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Higher Mathematics: Science of the Purely Structural

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An Aristotelian Realist Philosophy of Mathematics
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Abstract

The last three centuries have seen mathematics move well beyond its original subject-matter of quantity. It has discovered an entirely different subject-matter, structure or pattern. We survey the historical development with an eye to philosophically significant examples, then address the crucial philosophical question of characterizing precisely what ‘structure’ is.

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Notes and Bibliography

  1. M. Ascher, Ethnomathematics: A Multicultural View of Mathematical Ideas (Brooks/Cole, Pacific Grove, CA, 1991).

    Google Scholar 

  2. N. Biggs, E. Lloyd and R. Wilson ed., Graph Theory 1736–1936 (Oxford University Press, Oxford, 1976), 3–8.

    Google Scholar 

  3. H. Weyl, Symmetry (Princeton University Press, Princeton, NJ, 1952).

    Google Scholar 

  4. P. Benacerraf, What numbers could not be, Philosophical Review 74 (1965), 495–512.

    Article  Google Scholar 

  5. S. Shapiro, Philosophy of Mathematics: Structure and Ontology (Oxford University Press, New York, 1997).

    Google Scholar 

  6. M.D. Resnik, Mathematics as a Science of Patterns (Oxford University Press, Oxford, 1997).

    Google Scholar 

  7. D. Dennett, Real patterns, Journal of Philosophy 88 (1991), 27–51.

    Article  Google Scholar 

  8. H. Poincaré, Science and Hypothesis (Walter Scott, London, 1905), 20.

    Google Scholar 

  9. N. Bourbaki, The architecture of mathematics, American Mathematical Monthly 57 (1950), 221–232.

    Article  Google Scholar 

  10. S. Mac Lane, Mathematics: Form and Function (Springer, New York, 1986).

    Book  Google Scholar 

  11. G.H. Hardy, A Mathematician’s Apology (Cambridge University Press, Cambridge, 1940), 59–60.

    Google Scholar 

  12. M. Giaquinto, Visual Thinking in Mathematics (Oxford University Press, Oxford, 2007), 207–208.

    Book  Google Scholar 

  13. D.M. Armstrong, A Theory of Universals: Universals and Scientific Reasoning, vol. II (Cambridge University Press, Cambridge, 1978), 69.

    Google Scholar 

  14. S. Haack, Deviant Logic, Fuzzy Logic: Beyond the formalism (University of Chicago Press, Chicago, 1996).

    Google Scholar 

  15. D. Lewis, Parts of Classes (Blackwell, Oxford, 1991).

    Google Scholar 

  16. G. Hellman, Mathematics Without Numbers: Towards a Modal-Structural Interpretation (Oxford University Press, Oxford, 1989), 48–49.

    Google Scholar 

  17. S. Willard, General Topology (Addison-Wesley, Reading, Mass, 1970), 23.

    Google Scholar 

  18. C.S. Peirce, On the logic of number, American Journal of Mathematics 4 (1881), 85–95.

    Article  Google Scholar 

  19. A. Newstead and J. Franklin, On the reality of the continuum, Philosophy 83 (2008), 117–127.

    Article  Google Scholar 

  20. S. Hegarty, Aristotle’s notion of quantity and modern mathematics, Philosophical Studies (Ireland) 18 (1969), 25–35.

    Article  Google Scholar 

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© 2014 James Franklin

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Franklin, J. (2014). Higher Mathematics: Science of the Purely Structural. In: An Aristotelian Realist Philosophy of Mathematics. Palgrave Macmillan, London. https://doi.org/10.1057/9781137400734_5

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