25 ideas
18189 | ZFC could contain a contradiction, and it can never prove its own consistency [MacLane] |
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
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] |
19712 | Maybe there is plain 'animal' knowledge, and clearly justified 'reflective' knowledge [Vahid] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
19703 | Epistemic is normally marked out from moral or pragmatic justifications by its truth-goal [Vahid] |
19705 | 'Mentalist' internalism seems to miss the main point, if it might not involve an agent's access [Vahid] |
19706 | Strong access internalism needs actual awareness; weak versions need possibility of access [Vahid] |
19707 | Maybe we need access to our justification, and also to know why it justifies [Vahid] |
19709 | Internalism in epistemology over-emphasises deliberation about beliefs [Vahid] |
19704 | Externalism may imply that identical mental states might go with different justifications [Vahid] |
19710 | With a counterfactual account of the causal theory, we get knowledge as tracking or sensitive to truth [Vahid] |
19711 | Externalism makes the acquisition of knowledge too easy? [Vahid] |