Abstract
This article offers a new interpretation of Piaget’s decanting experiments, employing the mathematical notion of equivalence instead of conservation. Some reference is made to Piaget’s theories and to his educational legacy, but the focus in on certain of the experiments. The key to the new analysis is the abstraction principle, which has been formally enunciated in mathematical philosophy but has universal application. It becomes necessary to identity fluid objects (both configured and unconfigured) and configured and unconfigured sets-of-objects. Issues emerge regarding the conflict between philosophic realism and anti-realism, including constructivism. Questions are asked concerning mathematics and mathematical philosophy, particularly over the nature of sets, the wisdom of the axiomatic method and aspects of the abstraction principle itself.
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Notes
See, for example, Birkhoff and MacLane (1965).
I have discussed this a little further in Cable (2013).
See, for example, Strawson and Chakrabarti (2006).
As, for example, in Skemp (1971).
Frege (1980/1884, §64).
Einstein et al. (1923, p. 112).
Bickhard (1997, p. 29).
See Smith (1993, p. 35).
See, for example, Devlin (1992).
See, for example, Bennour and Vonèche (2009, p. 50).
See, for example, Smith (1993).
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Acknowledgments
I wish to acknowledge with thanks the help I have received in the preparation of this article from three reviewers of a previous version and from numerous members of the Mathematics Interest Group at Kings College London, especially Margaret Brown and Jeremy Hodgen.
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Cable, J. La Meme Chose: How Mathematics Can Explain the Thinking of Children and the Thinking of Children Can Illuminate Mathematical Philosophy. Sci & Educ 23, 223–240 (2014). https://doi.org/10.1007/s11191-013-9628-z
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DOI: https://doi.org/10.1007/s11191-013-9628-z