30 ideas
8920 | Equivalence relations are reflexive, symmetric and transitive, and classify similar objects [Lipschutz] |
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] |
14064 | If a statue is identical with the clay of which it is made, that identity is contingent [Gibbard] |
14066 | A 'piece' of clay begins when its parts stick together, separately from other clay [Gibbard] |
14067 | Clay and statue are two objects, which can be named and reasoned about [Gibbard] |
14069 | We can only investigate the identity once we have designated it as 'statue' or as 'clay' [Gibbard] |
14076 | Essentialism is the existence of a definite answer as to whether an entity fulfils a condition [Gibbard] |
14077 | Essentialism for concreta is false, since they can come apart under two concepts [Gibbard] |
14070 | A particular statue has sortal persistence conditions, so its origin defines it [Gibbard] |
14073 | Claims on contingent identity seem to violate Leibniz's Law [Gibbard] |
14065 | Two identical things must share properties - including creation and destruction times [Gibbard] |
14074 | Leibniz's Law isn't just about substitutivity, because it must involve properties and relations [Gibbard] |
14072 | Possible worlds identity needs a sortal [Gibbard] |
14078 | Only concepts, not individuals, can be the same across possible worlds [Gibbard] |
14079 | Kripke's semantics needs lots of intuitions about which properties are essential [Gibbard] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
14071 | Naming a thing in the actual world also invokes some persistence criteria [Gibbard] |