more on this theme
|
more from this text
Single Idea 10171
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
]
Full Idea
Relativist Structuralism must first assume the existence of an infinite set, otherwise there would be no model to pick, and arithmetical terms would have no reference.
Gist of Idea
The existence of an infinite set is assumed by Relativist Structuralism
Source
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4)
A Reaction
See Idea 10169 for Relativist Structuralism. They point out that ZFC has an Axiom of Infinity.
Related Idea
Idea 10169
Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price]
The
18 ideas
from E Reck / M Price
10166
|
ZFC set theory has only 'pure' sets, without 'urelements'
[Reck/Price]
|
10165
|
'Analysis' is the theory of the real numbers
[Reck/Price]
|
10164
|
Peano Arithmetic can have three second-order axioms, plus '1' and 'successor'
[Reck/Price]
|
10167
|
Structuralism emerged from abstract algebra, axioms, and set theory and its structures
[Reck/Price]
|
10168
|
Formalist Structuralism says the ontology is vacuous, or formal, or inference relations
[Reck/Price]
|
10172
|
Set-theory gives a unified and an explicit basis for mathematics
[Reck/Price]
|
10174
|
Mereological arithmetic needs infinite objects, and function definitions
[Reck/Price]
|
10169
|
Relativist Structuralism just stipulates one successful model as its arithmetic
[Reck/Price]
|
10171
|
The existence of an infinite set is assumed by Relativist Structuralism
[Reck/Price]
|
10173
|
A nominalist might avoid abstract objects by just appealing to mereological sums
[Reck/Price]
|
10170
|
While true-in-a-model seems relative, true-in-all-models seems not to be
[Reck/Price]
|
10176
|
Universalist Structuralism is based on generalised if-then claims, not one particular model
[Reck/Price]
|
10177
|
Universalist Structuralism eliminates the base element, as a variable, which is then quantified out
[Reck/Price]
|
10175
|
Three types of variable in second-order logic, for objects, functions, and predicates/sets
[Reck/Price]
|
10178
|
Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous
[Reck/Price]
|
10179
|
There are 'particular' structures, and 'universal' structures (what the former have in common)
[Reck/Price]
|
10181
|
Pattern Structuralism studies what isomorphic arithmetic models have in common
[Reck/Price]
|
10182
|
There are Formalist, Relativist, Universalist and Pattern structuralism
[Reck/Price]
|