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 second-order 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 | 10171 | The existence of an infinite set is assumed by Relativist Structuralism |
§4 | p.350 | 10170 | While true-in-a-model seems relative, true-in-all-models seems not to be |
§4 | p.351 | 10172 | Set-theory gives a unified and an explicit basis for mathematics |
§4 | p.352 | 10173 | A nominalist might avoid abstract objects by just appealing to mereological sums |
§4 | p.352 | 10174 | Mereological arithmetic needs infinite objects, and function definitions |
§5 | p.356 | 10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets |
§5 | p.356 | 10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model |
§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 |