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Single Idea 9925

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism ]

Full Idea

Usually structuralism and nominalism are considered rivals. But structuralism can also be the first step in a strategy of nominalist reconstrual or paraphrase.

Gist of Idea

Structuralism and nominalism are normally rivals, but might work together

Source

JP Burgess / G Rosen (A Subject with No Object [1997], II.C.0)

Book Ref

Burgess,J/Rosen,G: 'A Subject with No Object' [OUP 1997], p.147

A Reaction

Hellman and later Chihara seem to be the main proponents of nominalist structuralism. My sympathies lie with this strategy. Are there objects at the nodes of the structure, or is the structure itself platonic? Mill offers a route.

The 15 ideas from 'A Subject with No Object'

9918 Abstract/concrete is a distinction of kind, not degree [Burgess/Rosen]
9919 The old debate classified representations as abstract, not entities [Burgess/Rosen]
9921 'True' is only occasionally useful, as in 'everything Fermat believed was true' [Burgess/Rosen]
9922 If space is really just a force-field, then it is a physical entity [Burgess/Rosen]
9923 We should talk about possible existence, rather than actual existence, of numbers [Burgess/Rosen]
9924 Modal logic gives an account of metalogical possibility, not metaphysical possibility [Burgess/Rosen]
9925 Structuralism and nominalism are normally rivals, but might work together [Burgess/Rosen]
9926 A relation is either a set of sets of sets, or a set of sets [Burgess/Rosen]
9928 Mereology implies that acceptance of entities entails acceptance of conglomerates [Burgess/Rosen]
9927 Mathematics has ascended to higher and higher levels of abstraction [Burgess/Rosen]
9929 Much of what science says about concrete entities is 'abstraction-laden' [Burgess/Rosen]
9930 Abstraction is on a scale, of sets, to attributes, to type-formulas, to token-formulas [Burgess/Rosen]
9933 The paradoxes are only a problem for Frege; Cantor didn't assume every condition determines a set [Burgess/Rosen]
9932 The paradoxes no longer seem crucial in critiques of set theory [Burgess/Rosen]
9934 Number words became nouns around the time of Plato [Burgess/Rosen]