32 ideas
13886 | Later Frege held that definitions must fix a function's value for every possible argument [Frege, by Wright,C] |
19048 | Contextual definition shifted the emphasis from words to whole sentences [Quine] |
9845 | We can't define a word by defining an expression containing it, as the remaining parts are a problem [Frege] |
19047 | Bentham's contextual definitions preserved terms after their denotation became doubtful [Quine] |
10019 | Only what is logically complex can be defined; what is simple must be pointed to [Frege] |
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
9886 | Cardinals say how many, and reals give measurements compared to a unit quantity [Frege] |
8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
9889 | Real numbers are ratios of quantities [Frege, by Dummett] |
18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
8764 | Categories are the best foundation for mathematics [Shapiro] |
10020 | Frege's biggest error is in not accounting for the senses of number terms [Hodes on Frege] |
10553 | A number is a class of classes of the same cardinality [Frege, by Dummett] |
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] |
9887 | Formalism misunderstands applications, metatheory, and infinity [Frege, by Dummett] |
8751 | Only applicability raises arithmetic from a game to a science [Frege] |
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] |
9891 | The first demand of logic is of a sharp boundary [Frege] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
19046 | Empiricism improvements: words for ideas, then sentences, then systems, then no analytic, then naturalism [Quine] |
19049 | In scientific theories sentences are too brief to be independent vehicles of empirical meaning [Quine] |
9890 | The modern account of real numbers detaches a ratio from its geometrical origins [Frege] |
11846 | If we abstract the difference between two houses, they don't become the same house [Frege] |
19050 | Holism in language blurs empirical synthetic and empty analytic sentences [Quine] |