Ideas from 'Structuralism' by Geoffrey Hellman [2007], by Theme Structure

[found in 'Oxf Handbk of Philosophy of Maths and Logic' (ed/tr Shapiro,Stewart) [OUP 2007,978-0-19-532592-8]].

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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism

8921	Structuralism is now common, studying relations, with no regard for what the objects might be

			Full Idea: With developments in modern mathematics, structuralist ideas have become commonplace. We study 'abstract structures', having relations without regard to the objects. As Hilbert famously said, items of furniture would do.
			From: Geoffrey Hellman (Structuralism [2007], §1)
			A reaction: Hilbert is known as a Formalist, which suggests that modern Structuralism is a refined and more naturalist version of the rather austere formalist view. Presumably the sofa can't stand for six, so a structural definition of numbers is needed.

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism

8922	Maybe mathematical objects only have structural roles, and no intrinsic nature

			Full Idea: There is the tantalizing possibility that perhaps mathematical objects 'have no nature' at all, beyond their 'structural role'.
			From: Geoffrey Hellman (Structuralism [2007], §1)
			A reaction: This would fit with a number being a function rather than an object. We are interested in what cars do, not the bolts that hold them together? But the ontology of mathematics is quite separate from how you do mathematics.