The emergence of objects from mathematical practices

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.

[1]  G. Lakoff,et al.  Metaphors We Live By , 1980 .

[2]  C. Batanero,et al.  CLARIFYING THE MEANING OF MATHEMATICAL OBJECTS AS A PRIORITY AREA OF RESEARCH IN MATHEMATICS EDUCATION , 2004 .

[3]  G. Santi.  Objectification and semiotic function , 2011 .

[4]  Willard Van Orman Quine,et al.  The Pursuit of Truth , 1990 .

[5]  A. Norton.  The Hutchinson dictionary of ideas , 1994 .

[6]  Raymond Duval.  A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics , 2006 .

[7]  Penelope Maddy,et al.  Realism in mathematics , 1991 .

[8]  H. Blumer,et al.  Symbolic Interactionism: Perspective and Method , 1988 .

[9]  W. Quine.  Ontological Relativity and Other Essays , 1969 .

[10]  S. Parada,et al.  Revista Latinoamericana de Investigación en Matemática Educativa , 2014 .

[11]  Piotr Błaszczyk.  On the Mode of Existence of the Real Numbers , 2005 .

[12]  Stewart Shapiro,et al.  Structure and Ontology , 1989 .

[13]  A. Sfard.  On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin , 1991 .

[14]  Ed Dubinsky,et al.  APOS: A Constructivist Theory of Learning in Undergraduate Mathematics Education Research , 2001 .

[15]  Paul Cobb,et al.  The emergence of mathematical meaning : interaction in classroom cultures , 2009 .

[16]  Murad Jurdak,et al.  Contrasting Perspectives and Performance of High School Students on Problem Solving in Real World, Situated, and School Contexts , 2006 .

[17]  D. Bloor.  Wittgenstein: A Social Theory of Knowledge , 1985 .

[18]  A. Sfard.  Thinking as Communicating: Human Development, the Growth of Discourses, and Mathematizing , 2008 .

[19]  Derek Allan Holton,et al.  The teaching and learning of mathematics at university level : an ICMI study , 2001 .

[20]  Vicenç Font,et al.  Metaphors in mathematics classrooms: analyzing the dynamic process of teaching and learning of graph functions , 2010 .

[21]  M. Trobok.  Platonism in the Philosophy of Mathematics , 2006 .

[22]  S. Shapiro.  Philosophy of mathematics : structure and ontology , 1997 .

[23]  J. Godino,et al.  Why is the learning of elementary arithmetic concepts difficult? Semiotic tools for understanding the nature of mathematical objects , 2011 .

[24]  M. Balaguer.  Fictionalism in the Philosophy of Mathematics , 2012 .

[25]  G. Lakoff,et al.  Where mathematics comes from : how the embodied mind brings mathematics into being , 2002 .

[26]  Boaz Miller.  A Social Theory of Knowledge , 2011 .

[27]  E. Purington.  Mathematics 10 , 1967, IEEE Spectrum.

[28]  Stewart Shapiro,et al.  Thinking about Mathematics: The Philosophy of Mathematics , 2000 .

[29]  Núria Planas,et al.  THE OBJECT METAPHOR AND SYNECDOCHE IN MATHEMATICS CLASSROOM DISCOURSE , 2010 .

[30]  Gordon P. Baker,et al.  Wittgenstein, rules, grammar, and necessity , 1985 .

[31]  Y. Engeström,et al.  Learning by expanding: An activity-theoretical approach to developmental research , 2014 .

[32]  Nathalie Sinclair,et al.  A constructive response to `Where mathematics comes from' , 2003 .

[33]  Paul Ernest.  Social Constructivism as a Philosophy of Mathematics , 1997 .

[34]  Hartry Field,et al.  Realism, Mathematics, and Modality , 1989 .

[35]  G. Schubring,et al.  The Ethics of Being and Knowing: Towards a Cultural Theory of Learning , 2008 .

[36]  S. Korner,et al.  The philosophy of mathematics , 1960 .

[37]  Thomas Tymoczko,et al.  Mathematics, science and ontology , 1991, Synthese.

[38]  P. Hacker,et al.  Wittgenstein, rules, grammar and necessity, vol. 2 of an Analytical Commentary of the Philosophical investigations , 1988 .

[39]  James J. Kaput,et al.  Representations, inscriptions, descriptions and learning: A kaleidoscope of windows , 1998 .

[40]  Luis Radford,et al.  The Anthropology of Meaning , 2006 .

[41]  P. Kleingeld,et al.  The Stanford Encyclopedia of Philosophy , 2013 .

[42]  Ed Dubinsky,et al.  A Framework for Research and Curriculum Development in Undergraduate Mathematics Educationjpeg , 2009 .

[43]  Hartry Field,et al.  Science without Numbers , 1983 .

[44]  M. Bunge.  Emergence and Convergence: Qualitative Novelty and the Unity of Knowledge , 2003 .

[45]  M. Bedau,et al.  Weak emergence : Causation and emergence , 1996 .

[46]  Alan H. Schoenfeld,et al.  Research in Collegiate Mathematics Education. I , 1994 .

[47]  Gert Schubring,et al.  Semiotics in mathematics education : epistemology, history, classroom, and culture , 2008 .

[48]  Hilary Putnam,et al.  The Philosophy of Mathematics: , 2019, The Mathematical Imagination.

[49]  C. Batanero,et al.  The onto-semiotic approach to research in mathematics education , 2007 .

[50]  Vicenç Font,et al.  The role of intuition in the solving of optimization problems , 2010 .

[51]  Vicenç Font,et al.  The problem of the particular and its relation to the general in mathematics education , 2008 .

[52]  Candia Morgan,et al.  The Interpretative Nature of Teachers ’ Assessment of Students ’ Mathematics : Issues for Equity , 2002 .

[53]  J. Kilpatrick,et al.  Mathematics Education as a Research Domain: A Search for Identity , 1998 .

[54]  R. Goodstein,et al.  Remarks on the Foundations of Mathematics , 1957, The Mathematical Gazette.