Ideas from 'Review of Chihara 'Struct. Accnt of Maths'' by John P. Burgess [2005], by Theme Structure

[found in 'Philosophia Mathematica' (ed/tr -) [- ,]].

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6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
Set theory is the standard background for modern mathematics
                        Full Idea: In present-day mathematics, it is set theory that serves as the background theory in which other branches of mathematics are developed.
                        From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], 1)
                        A reaction: [He cites Bourbaki as an authority for this] See Benacerraf for a famous difficulty here, when you actually try to derive an ontology from the mathematicians' working practices.
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / a. Structuralism
Structuralists take the name 'R' of the reals to be a variable ranging over structures, not a structure
                        Full Idea: On the structuralist interpretation, theorems of analysis concerning the real numbers R are about all complete ordered fields. So R, which appears to be the name of a specific structure, is taken to be a variable ranging over structures.
                        From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], 1)
                        A reaction: Since I am beginning to think that nearly all linguistic expressions should be understood as variables, I find this very appealing, even if Burgess hates it. Terms slide and drift, and are vague, between variable and determinate reference.
There is no one relation for the real number 2, as relations differ in different models
                        Full Idea: One might meet the 'Van Inwagen Problem' by saying that the intrinsic properties of the object playing the role of 2 will differ from one model to another, so that no statement about the intrinsic properties of 'the' real numbers will make sense.
                        From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], 5)
                        A reaction: There seems to be a potential confusion among opponents of structuralism between relations at the level of actual mathematical operations, and generalisations about relations, which are captured in the word 'patterns'. Call them 'meta-relations'?
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / e. Structuralism critique
If set theory is used to define 'structure', we can't define set theory structurally
                        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.
                        From: 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.
Abstract algebra concerns relations between models, not common features of all the models
                        Full Idea: Abstract algebra, such as group theory, is not concerned with the features common to all models of the axioms, but rather with the relationships among different models of those axioms (especially homomorphic relation functions).
                        From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], 1)
                        A reaction: It doesn't seem to follow that structuralism can't be about the relations (or patterns) found when abstracting away and overviewing all the models. One can study family relations, or one can study kinship in general.
How can mathematical relations be either internal, or external, or intrinsic?
                        Full Idea: The 'Van Inwagen Problem' for structuralism is of explaining how a mathematical relation (such as set membership, or the ratios of an ellipse) can fit into one of the three scholastics types of relations: are they internal, external, or intrinsic?
                        From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], 5)
                        A reaction: The difficulty is that mathematical objects seem to need intrinsic properties to get any of these three versions off the ground (which was Russell's complaint against structures).