47 ideas
19463 | Induction assumes some uniformity in nature, or that in some respects the future is like the past [Ayer] |
9542 | The best known axiomatization of PL is Whitehead/Russell, with four axioms and two rules [Russell/Whitehead, by Hughes/Cresswell] |
21720 | Russell saw Reducibility as legitimate for reducing classes to logic [Linsky,B on Russell/Whitehead] |
10044 | Russell denies extensional sets, because the null can't be a collection, and the singleton is just its element [Russell/Whitehead, by Shapiro] |
18208 | We regard classes as mere symbolic or linguistic conveniences [Russell/Whitehead] |
8204 | Lewis's 'strict implication' preserved Russell's confusion of 'if...then' with implication [Quine on Russell/Whitehead] |
9359 | Russell's implication means that random sentences imply one another [Lewis,CI on Russell/Whitehead] |
21707 | Russell unusually saw logic as 'interpreted' (though very general, and neutral) [Russell/Whitehead, by Linsky,B] |
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
10036 | In 'Principia' a new abstract theory of relations appeared, and was applied [Russell/Whitehead, by Gödel] |
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] |
18248 | A real number is the class of rationals less than the number [Russell/Whitehead, by Shapiro] |
8764 | Categories are the best foundation for mathematics [Shapiro] |
18152 | Russell takes numbers to be classes, but then reduces the classes to numerical quantifiers [Russell/Whitehead, by Bostock] |
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] |
10025 | Russell and Whitehead took arithmetic to be higher-order logic [Russell/Whitehead, by Hodes] |
8683 | Russell and Whitehead were not realists, but embraced nearly all of maths in logic [Russell/Whitehead, by Friend] |
10037 | 'Principia' lacks a precise statement of the syntax [Gödel on Russell/Whitehead] |
10093 | The ramified theory of types used propositional functions, and covered bound variables [Russell/Whitehead, by George/Velleman] |
8691 | The Russell/Whitehead type theory was limited, and was not really logic [Friend on Russell/Whitehead] |
10305 | In 'Principia Mathematica', logic is exceeded in the axioms of infinity and reducibility, and in the domains [Bernays on Russell/Whitehead] |
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] |
8684 | Russell and Whitehead consider the paradoxes to indicate that we create mathematical reality [Russell/Whitehead, by Friend] |
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] |
8746 | To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it [Russell/Whitehead, by Shapiro] |
8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
12033 | An object is identical with itself, and no different indiscernible object can share that [Russell/Whitehead, by Adams,RM] |
19459 | To say 'I am not thinking' must be false, but it might have been true, so it isn't self-contradictory [Ayer] |
19460 | 'I know I exist' has no counterevidence, so it may be meaningless [Ayer] |
19461 | Knowing I exist reveals nothing at all about my nature [Ayer] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
10040 | Russell showed, through the paradoxes, that our basic logical intuitions are self-contradictory [Russell/Whitehead, by Gödel] |
19464 | We only discard a hypothesis after one failure if it appears likely to keep on failing [Ayer] |
19462 | Induction passes from particular facts to other particulars, or to general laws, non-deductively [Ayer] |
21725 | The multiple relations theory says assertions about propositions are about their ingredients [Russell/Whitehead, by Linsky,B] |
23474 | A judgement is a complex entity, of mind and various objects [Russell/Whitehead] |
23455 | The meaning of 'Socrates is human' is completed by a judgement [Russell/Whitehead] |
23480 | The multiple relation theory of judgement couldn't explain the unity of sentences [Morris,M on Russell/Whitehead] |
18275 | Only the act of judging completes the meaning of a statement [Russell/Whitehead] |
23453 | Propositions as objects of judgement don't exist, because we judge several objects, not one [Russell/Whitehead] |