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6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / e. Structuralism critique

[objections to structuralism about mathematics]

23 ideas
If numbers are supposed to be patterns, each number can have many patterns [Frege]
Ordinals can't be defined just by progression; they have intrinsic qualities [Russell]
The identity of a number may be fixed by something outside structure - by counting [Dummett]
Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1 [Dummett]
The number 4 has different positions in the naturals and the wholes, with the same structure [Dummett]
Numbers can't be positions, if nothing decides what position a given number has [Bostock]
Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock]
We don't need 'abstract structures' to have structural truths about successor functions [Lewis]
If structures are relative, this undermines truth-value and objectivity [Hale/Wright]
The structural view of numbers doesn't fit their usage outside arithmetical contexts [Hale/Wright]
How could structures be mathematical truthmakers? Maths is just true, without truthmakers [Heil]
Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro]
If set theory is used to define 'structure', we can't define set theory structurally [Burgess]
Abstract algebra concerns relations between models, not common features of all the models [Burgess]
How can mathematical relations be either internal, or external, or intrinsic? [Burgess]
Different versions of set theory result in different underlying structures for numbers [Brown,JR]
Sets seem basic to mathematics, but they don't suit structuralism [Brown,JR]
The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price]
For mathematical objects to be positions, positions themselves must exist first [MacBride]
Structuralism is right about algebra, but wrong about sets [Linnebo]
In mathematical structuralism the small depends on the large, which is the opposite of physical structures [Linnebo]
Some questions concern mathematical entities, rather than whole structures [Koslicki]
If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan]