Mathematics Education in the Context of certain classical Debates in Philosophy and Mathematics

Michael Otte; Mircea Radu

doi:10.23925/1983-3156.2022v24i2p041-062

Authors

Michael Otte Universität Bielefeld
Mircea Radu Universität Bielefeld https://orcid.org/0000-0002-0337-4281

DOI:

https://doi.org/10.23925/1983-3156.2022v24i2p041-062

Keywords:

Language, Logic, Philosophy, Mathematics, Different forms of Complementarity

Abstract

The paper presents some illustrative turns in the history of the interactions between philosophy, logic, mathematics, and mathematical education since the 16th century. The underlying problem could be called the Aristotelian problem. Aristotle argued that any individual thing consists of a substantial form, which determines its general nature, and matter, which individuates the thing and makes it numerically distinct from other similar substances.

Author Biographies

Michael Otte, Universität Bielefeld

Possui mestrado em Matemática pela Universidade de Erlangen (1963), doutorado em Matemática pela Universidade de Goettingen (1967) e doutorado em Matemática pela Universitat Munster (Westfalische-Wilhelms) (1972). Professor aposentado da Universidade de Bielefeld (Alemanha), recentemente atua como professor colaborador na Universidade Federal de Mato Grosso - UFMT simultaneamente no Programa de Pós Graduação em Educação (PPGE) do Instituto de Educação (IE) e no Programa de Pós Graduação em Educação em Ciências e Matemática - PPGECEM da REDE AMAZÔNICA DE EDUCAÇÃO EM CIÊNCIAS E MATEMÁTICA- REAMEC. Tem experiência na área de Matemática, com ênfase em Educação Matemática, atuando principalmente nos seguintes temas: Educação Matemática, Filosofia da Matemática, História da Matemática, História da Filosofia Analítica, Semiótica e Complementaridade.

Mircea Radu, Universität Bielefeld

Mircea Radu teaches Mathematics at the Oberstufen-Kolleg des Landes Nordrhein-Westfalen in Bielefeld, Germany. His research work covers topics in the history and phylosophy of mathematics and mathematics education.

References

Bernays, P., 1976. Abhandlungen zur Philosophie der Mathematik, WBG Darmstadt.

Bolzano, B. 1981, Wissenschaftslehre (WL), 4 vols., Sulzbach: Wolfgang Schultz. Reprint Scientia Verlag Aalen.

Condillac, E.B. (2001). Essay on the origin of human knowledge. Transl. and ed. by H. Aarsleff , Cambridge University Press, Cambridge/UK.

Dummett, M., 1991, Frege, Harvard University Press, Cambridge, MA.

Effros, E. G. 1998, Mathematics as Language, in: H. G. Dales/G. Oliveri (eds.), Truth in Mathematics, Oxford.

Ermer, E., / K. Kiehl, in: The Economist, 11.11. 2010; also: J. Barkow, L. Cosmides & J. Tooby, (eds.), The Adapted Mind, Oxford 1992.

Euler, L., Vollständige Anleitung zur Algebra, Leipzig: Reclam, 1770.

Ferreirós, J., 1999, Labyrinth of Thought. Birkhäuser Verlag, Basel, Boston.

Field, Hartry, 1980, Science Without Numbers: A Defence of Nominalism, Princeton, N.J.: Princeton University Press.

Foucault, M., 1973, The Order of Things, Vintage Books, N.Y.

Fowler D., 1987, The Mathematics of Plato´s Academy, Oxford, Clarendon Press.

Frege, G., 1884, Grundlagen der Arithmetik, Breslau.

Grosholz, E., 1980, Descartes’ Unification of Algebra and Geometry, in. S. Gaukroger (ed.) Descartes.

Heijenoort, J. v. (1967), Logic as calculus and logic as language, Synthese 17, 324-330.

Henry, M., 1987, La Barbarie, Grasset Paris.

Hintikka, J., 1997, Lingua Universalis vs. Calculus Ratiocinator, Selected Papers, vol. 2, Kluwer, Dordrecht

Kahneman, Daniel, 2010, Thinking, Fast and Slow. Penguin UK.

Kant, I., (1992), The Critique of Pure Reason, N. Kemp Smith (Trans.), The Macmillan Press Ltd., London. (citation after the 1.(A) and 2.(B) edition).

Lakoff, G./M. Johnson, 1980. Metaphors We Live By, Chicago UP.

Lenhard, J./M. Otte, 2010. Two Types of Mathematization. In Bart Van Kerkhove, Jonas De Vuyst and Jean Paul Van Bendegem, Philosophical perspectives on mathematical practice. (pp. 301-330). London: College Publications.

Leibniz G., 1684, A New Method for Maxima and Minima, as Well Tangents, Which is not Obstructed by Fractional or Irrational Quantities.

Leibniz G., 1686. Discourse on metaphysics

Lenoir, T., 1979. Descartes and the Geometrization of Thought, Historia Mathematica. Vol.6, pp355-379.

Lovejoy, 1964/1936, The Great Chain of Being, Harvard Univ. Press.

Otte, M., 1989, The Ideas of Hermann Grassmann in the Context of the Mathematical and Philosophical Tradition since Leibniz, Historia Mathematica, 16, l-35,

Otte, M. 2003, Complementarity, Sets and Numbers, Edu. Studies in Math. 53, 203-228.

Otte, M. 2006, Proof-Analysis and Continuity, Foundations of Science, 11: 121-155.

Peirce, Ch. S.: CCL = The Cambridge Conferences Lectures of 1898, in: Ch. S. Peirce, Reasoning and the Logic of Things, ed. by K.L. Ketner, with an Introduction by K.L. Ketner and H. Putnam, Harvard UP, Cambridge/London 1992; CP = Collected Papers of Charles Sanders Peirce, Volumes I-VI, ed. by Charles Hartshorne and Paul Weiß, Cambridge, Mass. (Harvard UP) 1931-1935, Volumes VII-VIII, ed. by Arthur W. Burks, Cambridge, Mass. (Harvard UP) 1958 (quoted by no. of volume and paragraph)7

Radu, M. Basic skills versus conceptual understanding in mathematics education: The case of fraction division. Zentralblatt für Didaktik der Mathematik 34, 93–95 (2002). https://doi.org/10.1007/BF02655712

Rawls J. 2003, Future Pasts: The Analytic Tradition in Twentieth-Century Philosophy, ed. J. Floyd/S. Shieh, Oxford)

Richards, J., 1988, Mathematical Visions, Academic Press, Boston.

Rossi, P., 2000, Logic and the Art of Memory, The University of Chicago Press.

Rubenstein, R.,E., 2003, Aristotle’s Children, Harcourt Inc. N.Y.

Shea, W. R., 1991. The Magic of Numbers and Motion, Watson Publishing.

Stewart, M., 2006, The Courtier and the Heretic, Norton & Company N.Y

Tharp, L., 1989, »Myth and Mathematics«, in: Synthese, 81, pp. 167–201.

Thom, R., 1971, »Modern Mathematics: An Educational and Philosophic Error?«, in: American Scientist, Vol. 59, pp. 695–699.

Thom, R., 1972, »Modern Mathematics: Does it Exist?«, in: G. Howson (ed.), Developments in Mathematical Education, Cambridge, pp. 206 f.).

Thom, R., 1992, Leaving Mathematics for Philosophy, in: Casacuberta/Castallet (eds), Mathematical Research Today and Tomorrow, Springer Verlag, Heidelberg, pp 1-12.

Thurston, W., 1994, On proof and progress in mathematics. Bulletin of the AMS, vol. 30, 161-177.

Wertheimer, M., 1945, Productive Thinking, New York.

Wu, Hung-Hsi, 1999: Basic Skills Versus Conceptual Understanding – A Bogus Dichotomy. In: American Educator, Fall 1999.