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Single Idea 10169

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism ]

Full Idea

Relativist Structuralism simply picks one particular model of axiomatised arithmetic (i.e. one particular interpretation that satisfies the axioms), and then stipulates what the elements, functions and quantifiers refer to.

Gist of Idea

Relativist Structuralism just stipulates one successful model as its arithmetic

Source

E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4)

A Reaction

The point is that a successful model can be offered, and it doesn't matter which one, like having any sort of aeroplane, as long as it flies. I don't find this approach congenial, though having a model is good. What is the essence of flight?

The 18 ideas from E Reck / M 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]
10166 ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price]
10168 Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price]
10174 Mereological arithmetic needs infinite objects, and function definitions [Reck/Price]
10172 Set-theory gives a unified and an explicit basis for mathematics [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]
10169 Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price]
10170 While true-in-a-model seems relative, true-in-all-models seems not to be [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]
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]
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]