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Single Idea 10167

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism ]

Full Idea

Structuralism has emerged from the development of abstract algebra (such as group theory), the creation of axiom systems, the introduction of set theory, and Bourbaki's encyclopaedic survey of set theoretic structures.

Clarification

[Bourbaki is a notably unusual mathematician - worth investigating!]

Gist of Idea

Structuralism emerged from abstract algebra, axioms, and set theory and its structures

Source

E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)

A Reaction

In other words, mathematics has gradually risen from one level of abstraction to the next, so that mathematical entities like points and numbers receive less and less attention, with relationships becoming more prominent.

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]
10167 Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price]
10164 Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [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]
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]
10174 Mereological arithmetic needs infinite objects, and function definitions [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]