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Between Logic and Reality pp 29–57Cite as

Applied Mathematics in the Sciences

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Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 25))

Abstract

A complete philosophy of mathematics must address Paul Benacerraf’s dilemma. The requirements of a general semantics for the truth of mathematical theorems that coheres also with the meaning and truth conditions for non-mathematical sentences, according to Benacerraf, should ideally be coupled with an adequate epistemology for the discovery of mathematical knowledge. Standard approaches to the philosophy of mathematics are criticized against their own merits and against the background of Benacerraf’s dilemma, particularly with respect to the problem of understanding the distinction between pure and applied mathematics and the effectiveness of applied mathematics in the natural sciences and engineering. The evaluation of these alternatives provides the basis for articulating a philosophically advantageous Aristotelian inherence concept of mathematical entities. An inherence account solves Benacerraf’s dilemma by interpreting mathematical entities as nominalizations of structural spatiotemporal properties inhering in existent spatiotemporal entities.

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Notes

  1. 1.

    Plato’s myth of the divided line appears in Republic 509d–513e.

  2. 2.

    The problems of imitation of, participation in, and striving to attain approximation to archetypal Platonic Forms are discussed by Allen [1], Lee [25], Nehamas [31] and Sweeney [44].

  3. 3.

    See Balaguer [4] and Azzouni [3].

  4. 4.

    Ockham [32]. See also Boehmer [8].

  5. 5.

    Hilbert [19,20] and Jacquette [23, pp. 85–89].

  6. 6.

    Putnam [35, 36].

  7. 7.

    See, inter alia, Brouwer [9] and Heyting [18]. A useful introduction to Brouwer’s intutionism in its formal and philosophical aspects is found in Van Atten [45].

  8. 8.

    Maddy [26, 27, 28].

  9. 9.

    Field [14, 15, 16, 17].

  10. 10.

    See Chihara [12, especially pp. 108–13, 317–48].

  11. 11.

    Resnik [38, 39, 40] and Shapiro [41, 42].

  12. 12.

    Chihara [12, pp. 107–8, 163–217]. Also Chihara [11].

  13. 13.

    I offer a more extensive criticism of Chihara’s constructibility theory in my Jacquette [24] review of Chihara [12].

  14. 14.

    Quine [37, p. 400] speaks of the higher reaches of set theory as a “mathematical recreation… without ontological rights”.

  15. 15.

    Steiner [43].

  16. 16.

    Aristotle, Physics 263b3–9: “To the question whether it is possible to pass through an infinite number of units [i.e. intervals] either of time or of distance we must reply that in a sense it is and in a sense it is not. If the units [intervals] are actual it is not possible, if they are potential, it is possible.” Hume [21 especially pp. 27–65]. See Jacquette [23].

  17. 17.

    See Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre [10, pp. 181–82].

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Author information

Authors and Affiliations

  1. Universität Bern, Bern, BE, Switzerland

    Dale Jacquette

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  1. Dale Jacquette
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Correspondence to Dale Jacquette .

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Editors and Affiliations

  1. , Faculty of Philosophy, University of Rijeka, Slavka Krautzeka bb, Rijeka, HR-51000, Croatia

    Majda Trobok

  2. , Department of Philosophy, University of Maribor, Koroška cesta 160, Maribor, 2000, Slovenia

    Nenad Miščević

  3. , Faculty of Philosophy, University of Split, Teslina 12, Split, 21000, Croatia

    Berislav Žarnić

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Jacquette, D. (2012). Applied Mathematics in the Sciences. In: Trobok, M., Miščević, N., Žarnić, B. (eds) Between Logic and Reality. Logic, Epistemology, and the Unity of Science, vol 25. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2390-0_3

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