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Single Idea 8697

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory ]

Full Idea

If two children were brought up knowing two different set theories, they could entirely agree on how to do arithmetic, up to the point where they discuss ontology. There is no mathematical way to tell which is the true representation of numbers.

Gist of Idea

Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them

Source

report of Paul Benacerraf (What Numbers Could Not Be [1965]) by Michèle Friend - Introducing the Philosophy of Mathematics

Book Ref

Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.83

A Reaction

Benacerraf ends by proposing a structuralist approach. If mathematics is consistent with conflicting set theories, then those theories are not shedding light on mathematics.

The 14 ideas with the same theme [denial that mathematics is just set theory]:

16896 If numbers can be derived from logic, then set theory is superfluous [Frege, by Burge]
18161 The theory of classes is superfluous in mathematics [Wittgenstein]
8697 Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Benacerraf, by Friend]
8304 No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Benacerraf, by Lowe]
9906 If ordinal numbers are 'reducible to' some set-theory, then which is which? [Benacerraf]
17821 You can ask all sorts of numerical questions about any one given set [Yourgrau]
17815 We can't use sets as foundations for mathematics if we must await results from the upper reaches [Yourgrau]
10560 Set-theoretic imperialists think sets can represent every mathematical object [Fine,K]
10222 Mathematical foundations may not be sets; categories are a popular rival [Shapiro]
17827 Sets exist where their elements are, but numbers are more like universals [Maddy]
17830 Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy]
9643 Set theory may represent all of mathematics, without actually being mathematics [Brown,JR]
9644 When graphs are defined set-theoretically, that won't cover unlabelled graphs [Brown,JR]
23621 Numbers are properties, not sets (because numbers are magnitudes) [Hossack]