26 ideas
10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
10174 | Mereological arithmetic needs infinite objects, and function definitions [Reck/Price] |
10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
10172 | Set-theory gives a unified and an explicit basis for mathematics [Reck/Price] |
10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price] |
10169 | Relativist Structuralism just stipulates one successful model as its arithmetic [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] |
10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price] |
10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [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] |
10171 | The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price] |
17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
9287 | Bruno said that ancient Egyptian magic was the true religion [Bruno, by Yates] |