Platonism and anti-platonism: Why worry?

This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti-Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the 'no miracles' argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific 'realists' should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism.
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DOI 10.1080/02698590500051159
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Juha Saatsi (2007). Living in Harmony: Nominalism and the Explanationist Argument for Realism. International Studies in the Philosophy of Science 21 (1):19 – 33.
Russell Marcus (2013). Intrinsic Explanation and Field's Dispensabilist Strategy. International Journal of Philosophical Studies 21 (2):163 - 183.

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