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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 'Analysis' is the theory of the real numbers
2 p.343 Peano Arithmetic can have three second-order axioms, plus '1' and 'successor'
2 p.344 ZFC set theory has only 'pure' sets, without 'urelements'
2 p.346 Structuralism emerged from abstract algebra, axioms, and set theory and its structures
3 p.348 Formalist Structuralism says the ontology is vacuous, or formal, or inference relations
4 p.349 Relativist Structuralism just stipulates one successful model as its arithmetic
4 p.350 The existence of an infinite set is assumed by Relativist Structuralism
4 p.350 While true-in-a-model seems relative, true-in-all-models seems not to be
4 p.351 Set-theory gives a unified and an explicit basis for mathematics
4 p.352 Mereological arithmetic needs infinite objects, and function definitions
4 p.352 A nominalist might avoid abstract objects by just appealing to mereological sums
5 p.356 Universalist Structuralism is based on generalised if-then claims, not one particular model
5 p.356 Three types of variable in second-order logic, for objects, functions, and predicates/sets
5 p.358 Universalist Structuralism eliminates the base element, as a variable, which is then quantified out
5 p.359 Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous
6 p.362 There are 'particular' structures, and 'universal' structures (what the former have in common)
7 p.363 Pattern Structuralism studies what isomorphic arithmetic models have in common
9 p.374 There are Formalist, Relativist, Universalist and Pattern structuralism