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How Do You Apply Mathematics?

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Abstract

As far as disputes in the philosophy of pure mathematics goes, these are usually between classical mathematics, intuitionist mathematics, paraconsistent mathematics, and so on. My own view is that of a mathematical pluralist: all these different kinds of mathematics are equally legitimate. Applied mathematics is a different matter. In this, a piece of pure mathematics is applied in an empirical area, such as physics, biology, or economics. There can then certainly be a disputes about what the correct pure mathematics to apply is. Such disputes may be settled by the standard criteria of scientific theory selection (adequacy of empirical predications, simplicity, etc.) But what, exactly is it to apply a piece of pure mathematics? How is mathematics applied? By and large, philosophers of mathematics have cared more about pure mathematics than applied mathematics, and not a lot of thought has gone into this question. In this paper I will address the issue and some of its implications.

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Notes

  1. Talks based on this paper were given at the (online) workshop Disagreement in Mathematics, Free University of Amsterdam and the University of Hamburg, June 2021, and at Ohio State University, October 2021. I am grateful to members of the audiences for their helpful comments. I’m also very grateful to Hartry Field and Mel Fitting for their comments on earlier drafts of the paper. Finally, I’d like to thank three referees for this journal for their helpful comments.

  2. See, respectively, Bell (2013) and Colyvan (2001).

  3. Lakatos (1976).

  4. See Singh (1997).

  5. See, e.g., Klarreich (2018).

  6. Mathematical pluralism is defended at length in Priest (2013, 2019).

  7. For some discussion, see Gray (2017).

  8. See, e.g., Shapiro (2014), Priest (2019), Bell (1998) and Mortensen (2017).

  9. See, e.g., French (2019).

  10. There may well, of course, be important differences between the natural and the social sciences. As far as I can see, however, any such differences do no affect the points at issue here. Indeed, I choose an example from the social sciences precisely to show that the account of applied mathematics I am giving applies just as much to the social sciences as to the natural sciences.

  11. The basic idea of this is to be found in Priest (2005, 7.8).

  12. This account bears notable similarities to the “inferential conception” of applied mathematics, described and advocated by Bueno and Colyvan (2011) . (See esp. their diagram on p. 353.) A main difference is that they take the transitions involved to be between empirical phenomena and mathematical structures, whereas the account I give here takes them to be between statements about these things. In the last analysis, however, this difference may not be particulary significant.

  13. For a discussion of measurement in science, see Tal (2020).

  14. See, e.g., Chakravartty (2017, 2.1).

  15. Essentially this account is defended in Pincock (2004), where he calls it a structuralist account. The account is discussed by Bueno and Colyvan (2011), where they call it a mapping account. They argue there that it should be subsumed under what they call an inferential conception of applied mathematics.

  16. In just the way that the Tractatus reads off the structure of the world from the structure of the classical predicate calculus.

  17. I take this from Priest (2003).

  18. See, for example, Priest (1997, 2000).

  19. They can be found in Priest (2003, §7).

  20. On the genealogy of some mathematical concepts in certain human practices, see Lakoff and Núñez (2000); and for further discussion, see Kant and Sarikaya (2021).

  21. For an illustration from contemporary mathematical biology, see Montévil (2018) and Pérez-Escobar (2020).

  22. Perhaps most notably in Quine (1951). See Colyvan (2001, esp. 2.5). For more on Quine’s philosophy of mathematics see Hylton and Kemp (2019) and Priest (2010).

  23. This is sometimes known as ‘Quine’s indispensability argument’: abstract entities exist because they are indispensable for science. See, e.g., Colyvan (2001). Note that the plausibility of the ontological conclusion goes via the thought that we are justified in taking the statements to be true. The existence of a God is indispensable for Christian theology; but this provides no argument for the existence of God if the theological statements are not true.

  24. See Priest (2005).

  25. On the last of these, see Knuth (1974).

  26. On the difference between the two, see Priest (2021).

  27. Field (1980).

  28. In the way that non-finitary statements are used, according to Hilbert, in arithmetic. See Zach (2019).

  29. See Priest (2005, ch. 7).

  30. In the examples of §4, the empirical languages contained terms that refer to mathematical objects, though not quantification over them, e.g., with things such as \(\exists r\,r=\mu (I)\). The procedure I sketched carries over straightforwardly to such a syntax. The quantifiers are simply preserved in the abstracted pure mathematical statements (and back).

  31. So that they cannot be “factored out” with invariance under the appropriate transformations.

  32. See the discussion of conventionalism in Tal (2020).

  33. One has to be a bit careful as to how to spell this out, though. See the discussion of conservativity in §0.4 of the second edition of the book.

  34. Priest (2005, 7.8).

  35. See, e.g., Kroon and Voltolini (2019, §2).

  36. Wigner (1960).

  37. Pincock (2004), and Bueno and Colyvan (2011) take the discussion in fruitful directions. For some recent papers on the nature of applied mathematics see Issue 1 of European Journal for the Philosophy of Science 12 (2022).

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  1. Department of Philosophy, the Graduate Center, City University of New York, New York, USA

    Graham Priest

  2. Department of Philosophy, The University of Melbourne, Melbourne, Australia

    Graham Priest

  3. Department of Philosophy, Ruhr University of Bochum, Bochum, Germany

    Graham Priest

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Priest, G. How Do You Apply Mathematics?. Axiomathes 32 (Suppl 3), 1169–1184 (2022). https://doi.org/10.1007/s10516-022-09633-3

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