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Single Idea 9643

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

Full Idea

Maybe all of mathematics can be represented in set theory, but we should not think that mathematics is set theory. Functions can be represented as order pairs, but perhaps that is not what functions really are.

Gist of Idea

Set theory may represent all of mathematics, without actually being mathematics

Source

James Robert Brown (Philosophy of Mathematics [1999], Ch. 7)

Book Ref

Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.102

A Reaction

This seems to me to be the correct view of the situation. If 2 is represented as {φ,{φ}}, why is that asymmetrical? The first digit seems to be the senior and original partner, but how could the digits of 2 differ from one another?

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]