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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 A nominalist might avoid abstract objects by just appealing to mereological sums
§4 p.352 Mereological arithmetic needs infinite objects, and function definitions
§5 p.356 Three types of variable in second-order logic, for objects, functions, and predicates/sets
§5 p.356 Universalist Structuralism is based on generalised if-then claims, not one particular model
§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