29 ideas
10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
12452 | Our dislike of contradiction in logic is a matter of psychology, not mathematics [Brouwer] |
10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
15941 | For intuitionists excluded middle is an outdated historical convention [Brouwer] |
10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
18119 | Mathematics is a mental activity which does not use language [Brouwer, by Bostock] |
10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
18247 | Brouwer saw reals as potential, not actual, and produced by a rule, or a choice [Brouwer, by Shapiro] |
12451 | Scientific laws largely rest on the results of counting and measuring [Brouwer] |
18118 | Brouwer regards the application of mathematics to the world as somehow 'wicked' [Brouwer, by Bostock] |
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] |
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] |
12454 | Intuitionists only accept denumerable sets [Brouwer] |
12453 | Neo-intuitionism abstracts from the reuniting of moments, to intuit bare two-oneness [Brouwer] |
8728 | Intuitionist mathematics deduces by introspective construction, and rejects unknown truths [Brouwer] |
10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
10117 | Intuitonists in mathematics worried about unjustified assertion, as well as contradiction [Brouwer, by George/Velleman] |