25 ideas
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
10245 | One geometry cannot be more true than another [Poincaré] |
8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
8764 | Categories are the best foundation for mathematics [Shapiro] |
8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
11184 | Aristotelian essentialism is about shared properties, individuating essentialism about distinctive properties [Marcus (Barcan)] |
11181 | Aristotelian essentialism involves a 'natural' or 'causal' interpretation of modal operators [Marcus (Barcan)] |
11180 | Essentialist sentences are not theorems of modal logic, and can even be false [Marcus (Barcan)] |
11186 | 'Essentially' won't replace 'necessarily' for vacuous properties like snub-nosed or self-identical [Marcus (Barcan)] |
11185 | 'Is essentially' has a different meaning from 'is necessarily', as they often cannot be substituted [Marcus (Barcan)] |
11182 | If essences are objects with only essential properties, they are elusive in possible worlds [Marcus (Barcan)] |
11183 | The use of possible worlds is to sort properties (not to individuate objects) [Marcus (Barcan)] |
11187 | In possible worlds, names are just neutral unvarying pegs for truths and predicates [Marcus (Barcan)] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
11189 | Dispositional essences are special, as if an object loses them they cease to exist [Marcus (Barcan)] |