more on this theme     |     more from this thinker

Single Idea 10168

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

Full Idea

Formalist Structuralism endorses structural methodology in mathematics, but rejects semantic and metaphysical problems as either meaningless, or purely formal, or as inference relations.

Gist of Idea

Formalist Structuralism says the ontology is vacuous, or formal, or inference relations

Source

E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §3)

A Reaction

[very compressed] I find the third option fairly congenial, certainly in preference to rather platonist accounts of structuralism. One still needs to distinguish the mathematical from the non-mathematical in the inference relations.

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]