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Ideas of E Reck / M Price, by Text
[American, fl. 2000, Both at the University of Chicago.]
2000

Structures and Structuralism in Phil of Maths

§2

p.343

10165

'Analysis' is the theory of the real numbers

§2

p.343

10164

Peano Arithmetic can have three secondorder axioms, plus '1' and 'successor'

§2

p.344

10166

ZFC set theory has only 'pure' sets, without 'urelements'

§2

p.346

10167

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

§3

p.348

10168

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

§4

p.349

10169

Relativist Structuralism just stipulates one successful model as its arithmetic

§4

p.350

10170

While trueinamodel seems relative, trueinallmodels seems not to be

§4

p.350

10171

The existence of an infinite set is assumed by Relativist Structuralism

§4

p.351

10172

Settheory gives a unified and an explicit basis for mathematics

§4

p.352

10174

Mereological arithmetic needs infinite objects, and function definitions

§4

p.352

10173

A nominalist might avoid abstract objects by just appealing to mereological sums

§5

p.356

10176

Universalist Structuralism is based on generalised ifthen claims, not one particular model

§5

p.356

10175

Three types of variable in secondorder logic, for objects, functions, and predicates/sets

§5

p.358

10177

Universalist Structuralism eliminates the base element, as a variable, which is then quantified out

§5

p.359

10178

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

§6

p.362

10179

There are 'particular' structures, and 'universal' structures (what the former have in common)

§7

p.363

10181

Pattern Structuralism studies what isomorphic arithmetic models have in common

§9

p.374

10182

There are Formalist, Relativist, Universalist and Pattern structuralism
