

Personalised recommendations