33 ideas
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] |
15435 | If you think universals are immanent, you must believe them to be sparse, and not every related predicate [Lewis] |
15451 | I assume there could be natural properties that are not instantiated in our world [Lewis] |
15433 | Tropes are particular properties, which cannot recur, but can be exact duplicates [Lewis] |
15436 | Universals are meant to give an account of resemblance [Lewis] |
15438 | We can add a primitive natural/unnatural distinction to class nominalism [Lewis] |
15448 | The 'magical' view of structural universals says they are atoms, even though they have parts [Lewis] |
15449 | If 'methane' is an atomic structural universal, it has nothing to connect it to its carbon universals [Lewis] |
15439 | The 'pictorial' view of structural universals says they are wholes made of universals as parts [Lewis] |
15441 | The structural universal 'methane' needs the universal 'hydrogen' four times over [Lewis] |
15445 | Butane and Isobutane have the same atoms, but different structures [Lewis] |
15434 | Structural universals have a necessary connection to the universals forming its parts [Lewis] |
15437 | We can't get rid of structural universals if there are no simple universals [Lewis] |
15446 | Composition is not just making new things from old; there are too many counterexamples [Lewis] |
15440 | A whole is distinct from its parts, but is not a further addition in ontology [Lewis] |
15444 | Different things (a toy house and toy car) can be made of the same parts at different times [Lewis] |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
15450 | Maybe abstraction is just mereological subtraction [Lewis] |
15443 | Mathematicians abstract by equivalence classes, but that doesn't turn a many into one [Lewis] |
6017 | Nomos is king [Pindar] |