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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.
| 10185 | Set theory is the standard background for modern mathematics [Burgess] |
| 10184 | Structuralists take the name 'R' of the reals to be a variable ranging over structures, not a structure [Burgess] |
| 10186 | If set theory is used to define 'structure', we can't define set theory structurally [Burgess] |
| 10187 | Abstract algebra concerns relations between models, not common features of all the models [Burgess] |
| 10188 | How can mathematical relations be either internal, or external, or intrinsic? [Burgess] |
| 10189 | There is no one relation for the real number 2, as relations differ in different models [Burgess] |