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Single Idea 10176

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism ]

Full Idea

Universalist Structuralism is a semantic thesis, that an arithmetical statement asserts a universal if-then statement. We build an if-then statement (using quantifiers) into the structure, and we generalise away from any one particular model.

Gist of Idea

Universalist Structuralism is based on generalised if-then claims, not one particular model

Source

E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)

A Reaction

There remains the question of what is distinctively mathematical about the highly generalised network of inferences that is being described. Presumable the axioms capture that, but why those particular axioms? Russell is cited as an originator.

The 18 ideas from 'Structures and Structuralism in Phil of Maths'

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]