Abstract
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.
Similar content being viewed by others
Notes
Conceptual metaphors enable metaphorical expressions to be grouped together. A metaphorical expression, on the other hand, is a particular case of a conceptual metaphor. For example, the conceptual metaphor “the graph is a path” appears in classroom discourse through expressions such as “the function passes through the coordinate origin” or “if before point M the function is ascending and after it is descending then we have a maximum.” The teacher is unlikely to say to students that “the graph is a path” but, rather, will use metaphorical expressions that suggest this.
Some authors have argued that some of what are here called primary objects should be considered as objects. For example, Quine proposes referring to propositional objects rather than propositions (Quine, 1969). Another example can be found in some versions of nominalism, which consider that signs are the only mathematical objects that exist.
Depending on the text analysed, some of the elements in this configuration may be missing; this would be the case of epistemic configurations of axiomatic texts.
The notion of institutional cognition is associated with socio-cultural approaches to the construction of knowledge. If mathematical objects do not exist independently of people, but rather are produced intersubjectively through dialogue and convention, then this process by which objects are created constitutes a kind or form of cognition, namely institutional cognition.
The terms ostensive and non-ostensive can be considered as equivalent, respectively, to the terms actual and non-actual.
In phenomenological terms this would be the intentional object.
References
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.
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.
Balaguer, M. (2008). Fictionalism in the philosophy of mathematics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved 11 April 2012 from http://plato.stanford.edu/entries/fictionalism-mathematics/.
Barceló, R., Bujosa, J. M., Cañadilla, J. L., Fargas, M., & Font, V. (2002). Matemáticas 1. [Mathematics 1]. Barcelona: Almadraba.
Bedau, M. (1997). Weak emergence. Philosophical Perspectives, 11, 375–399.
Benacerraf, P. (1973). Mathematical truth. The Journal of Philosophy, 70(19), 661–679.
Błaszczyk, P. (2005). On the mode of existence of the real numbers. Analecta Husserliana, 88, 137–155.
Bloor, D. (1983). Wittgenstein. A social theory of knowledge. London: The Macmillan Press.
Blumer, H. (1969). Symbolic interactionism: Perspective and method. Berkeley: University of California Press.
Bunge, M. (2003). Emergence and convergence: Qualitative novelty and the unity of knowledge. Toronto: University of Toronto Press.
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.
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.
Cobb, P., & Bauersfeld, H. (Eds.). (1995). The emergence of mathematical meaning: Interaction in classroom cultures. Hillsdale: Lawrence Erlbaum Associates.
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.
Duval, R. (1995). Sémiosis et pensée humaine. Registres sémiotiques et apprentissages intellectuels [Semiosis and human thought. Semiotic registers and intellectual learning]. Berne: Peter Lang.
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1–2), 103–131.
Engeström, Y. (1987). Learning by expanding: An activity theoretical approach to developmental research. Helsinki: Orienta-Konsultit Oy.
Ernest, P. (1998). Social constructivism as a philosophy of mathematics. Albany: State University of New York Press.
Field, H. (1980). Science without numbers. Princeton, NJ: Princeton University Press.
Field, H. (1989). Realism, mathematics and modality. Oxford: Basil Blackwell.
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.
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.
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.
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.
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.
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.
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.
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.
Kaput, J. J. (1998). Representations, inscriptions, descriptions and learning: A kaleidoscope of windows. Journal of Mathematical Behaviour, 17(2), 266–281.
Lakoff, G., & Johnson, M. (1980). Metaphors we live by. Chicago: Chicago University Press.
Lakoff, G., & Nuñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.
Linnebo, Ø. (2009). Platonism in the philosophy of mathematics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved 11 April 2012 from http://plato.stanford.edu/entries/platonism-mathematics/.
Maddy, P. (1990). Realism in mathematics. Nueva York: Oxford University Press.
Malaspina, U., & Font, V. (2010). The role of intuition in the solving of optimization problems. Educational Studies in Mathematics, 75(1), 107–130.
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.
Norton, A. L. (Ed.). (1997). The Hutchinson dictionary of ideas. Oxford: Helicon Publishing.
O’Connor, T., & Wong, H. Y. (2006). Emergent properties. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved 11 April 2012 from http://plato.stanford.edu/entries/properties-emergent/.
Quine, W. (1969). Ontological relativity and other essays. New York: Columbia University Press.
Quine, W. (1990). Pursuit of truth. Cambridge: Harvard University Press.
Radford, L. (2006). The anthropology of meaning. Educational Studies in Mathematics, 61(1–2), 39–65.
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.
Resnik, M. D. (1988). Mathematics from the structural point of view. Revue International de Philosophie, 42(167), 400–424.
Rodriguez-Pereyra, G. (2008). Nominalism in metaphysics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved 11 April 2012 from http://plato.stanford.edu/entries/nominalism-metaphysics/.
Rozov, M. A. (1989). The mode of existence of mathematical objects. Philosophia Mathematica, s2–4(2), 105–111.
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.
Santi, G. (2011). Objectification and semiotic function. Educational Studies in Mathematics, 77(2–3), 285–311.
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.
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.
Sfard, A. (2008). Thinking as communicating. Human development, the growth of discourses, and mathematizing. New York, NY: Cambridge University Press.
Shapiro, S. (1997). Philosophy of mathematics. Structure and ontology. New York: Oxford University Press.
Shapiro, S. (2000). Thinking about mathematics. The philosophy of mathematics. Oxford: Oxford University Press.
Sinclair, N., & Schiralli, M. (2003). A constructive response to ‘Where mathematics comes from’. Educational Studies in Mathematics, 52(1), 79–91.
Smith, B. (1975). The ontogenesis of mathematical objects. Journal of the British Society for Phenomenology, 6(2), 91–101.
Tymoczko, T. (1991). Mathematics, science and ontology. Synthese, 88(2), 201–228.
Wittgenstein, L. (1953). Philosophical investigations. New York: The MacMillan Company.
Wittgenstein, L. (1978). Remarks on the foundations of mathematics (3rd ed.). Oxford: Basil Blackwell.
Yagisawa, T. (2009). Possible objects. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved 12 April 2012 from http://plato.stanford.edu/entries/possible-objects/.
Acknowledgment
The research reported in this article was carried out as part of the following projects: EDU2009-08120 and EDU2010-14947 (MICINN, Spain).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Font, V., Godino, J.D. & Gallardo, J. The emergence of objects from mathematical practices. Educ Stud Math 82, 97–124 (2013). https://doi.org/10.1007/s10649-012-9411-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10649-012-9411-0