24 ideas
23445 | Naïve set theory says any formula defines a set, and coextensive sets are identical [Linnebo] |
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
23447 | In classical semantics singular terms refer, and quantifiers range over domains [Linnebo] |
23443 | The axioms of group theory are not assertions, but a definition of a structure [Linnebo] |
23444 | To investigate axiomatic theories, mathematics needs its own foundational axioms [Linnebo] |
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] |
23446 | You can't prove consistency using a weaker theory, but you can use a consistent theory [Linnebo] |
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] |
23448 | Mathematics is the study of all possible patterns, and is thus bound to describe the world [Linnebo] |
8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
23441 | Logical truth is true in all models, so mathematical objects can't be purely logical [Linnebo] |
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] |
23442 | Game Formalism has no semantics, and Term Formalism reduces the semantics [Linnebo] |
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] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
6172 | The Inverted Earth example shows that phenomenal properties are not representational [Block, by Rowlands] |