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On Non-literal Approaches

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Book cover Explaining Beauty in Mathematics: An Aesthetic Theory of Mathematics

Part of the book series: Synthese Library ((SYLI,volume 370))

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Abstract

This chapter discusses in detail the idea that mathematical beauty should be reinterpreted to preserve mathematics’ stand among the sciences. Three reasons to reinterpret mathematical beauty are examined; the two cultures split, the epistemic character of mathematics, and its rational character. It shall be argued that the reasons for endorsing a non literal interpretation of mathematical beauty are rather weak. The discussion also examines the conceptions of mathematical beauty by Shaftesbury, Hutchenson and Gian-Carlo Rota.

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Notes

  1. 1.

    I must emphasize that the notation was not what made the proof hard to understand. The proof was hard to understand primarily because of its ideas were novel and it involved intricate and very abstract machinery that was alien to the field. That resulted in a notation that appeared obscure even to the specialist. But this simple need of having to learn a new notation makes technical subjects very opaque to the non-specialist. This is true not only for the lay person, but also for mathematicians themselves.

  2. 2.

    Rota does not mention it explicitly, but he is referring to G. H. Hardy’s view [33] that the unexpectedness and inevitability of a theorem or proof are the sources of mathematical beauty.

  3. 3.

    In a proof by contradiction, or reductio ad absurdum, one assumes the negation of the statement to be proven and shows that it leads to a contradiction.

  4. 4.

    It must be noted that intuitionism does not reject all proofs by reductio ad absurdum, but only those that intend to establish an existential claim, or those that intend to move from not-not-A to A. At any rate, the kind of proofs rejected by intuitionism can be considered characteristically non-enlightening.

  5. 5.

    Discussing the rationality of science or mathematics is beyond the scope of this book; here I address solely the issues relevant to the interpretation of beauty.

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    Ulianov Montano

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Montano, U. (2014). On Non-literal Approaches. In: Explaining Beauty in Mathematics: An Aesthetic Theory of Mathematics. Synthese Library, vol 370. Springer, Cham. https://doi.org/10.1007/978-3-319-03452-2_1

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