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Single Idea 10222
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
]
Full Idea
Foundationalists (e.g. Quine and Lewis) have shown that mathematics can be rendered in theories other than the iterative hierarchy of sets. A dedicated contingent hold that the category of categories is the proper foundation (e.g. Lawvere).
Gist of Idea
Mathematical foundations may not be sets; categories are a popular rival
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 3.3)
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.87
A Reaction
I like the sound of that. The categories are presumably concepts that generate sets. Tricky territory, with Frege's disaster as a horrible warning to be careful.
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]
|