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Single Idea 10186
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
]
Full Idea
It is to set theory that one turns for the very definition of 'structure', ...and this creates a problem of circularity if we try to impose a structuralist interpretation on set theory.
Gist of Idea
If set theory is used to define 'structure', we can't define set theory structurally
Source
John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1)
A Reaction
This seems like a nice difficulty, especially if, like Shapiro, you wade in and try to give a formal account of structures and patterns. Resnik is more circumspect and vague.
The
23 ideas
with the same theme
[objections to structuralism about mathematics]:
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8639
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If numbers are supposed to be patterns, each number can have many patterns
[Frege]
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9977
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Ordinals can't be defined just by progression; they have intrinsic qualities
[Russell]
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9627
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Different versions of set theory result in different underlying structures for numbers
[Zermelo, by Brown,JR]
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9829
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The identity of a number may be fixed by something outside structure - by counting
[Dummett]
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9828
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Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1
[Dummett]
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9192
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The number 4 has different positions in the naturals and the wholes, with the same structure
[Dummett]
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18116
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Numbers can't be positions, if nothing decides what position a given number has
[Bostock]
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18117
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Structuralism falsely assumes relations to other numbers are numbers' only properties
[Bostock]
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10815
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We don't need 'abstract structures' to have structural truths about successor functions
[Lewis]
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10629
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If structures are relative, this undermines truth-value and objectivity
[Hale/Wright]
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10628
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The structural view of numbers doesn't fit their usage outside arithmetical contexts
[Hale/Wright]
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18500
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How could structures be mathematical truthmakers? Maths is just true, without truthmakers
[Heil]
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10274
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Does someone using small numbers really need to know the infinite structure of arithmetic?
[Shapiro]
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10186
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If set theory is used to define 'structure', we can't define set theory structurally
[Burgess]
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10187
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Abstract algebra concerns relations between models, not common features of all the models
[Burgess]
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10188
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How can mathematical relations be either internal, or external, or intrinsic?
[Burgess]
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9628
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Sets seem basic to mathematics, but they don't suit structuralism
[Brown,JR]
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10171
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The existence of an infinite set is assumed by Relativist Structuralism
[Reck/Price]
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8926
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For mathematical objects to be positions, positions themselves must exist first
[MacBride]
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14083
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Structuralism is right about algebra, but wrong about sets
[Linnebo]
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14090
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In mathematical structuralism the small depends on the large, which is the opposite of physical structures
[Linnebo]
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14505
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Some questions concern mathematical entities, rather than whole structures
[Koslicki]
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17932
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If 'in re' structures relies on the world, does the world contain rich enough structures?
[Colyvan]
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