23 ideas
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
7557 | To solve Zeno's paradox, reject the axiom that the whole has more terms than the parts [Russell] |
10059 | In mathematic we are ignorant of both subject-matter and truth [Russell] |
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] |
7556 | A collection is infinite if you can remove some terms without diminishing its number [Russell] |
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] |
16978 | If conceivability is a priori coherence, that implies possibility [Tahko] |
7554 | Self-evidence is often a mere will-o'-the-wisp [Russell] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
16975 | Essences are used to explain natural kinds, modality, and causal powers [Tahko] |
16976 | Scientific essentialists tend to characterise essence in terms of modality (not vice versa) [Tahko] |
16977 | If essence is modal and laws are necessary, essentialist knowledge is found by scientists [Tahko] |