Abstract
Penelope Maddy (Defending the axioms: on the philosophical foundation of set theory. Oxford University Press, New York, 2011) suggests a new naturalistic account of the objectivity of mathematics, grounding it on the fruitfulness of mathematics both internally, in mathematics itself, and externally, in the applications of mathematical concepts to empirical sciences. In light of the notion of mathematical depth, I will try to defend Maddy’s epistemological project from Jeffrey Roland’s criticisms concerning the reliability of mathematical beliefs in her account and Maddy’s ontological agnosticism between Thin Realism and Arealism. Although Maddy’s presentation of mathematical depth could profit from certain clarifications, I suggest that it could nevertheless ground an account of the reliability of mathematical beliefs, in response to the first objection. With regard to the second, I will argue that Roland’s criticism is misplaced.
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Notes
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According to Roland, projectibility is a property of predicates that measures the degree to which past instances can be taken as guides to future ones.
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In Roland’s words, this means providing a “dissident epistemology” for science (Roland 2007, p. 432).
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As an example, consider that the requirement of a causal link between the world’s facts and beliefs, in the case of mathematical knowledge, is certainly not common among naturalists.
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For references and discussion on the distinction between intrinsic and extrinsic mathematical explanation, see, for example, Mancosu (2008).
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The role of intuition in philosophy is a topic of debate: for references and discussion, see, for example, the essays in Gendler (2010).
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Note that we are not denying our initial distinction between depth and fruitfulness, since we clearly stated that, despite this distinction, the depth of a mathematical notion, statement, or theory could be seen as constituting a condition for its fruitfulness.
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Maddy’s description of the objectivity of the mathematical depth seems to be robustly consistent with this sense of objective: see, for example, Maddy (2011, pp. 80–81).
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As McLarty clearly explains: “Maddy calls the existence claim [about sets] mathematical, since mathematicians routinely affirm it. She calls claims about possible existence, which do not occur in mathematics and are prominent in metaphysical discussion, metaphysical. She never argues against pursuing metaphysics and even the metaphysics of mathematics. She argues that we can understand what mathematics is and how it is justified by looking at mathematics and other sciences which mathematicians routinely do address, and not metaphysics.” (McLarty 2013, p. 386)
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Even the projectibility of predicates Roland (2009, p. 431) applied to terms denoting objects in mathematical statements could still be there, because in Maddy’s account the successful use of a mathematical notion, statement, or theory may be taken as a guide to future uses of the same notion, statement, or theory in mathematical practice. In Maddy’s view, successful mathematical practice relies on the knowledge of the history of mathematics and of the patterns of mathematical depth that we discover studying and practicing mathematics.
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References
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Acknowledgements
I am indebted to Gabriele Lolli, Marco Panza, and Andrea Sereni for their valuable suggestions and comments. I would also like to thank the audience of the conferences The Answers of Philosophy – 20th SIFA Anniversary Conference (Alghero 2012) and Philosophy of Mathematics: from Logic to Practice (Pisa 2012), where former versions of this chapter were presented, and two anonymous referees for helpful remarks.
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Imocrante, M. (2015). Defending Maddy’s Mathematical Naturalism from Roland’s Criticisms: The Role of Mathematical Depth. In: Lolli, G., Panza, M., Venturi, G. (eds) From Logic to Practice. Boston Studies in the Philosophy and History of Science, vol 308. Springer, Cham. https://doi.org/10.1007/978-3-319-10434-8_12
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