more on this theme     |     more from this text

Single Idea 10178

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

Full Idea

It is tempting to take a modal turn, and quantify over all possible objects, because if there are only a finite number of actual objects, then there are no models (of the right sort) for Peano Arithmetic, and arithmetic is vacuously true.

Gist of Idea

Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous

Source

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

A Reaction

[compressed; Geoffrey Hellman is the chief champion of this view] The article asks whether we are not still left with the puzzle of whether infinitely many objects are possible, instead of existent.

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]