Volume 1 · Number 2 · Pages 61–72
A Constructivist Approach to Experiential Foundations of Mathematical Concepts Revisited

Ernst von Glasersfeld

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Abstract

Purpose: The paper contributes to the naturalization of epistemology. It suggests tentative itineraries for the progression from elementary experiential situations to the abstraction of the concepts of unit, plurality, number, point, line, and plane. It also provides a discussion of the question of certainty in logical deduction and arithmetic. Approach: Whitehead’s description of three processes involved in criticizing mathematical thinking (1956) is used to show discrepancies between a traditional epistemological stance and the constructivist approach to knowing and communication. Practical implications: Reducing basic abstract terms to experiential situations should make them easier to conceive for students.

Key words: foundations of mathematics, concept formation, conceptual semantics

Citation

Glasersfeld E. von (2006) A constructivist approach to experiential foundations of mathematical concepts revisited. Constructivist Foundations 1(2): 61–72. http://constructivist.info/1/2/061

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References

Beth E. W. & Piaget J. (1961) Épistémologie mathématique et psychologie. Presses Universitaires de France, Paris. ▸︎ Google︎ Scholar
Bridgman P. B. (1961) The way things are. Compass, New York. ▸︎ Google︎ Scholar
Brouwer L. E. J. (1949) Consciousness, philosophy, and mathematics. Proceedings of the 10th International Congress of Philosophy, Volume 1, Part 2. North Holland, Amsterdam: 1235–1249. ▸︎ Google︎ Scholar
Brunschvicg L. (1981) Les étapes de la philosophie mathématique, 3rd edition. Blanchard, Paris. Originally published in 1912. ▸︎ Google︎ Scholar
Ceccato S. (1966) Un tecnico fra i filosofi, Volume 2. Marsilio: Padua. ▸︎ Google︎ Scholar
Confrey J. (1990) Splitting, similarity, and rate of change: A new approach to multiplication and exponential functions. Paper presented at the Annual Meeting of the American Educational Research Association. Boston MA. ▸︎ Google︎ Scholar
Davis P. J. & Hersh R. (1980) The mathematical experience. Birkhäuser, Boston. ▸︎ Google︎ Scholar
Deleuze G. & Guattari F. (1994) What is philosophy. Columbia Univ. Press, New York. ▸︎ Google︎ Scholar
Einstein A. (1954) Physics and reality. In: Einstein A., Ideas and opinions. Bonanza Books, New York: 290–323. Originally published in 1936 in: Journal of the Franklin Institute 221(3): 348–382. ▸︎ Google︎ Scholar
Euler L. (1770) Lettres a une princesse d’Allemagne. Steidel: Mietau/Leipzig. ▸︎ Google︎ Scholar
Frege G. (1884) Grundlagen der Arithmetik. English translation: Frege G. (1974) Foundations of arithmetic (translated by J. L. Austin). Blackwell, Oxford. ▸︎ Google︎ Scholar
Glasersfeld E. von (1981) An attentional model for the conceptual construction of units and number. Journal for Research in Mathematics Education 12: 83–94. ▸︎ Google︎ Scholar
Glasersfeld E. von (1989) Cognition, construction of knowledge, and teaching. Synthese 80(1): 121–140. http://www.vonglasersfeld.com/118
Glasersfeld E. von (1991) Abstraction, re–presentation, and reflection: An interpretation of experience and Piaget’s approach. In: Steffe L. P. (ed.) Epistemological foundations of mathematical experience. Springer, New York: 45–67. http://www.vonglasersfeld.com/130
Goblot E. (1922) Le système des sciences: Le vrai, l'intelligible et le réel. A. Colin, Paris. ▸︎ Google︎ Scholar
Hadamard J. (1954) The psychology of invention in the mathematical field. Dover, New York. ▸︎ Google︎ Scholar
Hersh R. (1986) Some Proposals for Reviving the Philosophy of Mathematics. In: Tymoczko T. (ed.) New directions in the philosophy of mathematics. Birkhäuser, Boston, Basel: 9–28. Originally published in 1979 in: Advances in Mathematics 31: 31–50. ▸︎ Google︎ Scholar
Husserl E. (1970) Philosophie der Arithmetik. Martin Nijhoff: The Hague. Originally published in 1887. ▸︎ Google︎ Scholar
James G. & James R. (1959) Mathematics dictionary. Van Nostrand: Princeton NJ. ▸︎ Google︎ Scholar
Kant I. (1902) Kritik der reinen Vernunft. Akademieausgabe, Volume 3. Preussische Akademie, Berlin. Original publication: 1787. ▸︎ Google︎ Scholar
Keys J. (alias G. Spencer Brown) (1972) Only two can play this game. Julian Press, New York. ▸︎ Google︎ Scholar
Lakatos I. (1976) Proofs and refutations: The logic of mathematical discovery. Cambridge University Press, Cambridge. ▸︎ Google︎ Scholar
Lakatos I. (1986) A renaissance of empiricism in the recent philosophy of mathematics? In: Tymoczko T. (ed.) New directions in the philosophy of mathematics. Birkhäuser, Boston, Basel: 29–49. Originally published in 1976 in: British Journal for the Philosophy of Science 27(3): 201–223. ▸︎ Google︎ Scholar
Lakoff G. & Johnson M. (1980) Metaphors we live by. Chicago University Press, Chicago. ▸︎ Google︎ Scholar
Lakoff. G. & Núñez R. E. (2000) Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books, New York. ▸︎ Google︎ Scholar
Lorenzen P. (1974) Konstruktive Wissenschaftstheorie. Suhrkamp, Frankfurt am Main. ▸︎ Google︎ Scholar
Mach E. (1970) The guiding principles of my scientific theory of knowledge and its reception by my contemporaries. In: S. Toulmin (ed.) Physical reality. Harper, New York: 44–53. German original published in 1910. ▸︎ Google︎ Scholar
Maddy P. (1997) Naturalism in mathematics. Clarendon Press, Oxford. ▸︎ Google︎ Scholar
Maturana H. R. (1980) Biology of cognition. In: Maturana H. R. & Varela F. J., Autopoiesis and cognition: The realization of the living. Boston Studies in the Philosophy of Science, Volume 42. Reidel, Dordrecht: 5–58. Originally published in 1970 as BCL Report 9.0. Biological Computer Laboratory. Department of Electrical Engineering, University of Illinois. ▸︎ Google︎ Scholar
Mill J. S. (1843) A system of logic, ratiocinative and inductive, being a connected view of the principles of evidence and the method of scientific investigation. London. ▸︎ Google︎ Scholar
Mittelstrass J. (1987) Philosophische Grundlagen der Wissenschaft. In: Pasternack G. (ed.) Philosophie und Wissenschaften: Das Problem des Apriorismus. Peter Lang, Frankfurt/Bern. ▸︎ Google︎ Scholar
Newman J. R. (1956) The world of mathematics (4 volumes). Tempus Books, New York. ▸︎ Google︎ Scholar
Piaget J. (1937) La Construction du réel chez l’enfant. Delachaux et Niestlé: Neuchâtel. ▸︎ Google︎ Scholar
Piaget J. (1972) Psychology and epistemology, Viking, New York. French original published in 1970 by Édition Denoël, Paris. ▸︎ Google︎ Scholar
Piaget J. (1977) Recherches sur l’abstraction réfléchissante, Volume I & II. Presses Universitaires de France, Paris. ▸︎ Google︎ Scholar
Quine W. V. (1969) Epistemology naturalized. In: Quine W. V., Ontological relativity and other essays. Columbia University Press, New York: 69–90. ▸︎ Google︎ Scholar
Rota G.-C. (1980) Introduction. In: Davis P. J. & Hersh R. (1980) The mathematical experience. Birkhäuser, Boston. ▸︎ Google︎ Scholar
Rotman B. (1987) Signifying nothing: The semiotics of zero. Stanford University Press, Stanford. ▸︎ Google︎ Scholar
Rotman B. (2000) Mathematics as sign: Writing, imagining, counting. Stanford University Press, Stanford. ▸︎ Google︎ Scholar
Steffe L. P., Glasersfeld E. von, Richards J. & Cobb P. (1983) Children’s counting types: Philosophy, theory, and application. Praeger, New York. ▸︎ Google︎ Scholar
Tymoczko T. (1986) Making room for mathematicians in the philosophy of mathematics. The Mathematical Intelligencer 8(3): 44–50. ▸︎ Google︎ Scholar
Tymoczko T. (1986) The four-color problem and its philosophical significance. In: Tymoczko T. (ed.) New directions in the philosophy of mathematics. Birkhäuser, Boston, Basel: 243–266. Originally published in 1979 in: The Journal of Philosophy 76(2): 57–83. ▸︎ Google︎ Scholar
Tymoczko T. (ed.) (1986) New directions in the philosophy of mathematics. Birkhäuser, Boston, Basel. ▸︎ Google︎ Scholar
Valéry P. (1974) Cahiers, Volume II. Gallimard, Paris. Originally published in 1935. ▸︎ Google︎ Scholar
Vico G.-B. (1858) De Antiquissima Italorum Sapientia, Stamperia de Classici Latini. Naples. Originally published in 1710. ▸︎ Google︎ Scholar
Whitehead A. N. (1956) Mathematics as an element in the history of thought. In: Newman J. R. (ed.) The world of mathematics, Volume 1. Simon & Schuster, New York. Reprinted in 1988 by Tempus, Microsoft Press. ▸︎ Google︎ Scholar
Wittenberg A. I. (1968) Vom Denken in Begriffen. Birkhäuser: Basel/Stuttgart. ▸︎ Google︎ Scholar

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