Volume 1 · Number 2 · Pages 61–72

< Previous Paper · Next Paper >

A Constructivist Approach to Experiential Foundations of Mathematical Concepts Revisited

Ernst von Glasersfeld

Download the full text in
PDF (143 kB)

> Citation > Similar > References > Add Comment


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


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

Export article citation data: Plain Text · BibTex · EndNote · Reference Manager (RIS)

Similar articles

Cariani P. (2012) Infinity and the Observer: Radical Constructivism and the Foundations of Mathematics

Glasersfeld E. von (2010) Why People Dislike Radical Constructivism

Glasersfeld E. von (2005) Thirty Years Constructivism

Glasersfeld E. von (2009) Sketches from Partial Memories

Glasersfeld E. von (2006) Review of: Holger Lindemann (2005) Konstruktivismus und Pädagogik. Grundlagen, Modelle, Wege zur Praxis


Beth E. W. & Piaget J. (1961) Épistémologie mathématique et psychologie. Presses Universitaires de France, Paris. << Google Scholar

Bridgman P. B. (1961) The way things are. Compass, New York. << Google 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. << Google Scholar

Brunschvicg L. (1981) Les étapes de la philosophie mathématique, 3rd edition. Blanchard, Paris. Originally published in 1912. << Google Scholar

Ceccato S. (1966) Un tecnico fra i filosofi, Volume 2. Marsilio: Padua. << Google 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. << Google Scholar

Davis P. J. & Hersh R. (1980) The mathematical experience. Birkhäuser, Boston. << Google Scholar

Deleuze G. & Guattari F. (1994) What is philosophy. Columbia Univ. Press, New York. << Google 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. << Google Scholar

Euler L. (1770) Lettres a une princesse d’Allemagne. Steidel: Mietau/Leipzig. << Google Scholar

Frege G. (1884) Grundlagen der Arithmetik. English translation: Frege G. (1974) Foundations of arithmetic (translated by J. L. Austin). Blackwell, Oxford. << Google 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. << Google Scholar

Glasersfeld E. von (1989) Cognition, construction of knowledge, and teaching. Synthese 80(1): 121–140. Available at 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. Available at http://www.vonglasersfeld.com/130

Goblot E. (1922) Le système des sciences: Le vrai, l'intelligible et le réel. A. Colin, Paris. << Google Scholar

Hadamard J. (1954) The psychology of invention in the mathematical field. Dover, New York. << Google 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. << Google Scholar

Husserl E. (1970) Philosophie der Arithmetik. Martin Nijhoff: The Hague. Originally published in 1887. << Google Scholar

James G. & James R. (1959) Mathematics dictionary. Van Nostrand: Princeton NJ. << Google Scholar

Kant I. (1902) Kritik der reinen Vernunft. Akademieausgabe, Volume 3. Preussische Akademie, Berlin. Original publication: 1787. << Google Scholar

Keys J. (alias G. Spencer Brown) (1972) Only two can play this game. Julian Press, New York. << Google Scholar

Lakatos I. (1976) Proofs and refutations: The logic of mathematical discovery. Cambridge University Press, Cambridge. << Google 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. << Google Scholar

Lakoff G. & Johnson M. (1980) Metaphors we live by. Chicago University Press, Chicago. << Google Scholar

Lakoff. G. & Núñez R. E. (2000) Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books, New York. << Google Scholar

Lorenzen P. (1974) Konstruktive Wissenschaftstheorie. Suhrkamp, Frankfurt am Main. << Google 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. << Google Scholar

Maddy P. (1997) Naturalism in mathematics. Clarendon Press, Oxford. << Google 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. << Google 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. << Google Scholar

Mittelstrass J. (1987) Philosophische Grundlagen der Wissenschaft. In: Pasternack G. (ed.) Philosophie und Wissenschaften: Das Problem des Apriorismus. Peter Lang, Frankfurt/Bern. << Google Scholar

Newman J. R. (1956) The world of mathematics (4 volumes). Tempus Books, New York. << Google Scholar

Piaget J. (1937) La Construction du réel chez l’enfant. Delachaux et Niestlé: Neuchâtel. << Google Scholar

Piaget J. (1972) Psychology and epistemology, Viking, New York. French original published in 1970 by Édition Denoël, Paris. << Google Scholar

Piaget J. (1977) Recherches sur l’abstraction réfléchissante, Volume I & II. Presses Universitaires de France, Paris. << Google Scholar

Quine W. V. (1969) Epistemology naturalized. In: Quine W. V., Ontological relativity and other essays. Columbia University Press, New York: 69–90. << Google Scholar

Rota G.-C. (1980) Introduction. In: Davis P. J. & Hersh R. (1980) The mathematical experience. Birkhäuser, Boston. << Google Scholar

Rotman B. (1987) Signifying nothing: The semiotics of zero. Stanford University Press, Stanford. << Google Scholar

Rotman B. (2000) Mathematics as sign: Writing, imagining, counting. Stanford University Press, Stanford. << Google Scholar

Steffe L. P., Glasersfeld E. von, Richards J. & Cobb P. (1983) Children’s counting types: Philosophy, theory, and application. Praeger, New York. << Google Scholar

Tymoczko T. (1986) Making room for mathematicians in the philosophy of mathematics. The Mathematical Intelligencer 8(3): 44–50. << Google 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. << Google Scholar

Tymoczko T. (ed.) (1986) New directions in the philosophy of mathematics. Birkhäuser, Boston, Basel. << Google Scholar

Valéry P. (1974) Cahiers, Volume II. Gallimard, Paris. Originally published in 1935. << Google Scholar

Vico G.-B. (1858) De Antiquissima Italorum Sapientia, Stamperia de Classici Latini. Naples. Originally published in 1710. << Google 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. << Google Scholar

Wittenberg A. I. (1968) Vom Denken in Begriffen. Birkhäuser: Basel/Stuttgart. << Google Scholar

Comments: 0

To stay informed about comments to this publication and post comments yourself, please log in first.