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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory

[denial that mathematics is just set theory]

14 ideas
If numbers can be derived from logic, then set theory is superfluous [Frege, by Burge]
The theory of classes is superfluous in mathematics [Wittgenstein]
Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Benacerraf, by Friend]
No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Benacerraf, by Lowe]
If ordinal numbers are 'reducible to' some set-theory, then which is which? [Benacerraf]
You can ask all sorts of numerical questions about any one given set [Yourgrau]
We can't use sets as foundations for mathematics if we must await results from the upper reaches [Yourgrau]
Set-theoretic imperialists think sets can represent every mathematical object [Fine,K]
Mathematical foundations may not be sets; categories are a popular rival [Shapiro]
Sets exist where their elements are, but numbers are more like universals [Maddy]
Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy]
Set theory may represent all of mathematics, without actually being mathematics [Brown,JR]
When graphs are defined set-theoretically, that won't cover unlabelled graphs [Brown,JR]
Numbers are properties, not sets (because numbers are magnitudes) [Hossack]