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Concept Formation and Concept Grounding

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Abstract

Recently Carrie S. Jenkins formulated an epistemology of mathematics, or rather arithmetic, respecting apriorism, empiricism, and realism. Central is an idea of concept grounding. The adequacy of this idea has been questioned e.g. concerning the grounding of the mathematically central concept of set (or class), and of composite concepts. In this paper we present a view of concept formation in mathematics, based on ideas from Carnap, leading to modifications of Jenkins’s epistemology that may solve some problematic issues with her ideas. But we also present some further problems with her view, concerning the role of proof for mathematical knowledge.

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Notes

  1. Jenkins (2005), Jenkins (2008). In our account of Jenkins’s theory below, we primarily follow Jenkins (2005). Chapter four in Jenkins (2008) is a recapitulation of the paper. The main difference is some more details and more elaborated definitions.

  2. One exception is Tennant (2010).

  3. Roland (2010), and Forrai (2011). There are other critical voices, e.g. Bradley (2011), who questions Jenkins’s analysis of the a priori, but they will not be treated here.

  4. See Sjögren (2011b).

  5. See Jenkins (2008), pp. 138-144, and Jenkins (2005) for a more comprehensive description of the hypotheses.

  6. Since these ideas have been presented in other contexts, we will be rather brief here. See Sjögren (2011a), and Bennet and Sjögren (2013).

  7. See Lear (1982), and Aristotle, Phys., Book II, 193b−194a.

  8. See Sjögren (2011a) for examples and details.

  9. See Kleiner (1989) for some remarks on the function concept, and Kline (1972) for a more comprehensive treatment.

  10. See Carnap’s treatment of probability in Carnap (1945), his first paper mentioning explications, and Carnap (1950) for conditions explicata must satisfy.

  11. Note that we are concerned here with extensional uniqueness or nonuniqueness. That some explications are (extensionally) unique may in some cases even be provable. See Sjögren (2011a), pp. 19-21, for a discussion concerning this issue.

  12. Multiplication is still more complicated, but we will not dwell upon this here. See Bennet and Sjögren (2013) for comments.

  13. In fact, Gödel’s proof assumes Peano arithmetic being even ω-consistent, which is a strictly stronger notion than pure consistency, while consistency is enough for Rosser’s proof of the Gödel-Rosser incompleteness result. For details see, e.g., Franzén (2005).

References

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Acknowledgments

The authors wish to thank the anonymous referees for valuable comments.

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Authors and Affiliations

  1. School of Engineering Science, University of Skövde, P.O. Box 408, SE-541 28, Skövde, Sweden

    Jörgen Sjögren

  2. Department of Pedagogical, Curricular and Professional Studies, University of Gothenburg, Box 100, SE-405 30, Gothenburg, Sweden

    Christian Bennet

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  1. Jörgen Sjögren
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  2. Christian Bennet
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Correspondence to Jörgen Sjögren.

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Sjögren, J., Bennet, C. Concept Formation and Concept Grounding. Philosophia 42, 827–839 (2014). https://doi.org/10.1007/s11406-014-9528-8

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  • DOI: https://doi.org/10.1007/s11406-014-9528-8

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